April 2, 2015 PHY2406S String Theory (2014-15) Department of Physics, University of Toronto Instructor: Prof

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April 2, 2015 PHY2406S String Theory (2014-15) Department of Physics, University of Toronto Instructor: Prof. Amanda W. Peet Notes typed with help from Ian T. Jardine N.B.: material draws heavily on BLT, BBS, Pol. texts.

SYLLABUS

Jan.09 Motivations for string theory. Running of couplings. Anomaly cancellation and gauge- gravity unification. Relativistic point particle theory. Jan.16 Worldlines vs worldsheets. Symmetries in various D and the role of BPS states. Clas- sical bosonic string theory. Nambu-Goto and Polyakov worldsheet actions. Conformal reparametrizations, Weyl invariance, and conformal gauge. Jan.23 Closed bosonic string oscillator expansions and the Virasoro algebra. Open bosonic string oscillator expansions, Virasoro, and D-branes. Quantum bosonic string theory. Canonical quantization and negative norm states. Jan.30 Critical dimension and mass formula for bosonic strings. Light cone quantization. Path integral quantization and Fadeev-Popov (b, c) ghosts. Feb.06 CFT basics. Radial quantization. The operator product expansion. Action of Virasoro generators on states: highest weights and descendants. Correlation functions. Feb.13 Properties of (b, c) ghosts. BRST invariance and unitarity. Applying BRST to the bosonic string. The Virasoro-Shapiro amplitude. Feb.23 Spinors in diverse dimensions. Green-Schwarz formalism. Neveu-Schwarz-Ramond formalism. ZPEs, physical state conditions and the GSO projection. SCFT, (β, γ) ghosts, and BRST. Feb.27 KK reduction in QFT, T-duality for closed bosonic strings, T-duality for closed open strings, D-branes, unit conventions and dimensional reduction, spacetime fields exerted by BPS object in string/M theory, superstring dualities and the deerskin diagram. Mar.06 The σ model and its beta-functions as the equations of motion for spacetime fields. Mar.20 Compactification (eg Calabi-Yau manifolds, orientifold planes, orbifolds); inflation, the string theory Landscape and the field theory Swampland. Mar.27 AdS/CFT: holography, and applications to condensed matter and the quark-gluon plasma. Apr.03 Black hole entropy and the information paradox, black rings, fuzzballs, and firewalls. Contents

1 Motivations for string theory 4 1.1 Running of couplings ...... 4 1.2 Anomaly cancellation and gauge/gravity uniﬁcation ...... 5 1.3 Relativistic point particle theory ...... 6 1.4 Worldlines vs worldsheets ...... 8 1.5 Symmetries in various D and the role of BPS states ...... 9

2 Classical Bosonic String Theory 11 2.1 Conventions ...... 11 2.2 Nambu-Goto and Polyakov actions ...... 12 2.3 Conformal reparametrizations, Weyl invariance, and conformal gauge . . . . 14 2.4 Closed string oscillator expansions and the Virasoro algebra ...... 16 2.5 Open string oscillator expansions, Virasoro, and D-branes ...... 18

3 Quantum Bosonic String Theory 21 3.1 Canonical quantization and negative norm states ...... 21 3.2 Light cone quantization ...... 26 3.3 Path integral quantization and Fadeev-Popov ghosts ...... 27

4 Conformal Field Theory 32 4.1 Radial quantization ...... 32 4.2 The operator product expansion ...... 37 4.3 Action of Virasoro generators on states: highest weights and descendants . . 38 4.4 Correlation functions ...... 41

5 Ghosts, BRST invariance, and scattering amplitudes 42 5.1 Properties of (b,c) ghosts and the critical dimension ...... 42 5.2 BRST invariance and unitarity ...... 44 5.3 Applying BRST to the bosonic string ...... 45 5.4 The Virasoro-Shapiro amplitude ...... 50

6 Superstrings 53 6.1 Spinors in diverse dimensions ...... 53 6.2 Green-Schwarz formalism ...... 55 6.3 Neveu-Schwarz-Ramond formalism ...... 56 6.4 ZPEs, physical state conditions and the GSO projection ...... 57 6.5 SCFT, (β, γ) ghosts, and BRST ...... 58

7 T-duality, D-branes and superstring duality 64 7.1 KK reduction in QFT ...... 64 7.2 T-duality for closed bosonic strings ...... 65 7.3 T-duality for open bosonic strings ...... 67 7.4 Unit conventions and dimensional reduction ...... 69 7.5 Spacetime ﬁelds exerted by BPS objects in string/M theory ...... 72

1 7.6 Superstring dualities and the deerskin diagram ...... 73

8 Sigma Models and String Theory 75 8.1 Sigma Model couplings ...... 76 8.2 Background ﬁeld method and Riemann normal coordinates ...... 77 8.3 Graviton beta function ...... 79 8.4 Kalb-Ramond ﬁeld beta function ...... 81 8.5 Dilaton beta function ...... 82 8.6 Weyl anomaly at one loop ...... 82 8.7 Nailing the dilaton ...... 87

