Holography of De Sitter Space and Disordered Systems

HOLOGRAPHY OF DE SITTER SPACE AND DISORDERED SYSTEMS

A DISSERTATION SUBMITTED TO THE DEPARTMENT OF PHYSICS AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

George Konstantinidis Coss May 2015

This work is licensed under a Creative Commons Attribution- Noncommercial 3.0 United States License. http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/zp606yx6742

ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Stephen Shenker, Primary Adviser

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Sean Hartnoll

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Eva Silverstein

Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost for Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives.

iii Abstract

This dissertation is devoted to the study of models that capture the intricate dynamics of two different physical setups: exponentially expanding universes and disordered systems. We explore late time divergences in the perturbative corrections of wavefunctions of interacting light fields on a fixed de Sitter background. The divergences are holographically interpreted as shifts in the conformal weights of dual CFT operators. We then compute functional determinants in a Euclidean CFT for various non-perturbative deformations. According to the dS/CFT correspondence, these functional determinants calculate the late time Hartle-Hawking wavefunctional of asymptotically de Sitter space in higher spin gravity as a function of the profile of the fields in the bulk. Numerical experiments suggest that upon fixing the average of the bulk scalar profile, the wavefunction becomes normalizable in all the other (infinite) directions of the deformations we study. For disordered systems, we investigate the extent to which quiver quantum mechanics models encode the complex dynamics of multicentered black holes in string theory. In a certain limit of the quiver system we display the emergence of a conformal symmetry, which mimics the emergence of conformal symmetry in the near-horizon geometries of extremal black holes. Finally, we take a Newtonian multiparticle limit of the quiver system away from the conformal regime. We study the dynamics of the system numerically to look for signs of ergodicity breaking, cages, and transitions to chaos.

iv Acknowledgements

I would like to first thank Dionysios Anninos. Dio, thank you for being patient, sup- portive, inspirational and fun, and thank you for teaching me so many great lessons about life and physics, but most of all thank you for believing in me from day one. This dissertation would not have existed without the support of my parents Con- stantine and Sofi, my brother Lazaros, and my grandparents Alice, Calliope, George, and Lazaros. I am very grateful to my thesis advisor Stephen Shenker for his sound and valuable advice in helping me find my way and graduate, as well as funding my research. I thank Sean Hartnoll for his unwavering support and for making me feel I can always count on him. I am also grateful to other faculty who provided guidance and funding including Savas Dimopoulos and Eva Silverstein. I thank my collaborators, Tarek Anous, Paul de Lange, Frederik Denef, Daniel Freedman, Raghu Mahajan and Edgar Shaghoulian. For all their help along the way, I thank Abdul- lah Al Akeel, Konstantinos Askitopoulos, Taylor Barrella, Kurt Barry, Nathan Ben- jamin, Leslie Brian, Simon Foreman, Abe Fornis, Daniel Fragiadakis, Andrea Fuentes, Kostas Gounaras, Xinlu Huang, George Karakonstantakis, Myrto Karapiperi, Tina Kechakis, Panagiotis Kontogiannis, Stefanos Koulandrou, Dimitris Koutsogiorgas, Lampros Lamprou, Petros Lekakis, Dafni Leon, Anna Liaroutsou, Chelsea Liekhus- Schmaltz, Emily Livermore, Jonathan Maltz, Nikolas Manetas, Cecilia Michel, the Nakas family, Chaya Nanavati, Giannis Pagkalos, Ashley Perko, Giannis Platis, Elise Potts, Djordje Radicevic, Michalis Rossides, Verina Samoli, Lauren Smith, Douglas Stanford, Athina Taka, Filippos Theodorakis, Basak Tulga, Antonis Tzouanakos, Christina Varvia, Dimitris Vrodamitis, Tim Weiser, and Vivian Wong.

v Contents

Abstract iv

Acknowledgementsv

1 Introduction1

2 Wavefunctions in de Sitter space4 2.1 Introduction...... 4 2.2 The Schr¨odingerequation in a ﬁxed de Sitter background...... 7 2.2.1 Bunch-Davies wavefunction...... 8 2.2.2 A Euclidean AdS approach...... 9

2.2.3 Interaction corrections to ZAdS ...... 12 2.3 Self-interacting scalars in four-dimenions...... 13 2.3.1 Tree level contributions for the massless theory...... 13 2.3.2 Loop correction to the two-point function...... 15 2.3.3 Tree level contributions to the conformally coupled case.... 17 2.3.4 Comments for general massive fields...... 19 2.4 Gauge fields and gravity in four-dimensions...... 21 2.4.1 SU(N) gauge fields...... 21 2.4.2 Scalar QED...... 23 2.4.3 Gravity...... 24 2.5 3d CFTs and (A)dS/CFT...... 26

2.6 dS2 via Euclidean AdS2 ...... 29 2.6.1 Tree level corrections for the massless theory...... 30

vi 2.6.2 Loop corrections for the massless theory...... 31 2.7 Outlook...... 33

3 Higher Spin dS/CFT 35 3.1 Introduction...... 35 3.2 Wavefunctionals and the free Sp(N) model...... 40 3.2.1 Wavefunctional...... 41 3.2.2 Functional determinant for radial deformations...... 42 3.3 Simple examples of radial deformations...... 43 3.3.1 Single Gaussian...... 44 3.3.2 Gaussian Ring...... 45 3.3.3 Double Gaussian...... 47 3.4 Radial deformations of ﬂat R3 and pinching limits...... 48 3.4.1 Balloon Geometry...... 48 3.4.2 Conformal Flatness of Balloon Geometry...... 50 3.4.3 Wavefunctions and balloon geometries...... 51 3.5 Spherical harmonics and a conjecture...... 52 3.5.1 Three-sphere harmonics...... 52 3.6 Three-sphere squashed and massed...... 57 3.6.1 Squashed and massed...... 57 3.7 Basis change, critical Sp(N) model, double trace deformations.... 60 3.7.1 Perturbative analysis on R3 ...... 61 3.7.2 Large N saddles for uniform S3 proﬁles...... 62 3.7.3 Double trace deformations as convolutions...... 64 3 3.7.4 Euclidean AdS4 with an S boundary...... 65 3.8 Extensions of higher spin de Sitter holography?...... 66

3.8.1 AdS4 ...... 66

3.8.2 dS4 ...... 67

4 Conformal Quivers 70 4.1 Introduction...... 70 4.2 General Framework: Quiver quantum mechanics...... 74

vii 4.2.1 Full quiver theory...... 74 4.2.2 Some properties of the ground states...... 77 4.3 Coulomb branch and a scaling theory...... 78 4.3.1 Supersymmetric conﬁgurations...... 80 4.3.2 Scaling Theory...... 81 4.4 Conformal Quivers: emergence of SL(2, R)...... 83 4.4.1 Conditions for an SL(2, R) invariant action...... 84 4.4.2 Two particles...... 85 4.4.3 SL(2, R) symmetry of the full three particle scaling theory.. 86 4.4.4 Wavefunctions in the scaling theory...... 89 4.5 Melting Molecules...... 92 4.5.1 Two nodes - melting bound states...... 93 4.5.2 Three nodes - unstable scaling solutions...... 98 4.6 Outlook...... 100 4.6.1 Random Hamiltonians and emergent SL(2, R)?...... 100 4.6.2 Holographic considerations...... 101 4.6.3 Possible extensions...... 102

5 Supergoop Dynamics 105 5.1 Introduction...... 105 5.1.1 String glasses...... 105 5.1.2 Supergoop...... 107 5.1.3 Dynamics...... 109 5.2 General Framework...... 111 5.2.1 Supersymmetric multiparticles...... 111 5.2.2 Classical features and multicentered black holes...... 113 5.2.3 Three particles...... 115 5.2.4 Regime of validity...... 116 5.3 Classical Phase Space...... 117 5.3.1 Two particles are integrable...... 117 5.3.2 Three particles are chaotic...... 118

viii 5.4 Euler-Jacobi Ground States...... 119 5.4.1 Euler-Jacobi three body problem...... 119 5.4.2 Classical Ground States...... 120 5.4.3 Quantum Ground States...... 122 5.5 Euler-Jacobi Dynamics: classical integrability...... 125 5.5.1 Setup and coordinate systems...... 126 5.6 Beyond Euler-Jacobi: the stringy double pendulum...... 128 5.6.1 Collinear dynamics...... 129 5.6.2 Poincar´eSections...... 130 5.7 Trapping...... 133 5.7.1 Setup and Energetics...... 133 5.7.2 A trap...... 134 5.7.3 Topology of the potential landscape...... 135 5.8 Holography of Chaotic Trajectories?...... 135

A De Sitter Wavefunction 139 A.1 A quantum mechanical toy model...... 139 A.1.1 Path integral perturbation theory...... 140 A.2 The bulk-to-bulk propagator...... 143

B Higher Spin de Sitter 146 B.1 Conformal Transformation from round S3 to ﬂat R3 ...... 146 B.1.1 Coordinate transformation and Weyl rescaling...... 146 B.1.2 Numerical Error...... 147 B.1.3 Balloon Geometries...... 147 B.2 Review of the squashed sphere...... 148 B.3 Perturbative Bunch-Davies modes for m2`2 = +2...... 149

B.3.1 Continuation from Euclidean AdS4 ...... 150 B.3.2 Wavefunctional...... 151 B.4 Wavefunctionals for bulk gauge ﬁelds...... 152

ix C Conformal Quivers 155 C.1 Notation...... 155 C.2 Superpotential corrections of the Coulomb branch...... 156 C.3 Finite D-terms...... 157 C.3.1 Non-linear D-term solutions for two-node quivers...... 158 C.4 Three node Coulomb branch...... 160 C.4.1 Second order Lagrangian for three-node quiver...... 161 C.4.2 Coulomb branch potential...... 166 C.5 Thermal determinant...... 167

D Supergoop Dynamics 169 D.1 Two-Body Problem...... 169

Bibliography 172

x List of Tables

xi List of Figures

2.1 Witten diagrams for the order λ contributions to ZAdS[ϕ~k, zc] in the theory with φ4 interaction...... 13

3.1 Examples of the radial deformations (3.3.1) on the left, (3.3.2) in the middle and (3.3.3) on the right. We have suppressed the polar coordinate θ of the S2 but kept the azimuthal direction...... 44 2 3.2 Plot of |ΨHH (λ, A)| for N = 2 for the Gaussian proﬁle (3.3.1) with

λ = 1 using lmax = 45. The solid blue line is an interpolation of the numerically determined points (shown in red). The wavefunction grows and oscillates in the negative A direction...... 45 2 3.3 Left: Density plot of |ΨHH (λ, a, A)| for N = 2 for the proﬁle (3.3.2) as

a function of A (vertical) and λ (horizontal) for a = 5 using lmax = 45. Again, the wavefunction grows and oscillates in the negative A and 2 positive λ directions. Right: Plot of |ΨHH (λ, a, A)| for the proﬁle (3.3.2) as a function of λ for A = −0.022 and a = 5...... 46 2 3.4 Left: Density plot of |ΨHH (λi, ai,Ai)| for N = 2 for the double Gaus-

sian proﬁle (3.3.3) as a function of A1 (x-axis) and A2 (y-axis) for

a1 = 0, a2 = 5, λ1 = λ2 = 1 using lmax = 45. The wavefunction

grows and oscillates for negative A1 and A2. Right: Density plot of 2 |ΨHH (λi, ai,Ai)| for the double Gaussian proﬁle (3.3.3) as a function

of λ1 (x-axis) and λ2 (y-axis) with A1 = −1, A2 = −1/100, a1 = 0 and

a2 = 5...... 47 3.5 The “balloon” deformation of R3, deﬁned by (3.4.1) and (3.4.3), represented schematically for positive ζ...... 49

xii 3.6 Plot of g(x) (left) and dg(x)/dx (right) as obtained by numerically solving equation 3.4.4( ζ∗ = 1741.51, ζ = 9ζ∗/10, γ = 1.36612, a = −95, λ = 30)...... 50 2 3.7 Density plot of the |ΨHH (ζ, m)| for N = 2 as a function of the pinching parameter ζ (vertical axis) and an overall mass deformation m

(horizontal axis) using lmax = 45, ζ∗ = 1741.51, γ = 1.36612, a = −95 and λ = 30...... 53 2 ∗ 3.8 Left: |ΨHH (ζ, m)| for N = 2 as a function of m for ζ = −ζ /2 (red dots), ζ = 0 (blue squares), ζ = ζ∗/2 (green diamonds) and ζ = 9ζ∗/10

(black triangles) using lmax = 45, ζ∗ = 1741.51, γ = 1.36612, a = −95 and λ = 30. Right: Same as left but for diﬀerent plot range...... 53 2 3.9 Left: Plot of |ΨHH (c1)| for N = 2 for the ﬁrst harmonic mapped to 3 2 R given in (3.5.2) using lmax = 45. Right: Plot of log |ΨHH (c1)| for

N = 2 using lmax = 45...... 55

3.10 Plot of log |ΨHH (ck)| for N = 2 for the ﬁrst nine spherical harmonics 3 mapped to R given in (3.5.2) using lmax = 45. Notice that only the

zeroth harmonic is non-normalizable in the negative c0 direction.... 55 2 3.11 |ΨHH (A)| (left) and log|ΨHH (A)| (right) for N = 2 as a function of A, the overall size of the radial deformation in (3.5.4) which is constructed

to be orthogonal to the zero mode of the three-sphere (lmax = 45)... 56 2 3.12 Plot of |ΨHH [mS3 ]| given by expression (3.6.1) for N = 2...... 58 2 3.13 Left: Density plot of |ΨHH [ρ, mS3 ]| for N = 2. The fainter peak centered at the origin reproduces perturbation theory in the empty de

Sitter vacuum. Horizontal lines are lines of constant mS3 and vertical 2 lines are lines of constant ρ. Right: Plot of |ΨHH [ρ, −2.25]| for N = 2. Notice that it peaks away from ρ = 0...... 59 2 3.14 Left: Density plot of |ΨHH [ρ, mS3 ]| for N = 2 for a slightly larger range. Notice the local de Sitter maximum visible in ﬁgure 3.13 is al-

ready too faint to be seen. Horizontal lines are lines of constant mS3

and vertical lines are lines of constant ρ. Right: Plot of 2 log |ΨHH [ρ, −4.5]| for N = 2...... 60

source σA over the whole three-sphere. We have normalized such that O(N) Zcrit [σA] = 1 at σA = 0...... 66

4.1 A 3-node quiver diagram which captures the ﬁeld content of the La-

grangian L = LV + LC + LW , each piece of which is given in (4.2.1), (4.2.2), and (4.2.5). This quiver admits a closed loop if κ1, κ2 > 0 and κ3 < 0...... 75 ˜ 4.2 Examples of H + a K eigenfunctions. Left: Plot of ψλ(x) for n ∈ ˜ (0.2, 5.2) in unit increments. Right: Plot of ψλ(x) for n ∈ (−5.7, −0.7) in unit increments...... 91 4.3 Thermal effective potentials of a two node quiver (θ = −1 and µ = 1). As the temperature is increased the system explores various thermal configurations of stable and metastable minima. From top left to bottom right the system is of type1 → 2 → 3 → 2 → 4a → 5...... 96 4.4 Thermal effective potentials of a two node quiver (for θ = −1 and µ = 1). Left: An example of phase type 4b. Right: A case where the potential of the supersymmetric minimum decreases as the temperature is increased. A similar observation was made for supersymmetric bound states in [208]...... 97 4.5 Thermal effective potentials of a two node quiver (θ = −1 and µ = 1). Asκ ˜ is increased we note that the first minimum disappears...... 99 4.6 Thermal potential along the scaling direction |qi| = λ κi for κ1 = κ2 = −κ3 = 1 at T = 0 (left) and T 6= 0 (right)...... 100 4.7 2-loop Feynman diagrams contributing to the δxδx term of the effective Lagrangian. Solid lines represent φ, while dotted lines correspond to ψ propagators...... 103

xiv 4.8 A schematic representation of a system in a mixed Higgs-Coulomb branch. The long arrows represent very massive strings. Note that

there is a closed loop connecting Γ1,Γ2 and Γ3...... 104

5.1 Examples of ground states for 100 electric Γe = (0, 1) plus 100 magnetic

Γm = (1, 0) particles...... 114

5.2 Left: Classical moduli space M in the δ1 − δ2 plane for θ3 6= 0. The nature of M for the diﬀerent regions is shown in ﬁgure 5.3. Right:

Classical moduli space for M with θ3 = 0. In regions i and ii the centers at z = a and z = −a are enclosed respectively...... 122

5.3 Classical moduli space in the δ1 − δ2 plane for θ3 6= 0. The order of the figures left to right starting at the top are the regions in figure 5.2... 123 5.4 Three node quiver with a closed loop (left) and without a closed loop (right)...... 125 5.5 The Euler-Jacobi flower. The red balls represent the fixed background centers and the blue line represents the classical trajectory of the probe. In this case, the trajectory precesses around only one of the fixed centers.126 5.6 Examples of closed phase space trajectories in the integrable probe regime. The plots show slices of phase space in the Cartesian coordinate system...... 128 5.7 Examples of open phase space trajectories in the chaotic regime. The plots show slices of phase space in the Cartesian coordinate system.. 129

xv 5.8 Poincar´esections of collinear setup with κ31 = 10, κ32 = −10, κ21 =

−10 and θ3 = −1, θ2 = 1 and energies E = {0.10, 0.20, 0.23, 0.26, 0.27, 0.30}. Note that the κ’s form a closed loop. The horizontal axis represents the position of particle 3 while the vertical axis represents its conjugate momentum. Any given plot is produced by varying the initial positions and momenta of the two probes subject to a ﬁxed total energy. The

pair (x3(t), p3(t)) is plotted every time the resulting trajectory of parti-

cle 2 crosses some ﬁducial point (x2(t) = xc) with positive momentum

(p2(t) > 0), i.e. roughly every time particle 2 completes a full cycle as it oscillates back and forth. In the quasi-integrable regime, different initial conditions correspond to different contours. The first Poincaré section shows a quasi-integrable behavior with two fixed points corresponding to the two low energy normal modes...... 132 5.9 The first row represents a low energy probe, which remains stuck in a subset of the phase space for seemingly arbitrarily long times. The second row represents an intermediate energy probe which illustrates the non-uniform escapes that occur from the low energy trapping behavior. We see that for a while it remains trapped in some subset of phase space, after which it escapes and gets stuck in some other subset of phase space. The final row represents a high energy probe which uniformly explores the molecule. The associated plots represent the percentage of the molecule explored as a function of the integration time, up to 15000 time steps in increments of 1500. The tapering off of the high energy probe is simply due to saturating the entire molecule. Below these points the increase is very uniform. The initial energy increases from 50% of the escape energy in the first row to 60% of the escape energy in the third row. These percentages, however, are very dependent on the parameters (e.g. κ, θ, etc.) in the problem...... 136

xvi 5.10 These contour plots show equipotential surfaces in the plane of a 2D molecule consisting of one hundred centers. From left to right, we have chosen κ = 1, κ = 1.5, κ = 3.5, and in all cases θ = −10. We observe that as the magnitude of κ increases, the minima (blue region), which initially lied near each center, are collectively expelled, forming an overall minimum that surrounds the molecule as a whole. For κ = 1, the trajectory remains close to the plane of the molecule and has been superimposed on the left contour plot (transparent white line). The axes label the x and y positions of the probe particle...... 137

2 B.1 Left: Comparison of |ΨHH (mS3 )| for N = 2 as obtained by calculating

ZCFT [mS3 ] analytically (blue line) and given in equation (3.6.1), and by numerically evaluating the functional determinant using the Dunne-

Kirsten regularization method with lmax = 45 (red dots). We have 2 normalized such that |ΨHH (mS3 )| = 1 at mS3 = 0. Right: Plot of the

percentage error as a function of the numerical cutoﬀ (at mS3 = −2.2). 147

C.1 Example of Feynman diagram contributing to DiDj from the superpotential...... 157

C.2 Plot of D0 for µ = −θ = κ = 1. Notice that the solution is real for all values of |q|...... 159

C.3 Left: Plot of <[D1] (blue) and <[D2] (violet). Right: Plot of =[D1]

(blue) and =[D2] (violet). Both plots are for µ = −θ = κ = 1. Notice that when complex, the solutions form a conjugate pair...... 159

C.4 Left: Plot of V evaluated on D0 for µ = −θ = κ = 1. Right: Plot of V evaluated on the perturbative D solution in violet, compared with

the full non-perturabative D0 in blue...... 160 2 C.5 1-loop Feynman diagrams contributing to the D1 term of the eﬀective Lagrangian...... 162

C.6 1-loop Feynman diagram contributing to the D1D2 term of the eﬀective Lagrangian...... 163

xvii C.7 1-loop Feynman diagrams contributing to the δx1δy2 term of the eﬀec- tive Lagrangian...... 163

C.8 1-loop Feynman diagrams contributing to the δx1δx2 term of the eﬀec- tive Lagrangian...... 164 a b C.9 1-loop Feynman diagrams contributing to the δx1 δx1 term of the eﬀective Lagrangian...... 165

D.1 Scatering angle...... 171

xviii Chapter 1

Introduction

Many experimental phenomena at very small scales are accurately described theoreti- cally using quantum field theory. As the name suggests, this theory assumes the world is composed of fluctuating fields which are quantized. Two simple examples are the electron and photon fields with their associated quanta. However, attempts to quan- tize the important gravitational field are obstructed by the non-renormalizability of the Einstein-Hilbert action. Renormalizability is an indication of our ability to use a theory to make predictions at smaller scales and higher energies. Therefore, the non- renormalizability of gravity implies that, at sufficiently high energies, we are unable to make any theoretical predictions. Nevertheless, the quantization of the gravitational field is a pressing problem. A consistent quantum theory of gravity will hopefully shed light on poorly understood experimental phenomena and theoretical paradoxes. These include the physics of black holes, dark energy, and the beginning and ultimate fate of our universe. The difficulties in quantizing gravity indicate that, at higher energies, new physics is required to provide a complete theoretical description. String theory is a possible completion. More technically speaking, string theory was initially understood as perturbative excitations of strings. Recently, a non-perturbative definition of string theory was provided in anti-de Sitter space by postulating its equivalence to a conformal field 5 theory of lower dimension. One example relates string theory on AdS5 × S to N = 4

1 CHAPTER 1. INTRODUCTION 2

super-Yang Mills theory, a supersymmetric, conformal gauge theory. The anti-de Sitter space/conformal field theory correspondence and the more general principle of relating theories of gravity to lower dimensional gauge theories appear under different names including AdS/CFT correspondence, holography, gauge/gravity duality and holographic duality. This dissertation will be focused on using the holographic principle to gain a better theoretical handle of a model of an expanding universe called de Sitter space, and disordered/glassy systems. For the former, our primary tool will be the conjectured de Sitter/conformal field theory (dS/CFT) duality, which is inspired by AdS/CFT. For disordered systems, we will use the fact that certain quantum mechanics models can capture aspects of the complex dynamics of bound black hole configurations. Experiments indicate that our universe experienced an exponential expansion at cosmologically early times called inflation, and is also currently undergoing an accelerated expansion. De Sitter space is the maximally symmetric geometry satisfying Einstein’s equations with a positive cosmological constant and plays a significant role in the theory of inflationary cosmology. Recent astronomical observations indicate our universe is entering a new asymptotically de Sitter phase, with a small positive value for the cosmological constant. Therefore, de Sitter space is more relevant to the evolution of our cosmos as it describes an expanding universe with positive cosmological constant, as opposed to anti-de Sitter space which requires a negative cosmological constant. In this context, one important problem is to understand the dynamics of quantum fields on the time-dependent de Sitter background. In Chapter2, we study a variety of fields in de Sitter space, and investigate the late-time evolution of their wavefunctions. We take a new approach at perturbatively analyzing well-known infrared divergences of light scalar fields, and use holography to argue against the existence of such divergences in pure Einstein theory. In Chapter3, we explore the validity of a conjectured holographic duality, between a Euclidean conformal field theory and a higher spin theory of gravity, called dS/CFT. In this case, the partition function of the CFT, which we find by numerically computing a functional determinant, is thought to be the same as the wavefunction CHAPTER 1. INTRODUCTION 3

of the gravity theory. Beyond the, global, wavefunction approach, one may want to use holography to understand an individual static patch observer in de Sitter space. The worldline data of a static patch observer is characterized by the emergence of an SL(2, R) symmetry. Interestingly, the static patch of four-dimensional de Sitter space is conformally 2 equivalent to AdS2 × S , whose SL(2, R) does not seem to reside within a larger structure containing a Virasoro algebra. It is unclear and worth investigating what a holographic dual may look like in cases when there is an SL(2, R) symmetry without a full Virasoro.

Motivated by these observations, and the additional importance of AdS2 as the near-horizon geometry of extremal black holes, in Chapter4 we consider the extent to which a quiver quantum mechanics system holographically captures the dynamics of AdS2 geometries and extremal black holes. More speciﬁcally, in a certain limit of the model, we show the emergence of an SL(2, R) symmetry. In Chapter5, we will consider multiparticle conﬁgurations of the quiver quantum mechanics system mentioned above, in order to study the low-energy dynamics of extremal black hole bound states. Stable bound states of extremal black holes have long relaxation times and a seemingly complicated free energy landscape. These general characteristics are notably shared by many disordered systems. Besides being poorly understood, disordered/glassy systems display many interesting phenomena. For example, when supercooled to form a glass, liquids often display diverging viscosities. Additionally, these systems are believed to break ergodicity. The question we will investigate is whether the Newtonian limit of a multiparticle quiver quantum mechanics system captures the complex dynamics of a disordered system, or at least some aspects of it. The work presented is based on the papers [1–4]. Chapter 2

Wavefunctions in de Sitter space

2.1 Introduction

The geometry of the inflationary epoch of our early universe was approximately de Sitter [5–9], and our universe is currently entering a de Sitter phase once again. It is thus of physical relevance to examine how to deal with quantum effects in a de Sitter universe. Such issues have been studied heavily in the past. The technical aspects of most calculations have involved the in-in/Schwinger-Keldysh formalism which is reviewed in [10], and focus on computing field correlations at a fixed time. Indeed, in the context of quantum cosmology we are interested in correlations of quantum fields at a given time rather than scattering amplitudes—which condition on events both in the far past as well as in the far future. A complementary approach is to build a perturbation theory for solutions of the Schrödingerequation itself. Knowledge of the wavefunction allows us to consider expectation values of a broad collection of observables, which in turn permits a richer characterization of the state [11]. Thus, an understanding of the wavefunction and its time evolution is of interest. Although generally complicated, there is one particular solution of the Schrödingerequation in a fixed de Sitter background which exhibits a simplifying structure. This solution is the Bunch-Davies/Hartle-Hawking wavefunction ΨBD [12–15], and its form strongly resembles that of the partition function in a Euclidean AdS background upon analytic continuation of the de Sitter length and

4 CHAPTER 2. WAVEFUNCTIONS IN DE SITTER SPACE 5

conformal time. This observation led to the elegant proposal of a close connection between dS and Euclidean AdS perturbation theory in [16] (see also [17,18]). It is our goal in this chapter to exploit the connection between dS and AdS to develop a more systematic perturbative framework for the construction of this wavefunction. We do this by considering a series of examples. The perturbative framework in an AdS spacetime has been extensively studied in the past [19–21] and is our primary calculational tool. Our examples involve self-interacting scalar fields, both massless and massive, as well as gauge fields and gravitons. We recast many of the standard issues involving infrared effects of massless fields1 in the language of the wavefunction. Many of these infrared effects exhibit correlations that grow logarithmically in the scale factor, as time proceeds and we display how such effects appear in the wavefunction itself. It is worth noting that though most calculations of ΨBD involve taking a late time limit, our approach requires no such limit and we construct ΨBD perturbatively for any arbitrary time. For massless scalar fields, the finite time dependence of the wavefunction at tree level is captured by the exponential integral function Ei(z), whose small argument behavior contains the logarithmic contributions. An interesting difference between the approach described here and the in-in formalism is that the two approaches use different propagators. For a massless scalar in 2 Euclidean AdS4, we use the Green’s function:

1 0 G (z, z0; k) = − (1 − kz)(1 + kz0)ek(z−z ) AdS 2k3L2 2kzc 0 e (1 − kz )(1 + kz)(1 + kz ) 0 − c e−k(z+z ) , (2.1.1) (1 + kzc) valid for z < z0. (For z0 < z, one simply exchanges the two variables.) The mathematical purpose of the second term is to enforce the Dirichlet boundary condition at the cutoff zc. It is perhaps more significant physically that the sum of the two terms is finite as k → 0. Thus, loop integrals using (2.1.1) do not produce infrared

1See [22–38] for an incomplete list of references on the topic of infrared issues in de Sitter space. 2The Euclidean AdS metric is ds2 = L2(dz2 + d~x2)/z2 and we work in momentum space. CHAPTER 2. WAVEFUNCTIONS IN DE SITTER SPACE 6

divergences at small k. The Green’s functions considered in the in-in formalism [37] are obtained by the continuation to dS4 of the ﬁrst term in square brackets. Its real part gives

1 G (η, η0; k) = (1 + k2ηη0) cos[k(η − η0)] + k(η − η0) sin[k(η − η0)] , (2.1.2) C 2k3`2 which is singular as k → 0. One of the main motivations of our approach is to connect our results with the idea [16,39,40] that Ψ BD (at late times) is holographically computed by the partition function of a conformal field theory. If this correspondence, known as the dS/CFT correspondence, is indeed true,3 infrared effects in de Sitter spacetime should be related to quantities in the putative conformal field theory itself. This could lead to a better understanding of possible non-perturbative effects. Moreover, in analogy with how the radial coordinate in AdS is related to some (as of yet elusive) cutoff scale in the dual CFT [46–49], it is expected that the scale factor itself is connected to a cutoff scale in the CFT dual to de Sitter space [50–54]. Our calculations may help elucidate such a notion. Of further note, having a better understanding of ΨBD at finite times allows us to compute quantum expectation values of fields within a single cosmological horizon, rather than metaobservables inaccessible to physical detectors.4 We begin in section 2.2 by explaining how solutions to the Schrödingerequa- tion can be captured by a Wick rotation to Euclidean time, hence establishing the connection between de Sitter and anti-de Sitter calculations. We then proceed in section 2.3 to examine a self-interacting scalar field with φ4 interactions in a fixed four-dimensional de Sitter background, whose contributions to the wave function contain terms that depend logarithmically on the conformal time η. In section 2.4 we discuss the case of gauge fields and gravitons. We argue that, to all orders in the tree-level approximation, no logarithms are present for a pure Einstein theory with a

3Recently several concrete realizations of this proposal have emerged [41–43] for theories of four- dimensional de Sitter space involving towers of interacting massless higher spin ﬁelds. Aspects of de Sitter holography are reviewed in [44, 45]. 4A complementary approach would be to compute quantities directly in the static patch of de Sitter [59, 60]. CHAPTER 2. WAVEFUNCTIONS IN DE SITTER SPACE 7

positive cosmological constant. We discuss our results in the context of holography in section 2.5. Finally, in section 2.6 we go to two-dimensional de Sitter space in order to compute loop eﬀects for a cubic self-interacting massless scalar. In appendix A.1 we set up a quantum mechanical toy model where the mathematics of our calculations is exhibited in a simple context.

2.2 The Schr¨odingerequation in a ﬁxed de Sitter background

The main emphasis of this section is to show that our method perturbatively solves the functional Schrödingerequation for a scalar field in the Bunch-Davies state. We will first provide the exact solution for a free field, and then show how the result can be obtained by continuation from Euclidean AdS as in [16]. We then treat interactions perturbatively.

We use conformal coordinates for dS(d+1),

`2 ds2 = −dη2 + d~x2 , ~x ∈ d , η ∈ (−∞, 0) . (2.2.1) η2 R

For simplicity we consider a self-interacting scalar but analogous equations will also hold for other types of ﬁelds. The action is:

(d−1) Z Z 2 ` dη 2 2 ` V (φ(η, ~x) SL = d~x (d−1) (∂ηφ(η, ~x)) − (∂~xφ(η, ~x)) − 2 . (2.2.2) 2 Rd |η| η

We specify the potential later, but we envisage the structure of a mass term plus φn interactions. It is convenient to take advantage of the symmetries of Rd and work in momentum space. Thus, we deﬁne:

Z ~ dk i~k·~x φ(η, ~x) = d e φ~k(η) . (2.2.3) Rd (2π) CHAPTER 2. WAVEFUNCTIONS IN DE SITTER SPACE 8

Henceforth we denote the magnitude of the momentum by k ≡ |~k|. Upon defining the canonical momenta π~k = −iδ/δφ~k conjugate to φ~k, we can write the Schrödinger equation governing wavefunctions Ψ[ϕ~k, η] in a fixed dSd+1 background:

2 !! X 1 |η|(d−1) `(d−1) k2 ` π π + ϕ ϕ + V˜ (ϕ ) Ψ[ϕ , η] 2 `(d−1) ~k −~k |η|(d−1) 2 ~k −~k |η| ~k ~k ~k∈Rd

= i ∂ηΨ[ϕ~k, η] . (2.2.4)

The variable ϕ~k is the momentum mode φ~k evaluated at the time η where Ψ is ˜ evaluated. The potential V (φ~k) is the Fourier transform of the original V (φ(η, ~x)); it has the structure of a convolution in ~k-space.

2.2.1 Bunch-Davies wavefunction

In principle, we can construct solutions to (2.2.4) by considering Feynman path integrals over the ﬁeld φ. We are particularly interested in the solution which obeys the Bunch-Davies boundary conditions. This state is deﬁned by the the path integral:

Z Y iS[φ~k] ΨBD[ϕ~k, ηc] = Dφ~k e , (2.2.5) ~k∈Rd

ikη in which we integrate over ﬁelds that satisfy φ~k ∼ e in the kη → −∞ limit and

φ~k(ηc) = ϕ~k at some fixed time η = ηc. The natural generalization of this state to include fluctuating geometry at compact slicing is given by the Hartle-Hawking wavefunction. The boundary conditions resemble those defined in the path integral construction of the ground state of a harmonic oscillator. As usual, physical expectation values are given by integrating over the wavefunction squared. For example, the n-point function of ϕ~k, all at coincident time ηc, is: R Q dϕ |Ψ [ϕ , η ]|2 ϕ . . . ϕ ~k ~k BD ~k c ~k1 ~kn hϕ~ . . . ϕ~ i = . (2.2.6) k1 kn R Q 2 ~k dϕ~k |ΨBD[ϕ~k, ηc]|

As a simple example we can consider the free massless field in a fixed dS4 background. In this case we can obtain ΨBD as the exact solution of the Schrödinger CHAPTER 2. WAVEFUNCTIONS IN DE SITTER SPACE 9

equation (2.2.4):

1/4 Y 2k3 i `2 k2 e−ikηc/2 ΨBD[ϕ~k, ηc] = exp ϕ~k ϕ−~k p . (2.2.7) π 2 ηc(1 − ikηc) (1 − ikηc) ~k

Although ηc can be considered to be an arbitrary point in the time evolution of the state, we are ultimately interested in the late time structure of the wave function.

At late times, i.e. small negative ηc we ﬁnd:

2 Z ~ 2 ` dk ik 3 log ΨBD[ϕ~k, ηc] = 3 − k ϕ~k ϕ−~k + .... (2.2.8) 2 (2π) ηc

Notice that the small ηc divergence appears as a phase of the wavefunction rather than its absolute value, i.e. it plays no role in the expectation values of the ﬁeld ϕ~k.

The late time expectation value of ϕ~k ϕ−~k is given by:

1 hϕ ϕ i = , (2.2.9) ~k −~k 2 `2 k3 which diverges for small k. The divergence stems from the fact that ΨBD is non- normalizable for the ~k = 0 mode.

2.2.2 A Euclidean AdS approach

When computing the ground state wavefunction of the harmonic oscillator from the path integral, one wick rotates time and considers a Euclidean path integral with boundary condition in the infinite past. Similarly, for the dS wavefunction, we can continue to Euclidean time z = −iη and consider a Euclidean path integral. Now, the path integral is over configurations that decay in the infinite Euclidean past, defined here as the limit z → ∞. If in addition we continue L = −i`, we see that the calculation becomes that of constructing the Euclidean partition function in a fixed

Euclidean AdS(d+1) background:

L2 ds2 = dz2 + d~x2 , ~x ∈ d , z ∈ (0, ∞) . (2.2.10) z2 R CHAPTER 2. WAVEFUNCTIONS IN DE SITTER SPACE 10

In other words we have that: ΨBD[ϕ~k, ηc] = ZAdS[ϕ~k, izc] (with L = −i`) at least in the context of perturbation theory in a fixed (A)dS background. The Euclidean path integral calculation incorporates, in principle at least, both classical and quantum effects. Let us ignore quantum effects temporarily and discuss how AdS/CFT works at the classical level. To be concrete, consider a massive scalar with quartic self-interaction. The classical action is

(d−1) Z Z ∞ 2 2 2 L dz 2 2 m L 2 λ L 4 S = d~x (d−1) (∂zφ) + (∂~xφ) + 2 φ + 2 φ . (2.2.11) 2 d z z 4z R zc

One seeks a solution φ(z, ~x) of the classical equation of motion that satisﬁes the boundary condition φ(z, ~x) → ϕ(~x) as z → zc. The cutoﬀ zc is needed to obtain correct results for correlation functions in the dual CFT. The classical solution is then substituted back in the action to form the on-shell action Scl[ϕ(~x)] which is a functional of the boundary data. In the classical approximation the partition function

−Scl[ϕ(~x)] is the exponential of the on-shell action i.e. ZAdS = e , and n-point correlation functions of the CFT operator dual to the bulk ﬁeld φ are obtained by taking n variational derivatives with respect to the sources ϕ(~x). Let us now perform the Euclidean version of the calculation that gives the result (2.2.7). For this purpose we ignore the quartic term in (2.2.11). In ~k-space, we wish to solve the previously mentioned boundary value problem5 captured by the classical equation of motion:

2 2 2 2 2 2 z ∂z − (d − 1)z∂z − (k z + m L ) φ~k(z) = 0 , φ~k(z = zc) = ϕ~k . (2.2.12)

The exponentially damped solution of the ODE involves the modiﬁed Bessel function

Kν(kz), and the solution of the boundary value problem can be neatly written as

d/2 √ z Kν(kz) 1 2 2 2 φ~k(z) ≡ K(z; k)ϕ~k = d/2 ϕ~k ν = d + 4m L . (2.2.13) zc Kν(kzc) 2

5There are many useful reviews of the AdS/CFT correspondence, including [55–57]. The present boundary value problem is discussed in Sec. 23.10 of [58]. CHAPTER 2. WAVEFUNCTIONS IN DE SITTER SPACE 11

This equation deﬁnes the important bulk-to-boundary propagator K(z, k). We follow the procedure outlined above and substitute the solution (2.2.13) into the action (2.2.11). After partial integration the on-shell action reduces to the surface term at z = zc:

1 Z d~k L(d−1) S [ϕ ] = − K(z; k) ∂ K(z; k) ϕ ϕ at z = z . (2.2.14) cl ~k 2 (2π)d z z ~k −~k c

Let’s restrict to the case of a massless scalar in AdS4 which is the case d = 3, ν = 3/2 of the discussion above. The Bessel function simpliﬁes greatly for half-odd-integer index, and the bulk-to-boundary propagator becomes:

(1 + kz)e−kz K(z; k) = −kz . (2.2.15) (1 + kzc)e c

The on-shell action then becomes:

Z ~ 2 2 1 dk L k zc Scl[ϕ~k] = 3 ϕ~k ϕ−~k . (2.2.16) 2 (2π) zc 1 + kzc

To discuss the AdS/CFT interpretation we need to take the small zc limit, which gives: Z ~ 2 1 dk L 2 3 2 3 Scl → 3 k zc − k zc + O(zc ) ϕ~kϕ−~k . (2.2.17) 2 (2π) zc 2 The ﬁrst term is singular as zc → 0, but the factor k ϕ~kϕ−~k is local in ~x-space. In fact it contributes a contact term δ(~x−~y) in the ~x-space correlation function. Such contact terms are scheme-dependent in CFT calculations and normally not observable. The remaining ﬁnite term has the non-local factor k3. It’s Fourier transform gives the 6 observable part of the 2-point correlator, hO3(x)O3(y)i ∼ 1/|~x − ~y| which is the power law form of an operator of scale dimension ∆ = 3. In AdS/CFT a bulk scalar 2 of mass m is dual to a scalar operator O∆ of conformal dimension ∆ = (d/2 + ν). It is more pertinent to discuss the relation between the Lorentzian and Euclidean signature results. In the free Lorentzian theory we can write ΨBD = exp(iSL) . Then upon continuation L → −i`, z → −iη, zc → −iηc, the Euclidean on-shell action CHAPTER 2. WAVEFUNCTIONS IN DE SITTER SPACE 12

(2.2.16) and its Lorentzian counterpart are related by

− SE ≡ −Scl → iSL . (2.2.18)

This is the expected relation for ﬁeld theories related by Wick rotation. Henceforth, the Euclidean signature AdS/CFT correspondence will be our primary method of computation. In this way we will be using a well developed and well tested formalism. After completion of a Euclidean computation, we will continue to de Sitter space and interpret the results as contributions to the late time wave function ΨBD.

2.2.3 Interaction corrections to ZAdS

We now consider the eﬀect of interactions in the bulk action, such as, for example, the φ4 term in (2.2.11). We treat the quantum ﬂuctuations using a background

field expansion φ = φcl + δφ. The classical field satisfies the non-linear classical equation of motion with Dirichlet boundary condition limz→zc φcl(~x,z) = ϕ(~x), while the fluctuation δφ vanishes at the cutoff. The partition function is then: Z − Scl −S[δφ,φcl] ZAdS[ϕ~k, zc] = e Dδφ e . (2.2.19)

Exact solutions of the non-linear classical equation are beyond reach, but the reason- ably eﬃcient perturbative formalism of Witten diagrams leads to series expansions in the coupling constant. For example, φcl = φ0 + λφ1 + ..., where φ0 solves the free equation of motion coming from the quadratic piece of the action and the full Dirichlet boundary condition.6

Witten diagrams without loops contribute to Scl, while those with internal loops appear in the perturbative development of the fluctuation path integral. The basic building blocks of Witten diagrams are the bulk-to-bulk Green’s function G(z, w; k) and the bulk-to-boundary propagator K(z; k). It is significant that G satisfies the

Dirichlet condition at the cutoﬀ, i.e. G(zc, w; k) = G(z, zc; k) = 0. In Appendix

6In Appendix A we present and develop a quantum mechanical toy model. This model is instructive because perturbative computations are quite feasible and their structure is closely analogous to those in our ﬁeld theories. CHAPTER 2. WAVEFUNCTIONS IN DE SITTER SPACE 13

B, these propagators are explicitly constructed for the main cases of interest in this chapter.

Witten diagrams for the order λ contributions to ZAdS[ϕ~k, zc] in the theory with φ4 interaction are depicted in Fig. 2.1.

Fig. 2.1: Witten diagrams for the order λ contributions to ZAdS[ϕ~k, zc] in the theory with φ4 interaction.

2.3 Self-interacting scalars in four-dimenions

4 We now discuss several contributions to ΨBD from interactions, mostly in φ theory.

As previously mentioned, we carry out calculations in Euclidean AdS4, then continue to dS4 by taking z = −iη, zc = −iηc and L = −i` . We use the metric (2.2.10) and action (2.2.11) in d = 3.

2.3.1 Tree level contributions for the massless theory

First we focus on the massless case m2L2 = 0. The relevant bulk-to-boundary propagator is given in (2.2.15). The tree-level contribution to ΨBD (left of Fig. 2.1) is captured by the integral:7

4 λ L4 Z ∞ Y dz − K(z; k ) = 8 z4 i zc i=1 4 2 2 2 3 kΣzc 3 λL kΣ + kΣ zc + kΣ (3kπ − kΣ) zc + 3kP zc − e kΣ kΣ3 zc Ei(−kΣzc) − 3 , (2.3.1) 8 3 kΣ zc (1 + k1zc)(1 + k2zc)(1 + k3zc)(1 + k4zc)

7 R ∞ −t The Ei(z) function is deﬁned as Ei(z) = − −z dt e /t. It has a branch cut along the positive real axis of z ∈ C. We are primarily interested in this function along the negative real axis and the negative imaginary axis, both away from the origin. CHAPTER 2. WAVEFUNCTIONS IN DE SITTER SPACE 14

where we have deﬁned the following quantities:

4 4 4 4 X X Y X 3 kΣ ≡ ki , kπ ≡ kikj , kP ≡ ki , kΣ3 ≡ ki . (2.3.2) i=1 i=1 i=1 i=1 j=i+1

We expand the result for small values of zc and analyze the divergent structure. In −3 2 −1 −2 summary, we ﬁnd terms of order ∼ zc as well as ∼ k zc divergence but no ∼ zc 3 term. Furthermore, we ﬁnd a ∼ k log k zc term.

Upon analytic continuation to dS4 the power law divergences become phases of the wavefunction. On the other hand, the logarithmic term contributes to the absolute value of ΨBD[ϕ~k, ηc] at small ηc:

log |ΨBD[ϕ~k, ηc]| = 4 Z ~ ~ ~ λ ` dk1 dk2 dk3 (kΣ3 log(−kΣ ηc) + ...) ϕ~ ϕ~ ϕ~ ϕ~ , (2.3.3) 24 (2π)3 (2π)3 (2π)3 k1 k2 k3 k4

P ~ where i ki = 0 due to momentum conservation. Thus we encounter contributions to |ΨBD[ϕ~k, ηc]| that grow logarithmically in the late time limit, |ηc| → 0. In fact, at late enough times the correction is no longer a small contribution compared to the λ = 0 pieces, and all subleading corrections will also begin to compete. In this way one recasts several of the infrared issues encountered when studying massless fields in the in-in/Schwinger-Keldysh formalism [10, 24]; now from the viewpoint of the wavefunction. Similar logarithmic terms are present at tree level in a cubic self-interacting massless theory, and their effect was noted in the context of non-Gaussian contributions to inflationary correlators in [64]. In this case one finds the late time correction:

4 Z ~ ~ λ ` dk1 dk2 log |ΨBD[ϕ~ , ηc]| = − (kΣ3 log(−kΣ ηc) + ...) ϕ~ ϕ~ ϕ~ , (2.3.4) k 6 (2π)3 (2π)3 k1 k2 k3

P ~ where i ki = 0 , and kΣ, kΣ3 are defined as in (2.3.2). In the case of slow roll inflation, these infrared effects are suppressed by the small slow roll parameters [16]. CHAPTER 2. WAVEFUNCTIONS IN DE SITTER SPACE 15

As was mentioned in the introduction, the dS/CFT proposal connects ΨBD to the partition function of a conformal ﬁeld theory. Here, one envisions some theory in de Sitter space that contains such light scalars in its spectrum, including the graviton (dual to the stress tensor of the CFT) and so on. In section 2.5 we will explore this connection and in particular, discuss a possible holographic interpretation of such divergences based on recent analyses of 3d CFT’s in momentum space [62,63].

2.3.2 Loop correction to the two-point function

It is of interest to understand the late time structure of loop corrections in the φ4 model. We will calculate the diagram on the right in Fig. 2.1, which corresponds to the following integral:

4 Z ∞ Z 3 λ L dz 2 d~p Iloop(k, zc) = − 4 K(z; k) 3 G(z, z; p) . (2.3.5) 4 zc z (2π)

To render the ~p-integral finite we must impose an ultraviolet cutoff. Recall that ~p is a coordinate momentum, such that the physical (proper) momentum at a given z is given by ~pph = z ~p/L . We impose a hard cutoff on ~pph, such that the ultraviolet cutoff of ~p is z-dependent, i.e. |~pUV | = |ΛUV L|/z . A large ΛUV expansion reveals 8 terms that diverge quadratically and logarithmcally in ΛUV :

3 λ L2 −|Λ L|2 + 2 log |Λ L| . (2.3.6) 8(2π)2 UV UV

To cancel the quadratic divergence, we can add a local counterterm:

Z dz d~k 3 λ L2 δ φ (z) φ (z) , δ = |Λ L|2 . (2.3.7) z4 (2π)3 ~k −~k 8(2π)2 UV

8It is worth comparing the divergence structure in (2.3.6) to a coincident point expansion of 1 3 the SO(4, 1) invariant Green’s function: G(u) ∼ L2 (2/u) F (3, 2, ; 4; −2/u). The argument u = [(z − z0)2 + (~x − ~x0)2]/2zz0 is an SO(4, 1) invariant variable. Near u = 0, we write z = z0 and 0 1 2 2 ~x = ~x + ~. In this limit G(u) ∼ L2 [−z / + 2 ln(/z) + ...] . This is precisely of the form (2.3.6), although the divergence is cut off by the physical length ~xph,UV = ~L/z. What we are suggesting is that the physical cutoff is a de Sitter invariant cutoff. CHAPTER 2. WAVEFUNCTIONS IN DE SITTER SPACE 16

Upon addition of the counterterm, the ~p-integral can be performed analytically rendering an expression containing the Ei(z) function that is only logarithmically divergent in |ΛUV L|. The remaining z-integral is complicated, but we are mainly interested in its small zc behavior, which we can extract. We ﬁnd the following terms divergent in the small zc expansion (to leading order in ΛUV ):

2 2 λ L 1 3k 3 − 2 log|ΛUV L| − 3 + + k log(zc k) , (2.3.8) 2(2π) zc 2zc

The logarithmic term contributes to the absolute value of the wavefunction upon analytic continuation to dS4:

`2 Z d~k λ log |Ψ [ϕ , η ]| = − k3 1 − log |Λ `| log(−η k) + ... ϕ ϕ . BD ~k c 2 (2π)3 (2π)2 UV c ~k −~k (2.3.9)

Notice that at late times, the width of the |ΨBD[ϕ~k, ηc]| for a fixed k mode narrows, which is physically sensible as the quartic part of the potential dominates compared to the kinetic term. To order λ, the “cosmological two-point correlation function” can be obtained from this wave function (including the contribution from (2.3.3)) via the general expression (2.2.6). The result closely resembles the late time two-point function computed, for example, in [37]. Notice that there is no need to impose an infrared cutoff when considering loop corrections of the wavefunction itself. As a final note, we could have also considered a slightly different subtraction where our counterterm also removes the logarithic divergence in |ΛUV L|. Evaluation of the integrals proceeds in a similar fashion leading to the following result upon continuation to dS4:

`2 Z d~k λ log |Ψ [ϕ ]| = − k3 1 + a k3 log(−η k) + ... ϕ ϕ , BD ~k 2 (2π)3 1 (2π)2 c ~k −~k (2.3.10) where a1 = −(−5 + 4γE + 4 log 2)/4 ≈ −0.02 . The result is now independent of the ultraviolet cutoﬀ altogether. CHAPTER 2. WAVEFUNCTIONS IN DE SITTER SPACE 17

2.3.3 Tree level contributions to the conformally coupled case

We now analyze a conformally coupled scalar in a ﬁxed Euclidean AdS4 background with m2L2 = −2. This case is of particular interest as it arises in the context of higher spin Vassiliev (anti)-de Sitter theories [65, 66]. The bulk-to-boundary propagator simpliﬁes to: z K(z; k) = ek(zc−z) , (2.3.11) zc and the free quadratic on-shell classical action is given by:

L2 Z d~k k 1 Scl = 3 ϕ~k ϕ−~k 2 − 3 . (2.3.12) 2 R3 (2π) zc zc

n For the sake of generality, we consider a self-interaction of the form λnφ(~x,z) /2n n with n = 3, 4,... For such a theory, the order λn tree level (ϕ~k) contribution requires computing integrals of the form:

Z ∞ dz z n 1 kΣ(zc−z) kΣ zc In(ki, zc) = 4 e = 3 e E(4−n)(kΣ zc) , (2.3.13) zc z zc zc

9 where En(z) is the exponential integral function and kΣ ≡ k1 + k2 + ... + kn. Ex- panding the integral reveals that logarithms will only occur in the small zc expansion for the case n = 3. For n = 3 we ﬁnd the following small zc expansion:

γE + log(kΣzc) kΣ(−1 + γE + log(kΣzc)) In=3(ki, zc) = − 3 − 2 zc zc 2 kΣ(−3 + 2γE + 2log(kΣzc)) 1 3 − − kΣ(−11 + 6γE + 6log(kΣzc)) . (2.3.14) 4zc 36

9 R ∞ −zt n The function En(z) = 1 dte /t for z ∈ C. It has a branch cut along the negative real axis. We are mostly interested in this function along the positive real axis and positive imaginary axis, both away from the origin. CHAPTER 2. WAVEFUNCTIONS IN DE SITTER SPACE 18

When we continue to dS4 by taking zc = −iηc and L = −i`, the leading contribution 3 to the order (ϕ~k) piece of the wavefunction at small ηc is given by:

λ `4 Z d~k d~k 1 π log Ψ(3) = − 3 1 2 ϕ ϕ ϕ −i γ − i log(−k η ) + , BD 3 3 ~k1 ~k2 ~k3 3 E Σ c 6 (2π) (2π) ηc 2 (2.3.15) ~ ~ ~ with k1 + k2 = −k3 due to momentum conservation (see [61] for related calculations). 3 We see that the absolute value of the wavefunction receives a ∼ 1/ηc divergent piece which is momentum independent (such that it becomes a contact term in position space). Interestingly, the cubic self-interaction of the conformally coupled scalar is absent in the classical Vasiliev equations [67,68]. As another example, consider the quartic coupling which is conformal in four- dimensions. We ﬁnd: 1 In=4(ki, zc) = 4 . (2.3.16) zc kΣ

Upon continuing to dS4 this gives a momentum-dependent contribution to the real part of the exponent of the wavefunction, but none to the phase. One can also consider loop corrections analogous to those computed in section 2.3.2. As an example we consider the one loop correction to the two-point function in the φ4 theory. The relevant Green function is given by:

wz G(z, w; k) = e−k(w+z) e2kz − e2kzc , z < w , (2.3.17) 2kL2 and similarly for z > w. The relevant integral is (2.3.5), though in this case there 2 is only a quadratic divergence ΛUV to be cancelled. A small zc expansion of the regulated integral reveals the following contribution to the wavefunction:

2 Z ~ ` dk k 3λ4 log |ΨBD[ϕ~k, ηc]| = − 3 2 1 − 2 log(−kηc) + ... ϕ~k ϕ−~k . (2.3.18) 2 (2π) ηc 4(2π)

Once again, we see that the wavefunction becomes narrower as time proceeds which is physically sensible. CHAPTER 2. WAVEFUNCTIONS IN DE SITTER SPACE 19

2.3.4 Comments for general massive ﬁelds

We discuss the non-interacting case (with λ = 0) but non-zero mass. The solutions of the Klein-Gordon equation are given by:

3/2 r z Kν(kz) 9 2 2 φ~k(z) = ϕ~k , ν ≡ + m L . (2.3.19) zc Kν(kzc) 4

Once again we have imposed that the solution vanishes at z → ∞. We are interested in the regime ν ∈ [0, 3/2], since this range corresponds to light non-tachyonic scalars q 9 2 2 in dS4 upon analytic continuation (such that ν = 4 − m ` ). Heavy particles in dS4 have pure imaginary ν. The on-shell action is found to be:

2 Z L d~k (3 − 2ν) k K(ν−1)(kzc) Scl = − 3 ϕ~k ϕ−~k 3 − 2 . (2.3.20) 2 (2π) 2 zc zc Kν(kzc)

For generic values of ν we can expand the action at small zc and ﬁnd:

2 Z ~ 2ν ! L dk 1 (3 − 2ν) 2π csc(πν) kzc Scl = − 3 ϕ~k ϕ−~k 3 − 2 + ... . (2.3.21) 2 (2π) zc 2 Γ(ν) 2

2ν The above diverges at small zc, in the region ν ∈ (0, 3/2), even for the ∼ k piece. In the context of AdS/CFT the boundary data for a scalar field with ∆−d = ν−d/2 6= 0 d−∆ must be “renormalized” via ϕ~k → zc ϕ~k to achieve finite correlation functions as the cutoff zc → 0. (See Sec. 23.10 of [58] for a discussion.) In the conformally coupled case discussed in section 2.3.3, ∆ = 2 and d = 3 such that ϕ~k → zc ϕ~k . This 2 3 renormalization absorbs the 1/zc divergence. Upon continuing to dS4 the ∼ 1/zc term 2(ν−3/2) in (2.3.20) becomes a phase and we find a factor (i|ηc|) which has a growing real part as |ηc| → 0.

A small zc expansion in the ν = 0 case reveals ∼ log kzc terms in addition to 3 the 1/zc divergence. Only the logarithmic term contributes to the absolute value of ΨBD[ϕ~k, η] upon continuing zc = −iηc and L = −i`. In addition, we have that for ν = 1 there are also logarithmic terms in the small zc expansion. These become phases upon continuing zc = −iηc. CHAPTER 2. WAVEFUNCTIONS IN DE SITTER SPACE 20

Tree-level diagrams

Once again, we can ask whether the presence of logarithmic contributions to the late time wavefunction occur for more general values of ν. The general bulk-to-boundary propagator is: z 3/2 K (kz) K(z; k) = ν . (2.3.22) zc Kν(kzc) n Consider again self-interactions of the simple form λnφ(~x,η) /2n. The tree level integrals of interest are:

4 Z ∞ n L λn dz Y I(ν)(z , k ) = K(z; k ) . (2.3.23) n c i 2n z4 i zc i=1

For generic ν, we will find that at small zc the non-local piece in momentum will be n(ν−3/2) accompanied by a divergent factor zc . Upon continuation to dS4, the local 3 pieces which go as 1/zc or 1/zc will become phases of the wavefunction. On the other n(ν−3/2) hand zc will not contribute a pure phase to the wavefunction. However, upon n computing a physical expectation value of (ϕ~k) by integrating over the tree level 2 n(3/2−ν) |ΨBD| one finds that it decays as |ηc| at late times. That the correlations decay in time for massive fields makes physical sense, since the particles dilute due to the expansion of space, and is consistent with a theorem of Weinberg [22].

On the other hand, an examination of the small z behavior of the Bessel Kν(kz) function:

(kz)ν Γ(−ν) (kz)2 (kz)−ν Γ(ν) (kz)2 K (kz) = 1 + + ... + 1 + + ... ν 21+ν 2(1 + ν) 21−ν 2(1 − ν) (2.3.24) reveals that logarithmic terms can only occur of special values of ν. They can only appear when the integrand of (2.3.23) contains terms that go as 1/z in its small z expansion, which integrate to a logarithm. For n = 3, we have already discussed the massless ν = 3/2 and conformally coupled ν = 1/2 cases at tree level, as well as the ν = 0 and ν = 1 cases at the free level. For general n, ν = 3/2 will still give rise to logarithmic contributions, as will ν = (3/2−3/n), where the logarithmic contributions P 6/n are of the form ∼ log( i ki zc)/zc and ν = (3/2 − 1/n), where the logarithmic CHAPTER 2. WAVEFUNCTIONS IN DE SITTER SPACE 21

P contributions are of the form ∼ log( i ki zc)/zc (in the range ν ∈ [0, 3/2]). In the latter case, the logarithmic contribution always appears as a phase upon analytic continuation to dS, not so for the former case. It would be interesting if these are the only values of ν that give logarithms to higher order in perturbation theory.

2.4 Gauge ﬁelds and gravity in four-dimensions

In this section we consider classical contributions to the de Sitter wave function for massless gauge fields and gravitons in a fixed dS4 background. Compared with scalars treated in earlier sections, there are significant changes in the structure of the wave functions because of the gauge symmetry.

2.4.1 SU(N) gauge ﬁelds

In the non-Abelian case, the Yang-Mills action is given by:

4 Z Z ∞ L dz a a µρ νσ a a abc b c SYM = d~x 4 Tr FµνFρσg g ,Fµν = ∂[µAν] + g f AµAν , (2.4.1) 4 3 z R zc where a = 1,...,N 2 − 1 is the adjoint index and f abc are the SU(N) structure constants. This action is conformally invariant at the classical level. This means that there will be no singular terms in 1/zc in AdS vertex integrals and thus no terms in the de Sitter wave function that are logarithmically sensitive to ηc. The reason for this is that the two inverse metrics in the action (2.4.1) soften the vertex integrals by a factor of z4 and cancel the 1/z4 from the metric determinant.

We perform calculations in AdS4 in the Az = 0 gauge, so only transverse spatial components of the gauge potential remain. In ~k-space these components are given by

~ ~ ~ ~ ∗ ~ Ai(z, k) = K(z; k)ci(k), kici(k) = 0, ci(k) = ci(−k) (2.4.2) K(z; k) = e−k(z−zc) . (2.4.3)

The bulk-to-boundary propagator is so simple because Ai obeys the same linearized equation as in ﬂat space. Although usually not written explicitly, the transverse CHAPTER 2. WAVEFUNCTIONS IN DE SITTER SPACE 22

2 projector Πij = δij − kikj/k is understood to be applied to spatial vector modes. As our first calculation, we obtain the contribution of the free gauge field. Metric factors cancel and we have the gauge-fixed action

Z Z ∞ 1 2 1 2 S = d~x (∂zAi) + (Fij) . (2.4.4) zc 2 4

After partial integration, as in Sec. 2.2.2, the on-shell action reduces to the surface term in ~k-space:

1 Z d~k S = − K(z, k)∂ K(z, k)c (~k)c (−~k) (2.4.5) 2 (2π)3 z i i 1 Z d~k = k(δ − k k /k2)c (~k)c (−~k) . (2.4.6) 2 (2π)3 ij i j i j

The result contains the ~k-space correlator of two conserved currents in the boundary

3d CFT. This structure, which contains no dependence on zc may be compared with its analogue in (2.3.11) for the conformally coupled scalar. The bulk ﬁelds φ and

Ai are both dual to CFT operators with ∆ = 2. There is only partial cancellation 2 of metric factors for the scalar, so the singular factor 1/zc remains. As discussed in Sec. 2.3.4, this factor can be absorbed by renormalization of sources in AdS, but it gives a late-time power law singularity in |ΨBD|. Next consider the tree-level three-point function. The relevant integral is straightforward and gives a result with no dependence on zc, namely:

T g f abc ijk (2.4.7) k1 + k2 + k3 where Tijk is the same antisymmetric tensor that appears in ﬂat space, namely:

~ ~ ~ ~ ~ ~ Tklm = (k1)lδkm − (k1)mδkl + (k2)mδkl − (k2)kδlm + (k3)kδlm − (k3)lδkm . (2.4.8)

Comparing (2.4.7) to the three point function of the conformally coupled scalar in

(2.3.14) we note the absence of logarithmic terms depending on zc. CHAPTER 2. WAVEFUNCTIONS IN DE SITTER SPACE 23

2.4.2 Scalar QED

Consider now a massive charged scalar ﬁeld coupled to a U(1) gauge ﬁeld, with Euclidean action:

Z Z ∞ 4 dz αβ ∗ 2 ∗ SSQED = L d~x 4 g (∂α + iAα)φ(∂β − iAβ)φ + m φ φ . (2.4.9) 3 z R zc

Properties of this theory were also considered in [32]. Transverse modes in ~k-space thus have the cubic interaction:

Z ~ ~ Z ∞ 2 dk1 dk2 dz ~ ~ ~ ∗ Sint = L Ai(z, k3)(k1 − k2)i φ~ (z) φ (z) , (2.4.10) 3 3 2 k1 ~k2 (2π) (2π) zc z

~ ~ ~ where momentum conservation requires k3 = −k1 −k2 . Again, a transverse projector is understood to be applied to spatial vector modes. Using this interaction vertex, we ﬁnd the following contribution to the partition function:

Z ~ ~ 2 dk1 dk2 ˜ ~ ~ ~ ˜ ~ ~ L ϕ~ ϕ~ Ai(k3)(k1 − k2)i Im2L2 , Ai(k) ≡ Ai(zc, k) . (2.4.11) (2π)3 (2π)3 k1 k2

For a scalar ﬁeld of mass m2L2 and bulk-to-boundary propagator K(z; k) the radial integral is Z ∞ dz −k(z−zc) 2 2 Im L = 2 e K(z; k1)K(z; k2) . (2.4.12) zc z We compare the two cases of massless and conformally coupled (m2L2 = −2) scalars with bulk-to-boundary propagators:

−kz (1 + kz)e z −k(z−zc) 2 2 2 2 Km L =0(z; k) = −kz ,Km L =−2(z; k) = e . (2.4.13) (1 + kzc)e c zc

Our motivation is to explore the appearance of log(kzc) terms in the 3-point function. For the massless case we ﬁnd:

1 k1k2 (k1+k2+k3)zc + + e k3Ei[−(k1 + k2 + k3)zc] zc k1+k2+k3 Im2L2=0 = (2.4.14) (1 + k1zc)(1 + k2zc) CHAPTER 2. WAVEFUNCTIONS IN DE SITTER SPACE 24

whose series expansion reveals a logarithmic term from the Ei(z)-function. For m2L2 = −2, the integral is elementary and gives:

Z ∞ e−(k1+k2+k3)(z−zc) 1 2 2 Im L =−2 = dz 2 = 2 . (2.4.15) zc zc (k1 + k2 + k3)zc

As in the case of the conformally coupled self-interacting scalar, we can absorb the 2 1/zc divergence into a renormalization of the boundary data.

After all is said and done, we ﬁnd a ∼ log(kzc) term in the 3-point function of the massless scalar but not in the conformally coupled case. The “practical” reason for the absence is the cancellation of the ∼ 1/z factors in the integrand of (2.4.15) due to the softer behavior of the scalar bulk-to-boundary propagators. It would be interesting to study the loop corrections to the wave function for these theories.

2.4.3 Gravity

It is a well known result that classical solutions in pure Einstein gravity with a positive cosmological constant Λ = +3/`2 have a uniform late time (small η) expansion. In four spacetime dimensions, this is given by [70]:

ds2 dη2 1 = − + g(0) + η2g(2) + η3g(3) + ... dxidxj , |η| 1 . (2.4.16) `2 η2 η2 ij ij ij

The independent data in this expansion is the conformal class

(0) (3) ω(~x) (0) (3) gij , gij ∼ e gij , gij . (2.4.17)

(3) The Einstein equations impose that gij is transverse and traceless with respect to (0) the boundary three-metric gij . Two of the phase space degrees of freedom reside (0) (3) in gij and the other two in gij . The Einstein equations also require that the term linear in η inside the parenthesis is absent. If g(0) and g(3) are appropriately related, the above solution will obey the Bunch-Davies boundary condition (this will require (3) gij to be complex). If Λ < 0 there is an analogous expansions of the same structure known as the CHAPTER 2. WAVEFUNCTIONS IN DE SITTER SPACE 25

Feﬀerman-Graham expansion [69]. The on-shell action for such solutions satisfying the Bunch-Davies boundary condition (i.e. that the three-metric vanishes at large z in EAdS) has been studied extensively [71]. Indeed, the on-shell classical action is given at some ﬁxed z = zc by:

Z Z ∞ Z √ 3 √ 1 i Sgr = 2 d~x dz g − 2 d~x hKi , (2.4.18) 8πGL M zc 8πGL ∂M where hij is the induced metric on the ﬁxed zc slice and Kij is the extrinsic curvature,

1 α K = L α g (z, ~x) , n = (z,~0) . (2.4.19) ij 2 n ij

The second term in (2.4.16), known as the Gibbons-Hawking term, is required for a well deﬁned variational principle. For the ﬁrst term we have used the on-shell condition R = −12/L2. We can evaluate the classical action (2.4.18) on the classical solutions obeying the

Euclidean AdS4 analogue of (2.4.16):

ds2 dz2 1 = + g(0) + z2g(2) + z3g(3) + ... dxidxj , z 1 . (2.4.20) L2 z2 z2 ij ij ij and expand in small zc. The expansion of the on-shell classical action contains only 3 2 divergences of the form 1/zc and 1/zc at small zc, but no 1/zc or log zc divergences [72]. The absence of a ∼ 1/z term in (2.4.20) is crucial for the logs to be absent in the 10 small zc expansion of the on-shell classical action. The divergent terms amount 11 to pure phases in ΨBD[gij, η] upon analytic continuation. (See [75] for a related discussion.) The important point is that there are no logarithmic divergences for small zc, which translates to the statement that the Bunch-Davies wavefunction exhibits 3 no ∼ k log(−ηc k) growth at tree level. 10Note that in odd space-time dimensions, there is a piece of the Feﬀerman-Graham expansion which contributes logarithmic terms to the phase of the wavefunction as well as local terms to its absolute value [16]. 11 2 2 2 2 i j There is a slight subtlety in assuming that the full solution ds /` = −dη /η + gij(~x,η)dx dx allowing for the expansion (2.4.16) can indeed by analytically continued to z = −iη at the non- (0) linear level. For small enough deviations away from the ﬂat metric gij = δij the bulk-to-bulk and bulk-to-boundary propagators allow for such a continuation. CHAPTER 2. WAVEFUNCTIONS IN DE SITTER SPACE 26

Thus the Feﬀerman-Graham expansion for dS4 seems to explain the absence of logarithms in the gravitational 3-point functions calculated, for example, in [73, 74]. This is in stark contrast to the case of the massless scalar.

2.5 3d CFTs and (A)dS/CFT

The dS/CFT correspondence proposes that the Bunch-Davies (or Hartle-Hawking) wavefunction of dS4 at late times is computed by the partition function of a three- dimensional Euclidean conformal field theory. It is closely related to the Euclidean AdS/CFT proposal, as we have tried to make clear above. In the AdS/CFT context the small zc cutoff is identified with a cutoff in the dual theory. This is due to the manifestation of the dilatation symmetry as the (z, ~x) → λ(z, ~x) isometry in the bulk.

For instance, bulk terms that diverge as inverse powers of zc (with even powers of k) are interpreted as local terms in the dual theory. On the other hand, the tree level zc-dependent logarithmic terms, such as those in the small zc expansion of (2.3.1), are not local in position space and yet seem to depend on the cutoﬀ. One may ask whether they have an interpretation from the viewpoint of a putative CFT dual. Recent analyses of CFT correlation functions in momentum space [62, 63] give a suggestive answer. Recall that the symmetries of CFTs have associated Ward identities, governing correlation functions. For concreteness we speciﬁcally consider the Ward identities, expressed in momentum space, constrainging the three point functions of a scalar operator O with weight ∆. The Ward identity for the dilatation symmetry is given by:

3 ! X 6 + (pj∂j − ∆) hO(~p1)O(~p2)O(~p3)i = 0 , (2.5.1) i=1 whereas for the special conformal transformations we have:

3 X 4 − 2∆ (~p )α ∂2 + ∂ hO(~p )O(~p )O(~p )i = 0 . (2.5.2) i i p i 1 2 3 i=1 i CHAPTER 2. WAVEFUNCTIONS IN DE SITTER SPACE 27

The Latin index labels a particular momentum insertion, O(~pi), whereas the Greek index labels a speciﬁc Euclidean component of ~pi. We have also removed the δ(~p1 +

~p2 + ~p3) conservation rule from the correlator. The solution to the above equations is most conveniently expressed as an integral over an auxiliary coordinate:

hO(~p1)O(~p2)O(~p3)i = Z ∞ ∆−3/2 1/2 c3 (p1p2p3) dz z K∆−3/2(z p1) K∆−3/2(z p2) K∆−3/2(z p3) , (2.5.3) 0 where Kν(z) is the modiﬁed Bessel function of the second kind. The above integral should look familiar; we have indeed encountered it in our previous analysis of massive ﬁelds. As previously noted, from the bulk point of view the conformal weight of a 2 2 p 2 2 scalar of mass m L in AdS4 is ∆ = 3/2 + ν where ν ≡ 9/4 + m L . The case

ν = 3/2, i.e. a massless scalar field in AdS4, corresponds to a marginal operator with ∆ = 3. Thus, the auxiliary variable z can be precisely identified with an AdS bulk coordinate, and the modified Bessel functions can be thought of as bulk-to-boundary propagators (2.3.22). The integral (2.5.3) is of course divergent for general ∆ near z = 0. Motivated by our bulk analysis, we chose a slightly different cutoff procedure from [62], where we instead cut the integral off at some small z = zc. Now from a CFT analysis, we find the appearance of logarithmic contributions, which will generally be cutoff dependent, to the three-point function of a scalar operator (this observation remains true even in the cutoff prescription chosen in [62]). Because these logarithmic terms contain a dependence on the cutoff scale zc, they are consequently referred to as anomalies in [62]. They may be present in the theory non-perturbatively. Thus, from the holographic point of view, terms logarithmic in zc that are associated to anomalies in the 3d CFT, such as those in the three-point function, will be present to all orders rather than part of a resummeable series. Let us note that we also observed such logarithms in higher point functions at special values of ν (e.g. ν = 3/2), where the analogous general CFT analysis is more cumbersome. In a similar fashion, the tree level logarithms we discussed for the Bunch-Davies wavefunction in section 2.3.4 (for ν = 0 or ν = 1) are related to a divergence in the Fourier transform of the two-point function of a weight ∆ = 3/2 or CHAPTER 2. WAVEFUNCTIONS IN DE SITTER SPACE 28

∆ = 5/2 scalar operator [62,76]. It is also possible, however, that the appearance of these logarithms are the result of small shifts in the conformal weights of certain operators in the 3d CFT. For instance, imagine that loop corrections (such as 1/N corrections in a large N CFT) shift ∆(0) by an order ∼ 1/N amount, i.e. ∆ = ∆(0) +α/N +O(1/N 2) with α ∼ O(1). The two point function in momentum space will then have a large N expansion:

(0) 2α k2(∆−3/2) = k2(∆ −3/2) 1 + log k + ... . (2.5.4) N

2(∆−3) From the bulk AdS4 point of view, we must include the factors zc to obtain the zc-dependent bulk partition function, as we discussed in section 2.3.4, such that the expansion becomes:

2(∆−3/2) 2(∆(0)−3/2) (zck) (zc k) 2α 3 = 3 1 + log(zck) + ... . (2.5.5) zc zc N

As we already noted, we can extrapolate the perturbative results in AdS to those in dS by continuing zc = −iηc and L = −i`. From the point of view of a putative dual CFT of dS, the zc = −iηc continuation corresponds to an analytic continuation of the cutoff itself. Though unusual from the point of view of field theory, it may be interesting to consider general properties of field theories with such imaginary cutoffs. Notice that the expansion (2.5.5) now contains ∼ log(−kηc) pieces which 12 are resummed to a power law behavior in ηc. With this interpretation we might view (2.3.10) as a small negative shift in the weight ∆ = 3 of the relevant operator dual to the bulk massless field, such that it becomes slightly relevant. On the other hand, the fact that the three-point function of a marginal scalar operator contains an anomalous logarithm suggests that the wavefunction has a non-trivial time evolution. In the case we consider, where it is due to a cubic self-interaction of a massless scalar

12A concrete realization occurs in the conjectured duality between the three-dimensional Sp(N) critical model [3,41] and the minimal higher spin theory in dS4. The bulk scalar has a classical mass m2`2 = +2 and is dual to a spin zero operator whose conformal weight is ∆ = 2 at N = ∞, but receives 1/N corrections [77] (related by N → −N to those of the critical O(N) model). Similar corrections will also occur for the extended dS/CFT proposals in [42, 43]. CHAPTER 2. WAVEFUNCTIONS IN DE SITTER SPACE 29

(see (2.3.4)), this might have been expected given that we are not perturbing about a stable minimum of the bulk scalar potential. However, this anomalous logarithm may disappear should we correct the propagators to reﬂect the negative shift in weight.

The CFT stress tensor operator Tij also weight ∆ = 3 and is thus also a marginal operator. In (A)dS/CFT it is dual to the bulk graviton. Absence of a Weyl anomaly in three dimensional CFTs can be expressed as the following property of the CFT partition function: ω(x) ZCFT [gij] = ZCFT [e gij] , (2.5.6) where ω(x) is a smooth function and we are removing local counterterms. The above implies that correlation functions of the stress tensor, given by variational derivatives with respect to gij, cannot depend on the Weyl factor of gij (in the absence of any other sources) and in particular cannot depend on the logarithm of the cutoﬀ. This strongly suggests, if we are to take the picture of dS/CFT seriously, that late time log ηc contributions to the wavefunction ΨBD[gij], such as the one describing pure Einstein theory, are absent to all orders in perturbation theory. This agrees with several computations of the cubic contribution [73, 74], as well as our general tree level argument in section 2.4.3, which are all devoid of such logarithmic terms. These observations, however, do not preclude the possibility of ΨBD[gij, η] peaking far from the de Sitter vacuum.

2.6 dS2 via Euclidean AdS2

We now proceed to study several perturbative corrections of the Bunch-Davies wavefunction about a ﬁxed dS2 (planar) background. We consider the massless scalar ﬁeld in Euclidean AdS2 whose action is:

1 Z Z ∞ L2 m2 L2 λ S = d~x dz (∂ φ(~x,z))2 + (∂ φ(~x,z))2 + φ(~x,z)2 + φ(~x,z)3 . E 2 z ~x z2 3 z2 R zc (2.6.1) The reason for reducing to two-spacetime dimensions is that the integrals needed 2 for order λ calculations are far simpler that for AdS4, although their mathematical CHAPTER 2. WAVEFUNCTIONS IN DE SITTER SPACE 30

structure and the physical issues are quite similar. We will focus on cubic interactions.

2.6.1 Tree level corrections for the massless theory

3 The simplest contribution to consider is the order λ (ϕ~k) contribution. For massless ﬁelds, the bulk-to-boundary K(z; k) and bulk-to-boundary G(z, w; k) propagators are given in appendix A.2. This correction is a tree level diagram involving three bulk- to-boundary propagators. In order to calculate it, we must evaluate the integral:

L2 λ Z ∞ dz L2 λ 1 zc kΣ − 2 K(z; k1)K(z; k2)K(z; k3) = − + e kΣ Ei(−zc kΣ) , 6 zc z 6 zc (2.6.2) where kΣ ≡ k1 + k2 + k3. In the limit of small zc we ﬁnd a ∼ log(zc kΣ) contribution.

Continuing to dS2 by taking L = −i` and zc = −iηc, the Bunch-Davies wavefunction at late times to order λ is given by:

log ΨBD = ! Z d~k k `2λ Z d~k i 1 − ϕ ϕ + 2 ϕ ϕ ϕ + k (γ + log(−η k ) , (2.6.3) ~k1 −~k1 ~k1 ~k2 ~k3 Σ E c Σ 2π 2 6 2π ηc

~ ~ ~ where we must impose k3 = −k1 −k2 due to momentum conservation. Once again, we note that the absolute value of the Bunch-Davies wave function receives a logarithmic contribution. 2 4 At order λ we have a ∼ (ϕ~k) contribution to the wave function which also involves an integration over the bulk-to-bulk propagator. The integrals can also performed to obtain a result that behaves (schematically) in the small zc limit as 2 2 ∼ λ k log kzc . The integral we need is:

2 4 Z λ L dz dw 2 2 G (z, w, ~q) K z; k1 K z; k2 K w; k3 K w; k4 , (2.6.4) 8 D z w CHAPTER 2. WAVEFUNCTIONS IN DE SITTER SPACE 31

2 where the domain of integration is D = [zc, ∞] . In the small zc limit, we ﬁnd:

2 4 λ L 1 1 2 1 2 − (s + q) (log[(s + q)zc]) − (t + q) (log[(t + q)zc]) + ... , (2.6.5) 8 zc 2 2

~ ~ ~ ~ ~ ~ ~ ~ where s ≡ |k1| + |k2| and t ≡ |k3| + |k4| and q ≡ |~q| = |k1 + k2| = |k3 + k4| (note that s, t > q by the triangle inequality).

2.6.2 Loop corrections for the massless theory

A tadpole diagram contributes to the wavefunction at order λ. The relevant integral is given by: L2 λ Z ∞ dz Z d~p 2 K(z, k = 0) G(z, z; p) . (2.6.6) 2 zc z 2π Note that K(z, k = 0) ≡ 1. To render the integral finite we impose a physical ultraviolet cutoff, which becomes a z-dependent cutoff pUV = ΛUV L/z for the coordinate momentum over which we are integrating. We can add a counterterm to the action of the form: Z Z ∞ dz S = δ L2 d~x φ(~x,z) . (2.6.7) ct z2 R zc

The constant δ can be selected to cancel the logarithmic divergence in ΛUV rendering the following result for the full integral (2.6.6):

L2 λ −1 + γ + log 2 E . (2.6.8) 4π zc

Upon continuation to dS2 this contributes only to the phase of the wavefunction. 2 2 At order λ we have two distinct loop corrections to the ∼ (ϕ~k) term. One involves attaching a tadpole to the tree level propagator whose ultraviolet divergence can be treated as above. The relevant integral is given by:

L4 λ2 Z dw dz Z d~p Itadpole(k, zc; L) = 2 2 G(w, w; p) G(z, w; 0) K(z; k) K(z; k) . 4 D w z 2π (2.6.9)

At small zc the above integral contains a ﬁnite term plus a logarithmic piece in zc. CHAPTER 2. WAVEFUNCTIONS IN DE SITTER SPACE 32

The result is:

Itadpole(k, zc; L) = 4 2 2 L λ −12 + π + 6 (γE + log 2) (γE + log 2) 2 − k (logkzc) + ... . (2.6.10) 2π 24zc 4 where the subleading pieces are at most logarithmic in zc. The other order λ2 contribution comes from a ‘sunset’ diagram, which is ultraviolet ﬁnite in two-dimensions and thus requires no regularization. It involves an integral of the form:

4 2 Z Z L λ dz dw d~p ~ Isunset(k, zc; L) = 2 2 G(z, w; p)G(z, w; |~p + k|) K(z; k) K(w; k) . 4 D z w R 2π (2.6.11) For ~k = 0, the above integral can be performed analytically and we ﬁnd:

L4 λ2 π2 − 8 Isunset(0, zc; L) = . (2.6.12) 4 zc 8π

We were not able to perform the full integral analytically, however a numerical evaluation reveals the following small zc expansion:

L4 λ2 I (k, z ; L) − I (0, z ; L) = (a k log kz + a k + ...) , (2.6.13) sunset c sunset c 2 1 c 2

2 with a1 ≈ +0.261 ... and a2 ≈ +0.58 ... The (ϕ~k) piece of the late time (absolute value of the) wavefunction to order λ2 is then:

log |ΨBD[ϕ~k, ηc]| = Z d~k k − + I (k, −iη ; −i`) + I (k, −iη ; −i`) ϕ ϕ . (2.6.14) 2π 2 sunset c tadpole c ~k −~k

2 Thus we see that at loop level there are logarithmic corrections to the (ϕ~k) piece of the wavefunction. For the sunset diagram, the loop correction required no ultraviolet cancelation and so the logarithmic term present in the result is free of any potential CHAPTER 2. WAVEFUNCTIONS IN DE SITTER SPACE 33

scheme dependence.

2.7 Outlook

In this chapter we have explored the late time structure of ΨBD in a de Sitter background, by computing its quantum corrections employing a perturbative framework heavily used in the AdS/CFT literature. We have identiﬁed several types of behavior, including the logarithmic growth in conformal time. Logarithmic growth commonly appears in the correlators computed in the in-in formalism. Furthermore, we have connected the late time properties of ΨBD to certain anomalies and shifts in conformal weights of a CFT putatively dual to a bulk de Sitter theory containing the types of ﬁelds and interactions we studied. There are several interesting avenues left to explore.

• Graviton loops: One would like to ﬁrmly establish the absence (or presence) of logarithmic growth for pieces of the wavefunction that depend on the metric only, both for a pure Einstein theory and more general theories of gravity, such as those with higher derivative terms.

• Higher spin holography: We found that cubic interactions for conformally cou-

pled scalars lead to an additional local cubic piece of ΨBD that was intricately related to a logarithmic phase. Such scalars are present in the higher spin Vasiliev theory, but the cubic scalar coupling is absent at the classical level [67,68]. At

loop level, however, there may be a contribution to the cubic piece of |ΨBD|, which can be computed in the dual CFT. The presence of such additional lo-

cal contributions may give interesting new contributions to ΨBD for large ﬁeld values. Similar considerations may also interesting for the alternate boundary condition dual to a ∆ = 1 scalar operator in the CFT.

• Resummation: We discussed a possible interpretation of the logarithmic growths as pieces of a series corresponding to a small shift in the conformal weight ∆ of an operator in the dual CFT. For a massless scalar with φ4 self-interactions, we saw that such a shift would cause the dual operator to be marginally relevant, CHAPTER 2. WAVEFUNCTIONS IN DE SITTER SPACE 34

∆ < 3, rather marginally irrelevant. It would be interesting to relate this picture of resummation to other proposals involving dynamical renormalization group methods (see for example the review [24]).

• Stochastic inflation: It would be interesting to relate our calculations and in- terpretations to the framework of stochastic inflation [29] which proposes a non-perturbative approach for interacting fields in a fixed de Sitter background. Another approach to study strongly coupled (conformal) field theories in a fixed de Sitter background is using the AdS/CFT correspondence where AdS has a de Sitter boundary metric, on which the CFT resides (see for example [79]). Chapter 3

Higher Spin dS/CFT

3.1 Introduction

As we have seen in the previous chapter, a natural object to consider when studying an asymptotically (approximately) de Sitter spacetime, such as the universe during the inflationary era, is the wavefunction [12] as a function of small fluctuations of the bulk fields. For a free massless scalar φ (such as the inflaton in slow roll inflation) in the Bunch-Davies vacuum state1 |Ei [14, 15, 82–88] in a fixed four-dimensional de Sitter background:

`2 ds2 = −dη2 + d~x2 , η ∈ (−∞, 0) , (3.1.1) η2 we found the late time Hartle-Hawking Gaussian wavefunctional at η = ηc → 0 [16]:

2 Z 3 ` d k 3 lim |ΨHH [ϕ(~x), ηc] | ∼ exp − 3 k ϕ~k ϕ−~k , (3.1.2) ηc→0 2 (2π) where ϕ~k are the Fourier components of the late time proﬁle ϕ(~x). Such a Gaussian wavefunction gives rise to the scale-invariant ﬂuctuations of the cosmic background radiation. Understanding the behavior of such a wavefunction for a large range of

1It might be more appropriate to call it the Bunch-Davies-Hartle-Hawking-Euclidean- Schomblond-Spindel-Chernikov-Tagirov-Mottola-Allen-Sasaki-Tanaka-Yamamoto-Critchley- Dowker-Candelas-Raine-Boerner-Duerr vacuum state.