9 Compactification 93 9.1 Hiding extra dimensions ...... 93 9.2 SUSY ...... 93 9.3 Brane world models ...... 94 9.4 Heterotic string theory on CY3s...... 94 9.5 Calabi-Yaus ...... 95 9.6 Geometry ...... 95 9.7 Hodge numbers ...... 96 9.8 Mirror symmetry and the conifold ...... 96 9.9 Dp-brane probes ...... 97 9.10 Example: 1 Dp probing N Dps...... 97 9.11 Forces between D-branes ...... 98 9.12 Open string BCFT ...... 98 9.13 What is an orbifold? ...... 99 9.14 Simple compact and non-compact orbifolds ...... 99 9.15 Spectra of states for orbifolds ...... 100 9.16 What is an orientifold? ...... 100 9.17 What are Op-planes good for? ...... 100 9.18 Flux compactifications ...... 101 9.19 Flux-ology ...... 101 9.20 Inflation in string theory ...... 102 9.21 KKLT and the Landscape ...... 102 9.22 Ur potential ...... 103

10 AdS/CFT 103 10.1 Origin of AdS/CFT ...... 103 10.2 Maldacena’s decoupling limit ...... 104 10.3 AdS=CFT ...... 104 10.4 Probes and IR/UV relations ...... 105 10.5 IMSY and applications of AdS/CFT ...... 105 10.6 Holographic dictionary ...... 106 10.7 Feﬀerman-Graham ...... 106 10.8 BF bound and GKP/W ...... 107 10.9 Holography and black holes ...... 107

2 10.10Maldacena’s eternal AdS BH ...... 108 10.11Less symmetric holography ...... 108 10.12Higher-spin/vector holography ...... 109 10.13Bulk locality ...... 109 10.14Geometrization of entanglement ...... 110

11 Black hole entropy and the information paradox 110 11.1 Cooking up D1 k D5 ...... 110 11.2 Cooking up W k D5 and W k D1 ...... 111 11.3 Problems with too few ingredients ...... 111 11.4 Why we started with D = 5 BH ...... 112 11.5 The harmonic function rule ...... 112 11.6 Cooking the D1-D5 system ...... 112 11.7 Adding the gravitational wave ...... 113 11.8 The D1-D5-W metric in 5D ...... 113 11.9 Bekenstein-Hawking entropy ...... 114 11.10Properties of SBH ...... 114 11.11Yes, extreme BH can have ﬁnite SBH ...... 115 11.12Four charges in four dimensions ...... 115 11.13The D2-D6-W-NS5 duality frame ...... 116 11.14SBH for the 4D 4-charge BH ...... 116 11.15The D-brane picture ...... 117 11.16Open string dynamics ...... 117 11.17String partition function ...... 118 11.18Degeneracy of states ...... 118 11.19Adding rotation [BMPV] ...... 118 11.20Rotating entropy agreement ...... 119 11.21d = 4 entropy counting ...... 119 11.22d = 4 microscopic entropy and non-extremality ...... 120 11.23Nonextremal entropy and greybody factors ...... 120 11.24String theory, D-branes, and SBH ...... 121 11.25Emission rates and the fuzzball programme ...... 121 11.26D1-D5 CFT ...... 122 11.27Firewalls ...... 122 11.28Avoiding ﬁrewalls ...... 123 11.29‘ER=EPR’ ...... 123

3 1 Motivations for string theory

1.1 Running of couplings As you may already know, coupling “constants” in Nature such as the fine structure constant 2 αEM = e /(4π~c) of electromagnetism are not actually constant. They run logarithmically with energy in four spacetime dimensions (D = 4). You will learn the details of why this happens by the time you have finished QFT II. For now, all you need is the concept that running of couplings is caused by the quantum phenomenon of vacuum polarization. For QED, electric charge is screened by polarization of all the virtual particle-antiparticle pairs that are constantly popping in and out of the QED vacuum. The result of screening is that you see more charge the deeper you dig: the coupling of electromagnetism grows with energy. Note that this is a wholly quantum mechanical effect. Unfortunately, the growth of αEM with energy makes perturbation theory less reliable. QCD behaves in a more physically sensible way: its SU(3) colour coupling becomes weaker at high energy. In order to do this, it has to anti-screen. The flip side of QCD weakening at high energy is confinement of colour at low-energy. If you can explain exactly how this happens, the Clay Mathematics Institute will dish out a million dollar prize! The story of the electroweak force, with hypercharge U(1), isospin SU(2), W ±,Z and Higgs bosons, will be described soon in your QFT II course in the section on spontaneous symmetry breaking.

Throughout this course we will work in particle physicist units where ~ = c = kB = 1. We will not set the Newton constant (or any other physical constant) to unity – this would not make sense from the perspective of dimensional analysis in quantum physics. Indeed, we can build a constant with dimensions of length out of GN , c and ~,

G 1/(D−2) ` ∼ N ~ (1.1) P,D c3

4 You can work out this formula by starting from the Einstein-Hilbert action principle for GR and remembering that the Ricci scalar has two derivatives in it. The Planck length is of the order of 10−33cm in D = 4. Converting this into a mass scale gives about 10−5g or 1019GeV. Einstein’s General Relativity is well-tested experimentally on distance scales from the microscopic to the cosmic. It does however have a big Achilles heel: it predicts the seeds of its own destruction. To be precise, singularity theorems of GR based on the Raychaudhuri equation for geodesic deviation predict that the Einstein equations will develop singularities, loci at which GR has nothing to say about the physics except that every local observable including spacetime curvature is infinite. The two most common types of singularity in GR that you will have already heard of are (a) the cosmological singularity at the Big Bang and (b) the singularity at the heart of every black hole. The Cosmic Censorship conjecture that proposed every singularity to be hidden behind an event horizon turned out to be incorrect, so we know that GR has inherent limitations even as a classical theory. The sickness of GR is actually inherent at the quantum level. It is very easy to see why. The only dimensionless coupling out of which we can build scattering amplitudes scales like a D−2 positive power of energy: αG ∼ GN E . This breaches unity at the Planck scale. What this implies in detail is something very serious: GR predicts probabilities for graviton scattering in excess of 100% at high energies. String theory is the only extant formulation of quantum gravity that insists from the outset that the theory obeys the correspondence principle. There is a Newtonian limit and it is provable without any handwaving. Other approaches simply assume that their nonperturbative formulation of “quantum gravity” somehow finagles this question, and try to justify it post hoc, without providing a calculable mechanism. These approaches may well end up defining quantum theories of gravity, but it is not clear that any of them has a classical limit that is our gravity.