35 CHAPTER 3. HIGHER SPIN DS/CFT 36

values for its arguments, which include the metric, and with the inclusion of quantum corrections is a basic problem in quantum cosmology. The perturbative Bunch-Davies wavefunction (3.1.1) was noticed to be a simple analytic continuation [16] of the partition function of a free massless field in a fixed Euclidean anti-de Sitter space. These observations, coupled with the correspondence between anti-de Sitter space and conformal field theory, motivate the proposal that at late times (or large spatial volume) ΨHH is computed by a statistical (and hence Euclidean) conformal field theory, in what has come to be known as the dS/CFT conjecture [16, 39, 40].2 In its weakest form dS/CFT conjectures that the Taylor coefficients of the logarithm of the late time Hartle-Hawking wavefunctional expanded # about the empty de Sitter vacuum at large N ∼ (`/`p) are the correlation functions of such a non-unitary CFT. Namely, at some late time cutoff η = ηc → 0 we have:

log ΨHH [ϕ(~x), ηc] = ∞ X 1 Z Z d3x ... d3x ϕ(~x ) . . . ϕ(~x ) hO(~x ) ... O(~x )i . (3.1.3) n! 1 n 1 n 1 n CFT n=1

The correlators hO(~x1) ... O(~xn)iCFT , where the operator O has been rescaled by an appropriate ηc dependent factor, compute late time bulk correlation functions with future boundary conditions [127]. The bulk late time profiles ϕ(~x) are taken to be infinitesimal such that ΨHH [ϕ(~x), ηc] is merely a generating function of late time correlators about ϕ(~x) = 0. In its strongest form, the claim is that the CFT is a non-perturbative definition of ΨHH for finite deviations away from the pure de Sitter vacuum and at finite N. Particularly, ΨHH is computed by the partition function of the putative CFT with sources turned on. The single-trace operators are dual to the bulk de Sitter fields. Abstractly speaking, if we could write down a complete basis B (which may include information about topology, geometry and matter) for the Hilbert space of our theory, ΨHH would be computing the overlap of the Hartle- Hawking state |Ei with a particular state |βi ∈ B. One could also consider computing

2See [45] for a discussion of several aspects of de Sitter space. Other proposals include [11,13,59, 89–102]. CHAPTER 3. HIGHER SPIN DS/CFT 37

the partition function with sources for more general multi-trace operators turned on. As we shall see this computes the overlap of |Ei with states which are not sharp eigenstates of the field operator or its conjugate momentum. More dramatically, if there are single-trace operators which are irrelevant they may correspond to an exit from the de Sitter phase (see for instance [52]). de Sitter space arises as a non-linear classical solution to four-dimensional higher spin gravity [65,66,103–106]. This theory has a tower of light particles with increasing spin, including a spinless bulk scalar with mass m2`2 = +2 and a spin-two graviton. The scalar potential has a minimum about the pure de Sitter solution and the kinetic terms of the higher spin particles carry the right signs and are canonical. Hence the de Sitter vacuum is perturbatively stable and free of tachyons and ghosts in this theory. Beyond perturbation theory, the late time Hartle-Hawking wavefunctional of asymptotically de Sitter space in higher spin gravity (at least when the topology at I+ is sufficiently simple) is conjectured to be computed by the partition function of the 2 Sp(N) model [107] (see also [108,109]), with N ∼ (`/`p) . This model comes in two flavors, either a free theory of N anti-commuting scalar fields transforming as vectors under the Sp(N) symmetry or as a critical theory obtained from the Sp(N) model by a double trace deformation [110]. The partition function of the critical model (at least at large N) as a functional of the sources of the single-trace operators computes the wavefunction in the ordinary field basis. On the other hand, the free theory computes the wavefunction in a slightly modified basis. The quantum mechanical analogue of this basis is given by eigenstates of the Hermitian operatorς ˆ = (βxˆ − αpˆ) with α, β ∈ R. Given the wavefunction in the coordinate basis ψ(x) we can compute in the ςˆ-basis by performing the transform:

i βx2 Z − −ςx iβx2 Z 1 α 2 1 − iςx ψ(ς) = √ dx e ψ(x) , ψ(x) = √ e 2α dς e α ψ(ς). 2πα 2πα (3.1.4) Normalizability in thex ˆ-basis implies normalizability in theς ˆ-basis and vice-versa. However, a node-less ψ(x) will not necessarily give a node-less ψ(ς). The free Sp(N) model computes the late time Hartle-Hawking wavefunction of CHAPTER 3. HIGHER SPIN DS/CFT 38

the bulk scalar in the eigenbasis of the late time operator [78]:

√ 2 ˆ 3 i δ N ηc σˆ = φ − ηc πˆφ , πˆφ ≡ −p , (3.1.5) det gij δφ

ˆ where φ is the bulk scalar field operator andπ ˆφ is the field momentum density op- 2 erator. We have taken |ηc| 1 as a late time cutoff and the combination gij/ηc represents the spatial metric at this time in Fefferman-Graham gauge. It is somewhat remarkable that there exists a basis for which the wavefunctional is computed by a free dual theory given that in the ordinary field basis it is computed by a strongly coupled theory. The partition function of the free Sp(N) theory on an R3 topology is an explicit resummation of the correlation functions and there are no non-perturbative phenomena. The remaining coordinates of the wavefunctional, which are late time profiles of the higher spin fields including the bulk graviton, are computed in the ordinary field basis for both the free and critical models. The wavefunctionals for the bulk scalar in the two different bases are related by a functional version of the transform (3.1.4). As shown in [78], in the large N limit performing the transform amounts to finding the saddles of a functional equation. One of these saddles (not necessarily the dominant one) gives the field basis wavefunction whose perturbative expansion agrees with that computed perturbatively in the bulk about the pure de Sitter solution. In most of what follows, we will perform computations in theσ ˆ-basis, which amounts to the calculation of a functional determinant. This is a considerably hard object to compute and we will limit ourselves to situations where we have conditioned all bulk fields except the metric gij and the scalar φ to have vanishing late time profiles. We should emphasize that this amounts to a very sharp conditioning and ultimately the situation could be altered by allowing higher spin deformations.3 The main body of this chapter is dedicated to the study of the partition function of the Sp(N) model for a large class of SO(3) preserving deformations of the bulk scalar and graviton. That is to say we study the late time wavefunction for bulk

3It might be worth mentioning that non-linear bulk solutions exist in higher spin gravity for which only the bulk metric and scalar are turned on [111]. CHAPTER 3. HIGHER SPIN DS/CFT 39

graviton and scalar conﬁgurations with late time proﬁle on an R3 topology:

ds2 = dr2 + f(r)2 r2 dΩ2 , φ = φ(r) . (3.1.6)

2 2 where dΩ2 is the round metric on an S whose SO(3) symmetry is the one preserved. We also study the analogous problem on an S3 topology, which amounts to a simple conformal transformation of the R3 case, where the late time proﬁles take the form:

ds2 = dψ2 + f(ψ)2 sin2 ψ dΩ2 , φ = φ(ψ) . (3.1.7)

This allows us to examine the behavior of the wavefunction of higher spin de Sit- ter space for inhomogeneous and anisotropic deformations which extend the rather uniform and homogenous deformations studied in [78]. In [78] it was found that the wavefunction in theσ ˆ-basis diverges as a function of uniform mode of the bulk scalar on the round metric on S3. One of the questions we would like to understand is whether such non-normalizabilities persist for less ‘global’ late time configurations such as the above SO(3) deformations. The above metric deformations are conformally equivalent to the flat/round metric on R3/S3. We find particularly striking numerical evidence, discussed in section 3.5, that upon fixing the uniform mode of the bulk scalar on S3 (in the conformal frame where it is endowed with the standard metric) all other directions of a general SO(3) late time deformation are normalizable. We have also analyzed a different geometric deformation, this time homogeneous but anisotropic, which does not keep the metric in the same conformal class. This is a squashing deformation α of the round metric on S3 expressed as an S1 fiber over an S2: 1 1 ds2 = dθ2 + cos2 θdφ2 + (dψ + sin θdφ)2 , (3.1.8) 4 1 + α along with a uniform late time profile for the bulk scalar. When α = 0 the metric (3.1.8) reduces to the standard metric on S3. Once again, so long as the scalar profile is kept fixed we find that the wavefunction is bounded in the α direction. We begin by briefly reviewing the Sp(N) model in section 3.2. In section 3.3, using technology developed in [112], we compute the wavefunction (in theσ ˆ-basis) CHAPTER 3. HIGHER SPIN DS/CFT 40

for some Gaussian radial deformations of the bulk scalar. This amounts to computing a functional determinant of a scalar field with a radially dependent mass term. In section 3.4 we compute the wavefunction for a radial deformation of the geometry, in the presence of a radial mass deformation, that takes it from the flat metric on R3 to a 2 2 2 2 2 general form ds = dr +r f(r) dΩ2 . In section 3.5 we compute the wavefunction for several harmonics on the three-sphere and linear combinations thereof and note that the wavefunction seems to diverge only when the zero harmonic becomes large and negative. We then discuss the behavior of the wavefunction on a squashed sphere with a uniform profile of the late time bulk scalar in section 3.6. In section 3.7 we make some general remarks about double trace deformations. We end by speculating on possible extensions of higher spin holography in section 3.8. Most of our calculations can carry over to the O(N) model and its AdS4 dual in higher spin gravity [113–116].

3.2 Wavefunctionals and the free Sp(N) model

We wish to study the Hartle-Hawking wavefunctional of an asymptotically de Sitter higher spin gravity for deformations of the bulk scalar and graviton away from the pure de Sitter solution. We will restrict the topology of space to be R3 or S3 and allow only deformations that decay sufficiently fast at infinity. In this section, we remind the reader of the Sp(N) theory and discuss how to compute its functional determinant for certain SO(3) invariant radial deformations. One motivation to do so is to understand the behavior of the wavefunction of higher spin de Sitter space for mass deformations that are more ‘localized’ than those studied in [78] which were uniform over the entire S3. It is also worth noting that computing analogous pieces of the Hartle-Hawking wavefunction for a simple toy model of Einstein gravity coupled to a scalar field with a simple potential would require significant numerical work even in the classical limit, let alone at finite N. One would have to find a complex solution of the Euclidean equations of motion that caps off smoothly in the interior and has the prescribed boundary values at large volume, and compute its on-shell action. CHAPTER 3. HIGHER SPIN DS/CFT 41

3.2.1 Wavefunctional

Recall that the action of the free Sp(N) model on a curved background gij with a i (0) A B source, m(x ), turned on for the J = ΩABχ χ ≡ χ · χ operator (dual to the bulk scalar) is given by:

1 Z √ R[g] S = d3x g Ω ∂ χA∂ χBgij + χAχB + m(xi)χAχB , 2 AB i j 8 {A, B} = 1, 2,...,N, (3.2.1) where N should be even. The ﬁelds χA are anti-commuting scalars that transform as Sp(N) vectors and ΩAB is the symplectic form. Notice that due to the presence of the conformal coupling, the action is invariant under a local Weyl transformation 2W (xi) i of the metric: gij → e gij, so long as we also rescale the source as m(x ) → e−2W (xi)m(xi). From the bulk point of view this amounts to performing a coordinate transformation η = e−W (xi)η, as can be seen by studying the Starobinski-Feﬀerman- Graham expansion [69,70,117] near η = 0:

ds2 dη2 1 = − + g (xi)dxidxj + . . . , φ = η ν(xi) + η2 µ(xi) + .... (3.2.2) `2 η2 η2 ij √ Notice that µ(xi) ≡ Nm(xi) is not the coefficient of the slowest falling power of η for bulk scalar. At the linearized level, the ν(xi) profile is dual to the vev of the J (0) operator in the presence of an infinitesimal m(xi) source. Computing the late time Hartle-Hawking wavefunctional in theσ ˆ-basis with σ = m(xi) amounts to computing the partition function of the Sp(N) theory with finite sources turned on. Given that it is a Gaussian theory, we can integrate out the anti-commuting χA fields and find:

N/2 i i 2 R[g] i lim ΨHH gij, m(x ), ηc = Zfree[gij, m(x )] = det −∇g + + m(x ) , ηc→0 8 (3.2.3) CHAPTER 3. HIGHER SPIN DS/CFT 42

where: N Z i Y A −S[χA,m(xi)] Zfree[gij, m(x )] ≡ Dχ e . (3.2.4) A=1

2W (xi) In the case of a metric gij = e δij that is conformally equivalent to the ﬂat metric on R3, it is convenient to compute the functional determinant in the conformal frame where gij is the ﬂat metric. This amounts to rescaling the source to:

mˆ (xi) = e2W (xi)m(xi) . (3.2.5)

We will use this fact in section 3.4.

3.2.2 Functional determinant for radial deformations

We have seen that for conformally flat metrics our problem reduces to computing a functional determinant: 2 i det −∇ 3 +m ˆ (x ) , (3.2.6) R where ∇2 is the Laplacian of the round metric on 3, namely ds2 = dr2 + r2dΩ2. R3 R 2 The above object is badly divergent unless we regulate it somehow. We will regulate it using a heat kernel or zeta function approach, both of which give the same answer. In fact, this precise problem has been studied by Dunne and Kirsten in [112] for functionsm ˆ (xi) which only depend on the radial coordinate, i.e.m ˆ (xi) =m ˆ (r), and which vanish sufficiently fast at infinity. It was shown that the zeta function regulated determinant is given by the following sum:

∞ R ∞ det [−∇2 + µ2 +m ˆ (r)] X dr r mˆ (r) log = (2l + 1) log T (l)(∞) − 0 . det [−∇2 + µ2] 2l + 1 l=0 (3.2.7) In the above, the factor (2l + 1) originates from the degeneracy of eigenfunctions on a two-sphere and T (l)(r) solves the equation:

2 d (l) (1 + l) I3/2+l(µr) d (l) (l) − 2 T (r) − 2 + µ T (r) +m ˆ (r)T (r) = 0 , (3.2.8) dr r I1/2+l(µr) dr CHAPTER 3. HIGHER SPIN DS/CFT 43

with boundary conditions T (l)(0) = 1 and dT (l)(0)/dr = 0. The parameter µ2 ∈ R is a constant mass parameter that we will set to zero. The derivation of the above formula employs the Gelfand-Yaglom theorem [118], which expresses the regulated functional determinant of a one-dimensional Schrödingeroperator in terms of a single boundary value problem. The problem of computing the logarithm of a ratio of functional determinants for purely radial operators reduces to an infinite number of Gelfand-Yaglom problems, one for each l, whose solutions need to be summed (this is the first piece on the right hand side of (3.2.7)) and regularized (this is the second piece on the right hand side of (3.2.7)). The applicability of the formula requiresm ˆ (r) to vanish faster than r−2 at infinity, and these are the only types of deformations for which we will compute the wavefunction in the latter sections. When implementing the above formula we must sum up to a certain cutoff l = lmax which we take to be lmax = 45. A discussion of how the error decreases with lmax is given in appendix B.1.

3.3 Simple examples of radial deformations

The purpose of this section is to exploit the general formula (3.2.7) for a simple set of radial functions. By studying ΨHH as a functional ofm ˆ (r) we can identify some qualitative features already observed in [78], such as regions where the wavefunction oscillates and grows exponentially, as well as some new ones. Furthermore, we can study its dependence on more detailed features of the localized deformation. The zeroes of the wavefunction in theσ ˆ-basis occur only when the effective potential V (r) = l(l + 1)/r2 +m ˆ (r) of the differential operator −∇2 +m ˆ (r) is negative for eff R3 some range of r. If the effective potential were positive for all r it could not have vanishing eigenvalues, and hence the wavefunction could not vanish. Thus, we expect all oscillations of the wavefunction in theσ ˆ-basis to occur in directions wherem ˆ (r) is negative for some range of r. Assessing the magnitude of the wavefunction as a functional ofm ˆ (r) is a more complicated task. We observe that the wavefunctional acquires increasingly high local R ∞ maxima between its oscillations only in regions where the quantity Imˆ ≡ 0 dr r mˆ (r) appearing in (3.2.7) becomes large and negative. CHAPTER 3. HIGHER SPIN DS/CFT 44

Fig. 3.1: Examples of the radial deformations (3.3.1) on the left, (3.3.2) in the middle and (3.3.3) on the right. We have suppressed the polar coordinate θ of the S2 but kept the azimuthal direction.

It is important to note that because we are working with the ﬂat metric on R3, which has no scale, our functional determinants will have an associated scaling symmetry given by r → r/λ andm ˆ (r) → mˆ (r/λ)/λ2. We should thus ﬁx the scaling when studying the functional determinant/wavefunction.

3.3.1 Single Gaussian

We ﬁrst considerm ˆ (r) to be given by a general single Gaussian proﬁle:

e−r2/λ2 Z ∞ A mˆ (r) = A 2 , dr r mˆ (r) = . (3.3.1) λ 0 2

See the left panel of ﬁgure 3.1 for an illustration of this deformation. Using equation

(3.2.7) we can explore |ΨHH (λ, A)|, where from now on whenever we write ΨHH it is implied as a late time wavefunctional. Using the scaling relation r → r/λ˜ andm ˆ (r) → mˆ (r/λ˜)/λ˜2 we can set λ = 1. In figure 3.2 we show a plot of the functional determinant. We immediately notice the same qualitative feature that was present for the constant mass deformation on a round S3 (displayed later on in figure 3.12). Namely, it oscillates and grows exponentially in the negative A direction. This is somewhat expected since our deformation is qualitatively similar to the mass deformation on the flat metric on R3 one gets by the conformal transformation of a constant mass on S3 (see appendix B.1). In particular, all oscillations occur for A < 0 CHAPTER 3. HIGHER SPIN DS/CFT 45

-20 -15 -10 -5 0

2 Fig. 3.2: Plot of |ΨHH (λ, A)| for N = 2 for the Gaussian proﬁle (3.3.1) with λ = 1 using lmax = 45. The solid blue line is an interpolation of the numerically determined points (shown in red). The wavefunction grows and oscillates in the negative A direction.

and the magnitude of the local maxima increases for increasing |A| for ﬁxed λ.

3.3.2 Gaussian Ring

We can also study the functional determinant of a proﬁle of the type:

mˆ (r) = A e−(r−a)2/λ2 r2 , ∞ Z A λ h 2 2 √ haii dr r mˆ (r) = 2e−a /λ λ a2 + λ2 + a π 2a2 + 3λ2 1 + Erf , 0 4 λ (3.3.2) √ which describes a Gaussian-like ring peaked around r ∼ 1/2 a + a2 + 4λ2. Erf[x] denotes the error function. See the middle panel of figure 3.1 for an illustration of this deformation. The factor of r2 is included to ensure that the profile is continuously differentiable near the origin. Again, using the scaling relation r → r/λ˜ andm ˆ (r) → mˆ (r/λ˜)/λ˜2 we either fix the value of λ, |a| or |A|. We show an example in figure 3.3 where we have fixed the value of a. CHAPTER 3. HIGHER SPIN DS/CFT 46

3.0

2.5

2.0

1.5

1.0

0.5

0.0 0.5 1.0 1.5

2 Fig. 3.3: Left: Density plot of |ΨHH (λ, a, A)| for N = 2 for the proﬁle (3.3.2) as a function of A (vertical) and λ (horizontal) for a = 5 using lmax = 45. Again, the wavefunction grows and oscillates in the negative A and positive λ directions. Right: 2 Plot of |ΨHH (λ, a, A)| for the proﬁle (3.3.2) as a function of λ for A = −0.022 and a = 5. CHAPTER 3. HIGHER SPIN DS/CFT 47

2 Fig. 3.4: Left: Density plot of |ΨHH (λi, ai,Ai)| for N = 2 for the double Gaussian proﬁle (3.3.3) as a function of A1 (x-axis) and A2 (y-axis) for a1 = 0, a2 = 5, λ1 = λ2 = 1 using lmax = 45. The wavefunction grows and oscillates for negative 2 A1 and A2. Right: Density plot of |ΨHH (λi, ai,Ai)| for the double Gaussian proﬁle (3.3.3) as a function of λ1 (x-axis) and λ2 (y-axis) with A1 = −1, A2 = −1/100, a1 = 0 and a2 = 5.

3.3.3 Double Gaussian

As a third example we consider a double Gaussian proﬁle:

2 2 2 2 2 −(r−a1) /λ −(r−a2) /λ mˆ (r) = r A1 e 1 + A2 e 2 . (3.3.3)

See the right panel of ﬁgure 3.1 for an illustration of this deformation. An example of 2 |ΨHH (λi, ai,Ai)| with a1 = 0 is shown in ﬁgure 3.4. Once again we observe a pattern of maxima encircled by regions where the wavefunction squared vanishes identically.

Furthermore, the wavefunction grows for increasingly negative values of A1 and A2. CHAPTER 3. HIGHER SPIN DS/CFT 48

3.4 Radial deformations of ﬂat R3 and pinching limits

In this section, we introduce and study a class of SO(3) preserving deformations of the flat metric on R3. We show that they are conformally equivalent to the flat metric on R3. Thus, the wavefunction can only depend on such deformations of the metric if we also turn on a radial mass mb(r). Turning on such a mass, we can then perform the symmetry transformation discussed at the end of section 3.2.1 to get a mass deformationm ˆ (r) on the flat metric on R3. We will pick a functional form that is related by the symmetry transformations of 3.2.1 to a constant mass deformation on a deformed three-sphere geometry (see appendix B.1 for details). Depending on the sign of a parameter, the deformed three-sphere will look either like a peanut or an inverse peanut, i.e. like bulbous pears inverted relative to one another and conjoined on their fatter ends. The partition function that we compute can then also be understood as the answer for the partition function on this deformed three-sphere with a constant mass deformation. The pinching limit will be when the waist of the peanut-shaped geometry vanishes. We emphasize that how we perceive these deformations of the late time metric depends greatly on what we decide are natural constant time slices, since there always exists a conformal frame where the late time metric is the flat one. Ideally, it would be useful to analyze a qualitatively similar geometric deformation that would take the original geometry outside its conformal class, but we must restrict to the former case in this section since we will be constrained by considering SO(3) preserving deformations. Section 3.6 will go beyond this restriction by considering a new conformal class.

3.4.1 Balloon Geometry

Consider the following class of SO(3) preserving metrics deﬁned on R3:

2 2 2 2 2 2 2 2 2 ds = dr + r fζ (r) dΩ2 , dΩ2 ≡ dθ + sin θdφ , (3.4.1) CHAPTER 3. HIGHER SPIN DS/CFT 49

Fig. 3.5: The “balloon” deformation of R3, deﬁned by (3.4.1) and (3.4.3), represented schematically for positive ζ.

∗ with r ∈ [0, ∞). Consider a family of smooth functions fζ (r) with ζ ≤ ζ for positive ∗ ζ that tend to unity both at large r and near r = 0. We require that fζ (r) vanishes at some r = r∗ for the critical value ζ = ζ∗. We furthermore impose that:

2 d 2 2 lim r fζ (r) = 2 . (3.4.2) ζ→ζ∗ 2 dr r=r∗

For positive ζ, the geometries described by (3.4.1) can be pictured as a two-sphere whose size at some ﬁnite ζ < ζ∗ grows, then shrinks and subsequently grows again. 3 If fζ (r) = 1 the geometry is of course nothing more than the ﬂat metric on R . As we approach ζ = ζ∗, the size of the two-sphere tends to vanish at r = r∗ eventually pinching the geometry into a warped three-sphere and a slightly deformed metric on R3. The choice (3.4.2) ensures there are no conical singularities at the pinching point. It would be of interest in and of itself to study geometries with conical singularities (see for example [119]). As a concrete example, we will take:

1 2 2 f (r)2 = 1 − ζ r2 + (γr)4 e−(r−a) /λ . (3.4.3) ζ ζ∗

The parameters a and λ are chosen and γ is tuned to obey the condition (3.4.2). Though ζ∗ is not an independent parameter it is useful to isolate in the expression. A schematic representation of this deformation, for positive ζ, is presented in ﬁgure CHAPTER 3. HIGHER SPIN DS/CFT 50

40 0.15

30 0.10 20 0.05 10

100 200 300 400 100 200 300 400 Fig. 3.6: Plot of g(x) (left) and dg(x)/dx (right) as obtained by numerically solving equation 3.4.4( ζ∗ = 1741.51, ζ = 9ζ∗/10, γ = 1.36612, a = −95, λ = 30).

3.5.

3.4.2 Conformal Flatness of Balloon Geometry

It is important to note that the geometry (3.4.1) is conformally ﬂat. This can be shown in a straightforward fashion. Consider a coordinate transformation r = g(x). It immediately follows that if the following ordinary diﬀerential equation:

dg(x) x = g(x)f (g(x)) , (3.4.4) dx ζ has a smooth solution for g(x) whose derivative is positive for all x > 0 then our metric becomes: dg(x)2 ds2 = dx2 + x2dΩ2 . (3.4.5) dx 2 Though we cannot solve the non-linear o.d.e analytically, we can easily evaluate it numerically and confirm for several cases that g(x) satisfies the necessary requirements. Hence, our metric (3.4.1) is indeed conformally equivalent to the flat metric on R3. In figure 3.6 we give a numerical example of this. This result is already of some interest even for the case of ordinary Einstein gravity. It informs us that upon conditioning that all other fields vanish at late times, the absolute value of the late time Hartle-Hawking wavefunction, |ΨHH [gij]|, is independent of any radial SO(3) preserving deformation of the late time metric. Indeed, from CHAPTER 3. HIGHER SPIN DS/CFT 51

the bulk perspective a smooth conformal transformation of the late time three-metric can be induced by a time diffeomorphism that preserves the Starobinski-Fefferman- Graham form. From a holographic perspective, the late time wavefunction is computed by the partition function of a three-dimensional conformal field theory and thus only depends on the conformal metric (recall there are no conformal anomalies in three dimensions).

3.4.3 Wavefunctions and balloon geometries

We now examine what happens to the functional determinant as we vary the waist parameter ζ for an example. We will also turn on a mass that would correspond to a uniform mass m on the deformed three-sphere (as discussed further in appendix B.1.3), which upon the conformal transformation discussed there becomes:

2 2 m (r) = m . (3.4.6) b 1 + r2

This is the mass deformation on the balloon geometry. The ﬁnal deformationm ˆ (x), to be used in the Dunne-Kirsten formula, is obtained by performing a conformal 3 2 2 2 2 rescaling of the balloon geometry to the ﬂat metric on R : ds = dx + x dΩ2. This requires a conformal rescaling of mb(r) to:

dg(x)2 2 2 mˆ (x) = m . (3.4.7) dx 1 + r(x)2

Thus, we will study the functional determinant as a function of m and ζ. In figures 3.7 and 3.8 we display our numerical results. As expected, at m = 0, nothing changes as we vary ζ since the balloon geometries are conformally flat. However, when we turn on m 6= 0 the wavefunction becomes sensitive to changes in ζ. Interestingly, decreasing the girth of the throat while keeping everything else fixed is favored, at least near m = 0. Thus, though supressed exponentially with respect to the local maximum of the wavefunction at m = 0, the wavefunction does not vanish in the pinching limit. It is tempting to speculate that such pieces of the wavefunction might CHAPTER 3. HIGHER SPIN DS/CFT 52

be connected to the fragmentation picture of [120].

3.5 Spherical harmonics and a conjecture

In this section, we present numerical evidence that when mapping the problem back to the three-sphere (using the discussion in appendix B.1), all profiles give a normalizable wavefunction upon fixing their average value over the whole three-sphere. Thus, it is conceivable that the only divergence of the wavefunction occurs precisely for large and negative values of a uniform profile over the whole three-sphere [78]; a single direction in an infinite dimensional configuration space! For instance, as we shall show below, by mapping the Gaussian profile (3.3.1) to the three-sphere and removing the zero mode from its expansion in terms of three-sphere harmonics, the resultant profile produces a normalizable wavefunction as a function of its amplitude.

3.5.1 Three-sphere harmonics

We now study some examples of SO(3) invariant deformations which correspond to harmonics of the three-sphere conformally mapped back to R3. These harmonics are the eigenfunctions of the Klein-Gordon operator on the three-sphere with metric 2 2 2 2 th 2 ds = dψ + sin ψ dΩ2. The k harmonic (independent of the S coordinates) is given by:

Fk(ψ) = ck csc(ψ) sin[ (1 + k)ψ ] , k = 0, 1, 2,.... (3.5.1)

As explained in appendix B.1, to evaluate the partition function for this deformation using the method of Dunne and Kirsten we must ﬁrst perform the coordinate transformation ψ(r) = 2 cot−1 r−1, and then scale the deformation by the inverse of the conformal factor that maps the three-sphere metric to the metric on R3. The ﬁnal radial deformation which is to be used in the Dunne-Kirsten formula is:

2 sin[ 2(1 + k)tan−1r ] mˆ (r) = c . (3.5.2) k k (r + r3) CHAPTER 3. HIGHER SPIN DS/CFT 53

2 Fig. 3.7: Density plot of the |ΨHH (ζ, m)| for N = 2 as a function of the pinching parameter ζ (vertical axis) and an overall mass deformation m (horizontal axis) using lmax = 45, ζ∗ = 1741.51, γ = 1.36612, a = −95 and λ = 30.

20 ì 1.0ìæòà ìæòà ìæòà òìò à ìòà òìàìæà æ ìæòà æ æ æ ìà òà ò 0.8 ìæ 15 ò ìòà ò æ ò ìà æ ìà à æ ò ì ò æ ì ìà æ ò æ à à 0.6 à ì ò ò ò ìæ ì ìà 10 ò à æ æ òà æ à ìò ì æ à ò ìæ 0.4 æ ò ò æ à ò ìà ì à ì æ ì ì à ò ò òæ 5 à ìà ò æ à æ æ ò ìàæ 0.2 ìà à ò ìæ ì æ òà ò ì òìàæ ì ò ì ò òæà æàìæòàìæòàìæòà à àæ òìàæ ìò ìæòàìæòàìò ìæòàìæòàìæòàìæòà æ æ òìòìòìòàìæòàìæòàìæòàìæòàìæòàìæòàìæòàìæàæàæ ìæòàìæòàìæòàìæòà ìæòàìæòàìæòàìæòàìò ìòà -5 -4 -3 -2 -1 1 -1.0 -0.5 0.5 1.0

2 ∗ Fig. 3.8: Left: |ΨHH (ζ, m)| for N = 2 as a function of m for ζ = −ζ /2 (red dots), ζ = 0 (blue squares), ζ = ζ∗/2 (green diamonds) and ζ = 9ζ∗/10 (black triangles) using lmax = 45, ζ∗ = 1741.51, γ = 1.36612, a = −95 and λ = 30. Right: Same as left but for diﬀerent plot range. CHAPTER 3. HIGHER SPIN DS/CFT 54

Taking k = 0 corresponds to the zero mode which has been previously studied in [78] and was found to be oscillatory and divergent as the coefficient ck goes to large negative values. In figures 3.9 and 3.10 we plot the partition functions for higher harmonics as a function of the coefficient ck. We notice that they are all well-behaved and normalizable, at least in the range we have explored. This motivates us to consider deformations which are linear combinations of spherical harmonics. Having looked at a deformation that is the linear combination of the zero mode with the first harmonic, as well as a deformation that is the linear combination of the first harmonic with the second harmonic, we notice that the partition function is not divergent so long as the coefficient of the zero mode is kept fixed. Postponing a more systematic study for the future, here we simply consider a modified version of the single Gaussian deformation given in (3.3.1). Previously, we had found that as the overall coefficient of the profile becomes large and negative, the partition function diverges. The new profile we will study is obtained by mapping the Gaussian profile to the three-sphere, subtracting off its zero mode, and mapping it back to R3. Notice that the Gaussian profile mapped to the three-sphere is constructed from an infinite number of harmonics, and here we are subtracting the piece that seems problematic from our analysis of harmonics and finite linear combinations thereof. Since F0(ψ) = c0 is simply a constant, the condition to be met is:

Z π r(ψ)2 + 12 dψ mˆ (r(ψ)) sin2 ψ = 0 , (3.5.3) 0 2

ψ −r2 2 2 where r(ψ) = tan( 2 ) andm ˆ (r) = Ae − 4a1/(r + 1) . In this case the integral can be done explicitly and we can solve for the coeﬃcient a1 analytically. The ﬁnal form of the single Gaussian radial deformation orthogonal to the zero mode of the three-sphere becomes: √ 2 8A (1 − e π Erfc[1]) mˆ (r) = A e−r − √ . (3.5.4) π (1 + r2)2

We plot the functional determinant as a function of A in ﬁgure 3.11. Interestingly, the partition function is once again well-behaved for large values of A. An analogous CHAPTER 3. HIGHER SPIN DS/CFT 55

1.0 -10 -5 5 10 0.8

0.6 -5

0.4 -10 0.2

-10 -5 5 10 -15

2 3 Fig. 3.9: Left: Plot of |ΨHH (c1)| for N = 2 for the ﬁrst harmonic mapped to R given 2 in (3.5.2) using lmax = 45. Right: Plot of log |ΨHH (c1)| for N = 2 using lmax = 45.

-40 -20 20 40 -40 -20 20 40 -40 -20 20 40 -50 -50 -100

-100 -100 -200

-150 -150 -300

-40 -20 20 40 -40 -20 20 40 -40 -20 20 40 -20 -50 -40 -50 -60 -100 -80 -100 -100 -150 -120 -150 -140 -200

-40 -20 20 40 -40 -20 20 40 -40 -20 20 40

-50 -50 -50

-100 -100 -100

-150 -150 -150 -200 -200

Fig. 3.10: Plot of log |ΨHH (ck)| for N = 2 for the ﬁrst nine spherical harmonics 3 mapped to R given in (3.5.2) using lmax = 45. Notice that only the zeroth harmonic is non-normalizable in the negative c0 direction. CHAPTER 3. HIGHER SPIN DS/CFT 56

1.0 -100 -50 50 100 0.8 -5

0.6 -10

0.4 -15 - 0.2 20 -25 -100 -50 50 100

2 Fig. 3.11: |ΨHH (A)| (left) and log|ΨHH (A)| (right) for N = 2 as a function of A, the overall size of the radial deformation in (3.5.4) which is constructed to be orthogonal to the zero mode of the three-sphere (lmax = 45).

analysis for the balloon geometries, where we ﬁx the zero-mode on the conformally related three-sphere, also results in the boundedness of the partition function in the ζ direction (ζ deﬁned in (3.4.3)). The above results motivate the conjecture:

The partition function of any SO(3) symmetric “radial” deformation for which the three-sphere zero mode harmonic is ﬁxed is bounded.

Notice that the three-sphere is chosen as the geometry in the conformal class on which to fix the uniform profile. For example, in the case of the peanut geometries one can show that fixing the uniform profile on the peanut is not sufficient to ensure that the partition function is bounded (though fixing some non-uniform profile would suffice). This is simply because keeping the uniform mass profile fixed to some negative value while taking ζ large and negative, which corresponds to the conformally related sphere getting fatter at the waist, implies that the uniform profile on the sphere is getting large and negative. Consistent with the rest of our observations, we see that the partition function is unbounded in this direction. Furthermore, the next section will provide evidence for a more general conjecture that would extend beyond the conformal class of the sphere. CHAPTER 3. HIGHER SPIN DS/CFT 57

3.6 Three-sphere squashed and massed

In this section, we would like to briefly revisit and extend some of the observations of [78] for a constant mass deformation on an S3. There, it was observed that in theσ ˆ-basis the wavefunction on an S3 with a uniform mass deformation oscillated and diverged at large negative mS3 . We explore in this section a new direction which is the squashing parameter of the round metric on the three-sphere (in the presence of a non-zero uniform scalar profile) and its effect on the zeroes and maxima. Unlike the previous SO(3) preserving deformations which were inhomogeneous, this SO(3) × U(1) preserving deformation is homogeneous yet anisotropic. Furthermore, the squashed three-sphere is not conformal to the ordinary round three-sphere. In fact, squashed spheres with different values of the squashing parameter belong to distinct conformal classes. Part of our motivation is to provide further evidence that the zeroes of the wavefunction are extended and that the local maxima of the wavefunction (other than the pure de Sitter one) will no longer necessarily peak about homogeneous and isotropic geometries. Additionally, and in the same spirit as the observations made in section 3.5, we find that upon fixing the value of the uniform profile over the whole squashed three-sphere the wavefunction is normalizable in the squashing direction.

3.6.1 Squashed and massed

Consider turning on a constant mass mS3 for the free Sp(N) model on the round metric on an S3 whose radius a is ﬁxed to one unless otherwise speciﬁed. The partition function is given by [78]:

m Z S3 √ √ −1 1 3ζ(3) π π 3 N log Zfree[mS ] = log 4 − 2 − dσ 1 − 4σ cot 1 − 4σ . 16 π 8 0 2 (3.6.1)

The analyticity of Zfree[mS3 ] in the complex mS3 -plane is ensured by the uniform convergence of the (regularized) inﬁnite product of analytic eigenvalues which deﬁnes it. For physical applications we restrict to real mS3 and note that this implies that 0 the Taylor series expanded about any mS3 = mS3 converges to the partition function CHAPTER 3. HIGHER SPIN DS/CFT 58

-3 -2 -1 1 2 3

2 Fig. 3.12: Plot of |ΨHH [mS3 ]| given by expression (3.6.1) for N = 2. above. Furthermore, the partition function has a zero if and only if one of the product eigenvalues in the functional determinant vanishes, which can only happen for mS3 < 0. As shown in ﬁgure 3.12, this wavefunction grows exponentially and oscillates in 4 the negative mS3 direction. It is worth noting that the mS3 dependent part of the phase of the wavefunction vanishes. The real part of log Zfree/N goes as |mS3 | for 3/2 large negative mS3 and −(mS3 ) for large positive mS3 .

It is worth studying what happens to the zeroes and local maxima of Zfree in the presence of an additional deformation. A computationally convenient deformation is to squash the round metric on S3 into that of a squashed sphere, which is a homogeneous yet anisotropic geometry. In this sense, this deformation is complementary to the inhomogeneous deformations we have been studying so far. We review the metric and eigenvalues of the squashed sphere with squashing parameter ρ in appendix B.2. Our method of regularization is a straightforward extension of heat kernel techniques used in section 3.2 of [78], and details can be found therein. In ﬁgure 3.13 we present a plot of the wavefunction as a function of the mass mS3 and the squashing parameter ρ (the round metric on S3 occurs at ρ = 0).

4Divergences of the Hartle-Hawking wavefunctional have been discussed in other circumstances such as Einstein gravity coupled to a scalar ﬁeld with a quadratic scalar potential and vanishing cosmological constant [121] or the wavefunction of dS3 on a toroidal boundary [122]. Their physical interpretation remains to be understood. For example, it might be a result of very sharp conditioning or an indication of an instability. CHAPTER 3. HIGHER SPIN DS/CFT 59

-2 -1 1 2 3

2 Fig. 3.13: Left: Density plot of |ΨHH [ρ, mS3 ]| for N = 2. The fainter peak centered at the origin reproduces perturbation theory in the empty de Sitter vacuum. Horizontal lines are lines of constant mS3 and vertical lines are lines of constant ρ. Right: Plot 2 of |ΨHH [ρ, −2.25]| for N = 2. Notice that it peaks away from ρ = 0.

We find that the local maxima are in general pushed away from ρ = 0 and the zeroes of the wavefunction are extended to enclose the local maxima. The fact that the zeroes are extended (codimension one in the (ρ, mS3 ) plane) is perhaps not sur- prising given that the zeroes in figure 3.12 arise from ΨHH [mS3 ] (which is purely real) changing sign. This feature should not disappear, at least for small perturbations to the equation which determines ΨHH [mS3 ]. Furthermore, the maxima of figure 3.12 are pushed even higher when squashing is allowed. This can be seen in the right hand side of figure 3.13, where one notices that the second local maximum of ΨHH [mS3 ] at ρ = 0 is pushed further up to some ρ > 0. This observation strongly suggests that away from the origin, and under such extreme conditioning of the late time profiles of the bulk fields, the wavefunction peaks in regions where the metric and perhaps even the higher spin fields are highly excited. On the other hand, the perturbative de Sitter saddle centered at the origin remains a true local maximum. CHAPTER 3. HIGHER SPIN DS/CFT 60

-1 1 2 3

-20

-40

-60

2 Fig. 3.14: Left: Density plot of |ΨHH [ρ, mS3 ]| for N = 2 for a slightly larger range. Notice the local de Sitter maximum visible in ﬁgure 3.13 is already too faint to be seen. Horizontal lines are lines of constant mS3 and vertical lines are lines of constant ρ. Right: Plot of 2 log |ΨHH [ρ, −4.5]| for N = 2.

3.7 Basis change, critical Sp(N) model, double trace deformations

In this section we further discuss the transformation to the field basis and clarify the role of double trace deformations. We would like to comment on the transform to the √ bulk field basis, which gives us the wavefunctional as a functional of ν ≡ Nσ˜ (with ν defined in (3.2.2) andσ ˜ being related the source of the single-trace operator dual to the bulk scalar), in the large N limit. As discussed in [78], one has to consider the free theory deformed by a relevant double trace operator f(χ · χ)2/(8N). We also keep a source −ifσ˜ turned on for the single-trace χ · χ operator. The parameter f ∈ C has units of energy. Performing a Hubbard-Stratonovich transformation by introducing an auxiliary scalar field σ we CHAPTER 3. HIGHER SPIN DS/CFT 61

ﬁnd:

1 Z √ Nσ2 S(f) = d3x g Ω ∂ χA∂ χBgij − ifσχ˜ AχB − − σ Ω χAχB . 2 AB i j f AB (3.7.1) Integrating out the χA ﬁelds, the partition function becomes:

√ Z Z 2 (f) − Nf R d3x g σ˜2 3 √ σ Z [˜σ] = e 2 Dσ exp N d x g + iσσ˜ Z [σ] , (3.7.2) crit 2f free where we have transformed variables σ → (σ − ifσ˜). Having interpreted Zfree[σ] = hσ|Ei = ΨHH [σ], we see that the above expression is a Fourier type transform of the σˆ-basis wavefunction. In order for this to become the actual basis changing transform (3.1.4) to the eigenbasis of the field operator, the constant f must be taken to infinity. The f → ∞ limit (where we are keeping the size of the three-sphere fixed) corresponds to sending the ultraviolet cutoff (of the infrared fixed point theory) to infinity, in the same sense as [123], or roughly speaking it corresponds to the late time limit in the bulk.5 We begin by performing a perturbative analysis for infinitesimal deformations of the free Sp(N) theory at large N on an R3.

3.7.1 Perturbative analysis on R3

For the sake of simplicity, we will put the theory on the ﬂat metric on R3, akin to studying perturbations in a small piece of I+.6 For reasons that will be clear

5Another way to think about this is keeping f fixed and taking the large size limit of the three- sphere that the CFT lives on. This is because the dimensionless quantity is |f|a where a is the size of the sphere. Indeed at late times the three-sphere grows large. 6The parallel story in anti-de Sitter space has been studied extensively [80, 124–126]. From the 2 2 2 2 2 bulk perspective in planar AdS4: ds = À(dz + d~x )/z , at least perturbatively about the empty AdS4 vacuum, the finite fA ∈ R double trace deformed theory computes correlation functions of the bulk scalar quantized with mixed boundary conditions. Near the boundary z → 0 of AdS4, the 2 2 2 bulk scalar with mass m À = −2 behaves as φ(z, ~x) ∼ α(~x)z + β(~x)z with α(~x) = fAβ(~x). This boundary condition is different from the conformally invariant one which sets either α(~x) or β(~x) to zero, corresponding to the free or critical O(N) models respectively. In de Sitter space we would consider a wavefunction of a scalar of mass m2`2 = +2 computed by imposing future boundary 2 conditions [16,17,127] φ(η, ~x) ∼ α(~x)η + β(~x)η (with α(~x) = fDβ(~x)) and Bunch-Davies conditions φ ∼ eikη for k|η| 1. At the level of perturbation theory this is computed by continuing the CHAPTER 3. HIGHER SPIN DS/CFT 62

momentarily we choose f = i|f| to be pure imaginary. We are interested in setting up a perturbative expansion about the σ ∼ 0 Gaussian peak of Zfree[σ]. From (3.7.2) we can compute the two point function of O ≡ χ · χ at large N in the double trace deformed Sp(N) theory on an R3. We do this by taking two variational derivatives of the logarithm of (3.7.2) with respect toσ ˜(xi) and evaluating atσ ˜(xi) = 0:

|f|2G(k) N hO O i = N ,G(k) ≡ hO O i = − , (3.7.3) ~k −~k f N + 2i|f|G(k) ~k −~k f=0 k where k is the magnitude of the momentum. For k |f| this becomes the two-point function of the free Sp(N) model, whereas for k |f| this becomes the two-point function of the critical Sp(N) model. Expanding for large |f| we ﬁnd:

N|f| N hO O i = −i − k + ... (3.7.4) ~k −~k f 2 4

Notice that the real part of the two-point function (3.7.4) is negative. Recalling (3.1.3), this is the Gaussian suppression of the Hartle-Hawking wavefunction near the de Sitter vacuum. Also, the local momentum independent term has become a phase for pure imaginary f. We can compare this to the bulk Hartle-Hawking wavefunction for a free m2`2 = +2 scalar in planar coordinates computed in (B.3.13) −1 of the appendix. This allows us to (roughly) identify |f| with the late time cutoﬀ |ηc| at large |f|. In a similar way we can compute the rest of the perturbative correlators of the critical Sp(N) theory [110]. Beyond such perturbative analyses, we must resort to a saddle point approximation which we now proceed to.

3.7.2 Large N saddles for uniform S3 proﬁles

We now put the theory on the round metric on S3. At large N, we can evaluate (3.7.2) by solving the saddle point equation (for σ = σ(˜σ) andσ ˜ uniform over the

Euclidean AdS4 partition function by z → −iη, À → i` and fA → ifD (where fD ∈ R for the ‘normalizable’ profile of the scalar field to be real). CHAPTER 3. HIGHER SPIN DS/CFT 63

1.0 2.0 8

0.8 1.5 6 0.6 1.0 4 0.4 0.5 2 0.2

-2 -1 1 2 -2 -1 1 2 -2 -1 1 2

whole three-sphere):

16πσ √ π √ + 16πiσ˜ = 1 − 4σ cot 1 − 4σ . (3.7.5) f 2

For a given solution Σi of (3.7.5), we can evaluate Zcrit[˜σ, Σi]. For example, there is a solution Σ0 where σ ∼ 0 whenσ ˜ ∼ 0 (with f → ∞). It is the piece of the wavefunctional evaluated from this saddle nearσ ˜ = 0 that reproduces the dS invariant perturbation theory in the Bunch-Davies state (about the pure de Sitter vacuum).

In ﬁgure 3.15 we plot Zcrit for the ﬁrst few Σi at large f = i|f|. These haveσ ˜ = 0 √ π near the subsequent zeroes of cot 2 1 − 4σ in (3.7.5). Notice that for all large N saddles Zcrit[˜σ] is peaked atσ ˜ = 0 but the saddles coming from the more negative σ peaks contribute more nearσ ˜ = 0. Away from the large N limit we must compute the integral in (3.7.2) without resorting to a saddle point approximation. If we restrict to uniform σ andσ ˜, we note that as we increase f = i|f| more and more of the growing negative σ peaks in

Zfree[σ] contribute to the integral before it is cutoff by the rapid oscillations due to the iNσ2/|f| piece. One can check that the integral grows for large and negativeσ ˜ upon fixing |f|. Some of the saddles in the large N limit should correspond to classical (complex) bulk solutions with a uniform late time profile of the scalar on the round metric of the three-sphere. Some of these solutions labelled by a continuous parameter, which only involve the bulk metric and scalar, were found by Sezgin and Sundell in [111]. CHAPTER 3. HIGHER SPIN DS/CFT 64

It is worth noting that though it may sound confusing that there are bulk solutions that have only the metric and scalar turned on (since all higher spin fields interact on an equal footing), this is natural from the CFT since turning on an SO(4) symmetric source for J (0) = χ · χ need not source the traceless higher spin currents due to symmetry reasons. The metric is non-vanishing since in effect we have also turned on a source for it by having the round metric on S3 at the boundary. At finite N, all these saddles mix quantum mechanically. Each ΨHH [˜σ, Σi] comes with a phase, so one should be careful when summing contributions from different saddles. Finally, it would be extremely interesting to understand the Lorentzian cosmologies associated to the wavefunction using the ideas developed in [128, 129] (see also [130]). In order to have a classical cosmology (or an ensemble of such cosmologies) we must ensure that the wavefunction takes a WKB form with a phase oscillating much more rapidly than its absolute value. At least in the large N limit, and for large |f|, this is ensured by the first term in (3.7.2).

3.7.3 Double trace deformations as convolutions

We can also consider keeping f finite and real. This defines a double-trace deformed field theory, in and of its own right, whose partition function Z(f)[σ0] can be computed in the large N limit. We also keep a uniform (on S3) source σ0 turned on for the single-trace χ · χ operator. This partition function is no longer computing overlaps between the Bunch-Davies vacuum and some late time field configuration which is ˆ an eigenstate of the field operator φ. Given that Zfree[σ] = hσ|Ei = ΨHH [σ], we see that Z(f)[σ0] is computing instead a convolution of the wavefunction in theσ ˆ-basis:7

" # Z Z (σ0 − σ)2 Z(f)[σ0] = Dσ exp N dΩ Ψ [σ] . (3.7.6) 3 2f HH

7Though we will not do so here, we can study higher multitrace deformations of the Sp(N) theory and arrive at similar expressions. Thus, the fact that multi-trace operators are irrelevant seems far less threatening than having low-spin, single-trace operators which are irrelevant and correspond to bulk tachyons. CHAPTER 3. HIGHER SPIN DS/CFT 65

We can also view Z(f)[σ0] as computing the overlap of the Hartle-Hawking state with the state: " # Z Z (σ0 − σ)2 |fi ≡ Dσ exp N dΩ |σi . (3.7.7) 3 2f

Notice that though the integral itself is convergent for ﬁnite f, the resulting function Z(f)[σ0] will grow exponentially at large negative σ0. One could also consider more generally a complex valued f ∈ C which would correspond to a kind of windowed Fourier transform.

3 3.7.4 Euclidean AdS4 with an S boundary

3 The situation can be contrasted with the case of Euclidean AdS4 (with an S bound- O(N) 3 ary) [126]. The partition function Zcrit [σA] of the critical O(N) model on an S , dual to Euclidean AdS4 in higher spin gravity, is obtained again from the free O(N) model by a double trace deformation in the limit fA → +∞:

Z Z 2 O(N) NfA R 2 σ O(N) 2 dΩ3σA Zcrit [σA] = lim e Dσ exp N dΩ3 − σAσ Zfree [σ] , fA→∞ 2fA

fA ∈ R . (3.7.8)

The ﬁrst term in front of the integral is local in the limit fA → +∞ and we can remove it by adding a counterterm. To ensure convergence of the integral we must choose an appropriate contour, which in this case is given by σ running along the O(N) imaginary axis (see for example [78]). Zfree [σ] is the partition function of the free O(N) model and is related to the free Sp(N) partition function by N → −N. Note O(N) that Zfree has poles precisely at the values where the wavefunction in theσ ˆ-basis (3.6.1) vanishes. At large N the integral (3.7.8) can be evaluated by a saddle point O(N) approximation. In ﬁgure 3.16 we display Zcrit [σA] for the case of a uniform source 3 fAσA over the whole S . In principle, this plot should be reproducible by computing the regularized on-shell Vasiliev action on the asymptotically Euclidean AdS4 Sezgin- Sundell solution [111]. CHAPTER 3. HIGHER SPIN DS/CFT 66

1.0

0.8

0.6

0.4

0.2

-0.4 -0.2 0.2

th O(N) Fig. 3.16: Plot of the N root of the ﬁnite part of Zcrit [σA] for a uniform source σA O(N) over the whole three-sphere. We have normalized such that Zcrit [σA] = 1 at σA = 0.

3.8 Extensions of higher spin de Sitter holography?

So far, our discussion was restricted to the minimal bosonic higher spin theory. A natural question that arises, particularly given possible interpretational issues of the wavefunction such as its (non)-normalizability, is whether this theory is part of a larger framework. We brieﬂy discuss possible extensions of higher spin de Sitter holography, inspired by the analogous situation in anti-de Sitter space.

3.8.1 AdS4

There exist parity violating deformations of the bulk equations of motion which de- form the original Vasiliev equations (in anti-de Sitter space) to a one-parameter family [65, 66, 104]. It was proposed that the dual description is given by coupling the theory to a Chern-Simons theory with level k, at least for simple enough topologies. The new parameter is given by the ’t Hooft coupling λ = N/k which is small when the dual is higher spin gravity. Such a theory was shown in the inﬁnite N limit with ﬁxed small λ [131–133] to have a spectrum of single-trace operators which is precisely that of the free U(N) model, namely a tower of higher spin currents which are conserved CHAPTER 3. HIGHER SPIN DS/CFT 67

√ (up to O(1/ N) corrections), in accordance with a bulk higher spin theory. As discussed in [132, 134] in the context of anti-de Sitter higher spin gravity, one can also endow the bulk higher spin fields with U(M) Chan-Paton factors, such that the fields all lie in the adjoint representation of the U(M). In the dual theory, this corresponds to adding a U(M) flavor symmetry to the U(N) model. If the flavor symmetry is weakly gauged, which can be achieved through the procedure described in [135], then one can form single-trace operators of the form Tr (ABAB . . . AB). The fields A and B transform in the (, ) and (, ) of the U(N) × U(M) gauge group respectively. As M/N increases the ‘glue’ between each TrAB (which are dual to higher spin fields in the bulk) becomes stronger. From the bulk point of view, it was suggested that the higher spin fields, now endowed with additional U(M) interactions, form bound states with binding energy that increases as we crank up M/N. These bound states would correspond to the Tr (ABAB . . . AB) operators. An appropriately supersymmetrized version of this story [132, 134] was conjectured to connect the higher spin (supersymmetric) theory to the ABJ model [136] where such long string operators are dual to bulk strings.

3.8.2 dS4

It is convenient in our discussion for the bulk to contain a spin-one gauge ﬁeld in its spectrum. We thus consider the non-minimal higher spin model with even and odd spins whose dual (at least at the level of correlation functions on R3) is a free/critical U(N) theory with N anti-commuting scalars transforming as U(N) vectors. We refer to this as the U˜(N) model. Given that the parity violating deformations of the Vasiliev equations in AdS retain the original ﬁeld content and reality conditions, it seems natural that they are present for the de Sitter theory as well. Thus, one might consider that such theories are dual to parity violating extensions of the U˜(N) theory obtained by adding a level k Chern-Simons term to the free anti-commuting complex scalars. The Lagrangian CHAPTER 3. HIGHER SPIN DS/CFT 68

of this theory is:

ik Z 1 Z S = − d3x Aa∂ Aa + f abcAaAbAc + d3x ∇ χA∇iχ¯A¯ . (3.8.1) CSM 4π ijk i i k 3 ijk i j k i

a A The fields Ai are (possibly complexified) U(N) gauge fields, the χ fields are anti- commuting complex scalars transforming in the fundamental of the U(N) gauge symmetry, and the ∇i derivative is covariant with respect to U(N) gauge transformations. Though classically this theory is conformally invariant, this need not be the case when we include loops, as the β-function might be non-vanishing. The ordinary U(N) Chern-Simons theory coupled to a vector of charged scalars has two exactly marginal deformations at infinite N as in [131, 137, 138]. These are the ’t Hooft A A¯ 3 coupling λ ≡ N/k and the coupling constant λ6 of the triple trace interaction (χ χ¯ ) . Though the Chern-Simons-U˜(N) theory (3.8.1) is non-unitary, it is conceivable that it is also has a vanishing β-function at large N [139]. This theory would also have an unchanged spectrum of single-trace operators at large N and small λ by the same arguments as those in [131,140]. 8 One can also consider adding U(M) Chan-Paton factors to the bulk higher spin de Sitter theory. This merely requires tensoring the ∗ algebra with that of M × M matrices. Once again, this will not affect the reality conditions on the higher spin fields and they will all transform in the adjoint of the U(M). This corresponds to adding a U(M) flavor symmetry to the U˜(N) vector model, which can be weakly gauged (see appendix B.4 for a discussion). The single-trace operators Tr (ABAB . . . AB) have increasingly real conformal weight. From the point of view of dS/CFT this would imply that the bulk theory has a tower of tachyonic bulk fields since the conformal

8 1 At finite k, the partition function on a non-trivial topology such as M = S × Riemmg will 2 grow as [141] (see also [142]): ZCFT ∼ exp (g − 1)N log k + O(N) where g is the genus of the M. This drastically favors higher topologies if interpreted as a probability, but it is unclear whether and how one should compare topologies and what the correct normalization for ΨHH is. It is interesting that at finite k one might also encounter monopole operators. In ABJM, such operators are dual to D0-branes in the bulk. It is unclear how they should be understood in the context of de Sitter space and higher spin gravity. For instance, they have a conformal weight that goes like k, which might suggest taking k to be complex or imaginary [143] in the de Sitter case. It is also worth noting that the potentially infinite wealth of topological data at I+ might be at odds with the finiteness of de Sitter entropy [144]. In Einstein gravity adding too much topology at I+ often results in bulk singularities [145]. The issue of topology in the context of dS/CFT is further discussed in [146]. CHAPTER 3. HIGHER SPIN DS/CFT 69

p 2 2 weight of a bulk field goes as ∆± ∼ 3/2 ± 9/4 − m ` . One might suspect that these will be the continuations of the higher spin bound states previously discussed for the anti-de Sitter case. Thus we see that even though the fundamental constituents (i.e. the higher spin particles) of such an extension of higher spin de Sitter gravity are not pathological (at least at the level of perturbation theory), they may form configurations which resemble tachyonic fields in de Sitter space. It is of interest to understand whether the late time behavior of such a theory can ever be asymptotically de Sitter [139]. From the CFT point of view these are highly irrelevant operators which are not conserved currents. For there to be a late time de Sitter phase, one would require that turning on such irrelevant deformations can flow the theory to a UV fixed point. Chapter 4

Conformal Quivers

4.1 Introduction

We will now shift our attention to the study of quiver quantum mechanics. Our goal is to investigate the extent to which such models capture the complicated dynamics of specific AdS2 geometries and disordered systems. When studying the extremal limit of black hole solutions in gravity one immediately encounters a rather unusual geometric feature. The geometry develops a throat which becomes infinitely deep at extremality. This can be seen in the simple example of a d-dimensional Reissner-Nordströmblack hole whose metric is:

2 2 2 1 2 r dr 2 2 ds = − 2 (r − r+)(r − r−) dt + + r dΩd−2 . (4.1.1) r (r − r+)(r − r−)

The inner and outer horizon lie at r = r− and r = r+ > r−. In the extremal limit r+ tends to r− and the proper distance between some finite distance r and the horizon diverges. One can isolate the geometry parametrically near the horizon within the infinitely deep throat and find a new solution to the Einstein-Maxwell system, namely d−2 the Bertotti-Robinson geometry AdS2 × S . The AdS2 structure is not contingent upon supersymmetry but rather extremality, i.e. that the temperature of the horizon vanishes. Turning on the slightest temperature will destroy the precise AdS2 structure and bring the horizon back to a finite distance. In the extremal near horizon region,

70 CHAPTER 4. CONFORMAL QUIVERS 71

the R × SO(d − 1) isometry group is enhanced to an SL(2, R) × SO(d − 1). While the Hawking temperature of the black hole vanishes, the entropy, given by one-quarter of the size of the horizon in Planck units, is still macroscopically large. It is interesting that a full SL(2, R) symmtetry (rather than just a dilatation symmetry) emerges in the near horizon region given that we are only losing a single scale, the temperature. From a holographic point of view, the surprise stems from the fact that in quantum mechanics invariance under dilatations does not generally imply invariance under the full SL(2, R) conformal transformations [147]. We can study such extremal, and in addition supersymmetric, black holes in string theory. They are given by wrapping branes around compact dimensions such that they are pointlike in the non-compact directions. The electric and magnetic charges of these branes are (schematically) given by the number of times they wrap around various compact cycles. A simple and beautiful example [148] is given by taking type 1 4 IIA string theory compactified on a small S ×T and wrapping N4 D4-branes around 4 the T and sprinkling N0 D0-particles on top. In the large N0 and N4 limit, this gives 3 an extremal D0-D4 black hole with an AdS2 × S near horizon. Using T -duality on the S1, we can map this system to a D1-D5 system in type IIB string theory. Since the circle becomes effectively non-compact in the IIB frame, our former black 3 hole now becomes a black string. This black string has an AdS3 × S near horizon with an SL(2, R)L × SL(2, R)R isometry group. We can express the AdS3 as a Hopf fibration of the real line over an AdS2 base space. The SL(2, R) isometry of the AdS2 base space is the left (or right) moving part of the isometries of the AdS3. The real line becomes the T -dual direction and thus we see how the original AdS2 naturally

ﬁts inside the AdS3. From the point of view of the AdS/CFT correspondence, the

AdS3 is dual to a certain two-dimensional CFT and the large black hole degeneracy is counted by the supersymmetric ground states of the CFT which only excite one of the two SL(2, R)’s. The isometry of the near horizon supersymmetric AdS2 reflects the remaining SL(2, R). There is another class of extremal black holes which arise in string theory which can be obtained by considering a Calabi-Yau compactification, with generic SU(3) holonomy, of eleven dimensional M-theory down to five non-compact dimensions. CHAPTER 4. CONFORMAL QUIVERS 72

Typically, such a Calabi-Yau will not have U(1) isometries that we can easily T -dualize along. Yet, wrapping branes around the supersymmetric cycles of the Calabi-Yau yields extremal five-dimensional black holes [149,150], which again have an SL(2, R) isometry in the near horizon. This AdS2 does not naturally fit inside an AdS3 with a two-dimensional CFT dual.1 Remarkably, attempts to count the microstates of such supersymmetric black holes have faced significant difficulties [157]. In fact, whenever the precise counting of microstates has been successful it has involved a Cardy formula [148,158,159]. Lacking the larger Virasoro structure, one may wonder where the ‘isolated’ SL(2, R) of these black holes originates and to what extent it is robust.

Perhaps an additional motivation for understanding such an ‘isolated’ AdS2 geometry is the emergence of an SL(2, R) symmetry in the worldline data of the static patch of de Sitter space [45, 59]. It is interesting to note that the static patch of 2 four-dimensional de Sitter space is conformally equivalent to AdS2 × S , whose ‘isolated’ SL(2, R) does not seem to reside within a larger structure containing a Virasoro algebra. One particular way the SL(2, R) isometeries of the black hole manifest themselves is in the worldine dynamics of D-particles propagating in the near horizon region. We might then ask whether there are microscopic models, such as matrix quantum mechanics models with a large number of ground states, whose eﬀective eigenvalue dynamics describe an SL(2, R) invariant multiparticle theory.2 Some insight into these issues can be provided by studying certain quiver quantum mechanics models, which capture the low energy dynamics of strings connecting a collection of wrapped branes [164–170] in a Calabi-Yau compactiﬁcation of type IIA string theory to four-dimensions. Under certain conditions these quiver theories

1 One might also consider this AdS2 as a degenerate limit of the warped AdS3/NHEK near horizon geometry of the rotating black hole, in the limit of vanishing angular momentum. The SL(2, R) might then be a global subgroup of the full symmetries associated to the duals of such geometries [151–156]. 2The original N × N Hermitean matrix models [160] in the double scaling limit has eigenvalue dynamics described by free fermions which naturally have an SL(2, R) symmetry. Of course, such models contain only the eigenvalue degrees of freedom due to the U(N) gauge invariance that allows for a diagonalization of the matrix, and hence do not have O(N 2) degrees of freedom. These models are dual to strings propagating in two dimensions (for some reviews see [161–163]). CHAPTER 4. CONFORMAL QUIVERS 73

have an exponential number of (supersymmetric) ground states whose logarithm goes as the charge of the branes squared, which is the same scaling as the entropy of a supersymmetric black hole in N = 2 supergravity. It has been argued [166,167] that the near horizon AdS2 of these supersymmetric black holes is related to the exponential explosion in the number of ground states in the quiver quantum mechanics. The states in question are referred to as pure-Higgs states since they reside in the Higgs branch of the quantum mechanics, where all the branes sit on top of each other. Interestingly, going to the Coulomb branch after integrating out the massive strings stretched between the wrapped branes, leads a non-trivial potential and velocity dependent forces governing the wrapped brane position degrees of freedom in the non-compact space. Moreover, whenever the Higgs branch has an exponentially large number of ground states, the Coulomb branch exhibits a family of supersymmetric scaling solutions [165, 166, 171] continuously connected to its origin. The equations determining the positions of the wrapped branes in such supersymmetric zero energy scaling solutions are reproduced in four-dimensional N = 2 supergravity [165], believed to be the appropriate description of the system in the limit of a large number of wrapped branes. In this chapter we will touch upon some of these issues. We do so by discussing two aspects of the Coulomb branch of such quiver theories, particularly those describing three wrapped branes containing scaling solutions.

First we derive the Coulomb branch Lagrangian of a three node quiver model (see figure 4.1), and establish the existence of a low energy scaling limit where the theory exhibits the full SL(2, R) symmetry of conformal quantum mechanics [172–175]. These scaling theories have velocity dependent forces, a non-trivial potential as well as a metric on configuration space. It is also worth noting that the full quiver quantum mechanics theory is itself not a conformal quantum mechanics (and most certainly not a two-dimensional conformal field theory). The emergence of a full SL(2, R) symmetry rather than only a dilatation symmetry in the scaling limit is not guaranteed, and is reminiscent of the emergence of a full SL(2, R) in the near horizon geometry of extremal black holes. CHAPTER 4. CONFORMAL QUIVERS 74

Second, we study the behavior of the Coulomb branch upon integrating out the strings in a thermal state, rather than in their ground state. At suﬃciently high temperatures, the Coulomb branch melts into the Higgs branch. This is reminiscent of the gravitational analogue where increasing the temperature of a black hole increases its gravitational pull, or a particle falling back into the ﬁnite temperature de Sitter horizon.

4.2 General Framework: Quiver quantum mechanics

In this section we discuss the quiver quantum mechanics theory and its Coulomb branch. These theories constitute the low energy, non-relativistic and weakly coupled sector of a collection of branes along the supersymmetric cycles of a Calabi-Yau three fold. The wrapped branes look pointlike in the four-dimensional non-compact Minkowski universe.

4.2.1 Full quiver theory

The N = 4 supersymmetric quiver quantum mechanics comprises the following fields: α α α α chiral multiplets Φij = {φij, ψij,Fij } and vector multiplets Xi = {Ai, xi, λi,Di}. The α α ψij are the fermionic superpartners of the φij, the λi are the fermionic superpartners α of the scalars xi, Ai is a U(1) connection, and Fij and Di are auxiliary scalar fields. α ¯ The Φij transform in the (1i, 1j) of the U(1)i × U(1)j. The index α = 1, 2,..., |κij| denotes the specific arrow connecting node i to node j (see figure 4.1). The chiral multiplets encode the low energy dynamics of strings stretched between the wrapped

D-branes of mass mi sitting at three-vector positions xi in the non-compact four- dimensions. The index i = 1, 2,...,N denotes the particular wrapped D-brane. The (i) (i) electric-magnetic charge vector, Γi = (QI ,PI ), of the wrapped branes depends on the particular cycles that they wrap, and the Zwanziger-Schwinger product of their (i) (j) (i) (j) charges are given by the κij = (PI QI − QI PI ). The κij count the number of CHAPTER 4. CONFORMAL QUIVERS 75

Fig. 4.1: A 3-node quiver diagram which captures the ﬁeld content of the Lagrangian L = LV + LC + LW , each piece of which is given in (4.2.1), (4.2.2), and (4.2.5). This quiver admits a closed loop if κ1, κ2 > 0 and κ3 < 0. intersection points in the internal manifold between wrapped branes i and j. In what follows we measure everything in units of the string length ls which we have set to one.

The Lagrangian L = LV + LC + LW for the three-node quiver quantum mechanics [165] describing the low energy non-relativistic dynamics of three wrapped branes which are pointlike in the (3 + 1) non-compact dimensions is given by:

3 X µi L = q˙ i · q˙ i + DiDi + 2iλ¯iλ˙ i − θiDi , (4.2.1) V 2 i=1 and,

3 X i 2 i i i i 2 i 2 ¯i i LC = |Dtφα| − q · q + siD |φα| + |Fα| + iψαDtψα i=1 √ ¯i i i ¯i i i − siψα(σ · q )ψα + i 2 siφαλ ψα − h.c. . (4.2.2)

In (4.2.1-4.2.2) and what follows we will mostly work with the relative degrees of freedom (i.e. xij ≡ xi − xj, Dij ≡ Di − Dj, etc.) since the center of mass degrees of freedom decouple and do not play a role in our discussion. The notation we use CHAPTER 4. CONFORMAL QUIVERS 76

1 2 3 is somewhat non-standard (for example (q , q , q ) ≡ (x12, x23, x13)) and is given in appendix C.1 along with all of our conventions. We note that the relative Lagrangian is only a function of two of the three vector multiplets, since q3 = q1 + q2, D3 = 1 2 3 1 2 D + D and λ = λ + λ . The si encode the orientation of the quiver. For the majority of our discussion we choose s1 = s2 = −s3 = 1, corresponding to a closed loop like the one in ﬁgure 4.1. The reduced masses µi (which we denote in superscript notation in (4.2.1-4.2.2)) are related to the masses mi ∼ 1/gs, where gs is the string coupling constant, of the wrapped branes sitting at the xi by:

m m m m m m µ1 = 1 2 , µ2 = 2 3 , µ3 = 1 3 . (4.2.3) m1 + m2 + m3 m1 + m2 + m3 m1 + m2 + m3

The superpotential, which is allowed by gauge invariance only when the quiver has a closed loop, is given by:

X 1 2 3 W (φ) = ωαβγφαφβφγ + higher order terms , (4.2.4) α,β,γ

(where we take coeﬃcients ωαβγ to be arbitrary) and contributes the following piece to the Lagrangian:

3 3 2 ! X ∂W (φ) i X ∂ W (φ) i j LW = F + h.c. + ψ ψ + h.c. . (4.2.5) ∂φi α i j α β i=1 α i,j=1 ∂φα∂φβ

We only consider cubic superpotentials and ignore the higher order terms. This is i i consistent so long as the φα are small, which in turn can be assured by taking the |θ | suﬃciently small [166]. The theory contains a manifest SO(3) global R-symmetry. In the absence of a superpotential, the Lagrangian is diagonal in the arrow (Greek) indices and thus the theory also exhibits a U(|κ1|) × U(|κ2|) × U(|κ3|) global symmetry under which the i i φα transform as U(|κ |) vectors. The superpotential explicitly breaks this symmetry down to the U(1)1 × U(1)2 × U(1)3 gauge symmetry. The theory can be obtained by dimensionally reducing an N = 1 gauge theory in four-dimensions to the (0 + 1)-dimensional worldline theory. It can also be viewed as CHAPTER 4. CONFORMAL QUIVERS 77

the dimensional reduction of the N = 2 two-dimensional σ-models studied extensively, for example, in [176]. In order for the theory to have supersymmetric vacua we also demand that the Fayet-Iliopoulos constants sum to zero: θ1 + θ2 + θ3 = 0. In units where ~ = 1 is dimensionless, and where we choose dimensions for which [t] = 1, the dimensions of energy are automatically set to [E] = −1. We also find the following dimensional assignments: [φ] = 1/2, [D] = −2, [x] = −1, [µ] = 3, [ωαβγ] = i −3/2, [ψ] = 0, [λ] = −3/2, [F ] = −1/2 and [θ] = 1. The mass squared of the φα fields 2 2 upon integrating out the auxiliary D fields is given by Mij = (|xij| + θi/mi − θj/mj). 2 2 i For physics whose energies obey E /M 1 we can integrate out the massive φα fields and study the effective action on the Coulomb branch. Finally, notice also that the coupling ωαβγ has positive units of energy and is thus strong in the infrared 3/2 i limit since the natural dimensionless quantity is ωαβγ/E . The contribution from φα loops grows as κi and thus the effective coupling constant at low energies is given by geff ∼ gsκ. In the large geff limit, the wrapped branes backreact and the appropriate description of the system is given by four-dimensional N = 2 supergravity [165].

4.2.2 Some properties of the ground states

An immediate question about the above model regards the structure of the ground i i ˆ i i states Ψg [Φα, Q ] of the theory, satisfying: H Ψg [Φα, Q ] = 0. Though an explicit i i expression for the full Ψg [Φα, Q ] remains unknown, the degeneracy of ground states has been extensively studied [165–170]. In particular, the degeneracy of ground states localized near Qi = 0, i.e. the ground states of the Higgs branch of the theory, were shown to grow exponentially in κi when the theory contains a superpotential, the quiver admits closed loops (e.g. κ1, κ2 > 0 and κ3 < 0) and the κi obey the triangle inequality (i.e. |κ2| + |κ3| ≥ |κ1| and cyclic permutations thereof). This growth is related to the exponential explosion in the Euler characteristic of the complete intersection manifold M [166] given by imposing the constraints from the F -term 3 i i i i i (δFα L|Fα=0 = 0) onto the D-term constraints (δD L|D =0 = 0). Since κ goes as 3As an example, we can take θ1, θ2 < 0. Then the complete intersection manifold is given by 3 3 1 2 κ1−1 κ2−1 setting φα = 0, imposing the κ F -term constraints: ωαβγ φαφβ = 0, inside a CP ×CP space 1 2 1 2 2 2 1 coming from the D-term constraints: |φα| = −θ and |φα| = −(θ − θ ). The space is a product CHAPTER 4. CONFORMAL QUIVERS 78

the charge squared of the associated U(1) gauge symmetry, the number of ground states scales in the same way as the Bekenstein-Hawking entropy of the associated black hole solutions in the large geﬀ limit. Though a complete match between the ground states of a single Abelian quiver model and the entropy of a BPS black hole in N = 2 supergravity is not known,4 the vast number of microstates makes these systems potentially useful candidate toy models to study features of extremal or near extremal black holes. i Other pieces of Ψg localized near Φα = 0, i.e. the (quantum) Coulomb branch of the theory, have also been studied [165]. Unlike the Higgs branch quiver with a closed loop, superpotential and an exponential growth in its number of ground states, it was found that the number of Coulomb branch ground states grows only polynomially in the κi. Interpreting the qi as the relative positions of wrapped branes, these ground states can be viewed as describing various multiparticle conﬁgurations, as we will soon proceed to describe in further detail. To each ground state in the quantum Coulomb branch there exists a corresponding ground state in the Higgs branch, but the converse is not true. Another way to view this statement is that whenever a given i i Ψg has non-trivial structure in the Q directions and peaks sharply about Φα = 0, i it will also have a non-trivial structure in the Φα directions and peak sharply about Qi = 0 but not vice versa.

4.3 Coulomb branch and a scaling theory

i i For large enough |q | we can integrate out the massive Φα’s (in their ground state) i from the full quiver theory (4.2.1). This can be done exactly given that the Φα appear quadratically in (4.2.1) whenever the superpotential vanishes. One ﬁnds the bosonic

k i of CP ’s since we have to identify the overall phase of the φα due to the U(1) gauge connection. 1 2 i When |κ | + |κ | − 2 ≥ |κ3|, which for large κ amounts to the κ satisfying the triangle inequality, κ1−1 κ2−1 the number of constraints become less or equal to the dimension of CP × CP , allowing for more complicated topologies for M. 4Indeed, there are several quiver diagrams with the same net charges and one might suspect that all such quivers are required to obtain the correct entropy of the supersymmetric black hole (see for example [177]). CHAPTER 4. CONFORMAL QUIVERS 79

quantum eﬀective Coulomb branch Lagrangian (up to quadratic order in q˙ i and Di):

2 3 i 1 X X si |κ | L = G q˙ i · q˙ j + Di Di − s |κi|Ad(qi) · q˙ i − + θi Di . (4.3.1) c.b. 2 ij i 2|qi| i=1 i=1

The terms linear in q˙ i and Di follow from a non-renormalization theorem [165], whereas the quadratic piece in q˙ i is derived in appendix C.4.1. Recall that the system is only a function of q1 and q2 since q3 = q1 + q2. The three-vector Ad is the vector potential for a magnetic monopole:

−y x Ad(x) = xˆ + yˆ , (4.3.2) 2r(z ± r) 2r(z ± r) and Gij is the two-by-two metric on conﬁguration space:

1 3 1 |κ1| 1 |κ3| 3 1 |κ3| ! µ + µ + 4 |q1|3 + 4 |q1+q2|3 µ + 4 |q1+q2|3 [Gij] = 3 2 3 . (4.3.3) 3 1 |κ | 2 3 1 |κ | 1 |κ | µ + 4 |q1+q2|3 µ + µ + 4 |q2|3 + 4 |q1+q2|3

Upon integrating out the auxiliary Di-ﬁelds, we obtain a multi-particle quantum mechanics with (bosonic) Lagrangian:

2 3 1 X X L = G q˙ i · q˙ j − s |κi|Ad(qi) · q˙ i − V (qi) . (4.3.4) c.b. 2 ij i i=1 i=1

That the quantum eﬀective Coulomb branch theory has a non-trivial potential V (qi) should be contrasted with other supersymmetric cases such as interacting D0-branes or the D0-D4 system [178] where the potential vanishes and the non-trivial structure of the Coulomb branch comes from the moduli space metric. The potential V (qi) is also somewhat involved and is given in appendix C.4.2. CHAPTER 4. CONFORMAL QUIVERS 80

4.3.1 Supersymmetric conﬁgurations

The supersymmetric conﬁgurations of the Coulomb branch consist of time independent solutions which solve the equations V (qi) = 0. For (4.3.1), this amounts to:

s |κi| s |κ3| i + 3 + 2θi = 0 , i = 1, 2 . (4.3.5) |qi| |q3|

In appendices C.2 and C.3 we review that these supersymmetric conﬁgurations are robust against corrections of the Coulomb branch theory from the superpotential and from integrating out higher orders in the auxiliary D ﬁelds.

Bound states

There are bound state solutions [165, 179] of (4.3.5) which are triatomic (or more generally N-atomic if dealing with N wrapped branes) molecular like configurations. Of the original nine degrees of freedom, three can be removed by fixing the center of mass. Then the bound state condition (4.3.5) fixes another two-degrees of freedom. Thus, bound state solutions have a four-dimensional classical moduli space. Due to the velocity dependent terms in the Lagrangian, the flat directions in the moduli space are dynamically inaccessible at low energies – the particles resemble electrons in a magnetic field. Several dynamical features of the three particles were studied in [4,180].

Scaling solutions

There are also scaling solutions [166] of (4.3.5) which are continuously connected to the origin |qi| = 0. They occur whenever the κi form a closed loop in the quiver diagram (e.g. κ1, κ2 > 0 and κ3 < 0) and obey the same triangle inequality (|κ2| + |κ3| ≥ |κ1| and cyclic permutations thereof) that the |qi| are subjected to. These solutions can be expressed as a series:

∞ i i X n |q | = |κ | anλ , λ > 0 . (4.3.6) n=1 CHAPTER 4. CONFORMAL QUIVERS 81

The coeﬃcient a1 = 1, while the remaining an can be obtained by systematically solving (4.3.5) in a small λ expansion, and will hence depend on θi. The moduli space of the scaling solutions is given by the three rotations as well as the scaling direction parameterized by λ. Though the angular directions in the moduli space are dynamically trapped due to velocity dependent forces, the scaling direction is not and constitutes a ﬂat direction even dynamically. n+1 n Requiring that the series expansion converges, i.e. (an+1λ )/(anλ ) 1, leads to the condition: 1 λ . (4.3.7) θ Because of this condition on the λ’s, one should be cautious when dealing with such scaling solutions. They occur in the near coincident limit of the branes where the bifundamentals that we have integrated out become light. In order for the mass of 2 2 the bifundamentals, Mij = (|xij| + θi/mi − θj/mj), to remain large we require:

θ 1/2 |qi| . (4.3.8) µ

i i i i α Taking µ = ν µb and |q | = |qb |/ν the inequalities (4.3.7) and (4.3.8) can be satisﬁed i i i i in the limit ν → ∞, with µb , qb , κ and θ ﬁxed and furthermore α ∈ (0, 1/2). Notice that (4.3.7) implies that the distances between particles in a scaling regime ∼ λκ is much less than the typical inter-particle distance of a bound state ∼ κ/θ.

4.3.2 Scaling Theory

To isolate the physics of the Coulomb branch in the scaling regime we take an infrared limit of the Lagrangian (4.3.1), pushing the qi near the origin and dilating the clock t. In particular, we would like the ∼ κ/|qi|3 part of the metric in conﬁguration space to dominate over the ∼ µ piece leading to:

κ1/3 |qi| . (4.3.9) µ CHAPTER 4. CONFORMAL QUIVERS 82

Additionally we must satisfy the inequalities (4.3.7) and (4.3.8). Again taking µi = i i i α α i i i i ν µb , q = qb /ν and in addition t = ν bt with fixed µb , qb , κ and θ , we can also satisfy (4.3.9) in the limit ν → ∞, so long as we also ensure α ∈ (1/3, 1/2).5 The rescaling of t is required to maintain a finite action in the scaling limit. In type √ II string compactifications µ ∼ 1/lP ∼ v/gsls, where lP is the four-dimensional Planck length and v is the volume of the Calabi-Yau in string units [165], therefore the ν → ∞ limit corresponds to a parametrically small string coupling. Furthermore the scaling throat deepens as we increase the mass of the wrapped branes. The rescaling above amounts simply to setting the θi and µi to zero in (4.3.1) and i i i replacing q and t with qb and bt. We call the remaining Lagrangian with vanishing θ and µi the scaling theory. Notice from equation (C.4.18) that the potential V (qi) in i i this limit becomes a homogeneous function of order one, i.e. V (ν qb ) = νV (qb ) and the linear in velocity term is retained.

Consistency of the small Di expansion

One can infer from the supersymmetry variations [165] that supersymmetric configurations have vanishing Di. Furthermore, as we have already noted, we have performed an expansion in small Di fields prior to integrating them out (see appendix C.3 for more details) in order to obtain the Coulomb branch. Thus, in order for a scaling theory to exist and be consistent with a small Di expansion, it must be the case that zero energy scaling configurations exist. Had we considered a two-particle theory, where no such scaling solutions can exist, taking a small q limit would be inconsistent with the small D expansion. That is because the dimensionless small quantity in our perturbation series is actually D/|q|2 and the supersymmetric configuration D = 0 occurs at |q| = −κ/2θ. Expanding the non-linear D equation (C.3.1) (see appendix C.3) in powers of ≡ D/|q|2, while imposing |q| (θ/µ)1/2, one finds the following consistency condition:

θ 1 κ 1 1 = + 1 − + O(2) . (4.3.10) µ |q|2 2µ |q|3 2

5Another limit one might imagine is given by: |qi| (κ/µ)1/3, |qi| κ/θ. In this case the metric on conﬁguration space remains ﬂat while the potential scales like ∼ 1/|q|2. CHAPTER 4. CONFORMAL QUIVERS 83

Indeed, the ﬁrst term on the right hand side is negligible by construction, since we took |q| (θ/µ)1/2 to keep the strings massive. Smallness of the second term in the equation would require |q|3 κ/µ, in contradiction with the condition (4.3.9) required to isolate the scaling theory.

We now consider the three-node case for si that admit a closed loop in the quiver. 2 2 2 For small 2 ≡ D /|q | (taking all masses the same and θ1 = θ2 = θ and again imposing |q2| (θ/µ)1/2) the equation of motion of the auxiliary D2-ﬁeld (in the form of (C.3.2)) is given by:

3θ |κ2| 1 = + δ − + O(2) (4.3.11) 2 2µ|q2|2 2µ|q2|3 2 2 2

|q2| |κ1| |κ3| where δ ≡ 1 − 2 |κ2| |q1| + |q3| measures how close the configuration is to the scaling solution. For sufficiently small δ ∼ 1 we can consistently satisfy (4.3.11) in addition to imposing the scaling inequalities (4.3.9). In this sense the scaling theory is in fact a theory of the deep infrared configurations residing parametrically near the zero energy scaling solutions. This is consistent with the dilation of time required to obtain the scaling theory.

4.4 Conformal Quivers: emergence of SL(2, R)

In this section we uncover that the bosonic scaling theory action, i.e. (4.3.4) with the θi and µi set to zero, has an SL(2, R) symmetry. This is the symmetry group of conformal quantum mechanics [175]. The group SL(2, R) is generated by a Hamilto- nian H, a dilatation operator D and a special conformal transformation K, with the following Lie algebra:

[H,D] = −2iH , [H,K] = −iD , [K,D] = 2iK . (4.4.1)

As we have already mentioned, the full SL(2, R) symmetry is not guaranteed by the existence of time translations and dilatations alone [181]. This is suggested by the fact that the H and D operators form a closed subalgebra of the full SL(2, R). The CHAPTER 4. CONFORMAL QUIVERS 84

presence of a full SL(2, R) is actually quite remarkable, particularly given the specific form of the scaling theory Lagrangian which has velocity dependent forces and a non-trivial potential. As discussed in the previous section, it is not true that, for any finite κi, µi and θi, the Coulomb branch can be described precisely by the scaling theory action. There will always be small corrections that break its manifest scaling symmetry: qi → γ qi, and t → t/γ. This is comforting, given that the full quiver theory has a finite number of ground states—yet a conformal quantum mechanics has a diverging number of arbitrarily low-energy states, with a density of states that behaves as dE/E. The corrections serve as a cutoff for the infrared divergence in the number of states, such that the Coulomb branch can fit consistently inside the full quiver theory (related discussions can be found in [167,182]).

4.4.1 Conditions for an SL(2, R) invariant action

The conditions under which an action will be SL(2, R) invariant (up to possible surface terms) have been studied extensively in [147,181,183]. Showing that a general theory with bosonic Lagrangian describing N degrees of freedom :

1 L = q˙i G q˙j − A q˙i − V (q) , i = 1, 2,...,N (4.4.2) 2 ij i has an SL(2, R) symmetry is equivalent to ﬁnding a solution to the following equations [183]:

2 ∇(iZj) = Gij , (4.4.3) i −Z ∂iV = V, (4.4.4)

2 Zi = ∂if , (4.4.5) j Z Fji = 0 ,Fij ≡ ∂[iAj] . (4.4.6)

Equations (4.4.3) and (4.4.4) ensure the existence of a dilatation symmetry. In particular, equation (4.4.3) implies that the metric on conﬁguration space allows for a CHAPTER 4. CONFORMAL QUIVERS 85

conformal Killing vector ﬁeld (also referred to as a homothetic vector ﬁeld). Equa- tions (4.4.5) and (4.4.6) ensure the that the action remains invariant under special conformal transformations, where f is an arbitrary function of the qi. Indices are raised and lowered with the metric Gij.

Interestingly, equation (4.4.5) imposes that the conformal Killing form Zi of the metric be exact, which is generically not the case. Hence the existence of a dilatation symmetry does not necessarily imply the symmetry of the full conformal group. Once a solution to (4.4.3-4.4.6) is found, the three conserved quantities are then given by [183]:

1 Q = tn+1 q˙i G q˙j − (n + 1)tnZi G q˙j + tn+1V (q) + F , n = −1, 0, 1 (4.4.7) n 2 ij ij n where F−1 = F0 = 0 and F1 = f. The charge Q−1 is the Hamiltonian, whereas Q0 and Q1 are related to dilatations and special conformal transformations, respectively. These three charges generate the SL(2, R) algebra (4.4.1) (up to factors of i) under the Poisson bracket. In what follows, we ﬁnd Zi and f for the scaling theory described in section 4.3.2.

4.4.2 Two particles

As a warm up, we can study a simple model consisting of a two node quiver with an equal number of arrows going to and from each node, with a total number κ > 0 arrows altogether. This is the theory describing, for example, the low energy dynamics of a wrapped D4-D0 brane system (see [178]). The bosonic Lagrangian for the relative position q = (qx, qy, qz) on the Coulomb branch, in the scaling limit, is given by:

κ q˙ 2 L = . (4.4.8) 2 |q|3

The above Lagrangian is also the non-relativistic limit of one describing a BPS particle 2 in an AdS2 × S background. The wordline theory has three degrees of freedom and −3 a diagonal metric on conﬁguration space: gij = κ |q| δij. In addition to the Hamiltonian H, the above theory has a dilatation operator D and special conformal CHAPTER 4. CONFORMAL QUIVERS 86

generator K given by:

i −9/2 9/2 i −1 D = i q ∂i + |q| ∂i |q| q ,K = 2 κ |q| , (4.4.9) such that the SL(2, R) algebra is satisﬁed. It is thus a simple example of a conformal quantum mechanics.

4.4.3 SL(2, R) symmetry of the full three particle scaling theory

For our particular problem, it is useful to note that the index structure of the relative coordinates, i.e. i = 1, 2, 3, is trivially tensored with the spatial index implicit in bold vector symbols (e.g. q = (qx, qy, qz)). Introducing Qα ≡ (q1, q2), with α = 1, 2,..., 6, i i our Lagrangian (4.3.4) (with µ and θ set to zero, and si that admit a closed loop) takes the form: 1 L = Q˙ α G Q˙ β − A(Q) Q˙ α − V (Qα) , (4.4.10) c.b. 2 αβ α where Gαβ and the six-dimensional vector potential can be extracted from (4.3.4). An expression for the vector potential is simple to write down and is given by:

(Q) 1 d 1 3 d 1 2 2 d 2 3 d 1 2 Aα = s1 |κ |A (q ) + s3 |κ |A (q + q ) , s2 |κ |A (q ) + s3 |κ |A (q + q ) . (4.4.11) where Ad(x) is the vector potential for a magnetic monopole and is given in (4.3.2). It is straightforward to check, using Mathematica for example, that the conditions α α α β (4.4.3-4.4.6) are indeed satisﬁed. We ﬁnd Z = −Q and f = 2 Q GαβQ . The explicit generators of the SL(2, R) are given by: √ 1 1 (Q) αβ (Q) α H = − √ ∂α G − iA G ∂β − iA + V (Q ) , (4.4.12) 2 G α β √ α 1 α D = i Q ∂α + √ ∂α GQ , (4.4.13) G α β K = 2 Q GαβQ , (4.4.14) CHAPTER 4. CONFORMAL QUIVERS 87

α (Q) where we have used that Q Aα = 0. The generator K simpliﬁes to:

|κ1| |κ2| |κ3| K = + + . (4.4.15) 2|q1| 2|q2| 2|q3|

We thus conclude that the bosonic scaling theory is an interacting multi-particle SL(2, R) invariant conformal quantum mechanics with velocity dependent forces. The theory admits a further SO(3) symmetry acting on the spatial three-vectors.

N-particle case

Though we do not prove it here, it is natural to conjecture that the N-particle scaling theory will be an N-particle conformal quantum mechanics. Indeed, for any closed loop of Ni-particles we expect that in the corresponding throat there will be a conformal quantum mechanics with D and K as in (4.4.12) except Qα runs over all the

3Ni bosonic degrees of freedom. The one loop metric on conﬁguration space in the Coulomb branch, i.e. the coeﬃcient of q˙ i · q˙ j, will be equivalent to that of Di Dj (which is far easier to compute) upon integrating out the massive strings for general quivers. Interestingly, when there are more than three nodes, there may be several distinct closed loops allowing for their own scaling throat and several distinct scaling throats may coexist, within its residing a decoupled conformal quantum mechanics theory.

Superconformal quantum mechanics

It is worth remarking that the Coulomb branch is in fact a supersymmetric quantum i mechanics with four supercharges QS and an SO(3) global R-symmetry group. This i follows from effective field theory. We are integrating out the heavy chiral Φα multiplet in its supersymmetric ground state. Thus, the low energy effective theory of the vector multiplet will be endowed with the four supercharges of the parent quiver theory. Of course, there could be anomalies that arise in the process. For instance, the discrete time reversal symmetry of the full quiver theory is violated in the Coulomb branch by the linear in velocity terms. This is an anomaly which does not spoil the supersymmetry of the Coulomb branch, and occurs due to a zero mode in the CHAPTER 4. CONFORMAL QUIVERS 88

i functional determinant of the ψα fermions. As shown in [184] for the two node case, one can compute the supercharges of the low energy eﬀective theory in a systematic fashion using perturbation theory. We have shown above that the bosonic Lagrangian of this theory exhibits an SL(2, R) symmetry. In order to establish that this extends to a superconformal quantum mechanics we must establish the existence of four supersymmetric special i i i conformal generators S . These will be given by the commutator S = i[QS,K]. As i i discussed in [147], one must also ensure that the commutator [QS,D] = −iQS is satisﬁed in order to have a superconformal system. This is guaranteed due to the manifest scale invariance of the supersymmetric action: qi → γ qi, λi → γ3/2 λi with t → t/γ.6 Hence the scaling theory is a superconformal multiparticle theory.