1.2 Anomaly cancellation and gauge/gravity unification String theory actually originated as a theory of the strong interactions, in the late 1960s. The rough idea of Susskind and others was to model two oppositely charged quarks within a meson connected via a flux tube by a relativistic string with endpoints, which is known as an open string.A closed string is a one-dimensional string without endpoints. The key fact we will derive about open strings is that they possess spin-one massless gauge fields in the spectrum. Similarly, the key fact about closed strings is that they possess massless spin-two particles in the spectrum, which interact like the graviton. This is why string theory has a chance to provide a unified theory of electromagnetic, nuclear, and gravitational interactions – a dream that eluded even Albert Einstein. Unfortunately, quantum relativistic open bosonic string theory turned out to have a tachyon in the spectrum, which quickly ruled it out as a physically realistic model of the strong interaction. Bosonic closed string theory also has a tachyon, with a more negative mass-squared than the bosonic open string. But if the ingredient of supersymmetry (SUSY) is also added to the mix, fully tachyon-free quantum interacting superstring theories can be found. Supersymmetry pairs bosonic and fermionic modes, and it is the only nontrivial extension of Poincaré(translation, rotation and Lorentz boost) symmetry in flat Minkowski spacetime. Internal gauge symmetry is also possible; by definition, this does

5 not mix at all with spacetime symmetries. String theory was later developed into a unified “theory of everything” encompass- ing all known forces and matter. This was predicated on a crucial discovery in 1984 by Michael B. Green and John H. Schwarz that quantum anomalies can be cancelled for five different tachyon-free superstring theories in ten spacetime dimensions. These five theories became known as Type I, Type IIA, Type IIB, Heterotic SO(32) and Heterotic E8 ×E8. You will meet anomalies later in your QFT II class; for now, you can think of them as quantum vi- olations of classical symmetries of the action. As you will see in QFT II, anomaly-producing Feynman diagrams can arise in D = 4 with three external gauge legs. In higher even dimensions, anomalies arise for Feynman diagrams involving a greater number of external gauge and/or graviton legs. In particular, in D = 10, hexagon anomaly diagrams arise, which must cancel independently for gauge, gravitational, and mixed anomalies. These requirements are sufficiently strict to permit very few solutions. The Type I and Type IIA/IIB theories are pure open and closed superstring theories in D = 10 respectively. The IIA theory is nonchiral (its two supersymmetries have opposite chirality) while the IIB theory is chiral (its two supersymmetries have the same chirality). The heterotic1 string theories are actually heteroses of bosonic strings and superstrings. They hybridize a bosonic string theory for left-handed string modes, which lives naturally in D = 26 spacetime dimensions, with a fermionic string theory for right-handed modes, which lives naturally in D = 10. The 16 leftover left-handed modes are recombined into a current algebra representing a spacetime gauge symmetry. Anomaly cancellation constraints for both gauge and gravitational interactions then require that the rank of the gauge group for heterotic strings be 496. There are only two (Lie) symmetry groups with this property: SO(32) and E8 × E8. Superstring theories all include the graviton. In the case of open string theory the graviton arises as a mode propagating in internal Feynman diagram loops. For the closed string theories it arises directly as the lightest mode in the spectrum. The most important thing about the graviton mode coming out of string theory is that it not only has the right mass and spin, but it also interacts like a graviton. This means that we can honestly call string theory a consistent quantum theory of gravity. But of course it has much more than just the massless graviton – the overtones (known as oscillators) change the physics at short distance in very important ways from Einstein’s low-energy effective field theory. The upshot is that string theory provides a way of unifying gauge and gravitational interactions together in a fashion that is consistent with quantum mechanics.

1.3 Relativistic point particle theory For non-relativistic point particles, you know how the drill goes. The basic dynamical variables are xi(t), where i = 1, 2,...D − 1. There is no issue about how to parametrize t, as all observers agree on time because of Galilean relativity. But for relativistic point particles we cannot just use the Newtonian kinetic energy: we have to use something that respects

1This word arises from the Greek ‘heterosis’. It is not the antonym of ‘homoerotic’!

6 Einsteinian relativity. The usual action that people write is proportional to the arc length, r Z dxµ dx S0 = −m dτ − µ , (1.2) particle dτ dτ where τ is the proper time. This action has the beneﬁt that, at low speeds, it reduces to the familiar non-relativistic action (up to an additive constant). The drawback of this action is twofold: the xµ(τ) are not independent functions, and the particle is assumed to be massive so that proper time can be used to parametrize the worldline. First, the dynamical variables must obey a mass shell constraint,

µ 2 2 2 p pµ = −E + |~p| = −m . (1.3) Second, suppose that we use this proper time story to try to ﬁnd the canonical momenta µ and Hamiltonian. We have pµ = mdx /dτ. But this means that the Hamiltonian H(τ) = µ 2 (pµp + m )/m which is zero by dint of the constraint equation. The fact that we have obtained a zero Hamiltonian is caused by the fact that we forgot to take account of a symmetry. Let us be more sophosticated about writing down our Lagrangian. Impose the mass shell constraint via a Lagrange multiplier e(λ) known as the einbein, where λ is an arbitrary worldline parameter, Z 1 dxµ dx 1 Se = dλ e−1(λ) µ − e(λ)m2 . (1.4) particle 2 dτ dτ 2 Note that this einbein action is invariant under

λ → λ0 , dλ e → e0 = e . (1.5) dλ0 Varying w.r.t. e(λ) gives the constraint as r ±1 dxµ(λ) dx (λ) e(λ) = µ , (1.6) m dτ dτ and the proper time gauge is one in which e(λ) = 1/m. Varying w.r.t. xµ(λ) gives the equations of motion d dxµ(λ) e−1(λ) = 0 , (1.7) dλ dλ which are equally valid for massive or massless particles. The canonical momenta are 1 dx (λ) p = µ (1.8) µ e(λ) dλ and so the Hamiltonian is 1 H = e pµp + m2 . (1.9) 2 µ µ µ µ ν This gives the correct Poisson Brackets: {x , pν}PB = δν , {x , x }PB = 0, and {pµ, pν}PB = 0.

7 1.4 Worldlines vs worldsheets What does string theory do differently? It assumes that the fundamental LEGO blocks of the universe are not zero-dimensional point particles but instead one-dimensional strings. This means that the path through spacetime is not a one-dimensional worldline described by xµ(λ), where λ is a parameter commonly chosen to be proper time for a massless particle or affine parameter for a massless particle. It is instead described by a worldsheet, which is the two-dimensional surface swept out in spacetime by a moving open or closed string. Topologically speaking, the worldsheet associated to an open string propagator would be equivalent to a finite-width strip while that for a closed string propagator would be equivalent to a finite-width cylinder. To describe string dynamics, we need a map from the worldsheet, parametrized by 2D coordinates (τ, σ), to spacetime: Xµ(τ, σ). We also need an action principle, which we will choose by analogy with the relativistic point particle to be the area swept out by the worldsheet. We will have much more to say about this story shortly. For now, we need to know the one big feature of the action principle for the Xµ(τ, σ) of string theory that makes life difficult: it possesses full 2D reparametrization invariance. The worldsheet action is fully invariant under coordinate transformations. Particle theory has a small echo of this symmetry, but it is far less powerful because it is only in two spacetime dimensions that the conformal symmetry group is infinite-dimensional. 2D conformal symmetry will give us a great deal of theoretical control over analyzing the quantum physics of strings. But we do have to learn how to write down a measure for the path integral of quantum string theory that does not double-count field configurations that are related to one another by worldsheet reparametrization. This involves a story known as Fadeev-Popov ghosts, which you can think of as a technical bookkeeping device for keeping track of reparametrization symmetry. The string is fundamental, which means that there is no medium doing any wiggling. Accordingly, there are no longitudinal oscillations of strings. There are (D − 2) physical oscillator modes in D spacetime dimensions. Now suppose that we want to make a superstring. Quite generally, in D dimensions, spinors do not have the same dimensionality as scalar fields like Xµ(τ, σ). For instance, the dimensionality of a Weyl spinor in D dimensions is 2[D/2]; note that this grows much faster than D. To match worldsheet bosonic and fermionic degrees of freedom requires fortunate accidents of available spinor representations, and this can only be done classically for string theory in D = 3, 4, 6, 10. (It is no accident that D − 2 for these cases is a power of 2.) As we will see later in this course, the quantum superstring is only consistent in D = 10, where there are eight real transverse Xi modes and eight real fermionic ψi modes to match them. There is no hand-waving way to derive the critical dimension – believe me, I have looked – we have to go through the full pain of quantizing superstrings to understand where it comes from. But I can give you a part of the answer right now. You might wonder why the groundstate of open and closed strings have spin. This is related to the existence of open and closed string tachyons, which are spinless and have negative m2. Indeed, the mass formula for an open string will turn out to be, schematically,

mc2 2 = (N − 1) . (1.10) T open

8 Here, N is the oscillator energy measured in units of the fundamental mode; it is a non- negative integer. The oscillation patterns are basically standing waves. For the closed string, we get travelling waves. Schematically,

mc2 2 = 2(NR + NL − 2) . (1.11) T closed where NL,R are non-negative integers representing the oscillator energies for the independent left- and right-handed modes in units of the fundamental. For momentum balance, we need NL = NR. Where do these mysterious mass deficits (the -1 and the -2) come from? The answer lies in the Casimir Effect. Quantum fields are like messy teenagers at heart: they prefer to fluctuate over all space. If you then exclude some part of space, you pay the price of an energy deficit, which is proportional to the volume of excluded space. The technical details for the strip diagram relevant to the open string give rise to a mass-squared deficit of -1, in units of the string tension 1 1 T ≡ ≡ . (1.12) 0 2 2πα 2π`s For the cylinder diagram relevant to the closed string, the same techniques gives a mass- squared deficit of -2. So this is why we get a gauge boson: we need one oscillator on the open string in order to make a massless mode. That oscillator points in a particular direction, and has (D − 2) components, indicating a vector. Similarly, we need two oscillators on the closed string (one on the left and one on the right) in order to make a massless mode that we can identify as a graviton. To be picky: the transverse traceless tensor part is identified with the graviton, the antisymmetric tensor part as the Kalb-Ramond field B, and the trace part as the dilaton field Φ.