Gravity

As discussed in the introduction, the emergence of a full SL(2, R) in the deep scaling regime is reminiscent of the emergence of a full SL(2, R) in the deep AdS2 throat in gravity, which could end at a horizon or cap oﬀ at the locations of entropy- less D-particles. The symmetry group manifests itself in the dynamics of wrapped branes moving in a ‘geometry’ resulting from integrating out the interconnecting strings and extends upon similar observations made for other multiparticle systems in [147, 181, 185]. However, we must point out that the conformal quantum mechanics obtainde here comes directly from the low energy dynamics of strings interacting with wrapped branes, i.e. the quiver theory, rather than from a gravity calculation of the moduli space of a multi-black hole system. It would in fact be interesting to repeat such gravitational Ferrel-Eardley type calculations [181, 185–188] for the low energy velocity expansion of the corresponding one-half BPS scaling solutions [179] in N = 2 supergravity. This might give an operational meaning to the AdS2/CQM correspondence [189–193] (particularly in the large gsκ limit and within the AdS2

6Though we have not presented the piece of the Lagrangian quadratic in λi, which will be of the form ∼ κ λ¯ λ/˙ |q|3 in the scaling region, the linear piece is ﬁxed by supersymmetry [165] to be (C.2.1), which is enough to read oﬀ the scaling dimension of λi. CHAPTER 4. CONFORMAL QUIVERS 89

scaling region). Recall that in the classical supergravity limit, the gravitational back- 7 reaction of a brane becomes parametrically small given that its mass goes like 1/lP . Furthermore, as emphasized in the introduction, our system is a quiver quantum mechanics that does not necessarily reside inside a larger two-dimensional conformal ﬁeld theory, and thus the SL(2, R) may be of the ‘isolated’ type.

4.4.4 Wavefunctions in the scaling theory

Given the existence of an SL(2, R) invariant scaling theory, one natural question is whether a quantum state will stay localized within the scaling regime or if its wavefunction will spread away from the scaling regime, or even fall back into the Higgs branch (where h|qi|i = 0). One particular direction in which a wavefunction might easily spread is the scaling direction where the potential is identically zero and where there are (classically) no velocity dependent magnetic forces. Studying the quantum mechanics problem of the full three-particle scaling theory is diﬃcult. Instead, we will look at the wavefunctions of the simpler wrapped D4-D0 model discussed in section 4.4.2. The zero angular momentum piece of the Schr¨odinger equation in spherical coordinates (i.e. qx = q cos θ sin φ, qy = q sin θ sin φ and qz = q cos φ, q ≡ |q|) is:

1 √ 1 ∂ ∂ − √ ∂ g gij ∂ ψ = − q5/2 q1/2 ψ = E ψ , (4.4.16) 2 g i j E 2κ ∂q ∂q E E

√ p 8 which can easily be solved: ψE(q) ∼ q exp ±2i 2κE/q . These energy eigenstates are non-normalizable at small q with respect to the covariant inner product: Z 3 √ ∗ hψ1|ψ2i = d q g ψ1(q) ψ2(q) . (4.4.17)

7This approach is reminiscent of attempts to match the Coulomb branch of the BFSS matrix model [194,195] with eleven-dimensional supergravity calculations. A basic diﬀerence is the presence of the ability to zoom into a deep AdS2 throat in the geometry and that the microscopic quiver model is vector like rather than matrix like. 8 lm E m m If we include the angular variables we would ﬁnd ψE = Ql (q)Yl (θ, φ) with Yl spherical E harmonics and Ql can be written in terms of Bessel functions. Here we only look at l = 0 modes, E ψE(q) = Q0 (q). CHAPTER 4. CONFORMAL QUIVERS 90

Non-normalizability of energy eigenstates is common for scale invariant theories [175], the non-normalizability means that energy eigenstates leak into the Higgs branch.

As in [175], we can consider instead eigenstates ψλ(q) of the H +aK operator with √ 1 eigenvalue λ ≡ a n + ∈ R, where a is a constant with appropriate dimensions. 2 √ If we deﬁne a variable x = q/(κ a), then the zero-angular momentum wavefunctions

ψλ(x) satisfy 1 ∂ ∂ 4 1 −x5/2 x1/2 + ψ = n + ψ , (4.4.18) 2 ∂x ∂x x λ 2 λ The normalizable wavefunctions are given by conﬂuent hypergeometric functions:

1 − n 3 4 ψ (x) = N e−2/x U , , , (4.4.19) λ 2 2 x where N is a finite normalization factor that depends on κ and a. Whenever n ∈ N, 1−n 3 4 the expression for U 2 , 2 , x simplifies significantly. We can also obtain asymptotic expressions near x = 0 and x = ∞:

1 − n 3 4 √ U , , = x−n/2 2n−1 x + O x3/2 , for x ∼ 0 , (4.4.20) 2 2 x √ √ π x 2 π = − + O x−1/2 , for x−1 (4.4.21)∼ 0 . 2 Γ [(1 − n)/2] Γ[−n/2]

The dependence on κ and a drops out of all expectation values of operators as a ˜ function of x labeled O(x), hence we can instead normalize the wavefunctions ψλ(x) ≡

ψλ(x)|κ,a=1 and reinstate the dependence on κ and a in expectation values of operators by looking at the dependence of O(x) on the variable x and how it transforms when √ ˜ we take q → xκ a. Examples of ψλ(x) are displayed in figure 4.2. To understand whether these states leak back into the Higgs branch, we consider for example hxiλ. This is only finite if n is an odd positive integer for which hxiλ = 8 √ (or equivalently hqiλ = 8κ a ), otherwise hxiλ is infinite. This can be seen from the √ fact that for positive and odd n, Γ[(1−n)/2] diverges and the O( x) term in the large x expansion (4.4.21) dissappears. However, for any λ(n) there exists a number s < 1 s such that hx iλ is finite. Similarly, when n is an odd positive integer, there exists a s number s > 1 such that hx iλ is infinite. Thus a large class of H + aK eigenstates do CHAPTER 4. CONFORMAL QUIVERS 91

é 2 é 2 g y l g y l

0.025 1.5 » 0.020»

1.0 0.015

0.010 0.5 0.005 x x 1 2 3 4 5 6 20 40 60 80 100 120

˜ Fig. 4.2: Examples of H + a K eigenfunctions. Left: Plot of ψλ(x) for n ∈ (0.2, 5.2) ˜ in unit increments. Right: Plot of ψλ(x) for n ∈ (−5.7, −0.7) in unit increments. not leak into the Higgs branch for large κ. Finding the wavefunctions in the three-node Coulomb branch is clearly more complicated. There are now six-degrees of freedom and a non-trivial potential V (qi). However, at the classical level, the scaling direction remains ﬂat and motion along it is uninhibited, neither by the potential nor by any velocity dependent magnetic forces. So if the wavefunction has any chance of spreading, it will do so along the scaling direction. Our previous analysis of the two particle wavefunctions, which deals essentially with the scaling direction, suggests that at least some H + aK eigenstates do not leak into the Higgs branch.9

SL(2, R) in the Higgs branch?

We end this section with an important question. The SL(2, R) symmetry we have been discussing so far resides in the Coulomb branch of the full quiver theory. In particular it resides in the scaling regime of the Coulomb branch which is connected to the Higgs branch at its tip, i.e. where all the qi become vanishingly small. How does the SL(2, R) structure act on (if at all) the Higgs branch degrees of freedom? Perhaps a way to answer this question is via the Higgs-Coulomb map of [196] (further

9We also note that the quantization of the classical solution space of scaling conﬁgurations was considered in [182], where it was found that the expectation value of the total angular momentum operator was non-vanishing. CHAPTER 4. CONFORMAL QUIVERS 92

developed in the context of quiver theories in [167]). Indeed, an SL(2, R) symmetry acting on the states residing in the Higgs branch would resonate closely with the appearance of an SL(2, R) isometry of the AdS2 in the near horizon region of the extremal black hole.

4.5 Melting Molecules

We have seen that the low energy excitations of the scaling regime in the Coulomb branch are described by a multiparticle SL(2, R) invariant quantum mechanics. It was noted that this is reminiscent of the SL(2, R) that appears in the near horizon geometry of an extremal black hole or the AdS2 geometry outside a collection of extremal black holes that reside within a scaling throat. If we heat up such a collection of black holes (for example by making the centers slightly non-extremal) they will fall onto each other due to gravity’s victory over electric repulsion. Naturally, then, we might ask what happens to our scaling theories and more generally the Coulomb branch configurations upon turning on a temperature?10 This amounts to integrating out the massive strings, i.e. the chiral multiplet, in a thermal state as opposed to their vacuum state. In this section we assess the presence of bound states and scaling solutions as we vary the temperature. Roughly speaking, this is the weak coupling version of the calculations in [208–211] where the multicentered solutions were studied at finite temperature.11 We find that the bound (but non-scaling) configurations persist at finite temperature until a critical temperature after which they either classically roll toward the origin or become metastable. This amounts to the ‘melting’ of the bound state. Scaling solutions on the other hand do not persist at

10Aspects of supersymmetric quantum mechanics at finite temperature are discussed in [197–199]. An incomplete list of more recent studies including numerical simulations is [200–207]. 11In fact, [208, 211] explored how the effective potential of a D-particle/black hole bound state changed as the temperature of the black hole was increased. Hence, a closer analogy to [208] would be to consider a four-node mixed Higgs-Coulomb branch as in figure 4.8. One node is ‘pulled’ away from the remaining three, such that one can integrate out all connecting heavy strings in some thermal state. The remaining three nodes, which we choose to contain a closed loop (and hence have exponentially many ground states), are in the Higgs branch which is also considered to be at some finite temperature. One could then study the Coulomb branch theory for the position degree of freedom of the far away wrapped brane as a function of the temperature. CHAPTER 4. CONFORMAL QUIVERS 93

finite temperature and instead the potential develops a minimum at the origin, even at small temperatures. As was alluded to, what we find is somewhat analogous to heating up a collection of extremally charged black holes, known as the Majumdar-Papapetrou geometries [212, 213], by making the masses of the centers slightly larger than their charges. Doing so will cause the black holes to collapse into a single center configuration upon the slightest deviation from extremality.

4.5.1 Two nodes - melting bound states

Consider the low energy dynamics of two wrapped branes, which is given by a two node quiver. There can be no gauge-invariant superpotential in this case since the quiver admits no closed loops. The relative Lagrangian is:

µ L = q˙ 2 + D2 + 2iλ¯λ˙ − θD , V 2 2 2 2 ¯ ¯ LC = |Dtφα| − |q| + D |φα| + iψαDtψα − ψαq · σψα √ ¯ α ¯ ¯ −i 2 φαψ λ − λψαφα , where Dtφα ≡ (∂t + iA) φα. The matter content is given by a chiral multiplet

Φα = {φα, ψα} and a vector multiplet Q = {A, q, D, λ} (A is a one-dimenional U(1) connection). The index α = 1, 2, . . . , κ (where κ > 0) is summed over. If we keep |q| constant and set to zero the fermionic superpartners λ, we can integrate out the φα and ψα ﬁelds to get the eﬀective bosonic action on the position degrees of freedom:

µ L = D2 − θD − κ log det −(∂ + iA)2 + |q|2 + D eﬀ 2 t 2 2 + κ log det −(∂t + iA) + |q| . (4.5.1)

If we wish to study the system at ﬁnite temperature T ≡ β−1, we must Wick rotate to Euclidean time t → itE and compactify tE ∼ tE + β. Notice we cannot set the zero mode of A to zero since one can have non-trivial holonomy around the thermal circle. CHAPTER 4. CONFORMAL QUIVERS 94

The φα ﬁelds have periodic boundary conditions and the ψα ﬁelds have anti-periodic boundary conditions around the thermal circle. The operator ∂tE has as eigenvalues the Matsubara frequencies ωn = 2πnT for the bosonic case and ωn = 2π(n + 1/2)T for the fermionic case, where n ∈ Z. Hence, it is our task to evaluate:

µ κ L(T ) = D2 − θD − log det 4π2(n + a)2T 2 + |q|2 + D eﬀ 2 β κ + log det 4π2(n + 1/2 + a)2T 2 + |q|2 , (4.5.2) β

(T ) where we have defined a ≡ A/2πT . Notice that Leff depends on the connection A which we eventually need to path-integrate over. Upon evaluating the determinant (see appendix C.5 for details), one finds the effective thermal potential is given by:

√  |q|2+D  cosh T − cos(2πa) µ 2   V (T, D, a) = − D + θD + κ T log   , (4.5.3) 2  |q|  cosh T + cos(2πa)

2 which has the correct low temperature limit limT →0 V (T, D, a) = −µD /2 + θD − κ|q|+κp|q|2 + D, in accordance with the results of [165]. Our task is to now perform the path integral of e−βV (T,a) over D and a. In doing so, we must be careful to ensure that D > −|q|2 such that integrating out the string is justifiable. Differentiating (4.5.3) with respect to a we find that V (T, D, a) has a minimum at cos(2πa) = 1 for all T and D > −|q|2. Solving the saddle point equations for D must be done numerically. The potential V has two scaling symmetries associated to it. Configurations related by:

T → γ T , |q| → γ |q| ,D → γ2 D , µ → γ−2δ µ , θ → δ θ , κ → γδ κ , (4.5.4) have respective potentials related by: V (µ, θ, κ, |q|,D,T ) → δγ2V (µ, θ, κ, |q|,D,T ). CHAPTER 4. CONFORMAL QUIVERS 95

The qualitative features of the potential thus only depend on scale invariant quantities: µ 1/2 µ 1/2 µ 1/2 κ˜ ≡ κ , T˜ ≡ T , |q˜| ≡ |q| , (4.5.5) |θ|3 |θ| |θ| ˜ µ and the scaling-invariant potential is given by V ≡ V |θ|2 . Note that our finite temperature analysis breaks down for T ∼ 1 in string units, where the effects of massive string modes start to kick in. However, we may still consider large T˜ as long as we restrict ourselves to parameter regions where our approximation remains valid, e.g. T 1 and µ/|θ| 1. We use the scaling symmetries to fix µ = |θ| = 1 and study the thermal phases of the theory as a function of T˜ andκ ˜, noting that we can always extract physical quantities by properly rescaling µ and θ.

Thermal phases

Having derived the thermal effective potential we can now describe the different thermal configurations for the two-node quiver. The physically distinguishable phases have potentials V˜ with the following distinct properties:

1. A single stable minimum away from |q˜| = 0 .

2. Two non-degenerate minima away from |q˜| = 0 .

3. Two degenerate minima away from |q˜| = 0 .

4. Two minima with one at |q˜| = 0 where:

(a) the minimum at |q˜| = 0 is the global minimum, (b) the minimum at |q˜| > 0 is the global minimum.

5. A single minimum at |q˜| = 0 .

Examples of each of these thermal configurations are displayed in figures 4.3 and 4.4. Depending on the location in the (˜κ, T˜)-plane, the system will be driven (either through thermal activation or quantum tunneling) to the most stable configuration. If the system is in a metastable configuration, such a process can take exponentially CHAPTER 4. CONFORMAL QUIVERS 96

15, T 0.75 9Κ = 9Κ 15, T 0.77= V V

0.8 0.30 0.25 0.6 0.20 0.4 0.15 0.10 0.2 0.05 ÈqÈ ÈqÈ 2 4 6 8 10 12 2 4 6 8 10 12

9Κ 15, T 0.777= 9Κ 15, T 0.78= V 0.14 V 0.12 0.10 0.10 0.08 0.05 0.06 0.04 ÈqÈ 0.02 2 4 6 8 10 12 ÈqÈ -0.05 2 4 6 8 10 12

9Κ 15, T 0.8= 9Κ 15, T 0.84= V V ÈqÈ ÈqÈ 2 4 6 8 10 12 2 4 6 8 10 12 -0.1 -0.5 -0.2 -0.3 -1.0 -0.4 -0.5 -1.5 -0.6

Fig. 4.3: Thermal effective potentials of a two node quiver (θ = −1 and µ = 1). As the temperature is increased the system explores various thermal configurations of stable and metastable minima. From top left to bottom right the system is of type1 → 2 → 3 → 2 → 4a → 5. long. At high enough temperatures, the system will eventually fall back to |q˜| = 0 where the Higgs and the Coulomb branch meet, mimicking gravitational collapse or the melting of the molecule. Note that for bound states in N = 2 supergravity, the effective potential of a small probe around a large hot black hole, as studied in [208], never exhibited two minima away from |q˜| = 0 (that is we never noticed potentials of types2-4b). Since CHAPTER 4. CONFORMAL QUIVERS 97

9Κ 70, T 1.43= 9Κ 6.2, T 0.1= V V 9Κ 6.2, T 0.5= 4 V 5 0.25 4 3 0.20 3 2 0.15 2 0.10 1 0.05 1 ÈqÈ 2 4 6 8 10 12 ÈqÈ ÈqÈ -0.05 20 40 60 80 2 4 6 8 10 12

Fig. 4.4: Thermal eﬀective potentials of a two node quiver (for θ = −1 and µ = 1). Left: An example of phase type 4b. Right: A case where the potential of the supersymmetric minimum decreases as the temperature is increased. A similar observation was made for supersymmetric bound states in [208].

supergravity is a good effective description at large geff ∼ gsκ, it would be interesting to see if this behavior matches qualitatively in the Coulomb branch asκ ˜ is increased. An example of this is shown in figure 4.5.

High temperature behavior: an entropic eﬀect

A way to think about the high temperature melting transition is to integrate out the 2 2 auxiliary D-fields first, which induces a quartic interaction (|φα| ) /µ for the bifundamentals. Since this interaction is relevant, it will play a minor effect at sufficiently large temperatures, namely for T µ−1/3. Thus, at high temperatures we are dealing with a collection of κ non-interacting complex bosonic and fermionic degrees of freedom with square masses (|q|2 + θ/m) and |q|2 respectively. In addition, due to the U(1) connection we must only consider the gauge invariant states: the spectrum is constrained to those states annihilated by the U(1) charge operator. A particularly convenient gauge is the temporal gauge, A = 0. At some fixed high temperature, the number of gauge invariant modes that aren’t Boltzmann suppressed increases with decreasing mass and thus the dominant contribution to the free energy will come from for the lightest possible mass, i.e. |q| = 0. This can be viewed as an entropic effect [214]. CHAPTER 4. CONFORMAL QUIVERS 98

4.5.2 Three nodes - unstable scaling solutions

A similar analysis for three nodes gives rise to the following effective thermal potential: √  i 2 i  |q | +siD i 3 cosh − cos(2πa ) X µi T V (T,Di, ai) = − DiDi + θiDi + |κi| T log   .  i  2  |q | i  i=1 cosh T + cos(2πa ) (4.5.6) Our main interest to understand the behavior of the scaling potential at finite temperature, and in particular the flat scaling direction. Thermal effects will kick in i when |q | . T . Since zero temperature scaling solutions occur for arbitrarily small |qi|, it is sufficient to perform a small temperature expansion. The dimensionless quantities capturing the thermal transition are |qi|/T . As in the two-node case, one must integrate out the U(1) connections ai and the Di. For the case where the κ1 = κ2 = −κ3 > 0 we can identify a critical point along the scaling direction |qi| = λ|κi|: a1 = 1, a2 = 0, D1 = 0 and D2 = 0. On this saddle the potential becomes: 3 X λ|κi| V ∗(T ) = 2 T |κi| log tanh , (4.5.7) 2T i=1 which is negative definite. Of course there could be more dominant saddles that make the potential even lower than the above. So more generally, this procedure must be done numerically, even in the small temperature expansion, as the equations governing the connections are intractable analytically. It amounts to numerically minimizing a function of four variables, namely a1, a2, D1 and D2 (recall that a3 = a1 + a2 and D3 = D1 + D2). Since we are interested in the scaling branch we can set θi = µi = 0 in our analysis.12 The resulting potential (with θi = µi = 0) then exhibits the following scaling relation:

κi → δ κi ,T → γ T , |qi| → γ |qi| ,Di → γ2 Di , (4.5.8)

12The reason we set θi = 0 is for computational simplicity and numerical clarity. We could have started a Coulomb branch with non-zero θi’s which has scaling solutions for |qi| = λ κi + O(λ2) in the limit λ → 0 and found similar results. CHAPTER 4. CONFORMAL QUIVERS 99

9Κ 14., T 0.77= 9Κ 14.4, T 0.77= V V 0.1 ÈqÈ 0.05 2 4 6 8 10 12 ÈqÈ -0.1 2 4 6 8 10 12 -0.2 -0.05 -0.10 -0.3 -0.15 9Κ 15.6, T 0.77= 9Κ 14.8, T 0.77= V V

0.15 0.6

0.10 0.4

0.05 0.2

ÈqÈ ÈqÈ 2 4 6 8 10 12 2 4 6 8 10 12

Fig. 4.5: Thermal effective potentials of a two node quiver (θ = −1 and µ = 1). As κ˜ is increased we note that the first minimum disappears. such that V (κi, |qi|,Di,T ) → γ δ V (κi, |qi|,Di,T ). From the scaling symmetries we observe that the numerical value of the temperature is of little meaning, all that matters for the thermal phase structure is whether or not it vanishes. This is to be expected since we are dealing with a scale invariant system. In figure 4.6 we display the potential at T = 0 (left) and T 6= 0 (right) in the scaling direction |qi| = λ |κi| for κ1 = κ2 = −κ3 = 1. At T = 0, we naturally find that it is vanishing for all λ. At T 6= 0 we see that for radial values larger than the temperature, the scaling direction is still effectively flat, but for radii of order the temperature the thermal potential quickly falls. The numerics are in good agreement with the analytic expression (4.5.7). Hence, at finite temperature the system falls back into its Higgs branch. It would be interesting to explore this phenomenon in more generality for a larger number of nodes. CHAPTER 4. CONFORMAL QUIVERS 100

V V 1 1

0 Λ 0 Λ 0.5 1.0 1.5 2.0 1 2 3 4 5

-1 -1

-2 -2

-3 -3

-4 -4 Fig. 4.6: Thermal potential along the scaling direction |qi| = λ κi for κ1 = κ2 = −κ3 = 1 at T = 0 (left) and T 6= 0 (right).

4.6 Outlook

We have observed the emergence of a full SL(2, R) symmetry from a quiver quantum mechanics model which itself is not a conformal quantum mechanics. The SL(2, R) manifested itself in the eﬀective theory of the position degrees of freedom of the D-particles, once the heavy strings were integrated out. Supersymmetry played an important role in the previous discussion given that the form of the quiver Lagrangian (4.2.1) is heavily constrained by it. But as was mentioned in the introduction, the SL(2, R) symmetry appears for geometries that need not be supersymmetric. In this ﬁnal speculative section, we consider the idea that randomness (in the Wigne- rian sense) is behind the SL(2, R) and observe that dilatations imply special conformal transformations for one dimensional systems whose dilatation symmetry is geometrized into an additional radial direction. Finally, we discuss some possible extensions.13

4.6.1 Random Hamiltonians and emergent SL(2, R)?

It would be interesting to understand whether the emergence of such an SL(2, R) from systems with large numbers of degrees of freedom, such as matrix quantum

13We would like to acknowledge Frederik Denef, Diego Hofman and Sean Hartnoll for many interesting discussions leading to these ideas. CHAPTER 4. CONFORMAL QUIVERS 101

mechanics, could be more general. For instance, if there are a large number of almost degenerate vacua in the putative microscopic dual of AdS2, one might imagine an approximate scale invariance emerging at large N due to the formation of almost continuous bands of low energy eigenvalues. More generally, if the Hamiltonian is sufficiently complicated due to the multitude of internal cycles being wrapped by the branes, one might imagine drawing it from a random ensemble HN of N × N Hermitean matrices. We can then formulate the following problem. Draw an N × N Hermitean matrix H ∈ HN and assess (under some as of yet unspecified measure, perhaps the Frobenius norm is a possibility) how well the matrix equations (4.4.1) can be satisfied, given arbitrary Hermitean matrices D and K. In particular, are the equations better satisfied as we increase N.14 The emergence of an SL(2, R) from a random ensemble of Hamiltonians, if true, would be similar in spirit to the emergence of the Wigner distribution of eigenvalue spacings from random matrices that is almost universal to quantum systems that become chaotic in the classical limit. Interestingly, an ensemble of random Hamiltonians with a scale invariant distribution of eigenvalues was found in [215]. We hope to return to this question in the future.

4.6.2 Holographic considerations

If one assumes that there exists a gravitational dual to the theory, and in addition that a scaling symmetry t → λat exists along with a radial transformation z → λ2z one obtains the two-dimensional metric (see for example [216]):

dt2 dz2 ds2 = − + , (4.6.1) za z2 which is nothing more than AdS2 (albeit in unusual coordinates). The isometry group of AdS2 is SL(2, R) and thus, holographically a dilatation symmetry seems to imply the existence a full SL(2, R) symmetry. This is not true in higher dimensions unless

14Of course, given the fact that SL(2, R) has no ﬁnite dimensional unitary representations, the equations can only be satisﬁed exactly for N = ∞. CHAPTER 4. CONFORMAL QUIVERS 102

one also assumes Lorentz invariance of the boundary metric. Another feature of conformal quantum mechanics that contrasts with higher dimensional conformal field theory is that there is no normalizable SL(2, R) invariant ground state. On the other hand, d-dimensional conformal field theories have an SO(d, 2) invariant ground state wavefunctional. We view this as a hint that what- ever the holographic description of AdS2 is, it is not necessarily a conformal quantum mechanics off the bat. Indeed, the quiver quantum mechanics whose ground state degeneracies count a large fraction of the microstates of a black hole with an AdS2 near horizon are not conformal quantum mechanics in and of themselves. Instead, the SL(2, R) symmetry of AdS2 might only become exact in some kind of large N 15 limit, as opposed to the SO(d, 2) symmetries of AdSd+1 which should persist at finite N (assuming the β-function of the dual CFT vanishes at finite N as is the case for N = 4 SYM).

4.6.3 Possible extensions

As a final note, we would like to mention some open questions and extensions to our discussion. We have left the question of superpotential corrections to the Coulomb branch of the three-node quiver untouched. Though the superpotential will not affect the ground state energy or existence of scaling solutions, it will correct the higher powers in the velocity expansion and it would be interesting to understand whether the SL(2, R) symmetry is preserved by such corrections. The quartic interaction 2 2 coming from integrating out the D-terms, (|φα| ) /µ, is dominated by an expansion in cactus diagrams in the large κ limit. There is no such cactus diagram expansion for the interaction coming from integrating out the F -term, the quartic bosonic interaction being: P ω ω∗ φi φ¯i φj φ¯j . Thus we are confronted with how (if at all) does α,β,γ,β,˜ γ˜ αβγ αβ˜γ˜ β β˜ γ γ˜ one organize a large κ expansion in this case. By dimensional analysis, and inspection of the two-loop Feynman diagrams (see figure 4.7), we expect the velocity squared piece of the Coulomb branch Lagrangian in the scaling regime to become (up to O(1)

15This SL(2, R) might persist in a perturbative treatment of a supergravity solution but not at ﬁnite N, unless of course it resides in a much larger Virasoro structure. CHAPTER 4. CONFORMAL QUIVERS 103

Fig. 4.7: 2-loop Feynman diagrams contributing to the δxδx term of the eﬀective Lagrangian. Solid lines represent φ, while dotted lines correspond to ψ propagators. coeﬃcients):

1 (|κ| |ω|)2 |κ|5|ω|4 δL ∼ |κ| ˙q · ˙q + + O , (4.6.2) c.b. |q|3 |q|6 |q|9

2 −3 P 2 where |ω| ∼ |κ| α,β,γ |ωαβγ| . In order for the superpotential contribution to be subleading we would further require:

2/3 |q| (|κ|ωαβγ) , (4.6.3) in addition to the condition (4.3.9) that forces the system into the scaling regime. Thus, we clearly see that the eﬀect of the superpotential (a relevant deformation) becomes important in the deep infrared region of the scaling regime, where detailed stringy physics begins to manifest itself and potentially destroys the SL(2, R). We have also left untouched the issue of larger numbers of nodes. In such a case one could consider mixed Higgs-Coulomb branches. For example, we could consider a four-node quiver where three nodes are in their Higgs branch and the remaining one residing far away in the Coulomb branch (see also [217,218]), as shown in ﬁgure 4.8. Given the exponentially large number of ground states in the Higgs branch of a three node closed loop quiver, heating such a system up might provide a useful toy model for a D-particle falling into a black hole. CHAPTER 4. CONFORMAL QUIVERS 104

Fig. 4.8: A schematic representation of a system in a mixed Higgs-Coulomb branch. The long arrows represent very massive strings. Note that there is a closed loop connecting Γ1,Γ2 and Γ3. Chapter 5

Supergoop Dynamics

5.1 Introduction

In Chapter4 we used a quiver quantum mechanics system to understand the emergence of ‘isolated’ SL(2, R) symmetries. We will now study a diﬀerent limit of the same theory in an attempt to capture the complicated dynamics of disordered/glassy systems.

5.1.1 String glasses

In the macroscopic world, where everything ﬂows and nothing abides, mayhem and disorder rule. Most dynamical systems are not integrable but chaotic. Most materials are not crystalline but amorphous. They are glasses, eternally in search of their true equilibrium. Although glassy systems are still notoriously resistant to fundamental theoretical understanding, much progress has been made over the past thirty years [222], and remarkable organizing principles have been uncovered. A beautiful early example is given by the simplest toy model of a spin glass, the Sherrington-Kirkpatrick model, for which the space of low temperature conﬁgurations organizes itself in a hierarchic treelike fashion [220,221]. String theory presents us with similarly, if not more, complex systems, on microscopic scales. These manifest themselves when studying the vast microstates of

105 CHAPTER 5. SUPERGOOP DYNAMICS 106

black holes or the vast number of flux compactifications, to give but two examples. Consider for instance (as reviewed in detail in [221]), a “3-charge” D4-brane wrapped 6 2 2 2 on a smooth four-cycle Σ inside a six-torus T = (T )1 × (T )2 × (T )3, and bound to n pointlike D0-branes. Denote the number of intersection points of Σ with the 2 A sub-tori (T )A by P , A = 1, 2, 3 — these are the D4-charges of the system. Then Σ has triple self-intersection product P3 ≡ 6 P 1P 2P 3, and of order P3 worldvolume deformation and flux degrees of freedom. In the regime n P3, the pointlike D0-brane degrees of freedom dominate the degeneracy Ω of supersymmetric ground states, and one easily computes:

r nP3 S = log Ω ≈ 2π . (5.1.1) micro 6

This agrees with the Bekenstein-Hawking entropy of the D4-D0 black hole. In its 5-dimensional uplifted version [148], the computation is in essence an application of the Cardy formula in the context of the AdS3/CFT2 correspondence [223]. 3 3 However, away from the Cardy regime, i.e. when P & n, the order P D4-degrees of freedom (deformations of Σ and turning on fluxes) become entropically dominant and the counting problem becomes far more intricate: Each choice of worldvolume fluxes induces a different, highly complex potential on the moduli space of deformations of Σ, and the ground states of the system correspond to the minima of this vast D-brane landscape. On the other hand, on the black hole side, an exponentially large number of molecule-like, multicentered stationary black hole bound state configurations appears [179], all with the same total charge, and they entropically dominate the single centered black hole [166]. In [208], it was argued in a probe analysis that this picture persists at least metastably for nonextremal black holes, up to a critical temperature where the single-centered D4-D0 black hole regains dominance (see also [209] for a closely related analysis). This transition between a single black hole and a zoo of metastable multi-black hole configurations is reminiscent of a structural glass transition and the rugged free energy landscape picture associated to the glass phase. Indeed, once in a particular multicentered configuration, it may take exponentially long for the system to find the CHAPTER 5. SUPERGOOP DYNAMICS 107

most entropic “true equilibrium” conﬁguration, as it proceeds through exponentially suppressed thermal and quantum tunneling, and furthermore far from the true equilibrium state, the preferred direction of local relaxation processes is likely to push the system into the direction other highly stable quasi-equilibrium states instead of the true maximal entropy equilibrium conﬁguration.

5.1.2 Supergoop

When sufficiently far separated and moving slowly close to a ground state configuration, the black hole constituents can be thought of as pointlike particles, moving in an approximately flat background, interacting with each other through specific static and velocity dependent interactions. These effective inter-particle interactions are highly constrained by the fact that these BPS systems preserve four supercharges: A nonrenormalization theorem [165] implies that once a metric has been fixed on the configuration space, the static and first order velocity dependent interactions are of a fixed form. For the flat metric:

N  N !2 X 1 2 X κpq H = (p − A ) + + θ + fermions , (5.1.2) 2m  p p |x − x | p  p=1 p q=1 p q where Ap is the vector potential produced at xp by a collection of Dirac monopoles of charge κpq at positions xq. The coefficients κpq = −κqp equal the electric-magnetic symplectic products between the charges of the centers p and q. The parameters θp and masses mp are fixed by the BPS central charge of each center. If the configuration space metric is not flat, the mp may depend on xp. As a result of this nonrenormalization theorem, exactly the same supersymmetric Lagrangians also appear in very different physical contexts where four supercharges are preserved and the low energy degrees of freedom can be identified with spatial positions. One example is a mixture of well-separated elementary particles obtained by wrapping D-branes around various internal cycles of a Calabi-Yau manifold, interacting with each other through gravitational, vector and scalar interactions. Clearly this can be viewed as an extreme limit of the multi-black hole systems considered CHAPTER 5. SUPERGOOP DYNAMICS 108

above, where the dyonic black holes have been replaced by dyonic particles. Another example are monopoles and dyons in N = 2 Yang-Mills theories [227]. A more remote example [165] is a collection of space-localized wrapped D-branes at weak string coupling in the substringy distance regime. Their low energy degrees of freedom are given by a (0+1)-dimensional supersymmetric quiver quantum mechanics, with a position 3-vector and a U(1) gauge symmetry for each singly wrapped brane (identified with the nodes of the quiver) and the lightest brane-brane stretched open string modes represented as bifundamental oscillator degrees of freedom (identified with the arrows of the quiver). When the branes are all well separated, i.e. when the quiver theory is on the Coulomb branch, the open string modes become very massive and can be integrated out. Again, the resulting effective theory for the position degrees of freedom must necessarily be of the form (5.1.2) fixed by supersymmetry.

The coefficients κpq are now identified with the net number of arrows between two nodes. Thus this type of supersymmetric multi-particle mechanics appears in many contexts, in widely different regimes. Much effort has been put into understanding and counting the supersymmetric ground states of such systems, in part because of their key role in physics derivations of BPS wall-crossing formulae [165,166,228–230]. How- ever, little has been said about excitations or dynamics for these systems. There are a few exceptions: [231] studied the classical and quantum dynamics of the two-particle system and found it was integrable, and in [208] the persistence of the black hole molecular configurations at finite temperature was studied. However, no studies of multi-particle dynamics or statistical mechanics have been done so far. In this chapter we wish to take steps in these directions. Besides the motivation for understanding D-brane and black hole statistical mechanics and their potentially interesting interpretation as holographic glasses, such studies would also be of intrinsic interest, as these systems are rather unusual in several aspects. Due to the special form of the potential (5.1.2), an N-particle bound state will have a 2(N − 1)-dimensional moduli space of zero energy ground state configurations folded in a very complicated way into the 3(N − 1)-dimensional full configuration space (factoring out the center of mass). Naively one might therefore CHAPTER 5. SUPERGOOP DYNAMICS 109

think that even at very low temperatures, the system would behave like a liquid, exploring large parts of the configuration space by flowing along the continuous valley of minimal energy configurations. As a simple example consider the case N = 2. The particle distance is fixed and the moduli space is a sphere. One might think a probability density initially localized near a point on that sphere would quickly dif- fuse out over it. However, due to the effective electron-monopole Lorentz interaction between the particles, this is not quite right, as diffusion is obstructed by magnetic trapping. Another way of understanding this is conservation of angular momentum: Monopole-electron pairs carry intrinsic spin directed along their connecting axis, of magnitude equal to half their symplectic product. Hence they behave like gyroscopes. They resist changing direction; kicks will just cause them to wobble. Thus it is natural to hypothesize that these supersymmetric multi-particle systems behave partly like a liquid and partly like a solid at very low temperatures — like goop. We will therefore refer to this peculiar state of matter as supergoop.

5.1.3 Dynamics

Many times we study Hamiltonian systems that are classically integrable. For this to be the case one requires the existence of at least N conserved charges (with all mutual Poisson brackets vanishing) for a system with a 2N-dimensional phase space. Examples include single one-dimensional particles with an arbitrary potential, since the energy is conserved, and two body problems with a central force. Phase space trajectories of a classically integrable system will map N-dimensional tori. Generally, however, our system will not be classically integrable and we must confront a chaotic system. The simplest example of a chaotic system is the double pendulum which has a four-dimensional phase space with a single conserved quantity: the energy. One then studies the phase space trajectories of the double pendulum as a function of increasing energy. For suﬃciently low energies the trajectories are constrained to live on a two- dimensional torus displaying quasi-integrable behavior. As the energy is increased this torus is deformed and eventually breaks apart into smaller tori. This process is seen to continue until there is no visible structure in the phase diagram, i.e. the system CHAPTER 5. SUPERGOOP DYNAMICS 110

tends toward ergodicity. What is perhaps most remarkable about the transition to chaos is that it occurs in a gradual fashion in which smaller and smaller islands of regular behavior are spawned before the system loses all manifest structure. A crucial question, especially for a system with many degrees of freedom, is quantifying when all ordered behavior disappears and how it depends on the parameters of the system (see for example [232,233]). It is our aim to begin a systematic study of the dynamical aspects of the un- derlying brane system on the Coulomb branch. In this chapter we mainly address the case of a three particle configuration. This is already a difficult non-integrable three-body problem. To render the problem tractable, we study first the classical ground states and subsequently the motion of a probe particle in a fixed background consisting of a two-centered bound state. Remarkably, we discover that the motion of the probe is classically integrable! This is due to the presence of an additional hidden conserved quantity and is somewhat reminiscent of the integrability of a Newtonian probe particle interacting gravitationally with a background of two fixed masses, as discovered by Euler and Jacobi. We then study the transition to chaos for a system of two probes in the presence of a heavy fixed particle, with the motion restricted to live on a line. This setup is directly analogous to the double pendulum, allowing us to exploit many of the tools developed for the study of the double pendulum. As for the double pendulum, by studying Poincarésections we observe the formation of islands in phase space and the eventual transition to global chaos with no apparent structure in phase space. Finally, we begin to address the far more intricate dynamics of a system with a large number of centers. We provide a brief exposition of the behavior of a probe particle inside a molecule with a given number of fixed centers. We observe highly complex trajectories that become trapped for long times in the sense that they do not explore the entire molecule. It is this latter sort of study, with many centers involved, that will bear more directly on the questions of glassiness; in this case the trapping problem is meant to address the glassy phenomena of ergodicity breaking and “caging” [234–236].

Note added: After submission, we became aware of [180,224–226], which elegantly CHAPTER 5. SUPERGOOP DYNAMICS 111

prove the classical integrability that we discuss in Section 5.5.

5.2 General Framework

Consider a system of branes wrapped on the cycles of a six-dimensional compact space, such that they are pointlike in the non-compact (3 + 1)-dimensions. The interactions between them are governed by strings whose ends reside on the branes themselves. The low energy physics is governed by an N = 4 supersymmetric quiver quantum mechanics [164]. The nodes of the quivers have gauge groups associated to them and the low energy string degrees of freedom are chiral multiplets transforming in the bifundamental between two given nodes. The position degrees of freedom xp of the branes are scalars in the vector multiplets of the gauge groups. It was shown in [165] that if we study the Coulomb branch of the branes, i.e. integrate out the massive chiral multiplets, a non-trivial potential is generated which governs the dynamics of the xp. The system of branes allows for a large family of bound states with fixed equilibrium distances as classical ground states. If the branes move too close the stretched strings become light and even tachyonic and the system enters the Higgs 1 phase. On the other hand, if there is a sufficiently large number nB of branes placed at a single point, such that the product of the string coupling gs and nB becomes large, the system is best described by closed string exchange and hence supergravity. In what follows we will consider the theory of supersymmetric multiparticle mechanics describing the Coulomb branch of the brane system. This has the advantage of being a simple setup which is interesting in and of its own right while reproducing many of the features of the multicentered configurations that exist in supergravity.

5.2.1 Supersymmetric multiparticles

The theory we consider is the multiparticle supersymmetric mechanics, which we refer to as supergoop, studied for example in [165, 179, 231, 237–246]. We simply state the Lagrangian of the system, referring to [165] for details.

1 In fact, the weak string coupling limit gs → 0 always pushes the system to the Higgs phase. CHAPTER 5. SUPERGOOP DYNAMICS 112

The supergoop Lagrangian is given by:

X mp X L = x˙ 2 + D2 + 2iλ¯ λ˙ + (−U D + A · x˙ ) 2 p p p p p p p p p p X ¯ ¯ + Cpqλpλq + Cpq · λpσλq , (5.2.1) p,q where λp is the fermionic superpartner to xp. The Dp ﬁelds are auxiliary non- dynamical scalars. We have deﬁned the functions:

X κpq 1 X U = + θ , A = − κ Ad(r ) + Ad(r ) , (5.2.2) p 2r p p 2 pq pq qp q pq q where: −y x Ad(x) = xˆ + yˆ (5.2.3) 2r(z ± r) 2r(z ± r) is the vector potential of a single magnetic monopole of unit charge at the origin and rpq ≡ xp − xq. For the Lagrangian to be supersymmetric we further require

κpq = −κqp. In the quiver quantum mechanics context, the κpq are the number of bifundamentals connecting two nodes. The supercharges are given by:

X γ p p ¯β X β ¯γ ¯β Qα = − i σαλγ ·(pp − Ap)+λαUp , Q = i σγ λq ·(pq − Aq)−λq Uq . (5.2.4) p q

∗ ¯α β The Weyl spinors obey (λα) ≡ λ and the σα are the usual Pauli matrices. The Hamiltonian of our system is deﬁned as:

X H = pp · x˙ p − L, pp ≡ mpx˙ p + Ap . (5.2.5) p

Upon integrating out the the D-terms we ﬁnd:

1 X X κpq H = (p − A )2 + U 2 + r · λ¯ σλ , (5.2.6) 2m p p p 2r3 pq pq pq p p p

forces. Furthermore, the system has three-body interactions due to the appearance 2 of Up in H. ¯β As usual our Hamiltonian H is related to the supercharges as {Qα, Q }D.B. = β −2iδαH. However this is most easily checked in the quantum mechanics context where pp → −i∇p and we replace the above Dirac bracket relation with the anticommutation relation: ¯β β {Qα, Q } = 2δαH, (5.2.7)

p ¯β −1 p α where {λα, λq } = mp δq δβ .

5.2.2 Classical features and multicentered black holes

To study the classical properties of this theory, we can turn the fermionic fields off and only consider the bosonic part of (5.2.1). Static BPS configurations occur when

Up = 0 for all p, i.e. when:

X κpq = −θ , ∀ p . (5.2.8) 2r p q pq

P Taking the sum over p of (5.2.8) we find that the θp’s must satisfy: p θp = 0. The solutions to equation (5.2.8) are bound states of particles, see for example figure 5.1. Given that a system of N particles has 3N-degrees of freedom which are constrained only by (N − 1) equations, the bound states have a (2N + 1)-dimensional classical moduli space M. This moduli space cannot be accessed dynamically at low temperatures however, due to the velocity dependent forces which constrain the particles to oscillate about a fixed location if given a small kick (just like an electron in the presence of a magnetic field). As the energy is increased, the rigid structure of the bound state is deteriorated and eventually lost completely. In the following sections we study the dynamical features of this system, with a particular focus on the three-particle system. Equation (5.2.8) is a familiar expression in supergravity. It recreates the integrability condition of [179] for multi-centered black hole bound states in four-dimensional CHAPTER 5. SUPERGOOP DYNAMICS 114

Fig. 5.1: Examples of ground states for 100 electric Γe = (0, 1) plus 100 magnetic Γm = (1, 0) particles.

N = 2 supergravity:

N X hΓp, Γqi = 2Im e−iαZ . (5.2.9) |x − x | p r=∞ q p q

I The above expression involves electric-magnetic charge vectors Γ = P ,QI with a duality invariant symplectic product given by:

˜ I ˜ ˜I hΓ, Γi = P QI − QI P . (5.2.10)

Expression (5.2.9) also involves a function called the central charge Zp(z) which depends on the vector multiplet scalars za of the supergravity theory, as well as the

iαp charge vector Γp. At spatial inﬁnity we can write Zp|r=∞ = mpe where mp is P the ADM mass of a BPS particle of charge Γp. If we denote Z = p Zp then the CHAPTER 5. SUPERGOOP DYNAMICS 115

parameter α in (5.2.9) is given by α = arg (Z|r=∞). Thus we may rewrite (5.2.9) as

N X hΓp, Γqi = m sin (α − α) . (5.2.11) 2r p p q pq

The supersymmetric multi-particle mechanics in (5.2.1) may be considered as a toy model for the dynamics of the multi-centered black hole bound states if we make the identiﬁcations

κpq = hΓp, Γqi and θp = −mp sin (αp − α) . (5.2.12)

5.2.3 Three particles

Much of the discussion that follows will concern the three particle system, which already exhibits rich dynamic and non-dynamic features. Here we describe some of the characteristic features of its zero energy conﬁgurations.

Classically, the (supersymmetric) ground states are found by setting U1 = U2 =

U3 = 0. Explicitly:

κ12 κ31 κ12 κ23 κ31 κ23 − = −θ1 , − + = −θ2 , − + = θ3 , (5.2.13) 2r12 2r31 2r12 2r23 2r31 2r23 with θ3 = −(θ1 + θ2). Notice that the third equation follows from the other two.

We also require that the three relative distances r12, r23 and r13 satisfy the triangle inequality. The three particles have nine position degrees of freedom and the above equations only constrain two of them. Factoring out the center of mass leaves us with (9 − 2 − 3) = 4 unconstrained degrees of freedom. Hence, even when three- particle bound states form, there is an inﬁnite classical moduli space of connected (and possibly also disconnected) ground states. In the case of a two-particle bound state the classical moduli space is simply a two-sphere of ﬁxed radius κ12/2θ1.

If we have κ31, κ12 and κ23 positive we ﬁnd there exist scaling solutions given by rij → λκij with λ → 0 with the κ’s obeying the triangle inequality [166]. Thus, in this CHAPTER 5. SUPERGOOP DYNAMICS 116

regime the particles can come arbitrarily close to each other with no cost in energy.2 Away from the scaling regime the solution to (5.2.13) corresponds to a bound state for which the particles may oscillate about a ﬁxed equilibrium radius upon small perturbations.

As an example, when θ3 = −3, θ2 = −2, θ1 = 5, κ12 = −1, κ13 = −1 and κ23 = 1, with the three particles living on a line and particle 3 between particles 2 and 1, we ﬁnd the solution:

1 √ 1 √ r = 7 + 19 ≈ 0.57, r = 8 − 19 ≈ 0.12 , 12 20 13 30 1 √ r = 1 + 19 ≈ 0.45 . (5.2.14) 23 12

Note that the triangle inequality is saturated since we have considered a collinear example. We should note, however, that there are clear instances where no solutions exist, such as when (κ13, κ23, θ3) > (0, 0, 0), for example.

5.2.4 Regime of validity

Since we are free to choose the set of αp,(5.2.12) does not really constrain the values of the θp in any way. There is, however, a restriction stemming from the requirement of the validity of the Coulomb branch description assuming our system comes out of integrating strings [165, 178]. The distances between particles must be smaller than the string scale, but larger than the ten dimensional Planck scale. For larger distances, the suitable description is given by the exchange of light closed strings (in which case supergravity is the reliable description). Furthermore, the velocities should be small compared to the speed of light to avoid higher derivative corrections to the non-relativistic Lagrangian (5.2.1).

Consider ﬁrst two particles with masses m1 and m2 and call the string length

2When thinking about scaling solutions in the gravitational context from the point of view of a far away observer, the scaling solutions are continuously connected to a single center and will look like a single centered black hole or particle. On the other hand, nearby observers will observe that the proper distance between the particles never shrinks to zero due to the formation of inﬁnite throats (at least at zero temperature). CHAPTER 5. SUPERGOOP DYNAMICS 117

2 3 2 ls. The restriction is found to be [165]: κ12 lsθ1 and µ ls θ1/κ12, where µ ≡ m1m2/(m1 + m2) is the reduced mass of two particles. Indeed, if κ12 lsθ1 the distance between the two particles in a bound state will be much larger than ls 2 3 2 and the appropriate description becomes that of supergravity. When µ ls θ1/κ12, the open strings between the branes become light and the appropriate description becomes that of the Higgs branch and eventually the fused D-brane system itself. For the multiparticle system, the Coulomb branch description is reliable so long as the inter-particle distances are suﬃciently large that the massive strings can be reliably p integrated out, i.e. rij ls |θi/mi − θj/mj| and suﬃciently small that we remain in the substringy regime, i.e. rij ls.

5.3 Classical Phase Space

Having discussed the general framework of the system under study, we now discuss some of its dynamical features, beginning with the classical phase space. Recall that the Hamiltonian of our system is given by:

1 X H = (p − A )2 + U 2 (5.3.1) 2m p p p p p

The Hamilton equations of motion are given by:

∇xp H = −p˙ p , ∇pp H = x˙ p . (5.3.2)

For N-particles we have a 3 × 2N = 6N dimensional phase space. As manifest conserved quantities we have the energy, the center of mass momentum, and the center of mass angular momentum.

5.3.1 Two particles are integrable

We review the classical properties of the two particle problem in appendix D.1. Recall that a classically integrable system with a 2N-dimensional phase space has at least N CHAPTER 5. SUPERGOOP DYNAMICS 118

conserved quantities with mutually vanishing Poisson brackets. Phase space trajectories for integrable systems reside on N-dimensional tori. In the case of two-particles we have a 12-dimensional phase space. There are six manifest conserved quantities given by the net momentum and angular momentum. As shown by D’Hoker and Vinet [239, 240], the presence of a conserved Runge-Lenz vector (D.1.2) leads to an enhanced SO(3, 1) symmetry. The angular momentum and Runge-Lenz vector are three-vectors, the Hamiltonian is a scalar and there exist two relations amongst the seven quantities, hence there is a total of (3 + 3 + 2) = 8 conserved quantities. Fac- toring out the center of mass yields a (maximally) super-integrable system.3 The super-integrability implies its equations can be separated in more than one coordinate system and one can solve for the quantum mechanical spectrum algebraically, as done in [231]. Further, it implies that trajectories in coordinate space follow paths which are closed in the case of bound orbits.

5.3.2 Three particles are chaotic

In the case of three-particles we have an 18-dimensional phase space and there is no longer a suﬃcient number of conserved quantities to render the system integrable. Thus, such systems will exhibit chaotic behavior. We may use several numerical tools to analyze the chaotic nature of such a system. For instance, we can study the Lyapunov exponent λ parameterizing the divergence of phase space trajectories with nearby initial conditions. Given two trajectories in phase space with initial separation δz0, the Lyapunov exponent is deﬁned by the limit:

1 δz(t) λ = lim lim log . (5.3.3) t→∞ δz0→0 t δz0

We can also study Poincar´esections in phase space. These are found by recording the location of a trajectory in a particular subspace of phase space each time it crosses some ﬁducial point (such as crossing the origin with positive velocity). These are particularly useful for lower dimensional systems such as the double pendulum,

3A super-integrable system [248] is a system with a 2N-dimensional phase space which has more than N conserved quantities. A maximally super-integrable system has 2N −1 conserved quantities. CHAPTER 5. SUPERGOOP DYNAMICS 119

where they clearly depict the breakdown of the integrable motion on a two-torus as the energy is increased (see chapter 11 of [232] for a discussion). We will discuss and examine the Poincar´esections of a collinear three particle system in section 5.6. Six of the phase space dimensions can be eliminated from net momentum conservation and factoring out the center of mass. We can also kill another four due to net angular momentum and energy conservation. The remaining 8-dimensions in phase space (as far as we know) are unconstrained by symmetries. Needless to say, systems with more than three particles will also display chaotic properties. A simpler setup, which we refer to as the Euler-Jacobi setup, is that of a probe particle moving in the background of two ﬁxed centers.

5.4 Euler-Jacobi Ground States

The simplest question we can ask about our system is what the (supersymmetric) ground states are, both classically and quantum mechanically. Classically there may be continuous moduli spaces of zero energy conﬁgurations. Quantum mechanically, given that the probe is a particle in the presence of a background magnetic ﬁeld, we expect the continuous classical moduli space to give rise to a degenerate set of quantum ground states due to Landau degeneracies.

5.4.1 Euler-Jacobi three body problem

We will consider a probe particle of mass m3 in the background of two ﬁxed centers unless otherwise speciﬁed. The background particles have masses m1 and m2 both very large compared to m3, charge vectors Γ1 and Γ2 with symplectic product κ12 > 0 and Fayet-Iliopoulos constant θ1 = −θ2 < 0. They sit along the z-axis at z =

±κ12/4θ1 ≡ ±a. By choosing m1 and m2 much larger than m3, the backreaction of the probe on the ﬁxed centers is suppressed by O(m3/m1, m3/m2). The probe also has charge vector γ3 and Fayet-Iliopoulos constant θ3. We may also consider the possibility of forming supersymmetric bound states between the probe and the ﬁxed centers. In such a case, a non-zero θ3 requires us CHAPTER 5. SUPERGOOP DYNAMICS 120

to modify the background condition θ1 = −θ2, since now the θ’s must add to zero.

We thus demand |θ3| |θ1|, |θ2| such that the correction to the positions of the original ﬁxed centers is of order O(θ3/θ1, θ3/θ2). The intersection products of the probe with the centers are given by κ31 and κ32. To avoid any large backreaction from the probe due to the κ interactions we further require that κ31/r31 and κ32/r32 are small compared to κ12/r12 ∼ θ2. The Hamiltonian governing the dynamics of the probe is given by:

2 1 2 1 κ31 κ32 Hprobe = (p3 − A3) + θ3 + + . (5.4.1) 2m3 2m3 2r31 2r32

Notice that the two scaling transformations:

(p3(t), x3(t); κij, θ3, m3, r12, t) −1 2 −1 → (σ p3(t), λ σ x3(t); λ κij, σ θ3, σ m3, λ σ r12, λ t) , (5.4.2) generate a family of solutions parameterized by λ and σ. Given a solution to the equation of motion for some O(1) parameters, we can exploit the scaling symmetries to map the solution to a rescaled one in the regime of validity for the Coulomb branch √ description as discussed in section 5.2.4. In particular, we require λ σ and λ σ which can be achieved for large σ. Notice that in this regime the velocity (which scales as σ−1) becomes parametrically small.

5.4.2 Classical Ground States

As noted in (5.2.8), the space of classical ground states M is given by setting Up = 0. Satisfying this condition gives rise to time independent classical bound states. In the probe limit, where the two background centers are ﬁxed, this amounts to solving the algebraic equation:

1 κ31 1 κ32 + = −θ3 , (5.4.3) 2 pρ2 + (z − a)2 2 pρ2 + (z + a)2 CHAPTER 5. SUPERGOOP DYNAMICS 121

where ρ2 = x2 + y2 and φ = tan−1(y/x). One can easily prove that the eﬀective magnetic ﬁeld B = ∇3 × A3 is always perpendicular to the tangent of M. This remains true for the moduli space of a probe in a background of more than two centers as well.

Consider ﬁrst the case θ3 = 0. We ﬁnd:

2 2 2 2 2 (a − z) κ32 − (a + z) κ31 ρ = 2 2 . (5.4.4) (κ31 − κ32)

For ρ(z) above to have solutions we choose κ31 > 0 > κ32 and furthermore |κ31| >

|κ32|, we ﬁnd a continuum of solutions between z = [z−, z+] where:

|κ32| ∓ |κ31| z± = ±a . (5.4.5) |κ31| ± |κ32|

Note that z− < −a < z+ < a and thus the θ3 = 0 surfaces enclose the ﬁxed charge at z = −a in this case. Similarly, for |κ31| < |κ32| the probe encloses only the center at z = a. For κ31 = −κ32 and θ3, the classical moduli space M becomes the z = 0 plane.

For θ3 6= 0 ﬁnding ρ(z) amounts to solving a quartic equation. In order to do so, it is convenient to go to prolate spheroidal coordinates:

ρ = ap(ξ2 − 1)(1 − η2) , z = aξη , φ = φ , (5.4.6) such that: 2 2 2aθ3(η − ξ ) = (κ31 + κ32)ξ + (κ31 − κ32)η . (5.4.7)

In the above we have implicitly used that η ∈ [−1, 1] and ξ ∈ [1, ∞]. We can easily ﬁnd a solution for η = η(ξ):

1 η(ξ) = δ − δ ± p(δ − δ )2 + 8(δ + δ )ξ + 16ξ2 , (5.4.8) 4 1 2 1 2 1 2 where δ1 ≡ κ31/(aθ3) and δ2 ≡ κ32/(aθ3). In ﬁgure 5.2 we show the diﬀerent qualitative types of M as a function of δ1 and δ2. The qualitative features of each region CHAPTER 5. SUPERGOOP DYNAMICS 122

Fig. 5.2: Left: Classical moduli space M in the δ1 − δ2 plane for θ3 6= 0. The nature of M for the different regions is shown in figure 5.3. Right: Classical moduli space for M with θ3 = 0. In regions i and ii the centers at z = a and z = −a are enclosed respectively. are shown in figure 5.3. Notice that upon defining the prolate coordinates (5.4.6) we have scaled out the distance r12 = a between the fixed centers. To obtain physical distances we simply multiply by r12 = a.

5.4.3 Quantum Ground States

From the classical point of view, our particle is nothing more than a charged particle in the presence of magnetic fields constrained to live on a surface. Thus, given a time independent supersymmetric bound state configuration we can compute the lowest Landau degeneracies dL by computing the degeneracy of states with vanishing energy for the constrained particle. Such a setup has been addressed for non-uniform magnetic fields everywhere normal to the surface [249], which is precisely the situation we find ourselves in. Following [165], the lowest Landau degeneracies are given by the total flux through the classical moduli space M. For instance, as we discuss below, the Landau degeneracy of the fixed background is given by κ21. Upon studying the phase diagram and corresponding M in figures 5.2 and 5.3, we find that the total CHAPTER 5. SUPERGOOP DYNAMICS 123

Region I 2

x 0

-1

-2 -4 -2 0 2 4 z

Region II Region III 4 4

2 2 x 0 x 0

-2 -2

-4 -4 -4 -2 0 2 4 -4 -2 0 2 4 z z

Region IV Region V 2 1.0

1 0.5 x

x 0 0.0

-0.5 -1

-1.0 -2 -2 -1 0 1 2 -3 -2 -1 0 1 2 3 z z

Fig. 5.3: Classical moduli space in the δ1 − δ2 plane for θ3 6= 0. The order of the ﬁgures left to right starting at the top are the regions in ﬁgure 5.2. CHAPTER 5. SUPERGOOP DYNAMICS 124

degeneracy is:

I, II and III : dtot = κ12 × |κ31| or dtot = κ12 × |κ32| , (5.4.9)

IV and V : dtot = κ12 × |κ31 + κ32| . (5.4.10)

For regions I, II and III, the degeneracy of states depends on which of the two background centers is encircled by M. Notice there is a jump in the number of ground states as we vary δ1 and δ2. Since we are in the probe limit, we expect these results to be correct up to order O(κ31/κ12) and O(κ32/κ12). ¯β From the supersymmetric quantum mechanics point of view, recall that {Qα, Q } = β 2δαH. In the absence of the probe, the ground state of the background is given by [165]: ˜¯α ˜ |bi = Ψα(~x1 − ~x2)λ |0i , λ ≡ λ1 − λ2 . (5.4.11)

The center of mass coordinate ~x0 ≡ (m1~x1 + m2~x2) /(m1 + m2) and center of mass spinor λ0 ≡ (m1λ1 + m2λ2) /(m1 + m2) drop out and thus |bi is naturally a function of the relative background position vector and spinor. The state |0i is annihilated by λ˜ and deﬁnes a three-dimensional Hilbert space through action of λ˜¯. There are

κ12 ground states filling a spin-(κ12 − 1)/2 multiplet. From the last term in the Hamiltonian (5.2.6), we observe that there exist spin-spin couplings between the background spinors λ1 and λ2 and the probe spinor λ3. It is convenient to introduce the relative spinors λ13 ≡ λ1 − λ3 and λ23 ≡ λ2 − λ3 and their associated vacua |023i and |013i, such that λ23|023i = 0 and so on. Both |023i and |013i have an associated three-dimensional Hilbert space and the general state must be a tensor product of all linear combinations of all such states, finally tensored with |bi. Given that the probe is sensitive to the background B-field, it will go into spin one-half states of |013i and

|023i aligning with the B-fields from the fixed particles at z = ±a. This will split the lowest Landau degeneracies. It would be interesting to compute the explicit ground state wavefunctions. For the sake of completeness we briefly mention another method to compute the number of ground states. One can associate a quiver diagram Q to the data (κij, θi) of a particular configuration [164,165]. It turns out that the dimension of the moduli CHAPTER 5. SUPERGOOP DYNAMICS 125

3 3

κ23 κ31 κ23 κ13

2 1 2 1 κ12 κ12

Fig. 5.4: Three node quiver with a closed loop (left) and without a closed loop (right). space M(Q, N, θ) of the quiver Q can be related to the number of BPS ground states. In particular, for the three body problem where each particle is a different species we have a quiver theory Q with N = (1, 1, 1), κ12 arrows between nodes 1 and 2, κ13 arrows between nodes 1 and 3 and κ23 arrows between nodes 2 and 3. The quiver diagram is presented in figure 5.4. The Fayet-Iliopoulos constants θv are additional parameters associated with each node v. Ground state degeneracies for similar setups to the one we are studying have been computed in [166]. Notice that scaling solutions can occur only for quivers with closed loops. In our problem, with κ12 > 0, we find that regions I, II and III correspond to quivers with closed loops and regions IV and V correspond to quivers with no closed loops.

5.5 Euler-Jacobi Dynamics: classical integrability

The equations governing the probe are dictated by the Hamiltonian in (5.2.5). There are two obvious constants of motion in this problem, namely the energy and the angular momentum in the direction of the line where the two centers are placed. If the system is to be rendered integrable, there must exist a third constant of motion. Such a constant of motion was found for the problem of a Newtonian probe interacting gravitationally with a background of two ﬁxed massive particles [250,251], also known as the Euler-Jacobi three-body problem. We will show that the analogous problem in the theory under consideration is also integrable. This was previously shown and discussed in [180,224–226]. CHAPTER 5. SUPERGOOP DYNAMICS 126

Fig. 5.5: The Euler-Jacobi flower. The red balls represent the fixed background centers and the blue line represents the classical trajectory of the probe. In this case, the trajectory precesses around only one of the fixed centers.

5.5.1 Setup and coordinate systems

Recall that we are considering two ﬁxed background centers sitting on the z-axis at z = ±κ12/4aθ1 ≡ ±a. Let us go to a cylindrical system with metric:

ds2 = dρ2 + ρ2dφ2 + dz2 , (5.5.1) where the Cartesian and cylindrical coordinates are related by x = ρ cos φ, y = ρ sin φ and z = z. One observes that the Lagrangian and Hamiltonian are independent of the φ coordinate which implies a symmetry. The conserved quantity of this symmetry is given by the angular momentum in the z-direction, such that the canonical momentum pφ = l is constant. The probe Hamiltonian (5.4.1) in cylindrical coordinates becomes:

2 1 0 ij 0 (U3(ρ, z)) Hprobe = (pi − Ai(ρ, φ, z)) g pj − Aj(ρ, φ, z) + , (5.5.2) 2m3 2m3

0 where the pi are the conjugate momenta in the cylindrical coordinates. The relation between conjugate momenta between the primed and unprimed coordinate systems 0 0j i is pi = pj ∂x /∂x . The third constant of motion is not manifest in cylindrical coordinates. One must CHAPTER 5. SUPERGOOP DYNAMICS 127

go to the prolate spheroidal coordinates (5.4.6) with metric:

dξ2 dη2 ds2 = a2(ξ2 − η2) + + a2(ξ2 − 1)(1 − η2)dφ2 . (5.5.3) (ξ2 − 1) (1 − η2)

Once in this coordinate system, we note that our Hamiltonian takes the following form: H + H H = ξ η , (5.5.4) probe ξ2 − η2

2 2 where Hξ depends only on ξ and pξ and Hη depends only on η and pη. Thus, we can write: 2 2 Hprobeξ − Hξ = Hη + Hprobeη ≡ G, (5.5.5) where G must be a constant of motion. More explicitly we have:

2 2 2 2 (ξ − 1) pφ ξ(κ31 − κ32) (κ31 − κ32) Hξ = pξ 2 + 2 2 + pφ 2 2 + 2 2 2a m3 2a m3(ξ − 1) 2a m3(ξ − 1) 8a m3(ξ − 1) θ ξ(κ + κ + aθ ξ) + 3 31 32 3 , 2am3 and

2 2 2 2 (1 − η ) pφ η(κ31 + κ32) (κ31 + κ32) Hη = pη 2 + 2 2 + pφ 2 2 + 2 2 2a m3 2a m3(1 − η ) 2a m3(η − 1) 8a m3(1 − η ) θ η(κ − κ − aθ η) + 3 31 32 3 . 2am3

We conclude that the probe-two-center problem of supergoop is integrable, providing another example to the distinguished list of integrable classical systems. In this system, one observes highly symmetric spatial trajectories, as illustrated in ﬁgure 5.5.1. CHAPTER 5. SUPERGOOP DYNAMICS 128

pz py 0.15 0.4 0.10

0.2 0.05

y 10 11 12 13 14 15 z -10 -5 5 10 -0.05

-0.2 -0.10

-0.15 -0.4 -0.20

Fig. 5.6: Examples of closed phase space trajectories in the integrable probe regime. The plots show slices of phase space in the Cartesian coordinate system.

5.6 Beyond Euler-Jacobi: the stringy double pendulum

If we move away from the probe approximation and allow backreaction with the fixed centers, our system is no longer integrable and begins to show chaotic features. For instance, one can study trajectories in phase space and see whether they are closed. One could also compute the Lyapunov coefficient of the system. In figures 5.6 and 5.7 we demonstrate the trajectories in phase space for the probe orbiting around both centers as we exit the probe limit. As we increase the number of degrees of freedom, the analysis of chaotic systems becomes increasingly challenging. The canonical example of a double pendulum, which already displays a significant set of features generic to chaotic systems at large, can be effectively analyzed with the use of Poincarésections. For a double pendulum, the phase space is four-dimensional with a single constant of motion – the total energy. Hence the system is not integrable. CHAPTER 5. SUPERGOOP DYNAMICS 129

pz py

0.4 0.6

0.4 0.2

0.2 z -100 -80 -60 -40 -20 y 40 60 80 100 120 -0.2 -0.2

- 0.4 -0.4

-0.6 -0.6

Fig. 5.7: Examples of open phase space trajectories in the chaotic regime. The plots show slices of phase space in the Cartesian coordinate system.

5.6.1 Collinear dynamics

Away from the probe limit, as we already noted, our system has an 18-dimensional phase space and becomes a complicated three-body problem. In order to study the transition to chaos of our system, it is instructive to find a setup that allows us to use the same tools used to analyze the double pendulum. This can be achieved by restricting the particles to be collinear, i.e. placing them on a line and only considering deformations along this direction. Notice that a system consisting of particles on a line will stay on the line so long as the velocities of the particles are parallel to the line itself. This is because the magnetic force v × B will vanish in this situation. Hence, the collinear system is a consistent truncation of our original Lagrangian (5.2.1). This is no longer true for the coplanar case. As was already discussed, we need at least three particles to find chaotic features. Three backreacting particles on a line have six degrees of freedom with a conserved energy, a situation closer to the triple pendulum. We can however take the mass of one of them to be much larger than the other two such that they behave as two probes in a fixed background. The probes are allowed to interact with each other since we P do not restrict the θi and κij in any way, except i θi = 0. The equations governing CHAPTER 5. SUPERGOOP DYNAMICS 130

the two probes can be extracted from the two-probe Hamiltonian:

2 2 2 2 p2 p3 1 κ21 κ23 1 κ31 κ32 Hcol = + + θ2 + + + θ3 + + . (5.6.1) 2m2 2m3 2m2 2x21 2x23 2m3 2x31 2x32

The above Hamiltonian is a good approximation for the three-particle system in the limit where m1 m2, m3. In this limit one particle becomes non-dynamical and the energy is fully contained in the motion of the two light particles. Thus there is a conserved quantity associated to the motion of the light particles and we are left with a three-dimensional phase space, which is also the dimensionality of the double pendulum phase space. ∗ ∗ The ground state (x21, x31) is found by setting U2 = U3 = 0. In addition to imposing the triangle inequality to fully specify the ground state, we must also declare the ordering of the three particles on the line. A given ground state is mapped to ∗ ∗ ∗ ∗ a family of ground states via the scaling relation (x21, x31; κij) → λ(x21, x31; κij). Slightly increasing the energy leads to small oscillations about the equilibrium position ∗ ∗ 2 (x21, x31). The linearized normal frequencies are the eigenvalues of the Ω matrix:

2 −1 −1 1 Ωjl = Mjk ∂k∂lHcol ,Mij ≡ δij , i, j, k = {2, 3} . (5.6.2) mi

The derivatives of Hcol are evaluated at the equilibrium point. Though the general formulae for the normal frequencies are quite involved they are readily computed.

As an example, the normal frequencies for κ13 = κ12 = −κ23 = 1, θ1 = −θ2 = −1,

θ3 = 0, and m2 = m3 = 1 are:

√ ( 2 2 5.73 ω± = 121 ± 13 73 ≈ . (5.6.3) 81 0.25

5.6.2 Poincar´eSections

When the system is integrable or quasi-integrable, i.e. for suﬃciently low energies, the trajectories in the four-dimensional phase space will reside on two-dimensional tori, since the system is simply given by two linearly coupled oscillators. Since it is hard CHAPTER 5. SUPERGOOP DYNAMICS 131

to visualize motion on the torus, we study instead particular snapshots of the system, known as Poincarésections (see [232, 233] for a more complete discussion). For a given energy, we can record (over many different initial conditions) the coordinate and conjugate momentum of one particle every time the other particle has positive momentum and crosses a particular point. The resulting contours in phase space, collectively known as a Poincarésection, display the transition from quasi-integrable to chaotic behavior in our system. For sufficiently low energies, the Poincarésections are given by two fixed points surrounded by a set of concentric contours. The fixed points correspond to motion in one of the two normal modes. It is useful to define the winding number w, which is the ratio of the number periods one particle completes for every full period completed by the other. At the linearized level away from the fixed point, the winding number w = ω1/ω2. If w is not a rational number, the trajectory will never quite return to its original position and thus fills one of the concentric contours. As the energy is increased, the winding number is detuned and eventually may even become rational. Hence, parts of the phase space acquire new fixed points with their own concentric contours. These correspond to nonlinear resonances. The last tori to break are those √ with the ‘most’ irrational w (the golden mean ( 5 − 1)/2 is the ‘most’ irrational number, as defined by the speed of convergence of its continued fraction expansion). The breaking of the original two islands into an increasing number is qualitatively similar to the case of a double pendulum. Eventually, there is essentially no visible structure left in the Poincarésection and we are in a regime of global chaos. For the same values of θ and m between the two particles, we find that chaos sets in before the escape energy θ2/2m is reached, while the normal frequencies in the linearized regime (i.e. before chaos sets in) depend on the κ values as well. The transition to chaos is in principle controlled by a complicated function of the intrinsic parameters κi, θi and mi of the system. We give an example of this in figure 5.8. Similar transitions to chaos are found for examples where the κ’s form closed and non-closed loops. In several examples where the κ’s form a closed loop and obey the triangle inequality, and the θ’s have the same sign, the formation of islands around fixed points representing nonlinear resonances seems to be far less manifest in the Poincarésections. In other CHAPTER 5. SUPERGOOP DYNAMICS 132

Fig. 5.8: Poincarésections of collinear setup with κ31 = 10, κ32 = −10, κ21 = −10 and θ3 = −1, θ2 = 1 and energies E = {0.10, 0.20, 0.23, 0.26, 0.27, 0.30}. Note that the κ’s form a closed loop. The horizontal axis represents the position of particle 3 while the vertical axis represents its conjugate momentum. Any given plot is produced by varying the initial positions and momenta of the two probes subject to a fixed total energy. The pair (x3(t), p3(t)) is plotted every time the resulting trajectory of particle 2 crosses some fiducial point (x2(t) = xc) with positive momentum (p2(t) > 0), i.e. roughly every time particle 2 completes a full cycle as it oscillates back and forth. In the quasi-integrable regime, different initial conditions correspond to different contours. The first Poincarésection shows a quasi-integrable behavior with two fixed points corresponding to the two low energy normal modes. CHAPTER 5. SUPERGOOP DYNAMICS 133

words, global chaos seems to set in much more quickly. We hope to study these issues systematically in the future.

5.7 Trapping

In this section, we envision a trapping problem. The setup consists of a localized bound state and another particle, which we take to be a probe, beginning inside the molecule. The probe begins its life at a random position well within the molecule. We explore the dynamical evolution of the probe as we vary the initial energy.

5.7.1 Setup and Energetics

Our setup will consist of a probe with charge γp = (1, 0) in the presence of a bulk molecule comprised of a number Nc of ﬁxed electric centers of charge γc = (0, κ). The positions of the electric centers will be obtained by drawing random points from a ball of radius Rmol using the algorithm in [254,255]. The classical probe Hamiltonian in this background is given by:

2 2 Nc ! (pp − Ap) 1 X κpi H = + + θ . (5.7.1) probe 2m 2m r p p p i=1 pi

Since all the background centers have the same charge, the κpi ≡ κ are all equal.

Also, to ensure that trapping occurs we require that κ and θp have opposite signs.

The zero energy conﬁgurations are given by setting the second term in Hprobe to zero. As usual, there is a classical moduli space M due to the fact that we have three probe coordinate degrees of freedom and we are solving only one equation. We could search for non-zero static minima of Hprobe, but a simple computation of the gradient of the potential shows that there are only zero energy minima. The minimal energy 2 required for the probe to reach inﬁnity is Emin = θp/2mp. 2 For probe energies Ein ≥ θp/2mp the probe can easily escape the molecule. For 2 energies in the range Ein < θp/2mp, we observe trapping. Our goal is to begin quantifying the amount of classical trapping. We do this by studying the fractional CHAPTER 5. SUPERGOOP DYNAMICS 134

volume fV (Ein, t) covered by the probe as a function of initial energy Ein and total trajectory time t. We estimate fV by studying how many centers the probe trajectory 1/3 approaches to within one-half of the average inter-particle distance ri.p. ∼ Rmol/Nc . We again stress that we keep the molecule and initial positions of our probe ﬁxed through all the trials, only varying the initial velocity of the probe. We take mp = 1,

θp = −10, Rmol = 20, Nc = 100 and κ = 1 for the presented data.

5.7.2 A trap

At low energies, we witness characteristic trapping: figure 5.9 shows one such example where a probe is confined to less than 20% of the molecule, exploring the same part of the molecule over and over again. We remind the reader that it is possible to be trapped in one region indefinitely and this behavior should not (necessarily) be looked at as a failure of not integrating for a long enough period of time. Indeed, as is seen in the Euler-Jacobi flower of figure 5.5.1, probes can remain in one part of a molecule for arbitrarily long periods of time. As we increase the energy, we see a transition that opens up more of phase space to the probe. Little pockets in the potential landscape form through which the probe particle can escape and begin exploring other regions of the molecule. Often this happens by sudden jumps, as illustrated in the middle row of figure 5.9. We have tracked the energies of the probe and the numerics are stable. The jump is not due to an erroneous kick in the integrator but rather appears to be due to small pockets through which the probe can escape given enough time. Finally, at high energies (e.g. around one half of the escape energy), the probe uniformly explores the entire molecule, as illustrated in the last row of figure 5.9. The parts of the molecule our probe gets trapped in are analogous to the “cages” that occur near the structural glass transition [234–236]; to explore “caging rearrangements,” which in this context would be collective motions of subsets of the molecule that allow the probe to escape its pocket and explore other portions of its phase space, it becomes necessary to unfreeze the dynamics of the background molecule. It is also interesting to study the Schrödinger equation for the probe in this background to see if the wavefunction exhibits trapping via some avatar of Anderson CHAPTER 5. SUPERGOOP DYNAMICS 135

localization.4

5.7.3 Topology of the potential landscape

To illustrate the potential landscape and gain some intuition for the motion of trajectories, we set up a co-planar molecule. Again, all centers in the molecule have equal charge and their positions are chosen by uniformly selecting Nc points on a disk of size Rmol [256]. A probe in this background will not remain in the plane due to the magnetic fields which will push it out. However, for a molecule where every center attracts the probe, at low energies the deviations from the plane are small relative to the size of the molecule, which can be made arbitrarily large. Thus, plotting equipotential contours over this two-dimensional molecule gives an accurate picture of the potential landscape which can be used to understand the trajectories. See figure 5.10 for such a comparison. For a fixed molecule size, as the magnitude of κ is increased relative to the magnitude of θ, the topology of the equipotentials changes by expelling the low energy part of the landscape to the outside of the molecule, as can be seen in figure 5.10. This mimics the change in topology of the moduli space in going from Region V to Region IV in figures 5.2 and 5.3. Topology changes in the moduli space of the three center setup also occurs as κ is increased while keeping θ fixed as well as the distance between the background centers.

5.8 Holography of Chaotic Trajectories?

We end our journey by discussing how the picture we are developing may fit into the broader context of holography. The usual interpretation of a large black hole in an asymptotically anti-de Sitter space is that we have prepared the dual CFT in some finite temperature state. On the other hand, the presence of a vast number of distinct entropically relevant multicentered black hole configurations inside an anti-de Sitter universe [258] implies a vast number of minima in the free energy of the dual CFT (as a function of configuration space). In particular, the usual assumptions of the

4Thanks to Douglas Stanford for discussions on the quantum dynamics. CHAPTER 5. SUPERGOOP DYNAMICS 136

% visited 100

20 à à à à à à à à à à à t 0 5000 10 000 15 000 % visited 100

40 à à à à 20 à à à à à à à t 0 5000 10 000 15 000 % visited 100 à à à à 80 à à

60 à

40 à à

20 à

à t 0 5000 10 000 15 000

Fig. 5.9: The first row represents a low energy probe, which remains stuck in a subset of the phase space for seemingly arbitrarily long times. The second row represents an intermediate energy probe which illustrates the non-uniform escapes that occur from the low energy trapping behavior. We see that for a while it remains trapped in some subset of phase space, after which it escapes and gets stuck in some other subset of phase space. The final row represents a high energy probe which uniformly explores the molecule. The associated plots represent the percentage of the molecule explored as a function of the integration time, up to 15000 time steps in increments of 1500. The tapering off of the high energy probe is simply due to saturating the entire molecule. Below these points the increase is very uniform. The initial energy increases from 50% of the escape energy in the first row to 60% of the escape energy in the third row. These percentages, however, are very dependent on the parameters (e.g. κ, θ, etc.) in the problem. CHAPTER 5. SUPERGOOP DYNAMICS 137

Fig. 5.10: These contour plots show equipotential surfaces in the plane of a 2D molecule consisting of one hundred centers. From left to right, we have chosen κ = 1, κ = 1.5, κ = 3.5, and in all cases θ = −10. We observe that as the magnitude of κ increases, the minima (blue region), which initially lied near each center, are collectively expelled, forming an overall minimum that surrounds the molecule as a whole. For κ = 1, the trajectory remains close to the plane of the molecule and has been superimposed on the left contour plot (transparent white line). The axes label the x and y positions of the probe particle. no-hair theorem fail since a set of macroscopic charges does not uniquely ﬁx the bulk solutions. In fact, the solutions are characterized by a large collection of multipole moments. Furthermore, though most of them do not constitute true ground states, they can be very long lived, decaying mostly through thermal and quantum tunneling. We can access information about the relaxation and response of the CFT by computing boundary-to-boundary correlators in the bulk. In the large frequency (or large mass) limit two-point functions have been associated to bulk geodesics which begin and end their life near the boundary of AdS [259–262]. Such geodesics will become highly complex and chaotic in the bulk due to the presence of the non-trivial black molecule, as evidenced by our simpler setup. In fact, a geodesic may become trapped in some very long lived unstable orbit before escaping back to the boundary. Thus, the two-point function expressed as a path integral over bulk trajectories and the applicability of the saddle point approximation may be a somewhat involved issue. This picture suggests that the linear response properties of the dual CFT, to the extent that they are captured by the two-point function in the geodesic approximation, in CHAPTER 5. SUPERGOOP DYNAMICS 138

the multicentered/glassy phase are rather different from those in a usual thermal state, where for example the motion of geodesics is integrable. The motion of a very massive probe or high energy graviton falling into the bulk corresponds to a point-like source cascading to lower energies (and covering larger size) in the CFT. Eventually the excitation returns back to a point-like source at some other point on the two- sphere where the CFT resides. From the bulk point of view this is when the particle dropped into the molecule returns back out. The chaotic nature of the bulk physics suggests chaotic behavior of the boundary theory itself. One may also consider the dynamical features at zero temperature for which asymptotically AdS3 multicentered configurations are known [263]. The possible presence of classical chaotic behavior of the bulk AdS3 should correspond to quantum dynamics in the dual CFT2. One effect of particular interest in chaotic systems is known as quantum scarring, where it has been observed that the wavefunction of a chaotic system peaks on closed classical trajectories [264]. We hope to explore these issues further in future work. Appendix A

De Sitter Wavefunction

A.1 A quantum mechanical toy model

We consider a simple quantum mechanical toy model that captures some of the essence and mathematics of our (A)dS calculations. The Hamiltonian governing the system is given in thex ˆ-eigenbasis by:

1 d2 m2x2 λ Hˆ = − + + x3 , = 1 , (A.1.1) 2 dx2 2 6 t2 ~ where x ∈ R and the time parameter t ∈ (−∞, 0) with t → 0 as the inﬁnitely late time limit of the system. The cubic interaction term is taken to be small and the dimensionless quantity α ≡ λ m−1/2 will serve as the small parameter in our perturbative analysis. The system has a scaling relation: x → ν1/2x, t → νt, m → m/ν and λ → λ/ν1/2, which we could use to set m = 1. Note that the cubic term is smaller than the quadratic term whenever:

λ m2hx2i−1/2 t2 , (A.1.2)

The above Hamiltonian is unbounded from below, but this will be of no concern at the perturbative level. Moreover, if the state of interest is normalizable at a given time, the Hermiticity of the above Hamiltonian is enough to ensure that it will remain

139 APPENDIX A. DE SITTER WAVEFUNCTION 140

normalizeable for all times. The Schr¨odingerequation governing the time evolution of a quantum state ψ is given by:

ˆ i ∂t ψ(x, t) = H ψ(x, t) . (A.1.3)

At λ = 0, we have that the ground state of the system is given by:

π 1/4 i m m ψ (x, t) = exp − t − x2 . (A.1.4) g m 2 2

The above state can be built from a Euclidean path integral with vanishing boundary conditions for x(t) in the inﬁnite Euclidean past τ → ∞, where τ ≡ −it. For such a state we have that hx2i1/2 ∼ 1/m1/2. We are interested in perturbations of the above wavefunction, i.e. solutions to the

Schr¨odingerequation that are continuously connected to ψg in the limit λ → 0.

A.1.1 Path integral perturbation theory

The quantum states of interest can be constructed via a Euclidean path integral: Z (λ) −SE (x) ψg (˜x, τc) = N Dx e , (A.1.5)

where SE is the Euclidean action governing the path integral:

Z 1 1 λ x3 S = dτ x˙ 2 + m2x2 + . (A.1.6) E 2 2 6 τ 2

As in the unperturbed case, the path integral is supplemented with the boundary conditions that x(τ) → 0 in the limit τ → ∞, and x(τc) =x ˜ (where τc > 0 is a late time cutoﬀ). We consider a solution to the classical equations of motion xcl obeying the prescribed boundary conditions, supplemented by a quantum ﬂuctuation δx. The path integral then splits as: Z (λ) −SE [xcl] −SE [δx] ψg (˜x, τc) = e Dδx e . (A.1.7) APPENDIX A. DE SITTER WAVEFUNCTION 141

2 Perturbatively, the solution can be expanded as xcl = x0 +λx1 +λ x2 +... We absorb the boundary dependence fully into the x0 term. Thus we have:

Z ∞ 0 m(τc−τ) 1 dτ 0 2 0 x0(τ) =x ˜ e , x1(τ) = − 0 2 x0(τ ) G(τ, τ ) , (A.1.8) 2 τc (τ ) and so on. The ‘bulk-to-bulk’ propagator G(τ, τ 0) obeys:

d2 − + m2 G(τ, τ 0) = δ(τ − τ 0) . (A.1.9) dτ 2

Explicitly:

1 0 0 G(τ, τ 0) = − e2mτc e−m(τ+τ ) − em(τ−τ ) , τ < τ 0 , (A.1.10) 2m and similarly for τ > τ 0. The classical action on such a solution is given by:

Z ∞ 1 λ dτ 3 − SE[xcl] = x0x˙ cl|τ=τc − 2 xcl . (A.1.11) 2 12 τc τ

It captures the tree-level diagrams of the perturbative expansion. As a concrete example, at order λ, the cubic inx ˜ contribution to the exponent of the (Euclidean) wavefunction is given by:

Z ∞ λ 3 dτ 3 λ 3 1 3mτc x˜ 2 K(τc, τ) = x˜ + 3e m Ei(−3m τc) , (A.1.12) 6 τc τ 6 τc where we have deﬁned the ‘bulk-to-boundary’ propagator:

0 0 0 m(τc−τ ) K(τc, τ ) ≡ lim ∂τ G(τ, τ ) = e . (A.1.13) τ→τc

A late time expansion of the cubic correction yields:

λ λ m + (γE + log(3 m τc)) + ... (A.1.14) 6τc 2

We see that there are 1/τc terms and log τc that grow and eventually violate the APPENDIX A. DE SITTER WAVEFUNCTION 142

perturbative assumption. To make contact with the ordinary Schr¨odingerequation, we must analytically continue τc = −itc. The ∼ 1/τc term then becomes a contribution to the phase of the wavefunction and plays no role in its absolute value. On the other hand, the logarithmic term retains real part upon analytic continuation of (λ) the time and thus contributes to the absolute value of ψg (˜x, tc). Furthermore, for times t m ∼ e−1/α the cubic correction of the wavefunction becomes comparable to the λ = 0 piece. As another example, we can consider a diagram involving a loop, namely a tadpole diagram contributing a linear inx ˜ piece to the exponent of the wavefunction. The correction is given by:

λ Z ∞ dτ emτc 2mτc 2 K(τc, τ) G(τ, τ) = 3 e Ei(−3mτc) + Ei(−mτc) . (A.1.15) 2 τc τ 4

As for the cubic correction, a small τc expansion reveals logarithmic terms. The presence of a non-vanishing tadpole also implies that the vev ofx ˆ is non-vanishing and in fact time dependent. A small t expansion renders to order λ:

λ hψ(λ)|xˆ|ψ(λ)i = (γ + log(−mt)) + ... (A.1.16) g g 4 m E

Thus, for λ > 0 and m > 0 and at small enough t the vev ofx ˆ drifts to negative values where it will become sensitive to the unbounded part of the potential. In this fashion, using as the basic propagators of our perturbation theory G and

K, we can build the quantum corrections of the ground state ψg(˜x, tc) at some time tc = −iτc . In this simple example, one can explicitly check that the corrected wavefunction indeed solves the time dependent Schr¨odinger equation (A.1.3) to the appropriate order in λ. APPENDIX A. DE SITTER WAVEFUNCTION 143

A.2 The bulk-to-bulk propagator

The Green’s function for the massless scalar in Euclidean AdS(d+1) satisﬁes the partial diﬀerential equation:

√ µν (d+1) ∂µ( gg ∂νG(z, w)) = −δ (z − w) . (A.2.1)

√ In this form the right side contains the naive δ-function, no 1/ g. The derivatives are taken with respect to the observation point zµ while wµ is the source point. We will enforce the symmetry G(z, w) = G(w, z). We really need the Green’s function in momentum space:

Z dd~k G(z, ~x; w, ~y) = ei~k·(~x−~y)G(z, w, k) . (A.2.2) (2π)d

This satisﬁes the second order ordinary diﬀerential equation:

L(d−1) (d − 1) ∂2 − ∂ − k2 G(z, w, k) = 0 z 6= w . (A.2.3) z z z z

First we choose a simple basis for the homogeneous modes of this equation. The basis contains exponentially damped modes, called φ2(z) below, as z → ∞, and exponentially growing modes, called φ1(z), obtained using the reﬂection symmetry z ↔ −z of the ODE. For the two cases D = 2, 4 we write:

kz −kz D = 2 φ1(z) = e φ2(z) = e , (A.2.4) kz −kz D = 4 φ1(z) = (1 − kz)e φ2(z) = (1 + kz)e . (A.2.5)

In [80] the bulk Green’s function was constructed using a diﬀerent choice of basis modes. The Green’s function for a second order ODE is commonly treated in texts on diﬀerential equations, and we have used Ch. 9 of [81]. The Green’s function is the product of modes in the two sectors z < w and z > w:

G(z, w, k) = Aφ1(z)φ2(w) + cφ2(z)φ2(w) z < w , (A.2.6) APPENDIX A. DE SITTER WAVEFUNCTION 144

= Bφ2(z)φ1(w) + cφ2(z)φ2(w) z > w . (A.2.7)

Note that we always choose the exponentially damped mode for the larger of the two variables. The coeﬃcients A, B are determined by the following conditions at the “diagonal” point z = w:

• G(z, w, k) is continuous at z = w ,

(d−1) • the ﬁrst derivative ∂zG(z, w, k) decreases by (z/L) as z increases through z = w .