1.5 Symmetries in various D and the role of BPS states Symmetry has proven to be an extremely powerful way of organizing our physics thoughts about Nature. We are already familiar with 10 spacetime symmetries from study of relativity: 4 spacetime translations, 3 spatial rotations, and 3 Lorentz boosts. Conservation laws associated to them ensure both linear and angular momentum conservation as well as correct CoM motion. The counting goes similarly in other spacetime dimensions, except that in D dimensions there are D translations, d=D−1 boosts, and d(d − 1)/2 rotations. (Note: the number of independent planes of rotation is [d/2] where [ ] denotes the integer part, e.g. the (x, y) and (z, w) planes in d=4 define two independent angular momenta.) Other symmetries, such as the U(1) gauge symmetry of electromagnetism, act on the fields directly. Field space, as distinct from spacetime, is usually referred to as the “internal space” for the field. The charge carried by a field can be thought of as like a handle pointing in field space, onto which a gauge boson can grab. Gauge fields can be in three distinct phases of physical behaviour. The most familiar is the “Coulomb phase”, resulting in an inverse-square law in D=4 as per intuition. Alterna- tively, like in QCD, the gauge field can be in a “confined phase”. The third possibility is a “Higgs phase” with spontaneous symmetry breaking.

9 Under a symmetry transformation, a field φa transforms as ∆φa = φa 0(x0) − φa(x) = φa 0(x0) + [−φa 0(x) + φa 0(x)] − φa(x) = [φa 0(x0) − φa 0(x)] + [φa 0(x) − φa(x)] a µ a = (∂µφ )δx + δφ (x) . (1.13) Notice that there is a transport term and a straight functional variation piece. Noether’s Theorem says that for every continuous symmetry there is a conserved current a ν µ ∂L ∆φ µ ∆x JA = a A − T ν A (1.14) ∂∂µφ ∆ω ∆ω where ∆φa = (∆φa/∆ωA)∆ωA and similarly for xν. The important thing about the infinitesimal parameters ∆ωA is that they are additive. If a field φ carries a representation R of a group U, it transforms as

0 A φ = Uφ = exp −i∆ω TA φ (1.15) where the TA are the inﬁnitesimal generators. The generators of the Lie algebra obey

C [TA,TB] = ifAB TC , (1.16) and the Jacobi identity

[TA, [TB,TC ]] + [TB, [TC ,TA]] + [TC , [TA,TB]] = 0 . (1.17) Poincaré symmetry transformations consist of translations, rotations, and Lorentz boosts. In D spacetime dimensions, the group is known as SO(D − 1, 1). When D=4, this can be made to look like a compact Lie group by Wick rotating to produce SO(4), which happens to be isomorphic as a group to SU(2) × SU(2). Now, we already know how to do SU(2) physics - the three generators are very familiar to us from Quantum Mechanics: they are the Ji from ladder operator fame. For more on the story of the representation theory of Poincaré,including why spin is an angular momentum for massive particles while massless particles are characterized by their helicity, pay attention in QFTII class PHY2404S. The generators of translations are known as the momentum generators Pµ. Note that in general spacetime dimension D, the [orbital] angular momentum is not a pseudovector but instead the space-space components of a two-index antisymmetric tensor. (Each rotation involves one plane perpendicular to its axis.) If we use a relativistic two-index antisymmetric tensor, we can actually pack both the rotation generators in (space-space components) and also the boost generators (time-space components). The orbital angular momentum is just what you might expect: Lµν = XµPν − XνPµ; the total angular momentum is Mµν = Lµν + Σµν, where Σµν are generators of spin angular momentum transformations which act nontrivially on any field with spin half or greater. The commutation relations for the Poincarégenerators then become

[Pµ,Pν] = 0 [Pµ,Mρσ] = +i (ηµρPσ − ηµσPρ) (1.18) [Mµν,Mρσ] = +i (ηνρMµσ − ηµρMνσ + ηµσMνρ − ηνσMµρ)

10 The supersymmetry algebra for point particles is described by extending the regular Poincar´egenerators, translations Pµ and the antisymmetric rotation/boost generators Mµν, A by SUSY generators Qα obeying

A B AB µ AB {Qα ,Qβ } = −2δ ΓαβPµ − 2iZ δαβ , (1.19) where all other commutators are zero and we have used the conventions of Polchinski2. Z is known as the central charge. There are also some commutators of Poincarégenerators A with the Qα which encode the fact that it transforms as a spinor. To show the power of the SUSY algebra, let us work in the rest frame of a massive particle. Suppose further for simplicity that we are in D = 0 + 1, i.e. doing QM, so that spinors have only one component (this is not an essential restriction). Then for two supersymmetries, the † AB AB simplest case of extended SUSY, the algebra says that {QA,QB} = −2δ M − 2iZ . We 1 2 † can easily simplify this by shifting basis to Q± := (Q ± iQ )/2; then we have {Q±,Q±} = † −M ∓ Z while {Q±,Q∓} = 0. Now comes the really cool part – we sandwich a physical † state |ψi around {Q±,Q±} = −M ∓ Z. Since the SUSY generators are physical operators, Q±|ψi is also a physical state, and it must have non-negative norm. This directly implies that M ≥ |Z| , (1.20) which is known as the Bogomolnyi-Prasad-Sommerfeld or BPS bound. It is extremely important and powerful, not least because it provides a lower bound on the mass of every state in the quantum theory. Also, the only way that the BPS bound can be saturated is for the physical state to possess unbroken supersymmetry, i.e., be annihilated by one or more SUSY generators. States with unbroken SUSY are known as BPS states. The best part about BPS states is that their mass-to-charge ratio can be followed reliably away from the perturbative regime. Quantum corrections to the mass and to the charge do typically occur, but the BPS bound derived directly from the SUSY algebra ensures that the ratio M/|Z| is protected from quantum corrections. This story is part of what helped the second superstring revolution to occur. One of the reasons string theory is so well constrained as a quantum theory in the ultraviolet is that it possesses an infinite dimensional symmetry algebra on the worldsheet. Handling the gauge symmetry associated to this is technically a bit more challenging than for spin-one U(1) gauge field theory that you studied in PHY2403F. This is why we will be introducing a bunch of new technologies. :D