For the ODE in the form (A.2.3), [81] speciﬁes that the jump is the reciprocal of the leading coeﬃcient as we have written. These conditions uniquely determine A, B, but not c since it multiplies a product of modes that is smooth across the diagonal.

• c is determined by enforcing the Dirichlet boundary condition G(z = zc, w, k) = 0 at the cutoﬀ.

It is easy to see that these conditions completely determine the Green’s function. In two bulk dimensions we have the expression:

1 φ1(zc)φ2(z)φ2(w) G(z, w, k) = φ1(z)φ2(w) − z < w , (A.2.8) 2k φ2(zc) 1 φ1(zc)φ2(z)φ2(w) = φ2(z)φ1(w) − z > w . (A.2.9) 2k φ2(zc) and in four bulk dimensions we have: 1 φ1(zc)φ2(z)φ2(w) G(z, w, k) = − 3 2 φ1(z)φ2(w) − z < w(A.2.10) , 2k L φ2(zc) 1 φ1(zc)φ2(z)φ2(w) = − 3 2 φ2(z)φ1(w) − z > w(A.2.11) . 2k L φ2(zc)

One very good check of these results comes enforcing the correct relation between the bulk-to-bulk and bulk-to-boundary propagators. This follows from the application of Green’s formula to the boundary value problem:

√ µν ∂µ gg ∂ν φ(z, ~x) = 0 , φ(zc, ~x) = ϕ(~x) . (A.2.12) APPENDIX A. DE SITTER WAVEFUNCTION 145

Green’s formula reads (note zc = wc)

Z ∞ Z d p dw d ~y g(w)(φ(w)wG(w, z) − G(w, z)wφ(w) (A.2.13) wc Z ∞ Z d p µν = dw d ~y∂µ( g(w)g (φ(w)∂νG(w, z) − G(w, z)∂wφ(w(A.2.14)) , wc and thus:

Z (d−1) d L − φ(z, ~x) = − d ~y ∂wG(w = wc, ~y; z, ~x)ϕ(~y) . (A.2.15) wc

To reach the last expression we use (A.2.1) and the fact that the PDE (A.2.12) has no bulk source, and we evaluate the second line at the boundary w = wc where the Dirichlet Green’s function vanishes. The main point is that the bulk-to-boundary propagator K(z, ~x) is the properly normalized radial derivative of the bulk-to-bulk Green’s function; the speciﬁc relation is

p ww K(z, ~x − ~y) = g(wc)g ∂wG(wc, ~y; z, ~x) . (A.2.16)

After Fourier transformation, the last expression of (A.2.13) exactly reproduces the solution of the linearized solution of the k space EOM (2.2.12) for both D = 2, 4. Appendix B

Higher Spin de Sitter

B.1 Conformal Transformation from round S3 to ﬂat R3

In this appendix, we remind the reader of the conformal relation between the three- sphere and the three-plane.

B.1.1 Coordinate transformation and Weyl rescaling

Consider the metric of the three-sphere:

2 2 2 2 ds = dψ + sin ψ dΩ2 . (B.1.1)

Upon a coordinate transformation, ψ(r) = 2 cot−1 r−1, the above metric maps to:

2 2 ds2 = dr2 + r2dΩ2 , (B.1.2) r2 + 1 2 which is conformally equivalent to the ﬂat metric on R3. According to our discussion in section 3.2.1, a constant mass source mS3 in the free Sp(N) theory on a three-sphere corresponds to the free Sp(N) theory on the ﬂat metric on R3 with the following source

146 APPENDIX B. HIGHER SPIN DE SITTER 147

100 15 80

10 60

40 5 20

0 -5 -4 -3 -2 -1 0 1 2 0 10 20 30 40 50

2 Fig. B.1: Left: Comparison of |ΨHH (mS3 )| for N = 2 as obtained by calculating ZCFT [mS3 ] analytically (blue line) and given in equation (3.6.1), and by numerically evaluating the functional determinant using the Dunne-Kirsten regularization method 2 with lmax = 45 (red dots). We have normalized such that |ΨHH (mS3 )| = 1 at mS3 = 0. Right: Plot of the percentage error as a function of the numerical cutoﬀ (at mS3 = −2.2).

for the χ · χ operators: 2 2 m 3 (r) = m 3 . (B.1.3) R r2 + 1 S

B.1.2 Numerical Error

As a check on our numerics, we display in figure B.1 a plot of the functional determinant using the Dunne-Kirsten method for the radial mass given in (B.1.3) laid over the analytic result for the three sphere written in (3.6.1). In the right hand side of the figure we display the error between the two as a function of the cutoff value of l for the sum in (3.2.7). For a maximum cutoff lmax = 200 the partition function is within 3 percent of the exact answer at mS3 = −2.2.

B.1.3 Balloon Geometries

We can use the same conformal transformation between the round metric on S3 and the ﬂat metric on R3 to show the conformal equivalence of the balloon geometry on APPENDIX B. HIGHER SPIN DE SITTER 148

an R3 topology discussed in section 3.4 and a diﬀerent geometry on an S3 topology:

2 2 2 2 2 2 −2 2 2 2 2 ds = dr + r fζ (r) dΩ2 = 2 cos (ψ/2) (dψ + sin ψ fζ (ψ) dΩ2) . (B.1.4)

Upon a conformal rescaling, we can see that this is just a deformed three-sphere:

2 2 2 2 2 2 2 2 ds˜ ≡ 2 cos (ψ/2) ds = dψ + sin ψ fζ (ψ) dΩ2 . (B.1.5)

The function fζ (ψ) serves to add a waist to the three-spheres along the ψ direction, with the parameter ζ controlling whether the sphere tapers or bulges. Thus, the mass deformation chosen in section 3.4 for the balloon geometry corresponds to a constant mass deformation on its conformally related deformed sphere. When mapping the balloon geometry to ﬂat space to apply the Dunne-Kirsten method for computing the partition function, we should understand that the answer we recover is also the answer for the partition function on this deformed sphere with a constant mass. The pinching limit of the balloon geometry maps into a pinch that tears the peanut-like geometry into two warped spheres.

B.2 Review of the squashed sphere

The metric of the squashed sphere, which is a homogeneous yet anisotropic space on an S3 topology, is given by:

1 1 ds2 = dθ2 + cos2 θdφ2 + (dψ + sin θdφ)2 , (B.2.1) 4 1 + α with ψ ∼ ψ + 4π. The geometry is an S1 ﬁber over S2 and consequently has an SO(3) × U(1) isometry group. The constant Ricci scalar and volume are given by:

2(3 + 4α) 2π2 R = ,V = √ . (B.2.2) (1 + α) 1 + α

At α = 0 we recover the S3 with enhanced SO(4) isometry group. It is convenient to parametrize the squashing parameter by ρ such that α = e2ρ − 1, and we will display APPENDIX B. HIGHER SPIN DE SITTER 149

our results using this parameter. The eigenvalues of the conformal Laplacian in the presence of a uniform mass term σ are given by:

1 λ = n2 − + α (n − 1 − 2q)2 + σ , q = 0, 1, . . . , n − 1 , n = 1, 2,..., n,q 4(1 + α) (B.2.3) with degeneracy n. Knowing the eigenvalues analytically allows for easier computation of the functional determinant.

B.3 Perturbative Bunch-Davies modes for m2`2 = +2

We brieﬂy review the Bunch-Davies zero modes of a free conformally coupled scalar 2 2 with m ` = +2 in a ﬁxed global dS4 background:

τ ds2 = −dτ 2 + `2 cosh2 dΩ2 . (B.3.1) ` 3

The action of the scalar is given by:

1 Z √ 2 S = − d4x −g gµν∂ φ∂ φ + φ2 . (B.3.2) φ 2 µ ν `2

Assuming φ = φ(τ) does not depend on the three-sphere coordinates, it is governed by the equation of motion:

2 τ φ(τ) + 3 tanh φ0(τ) + ` φ00(τ) = 0 . (B.3.3) ` `

The general solution to the above equation is given by:

τ τ φ(τ) = sech2 c + c sinh . (B.3.4) ` 2 1 `

The positive frequency modes of the Bunch-Davies vacuum are those which are regular on the lower Euclidean hemisphere θ ∈ [−π/2, 0] obtained by continuing the global APPENDIX B. HIGHER SPIN DE SITTER 150

τ 1 metric by ` → −iθ. This ﬁxes c1 = ic2 resulting in: τ τ φ (τ) = c sech2 1 + i sinh . (B.3.5) BD 2 ` `

We can expand the Bunch-Davies solution at late times to ﬁnd:

−2τ/` −τ/` φBD(τ) ∼ 2 c2 2 e + i e . (B.3.6)

B.3.1 Continuation from Euclidean AdS4

Another way to see this is by continuing the perturbative solutions of a conformally 2 2 coupled free scalar with mass m `A = −2 in a ﬁxed Euclidean AdS4 background:

2 2 2 2 ρ 2 ds = dρ + `A sinh dΩ3 . (B.3.7) `A which are smooth in the interior. These scalars obey the equation:

2 ρ 0 00 φ(ρ) + 3 coth φ (ρ) + À φ (ρ) = 0 . (B.3.8) À À

The smooth solution in the interior is given by:

2 ρ ρ φsmooth(ρ) =c ˜2 csch −1 + cosh . (B.3.9) `A `A

Near the boundary of Euclidean AdS the smooth solution behaves as:

−2ρ/`A −ρ/`A φsmooth(ρ) ∼ −2c ˜2 2 e − e . (B.3.10)

Notice that the Euclidean AdS and global dS metrics map onto each other under the transformation `A → i` and ρ → iτ + `π/2. Under this analytic continuation the smooth solution maps to the Bunch-Davies solution.

1Notice that the continuation τ/` → iθ with the same regularity condition would lead to a wavefunctional B.3.11 that diverges with the late-time proﬁle of the scalar. It is thus an inappropriate choice for a perturbatively stable vacuum state. APPENDIX B. HIGHER SPIN DE SITTER 151

B.3.2 Wavefunctional

We can compute the Hartle-Hawking wavefunctional using the complex solutions in

(B.3.5). We choose c2 such that the solution at some late time cutoff τ = τc ` has the real profile φ0. Evaluating the on-shell action of a free scalar field with mass m2`2 = 2 at late times and computing eiSφ gives:

`2π2 `2π2 Ψ [φ , τ ] ∼ exp − φ2 e2τc/` + ... exp −i φ2 e3τc/` + ... , HH 0 c 4 0 8 0 (B.3.11) where we have separated the phase from the magnitude and expanded in powers of eτ/`. Notice that the wavefunction is Gaussian suppressed and the phase diverges at late times. Picking a convenient overall normalization forσ ˆ, we define 2τc/` ˆ −3τc/` σˆ = e φ0 − α πˆφ and fix α = (e )/8 so that the late-time behavior of σ is purely fast-falling (this is the choice for which the late-time correlators in the bulk are computed by a CFT). We use the same convention forπ ˆφ as in (3.1.5). For τc ` we find:

`2π2 Ψ [σ, τ ] ∼ exp − σ2 + ... . (B.3.12) HH c 16 Notice that at the level of Gaussian wavefunctionals, the phase vanishes in theσ ˆ-basis. For the sake of completeness we also include below the wavefunctional of the m2`2 = +2 free scalar in planar coordinates ds2 = `2(−dη2 + d~x2)/η2. If its late time proﬁle at some cuttoﬀ η = ηc is given by φ~k then we have:

`2 Z d3k k i log ΨHH [φ~k, ηc] ∼ − 3 2 − 3 φ−~k φ~k . (B.3.13) 2 (2π) ηc ηc −2 ˆ 3 Transforming to theσ ˆ = ηc φ − ηc πˆφ basis as before, we ﬁnd:

`2 Z d3k 1 log Ψ [σ , η ] ∼ − σ σ . (B.3.14) HH ~k c 2 (2π)3 k −~k ~k APPENDIX B. HIGHER SPIN DE SITTER 152

This leads to a correlation function in position space given by:

1 hO(x)O(y)i ∼ , (B.3.15) |x − y|2 which is obtained by diﬀerentiating twice the logarithm of the wavefunction. This is the appropriate answer for an operator of conformal weight ∆ = 1, which for the Sp(N) theory corresponds to χ · χ.

B.4 Wavefunctionals for bulk gauge ﬁelds

We consider here the perturbative Hartle-Hawking wavefunctional for a bulk U(1) gauge ﬁeld with action:

1 Z √ S = − d4x −g F F µν ,F = ∂ A − ∂ A , (B.4.1) 4g2 µν µν µ ν ν µ in a ﬁxed de Sitter background ds2 = `2/η2 (−dη2 + d~x2). The putative dual CFT 2 would have a U(1) global symmetry. Working in the gauge Aη = 0, the on-shell action is simply given by a boundary piece at η = ηc:

1 Z S = d3xA ∂ A | , (B.4.2) on−shell 2g2 i η i η=ηc which is related by an analytic continuation η = iz to the on-shell action in Euclidean

AdS4 (see for example section 5.3 of [132]). As a simple example, consider a solution in the Aη = 0 gauge with ky = kz = 0. Then we can consistently set Ax = Ay = 0 and remain with the following solution satisfying the Bunch-Davies condition:

Z dk A = x A˜(kx)e−i|kx|(η−ηc)+ikxx , |η | 1 . (B.4.3) z (2π) z c

2As an example, the non-minimal bosonic Vasiliev theory, which includes all non-negative integer spins, contains a massless bulk U(1) gauge ﬁeld in its spectrum. The bulk gauged U(1) symmetry is dual to the global U(1) ﬂavor symmetry of the anti-commuting complex scalars in the CFT. APPENDIX B. HIGHER SPIN DE SITTER 153

∗ ˜(kx) ˜(−kx) Reality of the proﬁle at η = ηc requires Az = Az . The on-shell action at

η = ηc on the above complex solution is then:

1 Z dk i S = − x A˜(kx)(A˜(kx))∗|k | . (B.4.4) on−shell 2g2 2π z z x

The wavefunction is Gaussian suppressed as expected and the single power of k is in accordance with the conformal weight ∆ = 2 of the current operator dual to the bulk U(1) gauge ﬁeld. One can also consider adding a θ-term to the bulk action: Z θ 4 √ µνρσ i Sθ = i d x −g FµνFρσ , θ ∈ R . (B.4.5) 16π M

This term is independent of the metric. It is a total derivative and is equivalent to a

Chern-Simons term at η = ηc: Z θ 3 i Sθ = i d x ijkAi∂jAk|η=ηc . (B.4.6) 4π ∂M

Since the proﬁle of the gauge ﬁeld is real at η = ηc, the above is pure imaginary. Using the on-shell action we can construct the Bunch-Davies wavefunctional:

(θ) iSon−shell+iSθ ΨHH [Ai, ηc] ∼ e . (B.4.7)

(θ) 2 Notice that the absolute value squared |ΨHH [Ai, ηc]| is independent of θ. Although this would seem to suggest that the θ-term plays no role in the cosmological correlators (θ) 2 obtained from |ΨHH [Ai, ηc]| , it can still appear in computing observables involving ~ ~ the conjugate momentum of the gauge field, i.e. the electric field E = ∂ηA . Indeed, ~ in the Aη = 0 gauge, the wavefunction at zero E-field is given by: Z ˜ (θ) ~ (θ) ΨHH [E = 0, ηc] = DAi ΨHH [Ai, ηc] . (B.4.8)

Notice that we are now performing a path integral over a functional of Ai which includes the Chern-Simons term. Thus, the gauge ﬁeld becomes dynamical [135]. APPENDIX B. HIGHER SPIN DE SITTER 154

Much of our above discussion follows mostly unchanged when the U(1) gauge ﬁeld is replaced with a non-Abelian gauge ﬁeld. It would be interesting to understand how the topological dependence of the Chern-Simons term manifests itself in terms of cosmological expectation values. Appendix C

Conformal Quivers

C.1 Notation

For convenience, we have used a compact notation whereby latin superscripts denote relative degrees of freedom or arrow directions in the quiver, for example: 1 2 3 1 2 3 α β γ (q , q , q ) ≡ (x12, x23, x13) or (φα, φβ, φγ) ≡ (φ12, φ23, φ13). The supermultiplets i i i i in this notation are: Φα = (φα, ψα,Fα) and the relative vector multiplet is denotes as i i i i i 1 2 Q = (A , q , λ ,D ). The only exception to the rule is given by θ ≡ θ1, θ ≡ −θ3 and θ3 = 0.

Furthermore, we have: q3 = q1 + q2, D3 = D1 + D2, λ3 = λ1 + λ2. So with respect to the vector multiplet ﬁelds, the relative Lagrangian is only a function of q1, 2 1 2 1 2 i 2 P i 2 q , D , D , λ and λ . We also have: |φα| ≡ α |φα| .

The orientation of the quiver is encoded by the si. For three nodes we deal with the case of a quiver with a closed loop, and without loss of generality the particular 1 2 3 choice s1 = 1, s2 = 1 and s3 = −1, corresponding to κ , κ > 0 and κ < 0. An example of a quiver without closed loops is s1 = −1, s2 = 1, s3 = −1.

i i The spinors (ψα)a and (λ )a with a = 1, 2 transform in the 2 of the SO(3) and the anti- 12 i i i ab i x y z symmetric symbol = +1 is such that λ ψα ≡ (λ )a (ψα)b. The σ = (σ , σ , σ )

155 APPENDIX C. CONFORMAL QUIVERS 156

are the Pauli matrices: ! ! ! 0 1 0 i 1 0 σx = , σy = , σz = . (C.1.1) 1 0 −i 0 0 −1

¯i x i ¯i a x b i For example ψασ ψα ≡ (ψα) (σ )a (ψα)b. Finally the U(1) covariant derivative i i i Dtφα ≡ (∂t + iA ) φα.

We will occasionally revert back to the original xi notation, particularly in the appendices.

C.2 Superpotential corrections of the Coulomb branch

It is interesting to compute the eﬀects on the Coulomb branch dynamics due to quantum corrections from the superpotential. This amounts to a two-loop calculation. It is important to note that due to a non-renormalization theorem in [165], the linear piece of the N = 4 supersymmetric quantum mechanics Lagrangian is constrained to be of the form:

(1) X X ¯ ¯ L = (−Ui(x)Di + Ai(x) · x˙ i) + Cij(x)λiλj + Cij(x) · λiσλj , (C.2.1) i i,j with:

1 C = ∇ U = ∇ U = (∇ × A + ∇ × A ) ,C = 0 . (C.2.2) ij i j j i 2 i j j i ij

This form receives no corrections due to quantum eﬀects originating from the superpotential. This means, in particular, that the supersymmetric conﬁgurations of the

Coulomb branch which solve Ui = 0 for all i, both bound and scaling, are unmodiﬁed in the presence of a superpotential. On the other hand, the quadratic piece can receive quantum corrections in the presence of a superpotential. An example of a Feynman diagram contributing to the

DiDj term is given in ﬁgure C.1. APPENDIX C. CONFORMAL QUIVERS 157

Fig. C.1: Example of Feynman diagram contributing to DiDj from the superpotential.

C.3 Finite D-terms

When going to the Coulomb branch one integrates out the massive scalars, expands in small D/r2, solves for the D equations of motion and feeds the solution back into the action. We would like to show here that the supersymmetric configurations are in fact preserved for finite D. In the case of two node quivers, upon integrating out the chiral multiplet degrees of freedom (φα,Fα, ψα) one finds the following non-linear equations for D (we take

θ1 + θ2 = 0 and κ > 0):

κ µD − θ = , θ ≡ θ1 − θ2 . (C.3.1) 2p|q|2 + D

Given that the above equation is a cubic equation in D, one can find analytic solutions, which we are given in appendix C.3.1. It is straightforward to see that at |q| = −κ/(2θ), D = 0 is a solution to the non-linear equations. Plugging D = 0 back into the Lagrangian (which for zero velocity is the potential itself) shows that at |q| = −κ/(2θ) the system has a zero energy. Away from |q| = −κ/(2θ) the Lagrangian receives finite D corrections. It is not hard to convince one’s self, however, that the potential will never acquire another minimum due to finite D effects. We argue this in appendix C.3.1. i For three nodes the non-linear equations of D become (we take θ1 + θ2 + θ3 = 0 and pick the κ’s to form a closed loop in the quiver diagram):

|κ1| |κ3| µ1D1 + µ3D3 − θ1 = − , (C.3.2) 2p|q1|2 + D1 2p|q3|2 − D3 APPENDIX C. CONFORMAL QUIVERS 158

and cyclic permutations thereof. These equations can no longer be solved analytically. If we set the θ’s to zero we can see that at |qi| = λ |κi|, Di = 0 solves the linear equations and so the scaling solutions persist at finite D for zero θ. For non-zero θ’s, setting |qi| = λ |κi| + O(λ2) and expanding in small λ, we find that again Di = 0 is a consistent solution order by order in the λ perturbation theory. The existence of new bound states or scaling solutions which are non-supersymmetric due to finite D effects becomes far more intricate in this case. A preliminary numerical scan seems to suggest there are none and we hope to report further on this in the future.

C.3.1 Non-linear D-term solutions for two-node quivers

For two-nodes, the D-term equation is given by (C.3.1). This equation can be solved analytically for D. Supersymmetric bound states are found when κ θ < 0 at |q| = −κ/(2θ). In what follows, we choose κ > 0 and allow θ to have any sign. Equation (C.3.1) is equivalent to solving

θ 2 κ2 D − D + |q|2 − = 0 , (C.3.3) µ 4µ2 which can be turned into a depressed cubic of the form x3 + px + s = 0 using the 1 2 2θ subsitution D → x − 3 |q| − µ and identifying

1 θ 2 κ2 2 θ 3 p ≡ − |q|2 + , and s ≡ − + |q|2 + . (C.3.4) 3 µ 4µ2 27 µ

Since p < 0, the three roots of this equation may be written as [219]:

1 2θ r p 1 3sr 3 2π D = − |q|2 − + 2 − cos arccos − − k , (C.3.5) k 3 µ 3 3 2p p 3 for k = 0, 1, 2. For all roots to be real, the arguments of the arccos must be between 1/3 3 2 2 θ 3 κ2 [−1, 1], which implies 4p + 27s ≤ 0 or |q| ≥ − µ + 2 2µ2 . The k = 0 branch is real for all values of |q|. In ﬁgures C.2 and C.3 we show some plots of the solutions. In ﬁgure C.4 we display APPENDIX C. CONFORMAL QUIVERS 159

D0 0.2

q 0.5 1.0 1.5 2.0 2.5 3.0 -0.2

-0.4 È È

-0.6

-0.8

Fig. C.2: Plot of D0 for µ = −θ = κ = 1. Notice that the solution is real for all values of |q|.

Re Di Im Di -2 @ D 0.4 -4 @ D 0.2 q -6 0.5 1.0 1.5 2.0 2.5 3.0 -0.2 -8 q -0.4 È È 0.5 1.0 1.5 2.0 2.5 3.0

Fig. C.3: Left: Plot of <[D1] (blue) and <[D2] (violet). Right: Plot of =[D1] (blue) È È and =[D2] (violet). Both plots are for µ = −θ = κ = 1. Notice that when complex, the solutions form a conjugate pair. APPENDIX C. CONFORMAL QUIVERS 160

V V 0.35 0.35 0.30 0.30 0.25 0.25 0.20 0.20 0.15 0.15 0.10 0.10 0.05 0.05 q q 0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0

È È È È Fig. C.4: Left: Plot of V evaluated on D0 for µ = −θ = κ = 1. Right: Plot of V evaluated on the perturbative D solution in violet, compared with the full non- perturabative D0 in blue.

the effective potential for the D0 solution and its comparison to the effective potential obtained by expanding D to second order in D/|q|2 and then evaluating the potential on-shell. We notice that the position of the minimum is unaffected. Evaluating the potential on the D1 and D2 results in an unphysical complex potential for small values of |q|. Scanning the parameters for all possible combinations of signs of (κ, θ) results in no other bound states.

C.4 Three node Coulomb branch

In this appendix we present some details leading to the Coulomb branch Lagrangian of a three-node quiver. As was shown in [165], the N = 4 supersymmetric quantum mechanics of a U(1) vector multiplet has a Lagrangian whose linear piece in the velocity and D ﬁelds is completely ﬁxed by supersymmetry. Explicit expressions are given in (C.2.1) and (C.2.2).

In section C.4.1 below, we show that the coeﬃcient of the x˙ i · x˙ j term of the

Lagrangian is the same as that of the DiDj term, upon integrating out the chiral matter. In section C.4.2 we integrate out the auxiliary D ﬁelds in the Coulomb theory and obtain an expression for the eﬀective potential on the position degrees of freedom. APPENDIX C. CONFORMAL QUIVERS 161

C.4.1 Second order Lagrangian for three-node quiver

Recall the Lagrangian of a n-node quiver theory, after setting the fermionic λ ﬁelds to zero, is given by:1

X mi X L = |x˙ |2 + D2 − θ D + |φ˙ |2 + F 2 + iψ¯ ψ˙ 2 i i i i ij ij ij ij i j→i X 2 2 ¯ − |xij| + Dij |φij| + ψij σ · xij ψij . (C.4.1) j→i

For the case n = 3 there are three pairs of (i, j) to be considered: (1, 2), (2, 3) and (3, 1). We do not consider contributions from the superpotential in this appendix.

We take κ12 > 0, κ23 > 0 and κ31 > 0.

The φij propagator is given by:

i Dφij (ω) = 2 2 (C.4.2) ω − |xij| and the ψij propagator is given by:

! −i ω − zij −xij + iyij Dψij (ω) = 2 2 , (C.4.3) ω − |xij| −iyij − xij ω + zij where xij = (xij, yij, zij) (recall that each lower index denotes its corresponding node). We evaluate our momentum integrals on the imaginary axis, which is guaranteed to give the same result had we evaluated the integral along a real contour with the usual Feynman pole prescription, to lowest order in the the external momenta l. Higher order terms in the external momenta will diﬀer by factors of i.

1 For the purposes of this subsection we use the original notation, i.e. xi, λi, φij, ψij, xij ≡ (xi − xj). and so on. We also supress the Greek indices on the chiral multiplet ﬁelds. Also, we do not decouple the center of mass degrees of freedom and switch oﬀ the superpotential. APPENDIX C. CONFORMAL QUIVERS 162

2 Fig. C.5: 1-loop Feynman diagrams contributing to the D1 term of the eﬀective Lagrangian.

Di Dj term

First let us consider the diagonal terms and more speciﬁcally the contribution to

D1D1. The two diagrams that contribute are shown in ﬁgure C.5. The ﬁrst diagram, series expanded for small external momentum l, is given by

Z 2 1 2 dω κ12 l κ12 (−i) κ12 Dφ12 (ω)Dφ12 (ω + l) = 3 − 5 , (C.4.4) 2 2π 8|x12| 32|x12| while the second is given by

Z 2 1 2 dω κ31 l κ31 (−i) κ31 Dφ31 (ω)Dφ31 (ω + l) = 3 − 5 . (C.4.5) 2 2π 8|x31| 32|x31|

Therefore, ignoring the l2 terms, the total contribution to the eﬀective Lagrangian is

2 D1 κ31 κ12 L 2 = + . (C.4.6) eff,D1 3 3 8 |x31| |x12|

2 2 The D2 and D3 terms can be obtained by cyclic permutation.

Now let us consider the off diagonal terms and more specifically D1D2. The only diagram that contributes to 1-loop order is shown in figure C.6. This diagram differs only by a sign and a factor of 2 compared to (i) of figure C.5. Therefore, the D1D2 term of the effective Lagrangian is

κ12 Leff,D1D2 = −D1D2 3 . (C.4.7) 4|x12| APPENDIX C. CONFORMAL QUIVERS 163

Fig. C.6: 1-loop Feynman diagram contributing to the D1D2 term of the eﬀective Lagrangian.

Fig. C.7: 1-loop Feynman diagrams contributing to the δx1δy2 term of the eﬀective Lagrangian.

Other oﬀ diagonal terms can be obtained by cyclic permutation.

δxi · δxj term

In order to obtain the correction to the quadratic velocity terms in the Lagrangian, we consider fluctuations the bosonic degrees of freedom xi + δxi. Terms of this form can be divided in terms that are diagonal in the quiver such as δx1 · δx1 and others that are off diagonal such as δx1 · δx2. For each of these there is a further subdivision to terms that are diagonal in the vector component such as δx1δx1 and δx1δx2 and terms that are off diagonal in the vector component such as δx1δy1 and δx1δy2. We begin by considering terms that are off diagonal in the quiver index, and off diagonal in the vector index. For concreteness we will study δx1δy2. The diagrams that contribute are shown in figure C.7. The first diagram, series expanded for small external momentum l, is given by:

Z 2 dω x12y12 l x12y12 3 4x12y12(−i)iκ12 Dφ12 (ω)Dφ12 (ω + l) = −κ12 3 − 5 + O(l ) . 2π |x12| 4|x12| (C.4.8) APPENDIX C. CONFORMAL QUIVERS 164

Fig. C.8: 1-loop Feynman diagrams contributing to the δx1δx2 term of the eﬀective Lagrangian.

The second diagram, series expanded for small external momentum l, is given by:

Z dω κ (−i)i tr(σ D (ω)σ D (ω + l)) 12 2π 1 ψ12 2 ψ12 2 x12y12 lz12 l x12y12 3 = −κ12 − 3 + 3 + 5 + O(l ) . (C.4.9) |x12| 2|x12| 4|x12|

Summing the two, the only term that remains uncancelled is the term linear in l. Next we consider term oﬀ diagonal in the quiver index, but diagonal in the vector 1 1 index, for example δx1δx2. The contributing diagrams are shown in ﬁgure C.8. The

ﬁrst diagram is given by (C.4.8) with y12 replaced by x12. The second diagram gives:

Z dω κ (−i)i tr(σ D (ω)σ D (ω + l)) 12 2π 1 ψ12 1 ψ12 2 2 2 1 l 3 = −κ12 |y12| + |z12| 3 − 5 + O(l ) . (C.4.10) |x12| 4|x12|

The third diagram is given by: Z dω κ12 2iκ12 Dφ12 (ω) = . (C.4.11) 2π |x12|

Let us now consider terms that are diagonal in the quiver index. The Feynman a b diagrams contributing to δx1 δx1 are shown in figure C.9. One finds after similar calculations the following contribution to the effective Lagrangian: APPENDIX C. CONFORMAL QUIVERS 165

a b Fig. C.9: 1-loop Feynman diagrams contributing to the δx1 δx1 term of the eﬀective Lagrangian.

lz12δx1δy1 lx12δy1δz1 ly12δz1δx1 κ12 3 + 3 + 3 4|x12| 4|x12| 4|x12| 2 2 lz31δx1δy1 lx31δy1δz1 ly31δz1δx1 κ12l δx1 · δx1 κ31l δx1 · δx1 + κ31 3 + 3 + 3 − 3 − 3 . 4|x31| 4|x31| 4|x31| 8|x12| 8|x31| (C.4.12)

Quadratic Lagrangian

Having computed all relevant Feynman diagrams that amount to integrating out the

φij and ψij ﬁelds we can now write the resulting quadratic piece of the eﬀective Lagrangian:

(2) X mi 1 X |κij| L = | ˙x |2 + D2 + | ˙x − ˙x |2 + (D − D )2 . (C.4.13) eff 2 i i 8 |x |3 i j i j i i→j ij

Notice that the coeﬃcient of x˙ i · x˙ j indeed matches that of Di Dj . This can be simply generalized to the N-particle case, where the Lagrangian takes the same form as (C.4.13). As was noted before, the linear piece was ﬁxed by supersymmetry and given by (C.2.1) and (C.2.2)[165]. APPENDIX C. CONFORMAL QUIVERS 166

C.4.2 Coulomb branch potential

In this appendix we give the necessary formulae for the three-node Coulomb branch potential. Upon integrating out the chiral fields Φα in (4.2.1), while keeping the qi fixed and time independent, and expanding up to quadratic order in the Di fields, the D-dependent piece of the Lagrangian is:

3 X µi |κi| |κi| L = Di Di − θiDi − s Di + Di Di . (C.4.14) D 2 i 2|qi| 8|qi|3 i=1

3 1 2 1 2 Noting that D = D +D , we must integrate out D and D from LD. The resulting on-shell Lagrangian is simply the potential −V (qi). The equations of motion for Di, using s1 = s2 = −s3 = 1, are given by:

|κ1| |κ3| |κ3| s |κ1| s |κ3| µ1 + µ3 + + D1 + µ3 + D2 = θ1 + 1 + (C.4.15)3 4|q1|3 4|q3|3 4|q3|3 2|q1| 2|q3| |κ2| |κ3| |κ3| s |κ2| s |κ3| µ2 + µ3 + + D2 + µ3 + D1 = θ2 + 2 + (C.4.16)3 4|q2|3 4|q3|3 4|q3|3 2|q2| 2|q3|

i Writing the above equations as: aijD = ci, the solution is simply given by:

1 a12c2 − a22c1 2 a12c1 − a11c2 D = 2 ,D = 2 . (C.4.17) a12 − a11a22 a12 − a11a22

i Notice that the supersymmetric solution D = 0 is indeed given by c1 = c2 = 0 which is nothing more than equation (4.3.5).

Scaling Limit

In the scaling regime, where µi = 0, θi = 0, and κ1 > 0, κ2 > 0 and κ3 < 0 (or equivalently s1 = s2 = −s3 = 1), the potential can be written as:

1 α + β α + γ β + γ V (qi) = |q3|4 + |q2|4 + |q1|4 α + β + γ 2γ 2β 2α − |q1|2|q2|2 − |q1|2|q3|2 − |q2|2|q3|2 , (C.4.18) APPENDIX C. CONFORMAL QUIVERS 167

where |q1|3 |q2|3 |q3|3 α = , β = , and γ = − . (C.4.19) κ1 κ2 κ3

C.5 Thermal determinant

The thermal eﬀective potential for the two-node quiver at ﬁnite temperature can be derived from (4.5.1):

µ L = D2 − θD − κ ln det −(∂ + iA)2 + |q|2 + D + κ ln det −(∂ + iA)2 + |q|2 . eﬀ 2 t t

Wick rotating t 7→ itE and periodically identifying tE ∼ tE + β, introduces the thermal ensemble. The operator ∂tE has eigenvalues on the Matsubara frequencies: 1 ωn = 2πnT for bosons and ωn = 2π(n + 2 )T for fermions, with n ∈ Z. With a ≡ A/2πT and denoting |q|2 ≡ s this yields:

(T ) µ κ X L = D2 − θD − ln 4π2(n + a)2T 2 + s + D eff 2 β n∈Z κ X + ln 4π2(n + 1/2 + a)2T 2 + s , β n∈Z where we have now included a 1/β in the coefficient of the sums to cancel out the R β (T ) contribution from the integral over Euclidean time in SE = 0 dtE Leff . Taking a derivative with respect to s:

(T ) dL κ X κ X 1 eﬀ = f(n) = − ds β β 4π2(n + a)2T 2 + s + D n∈Z n∈Z κ X 1 + . (C.5.1) β 4π2(n + 1/2 + a)2T 2 + s n∈Z

We will treat the two sums separately using contour integration methods. We write

(C.5.1) as f(n) = −f1(n)+f2(n), denoting the two summands respectively. Consider:

I I πf1(z) cot(πz) = F (z)dz C C APPENDIX C. CONFORMAL QUIVERS 168

! X = 2πi ResF (z)|z=z+ + ResF (z)|z=z− + ResF (z)|z=n , n∈Z

√ √ i ± i i cot(∓ 2T s+D+πa) with z = −a ± s + D and ResF (z)| ± = ± √ and C a circle of 2πT z=z 4T s+D radius R around the origin with R → ∞. Given that F (z) decays rapidly enough, the integral evaluates to zero along C and we obtain:

X i i √ i √ − f1(n) = − √ cot s + D + πa − cot − s + D + πa . 4T s + D 2T 2T n∈Z (C.5.2) For the fermionic sum, the procedure is completely analgous, and (using cot(x + π/2) = − tan(x)) we ﬁnd: √ √ X i i s i s f (n) = − √ tan + πa − tan − + πa . (C.5.3) 2 4T s 2T 2T n∈Z

(T ) We need to integrate these two sums with respect to s to obtain Leff . Using the following identities: √ Z cot(a x + b) √ √ dx = 2 log sin(a x + b)/a (C.5.4) x √ Z tan(a x + b) √ √ dx = −2 log cos(a x + b)/b, (C.5.5) x we find: √  |q|2+D  cosh T − cos(2πa) (T ) µ 2   L = D − θD − κ T log   , (C.5.6) eff 2  |q|  cosh T + cos(2πa) as claimed. Appendix D

Supergoop Dynamics

D.1 Two-Body Problem

In this appendix we discuss the motion of a single probe particle in the background of a ﬁxed charge sitting at the origin. This was studied at length in [231]. The Hamiltonian of this system is given by

1 1 κ 2 H = (p − A)2 + + θ , p ≡ mx˙ + A (D.1.1) 2m 2m 2r and is conserved. For simplicity we choose κ > 0 and allow θ to be either positive or negative. Other than the Hamiltonian, this system admits two vector-valued conserved quantities known as the angular momentum L and the Runge-Lenz vector n. Explicitly

κ 1 L = x × (p − A) + x , and n = x + L × (p − A) . (D.1.2) 2r θ

Since this system is superintegrable, the probe particle’s trajectories can be found algebraically. First, notice that n·x˙ = 0 implying that n is perpendicular to the plane of motion of the probe particle. We use this fact to orient our axes such that n = |n|zˆ. It is

169 APPENDIX D. SUPERGOOP DYNAMICS 170

straightforward to show that

s 2mH κ2 |n| = L2 − , (D.1.3) θ2 4 which implies that |L| ≥ κ/2. With this choice of coordinates, the particle’s trajectory is constrained to lie in a plane of constant z. The magnitude of z can be obtained by 2 κ2 computing n · x = |n|z = − L − 4 /θ, giving

s |θ| L2 − κ2 z = − 4 . (D.1.4) θ 2mH

We have yet to choose an orientation for the x − y plane; we do so by aligning our coordinates such that Ly = 0 and Lx points in the positive x direction. The components of the angular momentum are given by

s s θ2 κ2 θ2 κ2 L = L2 − L2 − ,L = L2 − . (D.1.5) x 2mH 4 z 2mH 4

We can determine the particle’s trajectory explicitly by noticing that (D.1.2) implies that L · x = κ r/2 or

2 2 2 2 1 − e (x − x0) + 2`e(x − x0) + y = ` , (D.1.6) which is the equation for a conic section in cartesian coordinates. The quantities e and ` are the eccentricity and the semi-latus rectum of the conic section respectively and are given by

s 2L 2 θ2 κ2 4L2 − κ2 e = x = L2 − L2 − , ` = √ . (D.1.7) κ κ 2mH 4 κ 8mH

q √ 2 θ2 2 κ2 The quantity x0 ≡ L − 2mH L − 4 /(|θ| − 2mH) is the location of one of the foci of the conic section. APPENDIX D. SUPERGOOP DYNAMICS 171

Fig. D.1: Scatering angle.

The overall shape of a conic section is determined by its eccentricity, with elliptic orbits corresponding to e < 1, while parabolic and hyperbolic orbits correspond to e = 1 and e > 1 respectively. Intuition predicates that bound orbits should only happen for θ < 0, while θ > 0 gives rise to parabolic or hyperbolic orbits. For positive θ the Hamiltonian is bound such that 2mH ≥ θ2 which implies that e ≥ 1, thus verifying our intuition. For e < 1 the length of the semi-major axis a of the elliptic orbit is given by

2` κpH/2m a = 2 ∼ + O (1) (D.1.8) 1 − e Hescape − H

2 so as the energy approaches the escape energy Hescape = θ /2m (or as e approaches 1), the size of the bound orbit diverges. We end this appendix with a discussion of scattering for e ≥ 1. Since the trajectory of the particle is given by a conic section in the x − y plane, the scattering angle as deﬁned in ﬁgure D.1 is given by

√ ϕ = 2 arccot e2 − 1 . (D.1.9)

For parabolic orbits (e = 1), ϕ = π and we see that the particle completely back scatters. As we increase e the scattering angle decreases monotonically. Bibliography

[1] D. Anninos, T. Anous, D. Z. Freedman and G. Konstantinidis, arXiv:1406.5490 [hep-th].

[2] D. Anninos, T. Anous, P. de Lange and G. Konstantinidis, JHEP 1503, 066 (2015) [arXiv:1310.7929 [hep-th]].

[3] D. Anninos, F. Denef, G. Konstantinidis and E. Shaghoulian, “Higher Spin de Sitter Holography from Functional Determinants,” JHEP 1402, 007 (2014) [arXiv:1305.6321 [hep-th]].

[4] D. Anninos, T. Anous, F. Denef, G. Konstantinidis and E. Shaghoulian, “Super- goop Dynamics,” JHEP 1303, 081 (2013) [arXiv:1205.1060 [hep-th]].

[5] A. H. Guth, “The Inﬂationary Universe: A Possible Solution to the Horizon and Flatness Problems,” Phys. Rev. D 23, 347 (1981).

[6] A. D. Linde, “A New Inﬂationary Universe Scenario: A Possible Solution of the Horizon, Flatness, Homogeneity, Isotropy and Primordial Monopole Problems,” Phys. Lett. B 108, 389 (1982).

[7] D. Kazanas, “Dynamics of the Universe and Spontaneous Symmetry Breaking,” Astrophys. J. 241, L59 (1980).

[8] A. A. Starobinsky, “A New Type of Isotropic Cosmological Models Without Singularity,” Phys. Lett. B 91, 99 (1980).

[9] A. Albrecht and P. J. Steinhardt, “Cosmology for Grand Uniﬁed Theories with Radiatively Induced Symmetry Breaking,” Phys. Rev. Lett. 48, 1220 (1982).

172 BIBLIOGRAPHY 173

[10] S. Weinberg, “Quantum contributions to cosmological correlations,” Phys. Rev. D 72, 043514 (2005) [hep-th/0506236].

[11] D. Anninos and F. Denef, “Cosmic Clustering,” arXiv:1111.6061 [hep-th].

[12] J. B. Hartle and S. W. Hawking, “Wave Function of the Universe,” Phys. Rev. D 28, 2960 (1983).

[13] T. Hertog and J. Hartle, “Holographic No-Boundary Measure,” JHEP 1205, 095 (2012) [arXiv:1111.6090 [hep-th]].

[14] T. S. Bunch and P. C. W. Davies, “Quantum Field Theory in de Sitter Space: Renormalization by Point Splitting,” Proc. Roy. Soc. Lond. A 360, 117 (1978).

[15] N. A. Chernikov and E. A. Tagirov, “Quantum theory of scalar ﬁelds in de Sitter space-time,” Annales Poincare Phys. Theor. A 9, 109 (1968).

[16] J. M. Maldacena, “Non-Gaussian features of primordial fluctuations in single field inflationary models,” JHEP 0305, 013 (2003) [astro-ph/0210603].

[17] D. Harlow and D. Stanford, “Operator Dictionaries and Wave Functions in AdS/CFT and dS/CFT,” arXiv:1104.2621 [hep-th].

[18] I. Mata, S. Raju and S. Trivedi, “CMB from CFT,” JHEP 1307, 015 (2013) [arXiv:1211.5482 [hep-th]].

[19] D. Z. Freedman, S. D. Mathur, A. Matusis and L. Rastelli, “Correlation functions in the CFT(d) / AdS(d+1) correspondence,” Nucl. Phys. B 546, 96 (1999) [hep- th/9804058].

[20] E. D’Hoker and D. Z. Freedman, “General scalar exchange in AdS(d+1),” Nucl. Phys. B 550, 261 (1999) [hep-th/9811257].