2 Classical Bosonic String Theory

2.1 Conventions We will use the conventions of Polchinski’s “String Theory” textbooks. Mostly we will focus on Volume 1, but we will also make use of some Volume 2 material. In Lorentzian signature,

2We will use Polchinski’s conventions for units etc. throughout these lecture notes. For the SUSY algebra in D dimensions, see Appendix B of the second volume.

11 we use the {−, +,..., +} convention. Light-cone coordinates are deﬁned as 1 x± = √ x0 ± x1 . (2.1) 2 In Euclidean space with coordinates σ1, σ2, we can deﬁne complex coordinates

z = σ1 + iσ2 , z¯ = σ1 − iσ2 , (2.2) so that 1 1 ∂ = (∂ − i∂ ) , ∂ = (∂ + i∂ ) . (2.3) z 2 1 2 z¯ 2 1 2 zz¯ Then gzz¯ = 1/2, g = 2, with all other components zero, which implies

d2z ≡ 2dσ1dσ2 , (2.4) and 1 δ2(z, z¯) ≡ δ(σ1)δ(σ2) . (2.5) 2 Stokes’ Theorem is simple in 2D, Z I 2 z z¯ z z¯ d z (∂zv + ∂z¯v ) = (v dz¯ − v dz) (2.6) R ∂R Traditionally, the left-moving sector is called holomorphic and the right-moving sector is called anti-holomorphic. String theorists also tend to use a diﬀerent convention for the deﬁnition of the stress-energy tensor, with an extra factor of −2π, −4π δ T ab = √ S (2.7) −γ δγab

2.2 Nambu-Goto and Polyakov actions Previously we saw that there were two classical actions for (massive) point particles, the ’square-rooty’ action based on the geometric arc length and the quadratic einbein action which is easier to quantize. For classical strings, there are direct analogues known as the Nambu-Goto and Polyakov actions. The dynamical variables Xµ(τ, σ) are maps from the worldsheet to target space, i.e., ﬁelds living on the worldsheet. The Nambu-Goto action is simple: it measures the worldsheet area swept out by the string as it moves through spacetime,

1 Z p S = − dτdσ −h , (2.8) NG 2πα0 ab where3 µ ν hab = ∂aX ∂bX Gµν(X) (2.9)

3Note that this is not the Einstein tensor. It is the string frame metric tensor, which is distinct from the Einstein frame metric tensor.

12 is the induced metric on the worldsheet. It can also be written as 1 Z q S = dτdσ (X˙ · X˙ )(X0 · X0) − (X˙ · X0)2 (2.10) NG 2πα0 where ∂Xµ ∂Xµ X˙ µ = ,X0µ = . (2.11) ∂τ ∂σ The Polyakov action is given by 1 Z √ S = dτdσ −γγab∂ Xµ∂ XνG (X). (2.12) P 4πα0 a b µν where γab is a worldsheet metric. The bonus of the Polyakov action is that it is easier to quantize in Minkowski spacetime, since it is nice and quadratic in derivatives of the Xµ ﬁelds. Since the physics should be worldsheet reparameterization invariant, the worldsheet metric should not play any physical role, unlike the metric in the target space Gµν(X).As we will see in more detail later, this works out because gravity in 2D has no local degrees of freedom. The energy-momentum tensor deﬁned by −4π δS T ab = √ P . (2.13) −γ δγab gives, from the Polyakov action, −1 1 T = G (X) ∂ Xµ∂ Xν − γ ∂cXµ∂ Xν . (2.14) ab α0 µν a b 2 ab c

Varying the Polyakov action w.r.t. γab fixes it to be the induced metric hab, as you can check yourself. This then implies that the two action principles are classically equivalent. For the quantum theory, we will stick with the Polyakov action. Varying the Polyakov action w.r.t. Xµ gives the equation of motion √ ab µ ∂a −γγ ∂bX = 0 , (2.15) as long as we take care of the surface terms correctly. The variation of the Polyakov action obviously has two pieces: 1 Z +∞ Z ` √ 1 Z +∞ √ δS = dτ dσ −γδXµ∇2X − dτ − −γδXµ∂σX σ=` (2.16) P 0 µ 0 µ σ=0 2πα −∞ 0 2πα −∞ and these can be solved with two choices Xµ(τ, σ + `) = Xµ(τ, σ) (closed string) ∂L δXµ(σ = 0, `) = 0 (open string) (2.17) ∂Xµ0 These are the only options consistent with target space Poincaréinvariance. For the closed string, the function Xµ(τ, σ) has to be periodic in σ. For the open string we require Neumann boundary conditions, in order to respect target spacetime Poincaréinvariance. This ensures that the open string endpoints move freely and that momentum does not leak off. As we will find out in the later section on D-branes, there is another possibility to use Dirichlet boundary conditions for open strings, but this spontaneously breaks Poincaré,and typically involves momentum interchange between the open string and the D-brane.