[21] E. D’Hoker, D. Z. Freedman and L. Rastelli, “AdS / CFT four point functions: How to succeed at z integrals without really trying,” Nucl. Phys. B 562, 395 (1999) [hep-th/9905049]. BIBLIOGRAPHY 174

[22] S. Weinberg, “Quantum contributions to cosmological correlations. II. Can these corrections become large?,” Phys. Rev. D 74, 023508 (2006) [hep-th/0605244].

[23] L. Senatore and M. Zaldarriaga, “On Loops in Inﬂation,” JHEP 1012, 008 (2010) [arXiv:0912.2734 [hep-th]].

[24] D. Seery, “Infrared eﬀects in inﬂationary correlation functions,” Class. Quant. Grav. 27, 124005 (2010) [arXiv:1005.1649 [astro-ph.CO]].

[25] K. Freese, F. C. Adams, J. A. Frieman and E. Mottola, “Cosmology with De- caying Vacuum Energy,” Nucl. Phys. B 287, 797 (1987).

[26] V. K. Onemli and R. P. Woodard, “Superacceleration from massless, minimally coupled phi**4,” Class. Quant. Grav. 19, 4607 (2002) [gr-qc/0204065].

[27] A. Youssef and D. Kreimer, “Resummation of infrared logarithms in de Sit- ter space via Dyson-Schwinger equations: the ladder-rainbow approximation,” arXiv:1301.3205 [gr-qc].

[28] F. Gautier and J. Serreau, “Infrared dynamics in de Sitter space from Schwinger- Dyson equations,” Phys. Lett. B 727, 541 (2013) [arXiv:1305.5705 [hep-th]].

[29] A. A. Starobinsky and J. Yokoyama, “Equilibrium state of a selﬁnteracting scalar ﬁeld in the De Sitter background,” Phys. Rev. D 50, 6357 (1994) [astro- ph/9407016].

[30] D. Marolf and I. A. Morrison, “The IR stability of de Sitter: Loop corrections to scalar propagators,” Phys. Rev. D 82, 105032 (2010) [arXiv:1006.0035 [gr-qc]].

[31] M. Sasaki, H. Suzuki, K. Yamamoto and J. ’i. Yokoyama, “Superexpansionary divergence: Breakdown of perturbative quantum ﬁeld theory in space-time with accelerated expansion,” Class. Quant. Grav. 10, L55 (1993).

[32] T. Prokopec, N. C. Tsamis and R. P. Woodard, “Two Loop Scalar Bilinears for Inﬂationary SQED,” Class. Quant. Grav. 24, 201 (2007) [gr-qc/0607094]. BIBLIOGRAPHY 175

[33] A. M. Polyakov, “De Sitter space and eternity,” Nucl. Phys. B 797, 199 (2008) [arXiv:0709.2899 [hep-th]].

[34] N. C. Tsamis and R. P. Woodard, “Quantum gravity slows inﬂation,” Nucl. Phys. B 474, 235 (1996) [hep-ph/9602315].

[35] L. H. Ford, “Quantum Instability of De Sitter Space-time,” Phys. Rev. D 31, 710 (1985).

[36] S. B. Giddings and M. S. Sloth, “Fluctuating geometries, q-observables, and infrared growth in inﬂationary spacetimes,” Phys. Rev. D 86, 083538 (2012) [arXiv:1109.1000 [hep-th]].

[37] C. P. Burgess, L. Leblond, R. Holman and S. Shandera, “Super-Hubble de Sitter Fluctuations and the Dynamical RG,” JCAP 1003, 033 (2010) [arXiv:0912.1608 [hep-th]].

[38] C. P. Burgess, R. Holman, L. Leblond and S. Shandera, “Breakdown of Semi- classical Methods in de Sitter Space,” JCAP 1010, 017 (2010) [arXiv:1005.3551 [hep-th]].

[39] A. Strominger, “The dS / CFT correspondence,” JHEP 0110, 034 (2001) [hep- th/0106113].

[40] E. Witten, “Quantum gravity in de Sitter space,” hep-th/0106109.

[41] D. Anninos, T. Hartman and A. Strominger, “Higher Spin Realization of the dS/CFT Correspondence,” arXiv:1108.5735 [hep-th].

[42] C. -M. Chang, A. Pathak and A. Strominger, “Non-Minimal Higher-Spin DS4/CFT3,” arXiv:1309.7413 [hep-th].

[43] D. Anninos, R. Mahajan, D. Radicevic and E. Shaghoulian, “Chern-Simons- Ghost Theories and de Sitter Space,” arXiv:1405.1424 [hep-th].

[44] M. Spradlin, A. Strominger and A. Volovich, “Les Houches lectures on de Sitter space,” hep-th/0110007. BIBLIOGRAPHY 176

[45] D. Anninos, “De Sitter Musings,” Int. J. Mod. Phys. A 27, 1230013 (2012) [arXiv:1205.3855 [hep-th]].

[46] I. Heemskerk and J. Polchinski, “Holographic and Wilsonian Renormalization Groups,” JHEP 1106, 031 (2011) [arXiv:1010.1264 [hep-th]].

[47] J. de Boer, E. P. Verlinde and H. L. Verlinde, “On the holographic renormalization group,” JHEP 0008, 003 (2000) [hep-th/9912012].

[48] E. Kiritsis, W. Li and F. Nitti, “Holographic RG ﬂow and the Quantum Eﬀective Action,” arXiv:1401.0888 [hep-th].

[49] P. Mansﬁeld and D. Nolland, “One loop conformal anomalies from AdS / CFT in the Schrodinger representation,” JHEP 9907, 028 (1999) [hep-th/9906054].

[50] A. Strominger, “Inﬂation and the dS / CFT correspondence,” JHEP 0111, 049 (2001) [hep-th/0110087].

[51] F. Larsen, J. P. van der Schaar and R. G. Leigh, “De Sitter holography and the cosmic microwave background,” JHEP 0204, 047 (2002) [hep-th/0202127].

[52] A. Bzowski, P. McFadden and K. Skenderis, “Holography for inﬂation using conformal perturbation theory,” JHEP 1304, 047 (2013) [arXiv:1211.4550 [hep- th]].

[53] D. Das, S. R. Das and G. Mandal, “Double Trace Flows and Holographic RG in dS/CFT correspondence,” JHEP 1311, 186 (2013) [arXiv:1306.0336, arXiv:1306.0336 [hep-th]].

[54] J. Garriga and Y. Urakawa, “Inﬂation and deformation of conformal ﬁeld theory,” JCAP 1307, 033 (2013) [arXiv:1303.5997 [hep-th]].

[55] O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri and Y. Oz, Phys. Rept. 323, 183 (2000) [hep-th/9905111].

[56] E. D’Hoker and D. Z. Freedman, hep-th/0201253. BIBLIOGRAPHY 177

[57] H. Nastase, arXiv:0712.0689 [hep-th].

[58] D. Z. Freedman and A. Van Proeyen, “Supergravity,” Cambridge, UK: Cam- bridge Univ. Pr. (2012) 607 p

[59] D. Anninos, S. A. Hartnoll and D. M. Hofman, “Static Patch Solipsism: Con- formal Symmetry of the de Sitter Worldline,” Class. Quant. Grav. 29, 075002 (2012) [arXiv:1109.4942 [hep-th]].

[60] R. P. Bernar, L. C. B. Crispino and A. Higuchi, “Infrared-ﬁnite graviton two- point function in static de Sitter space,” arXiv:1405.3827 [gr-qc].

[61] D. Boyanovsky and R. Holman, “On the Perturbative Stability of Quantum Field Theories in de Sitter Space,” JHEP 1105, 047 (2011) [arXiv:1103.4648 [astro-ph.CO]].

[62] A. Bzowski, P. McFadden and K. Skenderis, “Implications of conformal invariance in momentum space,” JHEP 1403, 111 (2014) [arXiv:1304.7760 [hep-th]].

[63] C. Coriano, L. Delle Rose, E. Mottola and M. Serino, “Solving the Conformal Constraints for Scalar Operators in Momentum Space and the Evaluation of Feynman’s Master Integrals,” JHEP 1307, 011 (2013) [arXiv:1304.6944 [hep- th]].

[64] T. Falk, R. Rangarajan and M. Srednicki, “The Angular dependence of the three point correlation function of the cosmic microwave background radiation as predicted by inﬂationary cosmologies,” Astrophys. J. 403, L1 (1993) [astro- ph/9208001].

[65] M. A. Vasiliev, “More on equations of motion for interacting massless ﬁelds of all spins in (3+1)-dimensions,” Phys. Lett. B 285, 225 (1992).

[66] M. A. Vasiliev, “Higher spin gauge theories in four-dimensions, three-dimensions, and two-dimensions,” Int. J. Mod. Phys. D 5, 763 (1996) [hep-th/9611024]. BIBLIOGRAPHY 178

[67] E. Sezgin and P. Sundell, “Holography in 4D (super) higher spin theories and a test via cubic scalar couplings,” JHEP 0507, 044 (2005) [hep-th/0305040].

[68] A. C. Petkou, “Evaluating the AdS dual of the critical O(N) vector model,” JHEP 0303, 049 (2003) [hep-th/0302063].

[69] C. Feﬀerman and C. R. Graham, ”Conformal Invariants” Elie Cartan et les Mathematiques d’Aujordhui (Asterique, 1985) 95.

[70] A. A. Starobinsky, “Isotropization of arbitrary cosmological expansion given an eﬀective cosmological constant,” JETP Lett. 37, 66 (1983).

[71] K. Skenderis, “Lecture notes on holographic renormalization,” Class. Quant. Grav. 19, 5849 (2002) [hep-th/0209067].

[72] M. Henningson and K. Skenderis, “The Holographic Weyl anomaly,” JHEP 9807, 023 (1998) [hep-th/9806087].

[73] J. M. Maldacena and G. L. Pimentel, “On graviton non-Gaussianities during inﬂation,” JHEP 1109, 045 (2011) [arXiv:1104.2846 [hep-th]].

[74] A. Bzowski, P. McFadden and K. Skenderis, “Holographic predictions for cosmological 3-point functions,” JHEP 1203, 091 (2012) [arXiv:1112.1967 [hep-th]].

[75] G. L. Pimentel, “Inﬂationary Consistency Conditions from a Wavefunctional Perspective,” JHEP 1402, 124 (2014) [arXiv:1309.1793 [hep-th]].

[76] H. Osborn and A. C. Petkou, “Implications of conformal invariance in ﬁeld theories for general dimensions,” Annals Phys. 231, 311 (1994) [hep-th/9307010].

[77] E. Brezin and D.J. Wallace, Phys. Rev. B7 (1973) 1967; S-K. Ma, Phys. Rev. A7 (1973) 2172

[78] D. Anninos, F. Denef and D. Harlow, “Wave function of Vasiliev’s universe: A few slices thereof,” Phys. Rev. D 88, no. 8, 084049 (2013) [arXiv:1207.5517 [hep-th]]. BIBLIOGRAPHY 179

[79] D. Marolf, M. Rangamani and M. Van Raamsdonk, “Holographic models of de Sitter QFTs,” Class. Quant. Grav. 28, 105015 (2011) [arXiv:1007.3996 [hep-th]].

[80] S. S. Gubser and I. Mitra, Phys. Rev. D 67, 064018 (2003) [hep-th/0210093].

[81] R. Haberman, “Elementary Applied Partial Diﬀerential Equations” third edition, Prentice-Hall (1998) ISBN 0-13-263807-X

[82] E. Mottola, “Particle Creation in de Sitter Space,” Phys. Rev. D 31, 754 (1985).

[83] B. Allen, “Vacuum States in de Sitter Space,” Phys. Rev. D 32, 3136 (1985).

[84] C. Schomblond and P. Spindel, “Unicity Conditions of the Scalar Field Propa- gator Delta(1) (x,y) in de Sitter Universe,” Annales Poincare Phys. Theor. 25, 67 (1976).

[85] M. Sasaki, T. Tanaka and K. Yamamoto, “Euclidean vacuum mode functions for a scalar ﬁeld on open de Sitter space,” Phys. Rev. D 51, 2979 (1995) [gr- qc/9412025].

[86] J. S. Dowker and R. Critchley, “Eﬀective Lagrangian and Energy Momentum Tensor in de Sitter Space,” Phys. Rev. D 13, 3224 (1976).

[87] P. Candelas and D. J. Raine, “General Relativistic Quantum Field Theory-An Exactly Soluble Model,” Phys. Rev. D 12, 965 (1975).

[88] G. Boerner and H. P. Duerr, “Classical and quantum ﬁelds in de sitter space,” Nuovo Cim. A 64, 669 (1969).

[89] M. Alishahiha, A. Karch, E. Silverstein and D. Tong, “The dS/dS correspondence,” AIP Conf. Proc. 743, 393 (2005) [hep-th/0407125].

[90] X. Dong, B. Horn, E. Silverstein and G. Torroba, “Micromanaging de Sitter holography,” Class. Quant. Grav. 27, 245020 (2010) [arXiv:1005.5403 [hep-th]].

[91] X. Dong, B. Horn, S. Matsuura, E. Silverstein and G. Torroba, “FRW solutions and holography from uplifted AdS/CFT,” Phys. Rev. D 85, 104035 (2012) [arXiv:1108.5732 [hep-th]]. BIBLIOGRAPHY 180

[92] P. McFadden and K. Skenderis, “Holography for Cosmology,” Phys. Rev. D 81, 021301 (2010) [arXiv:0907.5542 [hep-th]].

[93] T. Banks and W. Fischler, “Holographic Theories of Inﬂation and Fluctuations,” arXiv:1111.4948 [hep-th].

[94] T. Banks, “Holographic Space-time from the Big Bang to the de Sitter era,” J. Phys. A A 42, 304002 (2009) [arXiv:0809.3951 [hep-th]].

[95] B. Freivogel, Y. Sekino, L. Susskind and C. -P. Yeh, “A Holographic framework for eternal inﬂation,” Phys. Rev. D 74, 086003 (2006) [hep-th/0606204].

[96] D. A. Roberts and D. Stanford, “On memory in exponentially expanding spaces,” arXiv:1210.5238 [hep-th].

[97] D. Harlow, S. H. Shenker, D. Stanford and L. Susskind, “The Three Faces of a Fixed Point,” arXiv:1203.5802 [hep-th].

[98] J. Garriga and A. Vilenkin, “Holographic Multiverse,” JCAP 0901, 021 (2009) [arXiv:0809.4257 [hep-th]].

[99] M. K. Parikh and E. P. Verlinde, “De Sitter holography with a ﬁnite number of states,” JHEP 0501, 054 (2005) [hep-th/0410227].

[100] D. Anninos and T. Hartman, “Holography at an Extremal De Sitter Horizon,” JHEP 1003, 096 (2010) [arXiv:0910.4587 [hep-th]].

[101] D. Anninos and T. Anous, “A de Sitter Hoedown,” JHEP 1008, 131 (2010) [arXiv:1002.1717 [hep-th]].

[102] D. Anninos, T. Anous, I. Bredberg and G. S. Ng, “Incompressible Fluids of the de Sitter Horizon and Beyond,” JHEP 1205, 107 (2012) [arXiv:1110.3792 [hep-th]].

[103] M. A. Vasiliev, “Consistent equation for interacting gauge ﬁelds of all spins in (3+1)-dimensions,” Phys. Lett. B243, 378-382 (1990). BIBLIOGRAPHY 181

[104] M. A. Vasiliev, “Higher spin gauge theories: Star product and AdS space,” arXiv:hep-th/9910096.

[105] C. Iazeolla, E. Sezgin, P. Sundell, “Real forms of complex higher spin ﬁeld equations and new exact solutions,” Nucl. Phys. B791, 231-264 (2008). [arXiv:0706.2983 [hep-th]].

[106] M. A. Vasiliev, “Free Massless Fields Of Arbitrary Spin In The De Sitter Space And Initial Data For A Higher Spin Superalgebra,” Fortsch. Phys. 35, 741-770 (1987).

[107] D. Anninos, T. Hartman and A. Strominger, “Higher Spin Realization of the dS/CFT Correspondence,” arXiv:1108.5735 [hep-th].

[108] D. Das, S. R. Das, A. Jevicki and Q. Ye, “Bi-local Construction of Sp(2N)/dS Higher Spin Correspondence,” JHEP 1301, 107 (2013) [arXiv:1205.5776 [hep- th]].

[109] G. S. Ng and A. Strominger, “State/Operator Correspondence in Higher-Spin dS/CFT,” arXiv:1204.1057 [hep-th].

[110] A. LeClair, M. Neubert, “Semi-Lorentz invariance, unitarity, and critical expo- nents of symplectic fermion models,” JHEP 0710, 027 (2007). [arXiv:0705.4657 [hep-th]].

[111] E. Sezgin and P. Sundell, “On an exact cosmological solution of higher spin gauge theory,” hep-th/0511296.

[112] G. V. Dunne and K. Kirsten, “Functional determinants for radial operators,” J. Phys. A 39, 11915 (2006) [hep-th/0607066].

[113] I.R. Klebanov and A.M. Polyakov, “AdS dual of the critical O(N) vector model,” Phys. Lett. B 550 (2002) 213 [arXiv:hep-th/0210114].

[114] E. Sezgin and P. Sundell, “Massless higher spins and holography,” Nucl. Phys. B 644 (2002) 303 [Erratum-ibid. B 660 (2003) 403] [arXiv:hep-th/0205131]. BIBLIOGRAPHY 182

[115] B. Sundborg, “Stringy gravity, interacting tensionless strings and massless higher spins,” Nucl. Phys. Proc. Suppl. 102, 113 (2001) [hep-th/0103247].

[116] S. Giombi and X. Yin, “Higher Spin Gauge Theory and Holography: The Three- Point Functions,” JHEP 1009, 115 (2010) [arXiv:0912.3462 [hep-th]].

[117] D. Anninos, G. S. Ng and A. Strominger, “Asymptotic Symmetries and Charges in De Sitter Space,” Class. Quant. Grav. 28, 175019 (2011) [arXiv:1009.4730 [gr- qc]].

[118] I. M. Gelfand and A. M. Yaglom, “Integration in functional spaces and it applications in quantum physics,” J. Math. Phys. 1, 48 (1960).

[119] M. Bordag, K. Kirsten and J. S. Dowker, “Heat kernels and functional determinants on the generalized cone,” Commun. Math. Phys. 182, 371 (1996) [hep-th/9602089].

[120] R. Bousso, “Proliferation of de Sitter space,” Phys. Rev. D 58, 083511 (1998) [hep-th/9805081].

[121] S. W. Hawking, “The Quantum State of the Universe,” Nucl. Phys. B 239, 257 (1984).

[122] A. Castro and A. Maloney, “The Wave Function of Quantum de Sitter,” JHEP 1211, 096 (2012) [arXiv:1209.5757 [hep-th]].

[123] K. G. Wilson, “Quantum ﬁeld theory models in less than four-dimensions,” Phys. Rev. D 7, 2911 (1973).

[124] E. Witten, “Multitrace operators, boundary conditions, and AdS / CFT correspondence,” hep-th/0112258.

[125] W. Mueck, “An Improved correspondence formula for AdS / CFT with multitrace operators,” Phys. Lett. B 531, 301 (2002) [hep-th/0201100]. BIBLIOGRAPHY 183

[126] S. S. Gubser and I. R. Klebanov, “A Universal result on central charges in the presence of double trace deformations,” Nucl. Phys. B 656, 23 (2003) [hep- th/0212138].

[127] D. Anninos, G. S. Ng and A. Strominger, “Future Boundary Conditions in De Sitter Space,” JHEP 1202, 032 (2012) [arXiv:1106.1175 [hep-th]].

[128] J. B. Hartle, S. W. Hawking and T. Hertog, “The Classical Universes of the No- Boundary Quantum State,” Phys. Rev. D 77, 123537 (2008) [arXiv:0803.1663 [hep-th]].

[129] J. B. Hartle, S. W. Hawking and T. Hertog, “No-Boundary Measure of the Universe,” Phys. Rev. Lett. 100, 201301 (2008) [arXiv:0711.4630 [hep-th]].

[130] S. Sarangi and S. -H. H. Tye, “The Boundedness of Euclidean gravity and the wavefunction of the universe,” hep-th/0505104.

[131] O. Aharony, G. Gur-Ari and R. Yacoby, “d=3 Bosonic Vector Models Coupled to Chern-Simons Gauge Theories,” JHEP 1203, 037 (2012) [arXiv:1110.4382 [hep-th]].

[132] C. -M. Chang, S. Minwalla, T. Sharma and X. Yin, “ABJ Triality: from Higher Spin Fields to Strings,” arXiv:1207.4485 [hep-th].

[133] S. Giombi, S. Minwalla, S. Prakash, S. P. Trivedi, S. R. Wadia and X. Yin, “Chern-Simons Theory with Vector Fermion Matter,” Eur. Phys. J. C 72, 2112 (2012) [arXiv:1110.4386 [hep-th]].

[134] S. Giombi and X. Yin, “The Higher Spin/Vector Model Duality,” arXiv:1208.4036 [hep-th].

[135] E. Witten, “SL(2,Z) action on three-dimensional conformal ﬁeld theories with Abelian symmetry,” In *Shifman, M. (ed.) et al.: From ﬁelds to strings, vol. 2* 1173-1200 [hep-th/0307041]. BIBLIOGRAPHY 184

[136] O. Aharony, O. Bergman and D. L. Jaﬀeris, “Fractional M2-branes,” JHEP 0811, 043 (2008) [arXiv:0807.4924 [hep-th]].

[137] J. Maldacena and A. Zhiboedov, “Constraining Conformal Field Theories with A Higher Spin Symmetry,” arXiv:1112.1016 [hep-th].

[138] J. Maldacena and A. Zhiboedov, “Constraining conformal ﬁeld theories with a slightly broken higher spin symmetry,” arXiv:1204.3882 [hep-th].

[139] work in progress

[140] L. Girardello, M. Porrati and A. Zaﬀaroni, “3-D interacting CFTs and generalized Higgs phenomenon in higher spin theories on AdS,” Phys. Lett. B 561, 289 (2003) [hep-th/0212181].

[141] S. Banerjee, S. Hellerman, J. Maltz and S. H. Shenker, “Light States in Chern- Simons Theory Coupled to Fundamental Matter,” arXiv:1207.4195 [hep-th].

[142] D. Radicevic, “Singlet Vector Models on Lens Spaces,” arXiv:1210.0255 [hep- th].

[143] E. Witten, “Analytic Continuation Of Chern-Simons Theory,” arXiv:1001.2933 [hep-th].

[144] T. Banks, “Some thoughts on the quantum theory of stable de Sitter space,” hep-th/0503066.

[145] L. Andersson and G. J. Galloway, “DS / CFT and space-time topology,” Adv. Theor. Math. Phys. 6, 307 (2003) [hep-th/0202161].

[146] Shamik Banerjee, Alexandre Belin, Simeon Hellerman, Arnaud Lepage-Jutier, Alexander Maloney, Dorde Radicevic, Stephen Shenker, “Topology of future in- ﬁnity in dS/CFT,” to appear.

[147] J. Michelson and A. Strominger, “The Geometry of (super)conformal quantum mechanics,” Commun. Math. Phys. 213, 1 (2000) [hep-th/9907191]. BIBLIOGRAPHY 185

[148] A. Strominger and C. Vafa, “Microscopic origin of the Bekenstein-Hawking entropy,” Phys. Lett. B 379, 99 (1996) [hep-th/9601029].

[149] J. C. Breckenridge, R. C. Myers, A. W. Peet and C. Vafa, “D-branes and spinning black holes,” Phys. Lett. B 391, 93 (1997) [hep-th/9602065].

[150] S. H. Katz, A. Klemm and C. Vafa, “M theory, topological strings and spinning black holes,” Adv. Theor. Math. Phys. 3, 1445 (1999) [hep-th/9910181].

[151] M. Guica and A. Strominger, “Wrapped M2/M5 Duality,” JHEP 0910, 036 (2009) [hep-th/0701011].

[152] D. Anninos, W. Li, M. Padi, W. Song and A. Strominger, “Warped AdS(3) Black Holes,” JHEP 0903, 130 (2009) [arXiv:0807.3040 [hep-th]].

[153] D. Anninos, “Hopﬁng and Puﬃng Warped Anti-de Sitter Space,” JHEP 0909, 075 (2009) [arXiv:0809.2433 [hep-th]].

[154] M. Guica, T. Hartman, W. Song and A. Strominger, “The Kerr/CFT Corre- spondence,” Phys. Rev. D 80, 124008 (2009) [arXiv:0809.4266 [hep-th]].

[155] M. Guica and A. Strominger, “Microscopic Realization of the Kerr/CFT Cor- respondence,” JHEP 1102, 010 (2011) [arXiv:1009.5039 [hep-th]].

[156] S. Detournay, T. Hartman and D. M. Hofman, “Warped Conformal Field The- ory,” Phys. Rev. D 86, 124018 (2012) [arXiv:1210.0539 [hep-th]].

[157] M.-x. Huang, A. Klemm, M. Marino and A. Tavanfar, “Black holes and large order quantum geometry,” Phys. Rev. D 79, 066001 (2009) [arXiv:0704.2440 [hep-th]].

[158] C. Vafa, “Black holes and Calabi-Yau threefolds,” Adv. Theor. Math. Phys. 2, 207 (1998) [hep-th/9711067].

[159] J. M. Maldacena, A. Strominger and E. Witten, “Black hole entropy in M theory,” JHEP 9712, 002 (1997) [hep-th/9711053]. BIBLIOGRAPHY 186

[160] E. Brezin, C. Itzykson, G. Parisi and J. B. Zuber, “Planar Diagrams,” Commun. Math. Phys. 59, 35 (1978).

[161] I. R. Klebanov, “String theory in two-dimensions,” In *Trieste 1991, Proceed- ings, String theory and quantum gravity ’91* 30-101 and Princeton Univ. - PUPT-1271 (91/07,rec.Oct.) 72 p [hep-th/9108019].

[162] P. H. Ginsparg and G. W. Moore, “Lectures on 2-D gravity and 2-D string theory,” In *Boulder 1992, Proceedings, Recent directions in particle theory* 277-469. and Yale Univ. New Haven - YCTP-P23-92 (92,rec.Apr.93) 197 p. and Los Alamos Nat. Lab. - LA-UR-92-3479 (92,rec.Apr.93) 197 p [hep-th/9304011].

[163] J. Polchinski, “What is string theory?,” hep-th/9411028.

[164] M. R. Douglas and G. W. Moore, “D-branes, quivers, and ALE instantons,” hep-th/9603167.

[165] F. Denef, “Quantum quivers and Hall / hole halos,” JHEP 0210, 023 (2002) [hep-th/0206072].

[166] F. Denef and G. W. Moore, “Split states, entropy enigmas, holes and halos,” JHEP 1111, 129 (2011) [hep-th/0702146 [HEP-TH]].

[167] I. Bena, M. Berkooz, J. de Boer, S. El-Showk and D. Van den Bleeken, “Scaling BPS Solutions and pure-Higgs States,” JHEP 1211, 171 (2012) [arXiv:1205.5023 [hep-th]].

[168] S. -J. Lee, Z. -L. Wang and P. Yi, “Quiver Invariants from Intrinsic Higgs States,” JHEP 1207, 169 (2012) [arXiv:1205.6511 [hep-th]].

[169] J. Manschot, B. Pioline and A. Sen, “From Black Holes to Quivers,” JHEP 1211, 023 (2012) [arXiv:1207.2230 [hep-th]].

[170] J. Manschot, B. Pioline and A. Sen, “On the Coulomb and Higgs branch formulae for multi-centered black holes and quiver invariants,” JHEP 1305, 166 (2013) [arXiv:1302.5498 [hep-th]]. BIBLIOGRAPHY 187

[171] I. Bena, C. -W. Wang and N. P. Warner, “Plumbing the Abyss: Black ring microstates,” JHEP 0807, 019 (2008) [arXiv:0706.3786 [hep-th]].

[172] R. Jackiw, “Introducing Scale Symmetry,” Physics Today 25, 23-27 (1972)

[173] C. R. Hagan, “Scale and Conformal Transformations in Galilean-Covariant Field Theory,” Phys. Rev. D 5, 377-388 (1972)

[174] U. Niederer, “The Maximal Kinematical Invariance Group of the Free Schr¨odingerEquation,” Helv. Phys. Acta 45, 802810 (1972)

[175] V. de Alfaro, S. Fubini and G. Furlan, “Conformal Invariance in Quantum Mechanics,” Nuovo Cim. A 34, 569 (1976).

[176] E. Witten, “Phases of N=2 theories in two-dimensions,” Nucl. Phys. B 403, 159 (1993) [hep-th/9301042].

[177] S. -J. Lee, Z. -L. Wang and P. Yi, “Abelianization of BPS Quivers and the Reﬁned Higgs Index,” arXiv:1310.1265 [hep-th].

[178] M. R. Douglas, D. N. Kabat, P. Pouliot and S. H. Shenker, “D-branes and short distances in string theory,” Nucl. Phys. B 485, 85 (1997) [hep-th/9608024].

[179] F. Denef, “Supergravity ﬂows and D-brane stability,” JHEP 0008, 050 (2000) [hep-th/0005049].

[180] A. Nersessian and V. Ohanyan, “Multi-center MICZ-Kepler systems,” Theor. Math. Phys. 155, 618 (2008) [arXiv:0705.0727 [math-ph]].

[181] R. Britto-Pacumio, J. Michelson, A. Strominger and A. Volovich, “Lectures on superconformal quantum mechanics and multiblack hole moduli spaces,” hep- th/9911066.

[182] J. de Boer, S. El-Showk, I. Messamah and D. Van den Bleeken, “Quantizing N=2 Multicenter Solutions,” JHEP 0905, 002 (2009) [arXiv:0807.4556 [hep-th]]. BIBLIOGRAPHY 188

[183] G. Papadopoulos, “Conformal and superconformal mechanics,” Class. Quant. Grav. 17, 3715 (2000) [hep-th/0002007].

[184] A. V. Smilga, “Perturbative corrections to eﬀective zero-mode hamiltonian in supersymmetric QED.” Nuclear Physics B 291 (1987): 241-255.

[185] J. Michelson and A. Strominger, JHEP 9909, 005 (1999) [hep-th/9908044].

[186] R. C. Ferrel and D. M. Eardley, “Slow-Motion Scattering and Coallescence of Maximally charged Black Holes,” Phys. Rev. Lett. 59, 14 (1987)

[187] G. W. Gibbons and P. J. Ruback, “The Motion Of Extreme Reissner-Nordstrom Black Holes In The Low Velocity Limit,” Phys. Rev. Lett. 57, 1492 (1986)

[188] A. Maloney, M. Spradlin and A. Strominger, “Superconformal multiblack hole moduli spaces in four-dimensions,” JHEP 0204, 003 (2002) [hep-th/9911001].

[189] A. Strominger, “AdS(2) quantum gravity and string theory,” JHEP 9901, 007 (1999) [hep-th/9809027].

[190] T. Azeyanagi, T. Nishioka and T. Takayanagi, “Near Extremal Black Hole En- tropy as Entanglement Entropy via AdS(2)/CFT(1),” Phys. Rev. D 77, 064005 (2008) [arXiv:0710.2956 [hep-th]].

[191] A. Sen, “State Operator Correspondence and Entanglement in AdS2/CF T1,” Entropy 13, 1305 (2011) [arXiv:1101.4254 [hep-th]].

[192] J. M. Maldacena, J. Michelson and A. Strominger, “Anti-de Sitter fragmentation,” JHEP 9902, 011 (1999) [hep-th/9812073].

[193] C. Chamon, R. Jackiw, S. -Y. Pi and L. Santos, “Conformal quantum mechanics

as the CFT1 dual to AdS2,” Phys. Lett. B 701, 503 (2011) [arXiv:1106.0726 [hep- th]].

[194] T. Banks, W. Fischler, S. H. Shenker and L. Susskind, “M theory as a matrix model: A Conjecture,” Phys. Rev. D 55, 5112 (1997) [hep-th/9610043]. BIBLIOGRAPHY 189

[195] K. Becker and M. Becker, “A Two loop test of M(atrix) theory,” Nucl. Phys. B 506, 48 (1997) [hep-th/9705091].

[196] M. Berkooz and H. L. Verlinde, “Matrix theory, AdS / CFT and Higgs-Coulomb equivalence,” JHEP 9911, 037 (1999) [hep-th/9907100].

[197] J. Fuchs, “Remarks On Supersymmetric Quantum Mechanics At Finite Tem- perature,” Annals Phys. 165, 285 (1985).

[198] A. Das, A. Kharev and S. V. Mathur, Phys. Lett. 181B, 299 (1986)

[199] P. Roy and R. Roychoudhury, “Remarks On The Finite Temperature Eﬀect In Supersymmetric Quantum Mechanics,” J. Phys. A 21, 3187 (1988).

[200] D. N. Kabat, G. Lifschytz and D. A. Lowe, “Black hole thermodynamics from calculations in strongly coupled gauge theory,” Int. J. Mod. Phys. A 16, 856 (2001) [Phys. Rev. Lett. 86, 1426 (2001)] [hep-th/0007051].

[201] D. N. Kabat, G. Lifschytz and D. A. Lowe, “Black hole entropy from nonperturbative gauge theory,” Phys. Rev. D 64, 124015 (2001) [hep-th/0105171].

[202] N. Kawahara, J. Nishimura and S. Takeuchi, “Phase structure of matrix quantum mechanics at ﬁnite temperature,” JHEP 0710, 097 (2007) [arXiv:0706.3517 [hep-th]].

[203] S. Catterall and T. Wiseman, “Towards lattice simulation of the gauge theory duals to black holes and hot strings,” JHEP 0712, 104 (2007) [arXiv:0706.3518 [hep-lat]].

[204] K. N. Anagnostopoulos, M. Hanada, J. Nishimura and S. Takeuchi, “Monte Carlo studies of supersymmetric matrix quantum mechanics with sixteen supercharges at ﬁnite temperature,” Phys. Rev. Lett. 100, 021601 (2008) [arXiv:0707.4454 [hep-th]].

[205] Y. -H. Lin, S. -H. Shao, Y. Wang and X. Yin, “A Low Temperature Expansion for Matrix Quantum Mechanics,” arXiv:1304.1593 [hep-th]. BIBLIOGRAPHY 190

[206] N. Iizuka, D. Kabat, S. Roy and D. Sarkar, “Black Hole Formation in Fuzzy Sphere Collapse,” Phys. Rev. D 88, 044019 (2013) [arXiv:1306.3256 [hep-th]].

[207] C. Asplund, D. Berenstein and D. Trancanelli, “Evidence for fast thermalization in the plane-wave matrix model,” Phys. Rev. Lett. 107, 171602 (2011) [arXiv:1104.5469 [hep-th]].

[208] D. Anninos, T. Anous, J. Barandes, F. Denef and B. Gaasbeek, “Hot Halos and Galactic Glasses,” JHEP 1201, 003 (2012) [arXiv:1108.5821 [hep-th]].

[209] B. D. Chowdhury and B. Vercnocke, “New instability of non-extremal black holes: spitting out supertubes,” JHEP 1202, 116 (2012) [arXiv:1110.5641 [hep- th]].

[210] B. D. Chowdhury, D. R. Mayerson and B. Vercnocke, “Phases of non-extremal multi-centered bound states,” arXiv:1307.5846 [hep-th].

[211] D. Anninos, T. Anous, F. Denef and L. Peeters, “Holographic Vitriﬁcation,” arXiv:1309.0146 [hep-th].

[212] S. D. Majumdar, “A class of exact solutions of Einstein’s ﬁeld equations,” Phys. Rev. 72, 390 (1947).

[213] A. Papapetrou, “Static solution of the equations of the gravitational ﬁeld for an arbitrary charge distribution,” Proc. Roy. Irish Acad. (Sect. A) A51, 191-204 (1947).

[214] E. P. Verlinde, “On the Origin of Gravity and the Laws of Newton,” JHEP 1104, 029 (2011) [arXiv:1001.0785 [hep-th]].

[215] A. Amir, Y. Oreg and Y. Imry, “Localization, anomalous diﬀusion and slow re- laxations: A random distance matrix approach,” [arXiv:1002.2123 [cond-mat.dis- nn]]

[216] Y. Nakayama, “A lecture note on scale invariance vs conformal invariance,” arXiv:1302.0884 [hep-th]. BIBLIOGRAPHY 191

[217] N. Iizuka, D. N. Kabat, G. Lifschytz and D. A. Lowe, “Probing black holes in nonperturbative gauge theory,” Phys. Rev. D 65, 024012 (2002) [hep- th/0108006].

[218] N. Iizuka and J. Polchinski, “A Matrix Model for Black Hole Thermalization,” JHEP 0810, 028 (2008) [arXiv:0801.3657 [hep-th]].

[219] Cubic Function, http://en.wikipedia.org/wiki/Cubic_function

[220] M. Mezard, G. Parisi and M. A. Virasoro, “Spin Glass Theory and Beyond,” vol. 9, Lecture Notes in Physics, (World Scientiﬁc, 1987). ISBN 9971501155. URL http://books.google.com/books?id=ZIF9QgAACAAJ.

[221] F. Denef, “TASI lectures on complex structures,” arXiv:1104.0254 [hep-th].

[222] L. Leuzzi and T. M.Vanieuwenhuizen, “Thermodynamics of the glassy state,” Taylor and Francis Group (2008) ISBN 9780750309974

[223] A. Strominger, “Black hole entropy from near horizon microstates,” JHEP 9802, 009 (1998) [hep-th/9712251].

[224] S. Krivonos, A. Nersessian and V. Ohanyan, Phys. Rev. D 75, 085002 (2007) [hep-th/0611268].

[225] S. Bellucci and V. Ohanyan, Phys. Lett. A 372, 5765 (2008) [arXiv:0802.1671 [hep-th]].

[226] S. Bellucci, S. Krivonos and V. Ohanyan, Phys. Rev. D 76, 105023 (2007) [arXiv:0706.1469 [hep-th]].

[227] H. Kim, J. Park, Z. Wang and P. Yi, “Ab Initio Wall-Crossing,” JHEP 1109, 079 (2011) [arXiv:1107.0723 [hep-th]].

[228] T. Dimofte and S. Gukov, “Reﬁned, Motivic, and Quantum,” Lett. Math. Phys. 91, 1 (2010) [arXiv:0904.1420 [hep-th]]. BIBLIOGRAPHY 192

[229] E. Andriyash, F. Denef, D. L. Jaﬀeris and G. W. Moore, “Wall-crossing from supersymmetric galaxies,” JHEP 1201, 115 (2012) [arXiv:1008.0030 [hep-th]].

[230] J. Manschot, B. Pioline and A. Sen, “Wall Crossing from Boltzmann Black Hole Halos,” JHEP 1107, 059 (2011) [arXiv:1011.1258 [hep-th]].

[231] S. G. Avery and J. Michelson, “Mechanics and Quantum Supermechanics of a Monopole Probe Including a Coulomb Potential,” Phys. Rev. D 77, 085001 (2008) [arXiv:0712.0341 [hep-th]].

[232] L. N. Hand and J. Finch, “Analytical Mechanics,” Cambridge University Press (1998) ISBN 0521573270

[233] V. Arnold, “Mathematical Methods of Classical Mechanics,” Springer-Verlag (1989) ISBN 0387968903

[234] E. Weeks, J.C. Crocker, A.C. Levitt, A. Schoﬁeld and D.A. Weitz, “Three- Dimensional Direct Imaging of Structural Relaxation Near the Colloidal Glass Transition,” Science 287, 627 (2000).

[235] E. Weeks and D.A. Weitz, “Properties of Cage Rearrangements Observed near the Colloidal Glass Transition,” Phys. Rev. Lett. 89, 095704 (2002).

[236] P. Charbonneau, A. Ikeda, G. Parisi and F. Zamponi, “Dimensional study of the caging order parameter at the glass transition,” Proceedings of the National Academy of Sciences of the United States of America 109, 15 (2012).

[237] D. -E. Diaconescu and R. Entin, “A Nonrenormalization theorem for the d = 1, N=8 vector multiplet,” Phys. Rev. D 56, 8045 (1997) [hep-th/9706059].

[238] E. D’Hoker and L. Vinet, “Spectrum (super)symmetries Of Particles In A Coulomb Potential,” Nucl. Phys. B 260, 79 (1985).

[239] E. D’Hoker and L. Vinet, “Hidden Symmetries And Accidental Degeneracy For A Spin 1/2 Particle In The Field Of A Dyon,” Lett. Math. Phys. 12, 71 (1986). BIBLIOGRAPHY 193

[240] E. D’Hoker and L. Vinet, “Constants Of Motion For A Spin 1/2 Particle In The Field Of A Dyon,” Phys. Rev. Lett. 55, 1043 (1985).

[241] L. Feher, P. A. Horvathy and L. O’Raifeartaigh, “Separating The Dyon Sys- tem,” Phys. Rev. D 40, 666 (1989).

[242] L. G. Feher and P. A. Horvathy, “Non-relativistic scattering of a spin-1/2 particle oﬀ a self-dual monopole,” Mod. Phys. Lett. A 3, 1451 (1988) [arXiv:0903.0249 [hep-th]].

[243] F. Bloore and P. A. Horvathy, “Helicity supersymmetry of dyons,” J. Math. Phys. 33, 1869 (1992) [hep-th/0512144].

[244] R. A. Coles and G. Papadopoulos, “The Geometry of the one-dimensional supersymmetric nonlinear sigma models,” Class. Quant. Grav. 7, 427 (1990).

[245] P. A. Horvathy, “The Biedenharn approach to relativistic Coulomb-type problems,” Rev. Math. Phys. 18, 311 (2006) [hep-th/0601123].

[246] E. Ivanov, S. Krivonos and O. Lechtenfeld, “New variant of N=4 superconformal mechanics,” JHEP 0303, 014 (2003) [hep-th/0212303].

[247] H. V. Mcintosh and A. Cisneros, “Degeneracy in the presence of a magnetic monopole,” J. Math. Phys. 11, 896 (1970).

[248] http://en.wikipedia.org/wiki/Superintegrable Hamiltonian system; http://en.wikipedia.org/wiki/Integrable system

[249] R. Alicki, J. R. Klauder and J. Lewandowski, “Landau-level ground state degeneracy and its relevance for a general quantization procedure,” Phys. Rev. A, vol. 48, Num. 4 (1993)

[250] H. Waalkens, H. R. Dullin and P. Richter, “The Problem of Two Fixed Centers: Bifurcations, Actions, Monodromy,” Physica D: Non-linear Phenomena, Vol. 196, Issue 3-4 (2004) BIBLIOGRAPHY 194

[251] L. Euler, Nov. Comm. Acad. Imp. Petropolitanae, 10, pp. 207-242, 11, pp. 152-184; M´emoiresde l’Acad. de Berlin, 11, 228-249.

[252] G. Marsaglia, “Choosing a point from the surface of a sphere,” The Annals of Mathematical Statistics, Vol. 43, No. 2, 645-646 (1972)

[253] E. W. Weisstein, “Sphere Point Picking,” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/SpherePointPicking.html

[254] F. Barthe, O. Guedon, S. Mendelson and A. Naor, “A Probabilistic Approach n to the Geometry of the `p -Ball,” Ann. Probab. 33, 480-513 (2005)

[255] E. W. Weisstein, “Ball Point Picking,” From MathWorld–A Wolfram Web Re- source. http://mathworld.wolfram.com/BallPointPicking.html

[256] E. W. Weisstein, “Disk Point Picking,” From MathWorld–A Wolfram Web Re- source. http://mathworld.wolfram.com/DiskPointPicking.html

[257] A. M. Childs, R. Cleve, E. Deotto, E. Farhi, S. Gutmann and D. A. Spielman, “Exponential algorithmic speedup by quantum walk,” arXiv:quant-ph/0209131

[258] D. Anninos, T. Anous, F. Denef and L. Peeters, “AdS Hot Halos,” to appear

[259] J. Louko, D. Marolf and S. F. Ross, “On geodesic propagators and black hole holography,” Phys. Rev. D 62, 044041 (2000) [hep-th/0002111].

[260] L. Fidkowski, V. Hubeny, M. Kleban and S. Shenker, “The Black hole singularity in AdS / CFT,” JHEP 0402, 014 (2004) [hep-th/0306170].

[261] G. Festuccia and H. Liu, “Excursions beyond the horizon: Black hole singularities in Yang-Mills theories. I.,” JHEP 0604, 044 (2006) [hep-th/0506202].

[262] P. Kraus, H. Ooguri and S. Shenker, “Inside the horizon with AdS / CFT,” Phys. Rev. D 67, 124022 (2003) [hep-th/0212277].

[263] J. de Boer, F. Denef, S. El-Showk, I. Messamah and D. Van den Bleeken, “Black hole bound states in AdS(3) x S**2,” JHEP 0811, 050 (2008) [arXiv:0802.2257 [hep-th]]. BIBLIOGRAPHY 195

[264] E. J. Heller, Phys. Rev. Lett. 16, 1515 (1984)