13 2.3 Conformal reparametrizations, Weyl invariance, and conformal gauge The Polyakov action possesses two important classes of symmetries: bulk global symmetries and local worldsheet symmetries. The global symmetries are Poincar´e:-

µ µ µ µ δX = a νX + b , aµν = −aνµ , δhab = 0 . (2.18)

The local symmetries are twofold. First, there is Weyl invariance,

δXµ = 0 , δhab = 2Λhab . (2.19)

Second, there is full worldsheet reparametrization invariance

µ a µ δX = −ξ ∂aX , δh = − (∇ ξ + ∇ ξ ) , √ ab a b√ b a a δ −h = −∂a ξ −h . (2.20)

Weyl invariance implies that the stress-energy tensor is traceless

a Ta = 0 , (2.21) while Diﬀ invariance implies that is is conserved,

ab ∇aT = 0 . (2.22)

What about adding other terms to the worldsheet action? There are only two other terms possible that are consistent with the above symmetries and possessing at most two derivatives. They are Z √ 2 S1 = λ1 d σ −h (2.23) Σ which is diff invariant but not Weyl invariant. There is also the Gauss-Bonnet term Z √ λ2 2 (2) S2 = d σ −h R = λ2 χ(Σ) , (2.24) 4π Σ which turns out to not contribute at all to the equations of motion because it is topological. The integrand is locally a total derivative because we are in 2D. While the Gauss-Bonnet term does not affect the EOM, it does affect the weighting of different worldsheets in the Feynman path integral. In particular, we see a weighting factor of e−λ2χ(Σ). We may use a mathematical fact χ(Σ) = 2 − 2g , (2.25) where g is the genus, to see that worldsheets of genus g contribute with weighting e−2(1−g)λ2 . The lowest genus zero term is weighted by 1/(eλ2 )2, the genus one term by 1, the genus two

14 term by (eλ2 )2, etc. Accordingly, we can think about the genus as a loop counting parameter in worldsheet perturbation theory. In two dimensions, the structure of gravity is very rigid: metrics are conformally ﬂat. For suitable worldsheets, the metric can be gauged away completely by using (a) reparametriza- Φ tion invariance to set hab = e ηab and (b) Weyl invariance to set Φ = 0. See p.18 of Blumenhagen-L¨ust-Theisen(BLT) for details of how this is done. In conformal gauge, the Polyakov action reduces to Z (conf.gauge) 2 µ SP = 2T d σ∂+X ∂−Xµ (2.26) and the equations of motion simplify enormously to give

µ µ − µ + X (τ, σ) = XR(σ ) + XL(σ ) . (2.27) In other words, the ﬁelds Xµ are straightforward travelling waves classically, for the closed string. For the open string we get straightforward standing waves. Vanishing of Tab on a classical solution is required by the equation of motion for the worldsheet metric. Expressed in linear combinations, this says

2 X˙ ± X0 = 0 , (2.28) or, in light cone coordinates,

T±± = −2πT (∂±X · ∂±X) = 0 ,T±∓ = 0 . (2.29)

This implies that ∂∓T±± = 0 and therefore

± T±± = T±±(σ ) . (2.30)

This has huge implications – it means there exists an inﬁnite number of conserved charges, ± obtained by integrating T±± against an arbitrary function of σ , e.g.

Z ` + + Lf = 2T dσf(σ )T++(σ ) (2.31) 0 and similarly for the anti-holomorphic sector. The Hamiltonian in conformal gauge is then

Z ` 2 2 H = T (∂+X) + (∂−X) . (2.32) 0

µ ˙ ˙ µ The canonical momentum is Π = ∂L/∂Xµ = T X , and the Poisson Brackets are as you would expect,

µ ν 0 {X (τ, σ),X (τ, σ )}P.B. = 0 , ˙ µ ˙ ν 0 {X (τ, σ), X (τ, σ )}P.B. = 0 , 1 {Xµ(τ, σ), X˙ ν(τ, σ0)} = ηµνδ(σ − σ0) . (2.33) P.B. T 15 The charges Lf above generate reparametrizations that stay within conformal gauge,

+ {Lf ,X(σ)}P.B. = −f(σ )∂+X(σ) . (2.34) So far we have just talked about the influence of local diffeo symmetries on worldsheet physics for classical strings. There is also the question of bulk Poincaréinvariance to think about. In conformal gauge, Z ` Pµ = T dσ (∂τ Xµ(σ)) (2.35) 0 and Z ` Jµν = T dσ (Xµ∂τ Xν − Xν∂τ Xµ) (2.36) 0 and these are conserved for closed strings by virtue of periodicity. For open strings, only those with Neumann boundary conditions have these two tensors conserved. We will explore the physics of Dirichlet-branes later in the course – stay tuned! You can check that the correct algebra is obtained for Poincarésymmetry:-

µ ν {P ,P }P.B. = 0 , µ ρσ µσ ρ µρ σ {P ,J }P.B. = η P − η P , µν ρσ µρ νσ νσ µρ νρ µσ µσ νρ {J ,J }P.B. = η J + η J − η J − η J . (2.37)

2.4 Closed string oscillator expansions and the Virasoro algebra Classically, the solution of the 2D Klein-Gordon equation is a superposition of left- and right-moving travelling waves,

µ µ µ X (z, z¯) = XR(τ − σ) + XL(τ + σ) , (2.38) where r 1 πα0 α0 X 1 Xµ(τ − σ) = (xµ − cµ) + pµ(τ − σ) + i αµe−2πin(τ−σ)/` , R 2 ` 2 n n n=06 r 1 πα0 α0 X 1 Xµ(τ + σ) = (xµ + cµ) + pµ(τ + σ) + i α¯µe−2πin(τ+σ)/` . (2.39) L 2 ` 2 n n n=06

µ µ In these expressions, x is the centre of mass position. Note that the αn are positive-frequency modes for negative n and negative-frequency for positive n. Requiring that Xµ(τ, σ) be real gives µ ∗ µ (αn) = α−n . (2.40) Now deﬁne rα0 αµ =α ¯µ = pµ ; (2.41) 0 0 2 then r n=+∞ 2π α0 X ∂ Xµ = e2πin(τ±σ) . (2.42) ± ` 2 n=−∞

16 We can evaluate the momenta and angular momenta associated to this string conﬁguration,

( ∞ ) X 1 P µ = pµ ,J µν = (xµpν − xνpµ) + −i αµ αν + αν αµ + h.c. , (2.43) n −n n −n n n=1 which give the Poisson brackets you would expect

µν {αm, αn} = −imη δm+n,0 , µν {α¯m, α¯n} = −imη δm+n,0 , {αm, α¯n} = 0 , {xµ, pν} = ηµν . (2.44) and for the Hamiltonian we obtain

+∞ π X H = (α · α +α ¯ · α¯ ) . (2.45) ` −n n −n n n=−∞ For the closed string, a complete set of functions is provided by plane waves: n o ± 2πimσ±/` fm(σ ) = e (2.46)

We already saw before that there is an infinite of conserved charges. So let us define ` Z ` L := − dσe−2πinσ/`T , n 2 −− 4π 0 ` Z ` L¯ := − dσe+2πinσ/`T ; (2.47) n 2 ++ 4π 0 then 1 X L = α · α , n 2 n−m m m 1 X L¯ = α¯ · α¯ . (2.48) n 2 n−m m m Notice that we have deliberately ignored zero-point energies in evaluating these sums. This is of course only a valid approximation for classical strings. In following chapters, we will revisit the zero-point energies and sum them up correctly. Recalling the orthonormality condition δ(σ − σ0) = (1/`) P e2πin(σ−σ0)/`, we find n∈Z 2 2π X T = − L e2πinσ/` , ++ ` n n 2 2π X T = − L¯ e2πinσ/` . (2.49) −− ` n n The reality condition for Xµ demands that

∗ Ln = L−n ,

17 ¯ ¯∗ Ln = L−n . (2.50) A nice consequence of the above deﬁnitions is that the Hamiltonian in conformal gauge (which we have been using throughout this subsection) is very simple, 2π H = L + L¯ . (2.51) ` 0 0 Rigid σ translations are generated by the constraint Z ` 0 2π ¯ T dσX · X = L0 − L0 , (2.52) 0 ` and since no point on the string is special, for momentum balance we require ¯ L0 = L0 . (2.53)

The inﬁnite set {Ln} obeys a Virasoro algebra

{Lm,Ln}P.B. = −i(m − n)Lm+n , ¯ ¯ ¯ {Lm, Ln}P.B. = −i(m − n)Lm+n , (2.54) When we do quantum string theory, we will ﬁnd that this algebra is modiﬁed by a central extension term proportional to (D − 26) (for the bosonic string). This will be required to vanish in order to have conformal invariance as a quantum symmetry of the theory.

2.5 Open string oscillator expansions, Virasoro, and D-branes For open strings, it is necessary to carefully distinguish between Neumann (N) and Dirichlet (D) boundary conditions, because they lead to physically diﬀerent mode expansions. With any given map function Xµ(τ, σ), the open string involved has two endpoints, requiring us to distinguish four cases: NN, DD, ND, and DN. We now discuss these cases in turn. For NN strings, the solution to the wave equation becomes 2πα0 √ X 1 nπσ (NN) Xµ(τ, σ) = xµ + pµτ + i 2α0 αµe−iπnτ/` cos( ) , (2.55) ` n n ` n=06 This time, because the string is open rather than closed, there is only one set of mode µ oscillators αn appearing in

r n=+∞ 1 π α0 X ∂ Xµ = (X˙ µ ± Xµ0) = αµe−iπn(τ±σ)/` . (2.56) ± 2 ` 2 n n=∞ µ Note that α0 is deﬁned diﬀerently than for the closed string because of a physically important factor of two, √ µ 0 µ α0 = 2α p . (2.57) For the open string the Poisson brackets are

µ ν µν {αm, αn}P.B. = −imη δm+n,0 ,

18 µ ν µν {x , p }P.B. = η . (2.58)

Now let us look at the Hamiltonian. Using the simple fact that

Xµ0(τ, σ) = −Xµ0(τ, −σ) , (2.59) a ‘doubling trick’ allows us to rewrite

Z ` Z ` 2 2 2 dσ (∂+X) + (∂−X) = dσ(∂+X) . (2.60) 0 −`

On σ ∈ [−`, +`], the plane waves eiπmσ/` are periodic. Then the Hamiltonian is

n=+∞ π X H = α · α . (2.61) 2` −n m−n n=−∞ The Virasoro generators are deﬁned by

` Z ` L = − dσ eiπmσ/`T + e−iπmσ/`T m 2 ++ −− 2π 0 ` Z ` = dσ eiπmσ/`(∂ X)2 + e−iπmσ/`(∂ X)2 2 0 + − 2π α 0 ` Z ` = dσ eiπmσ/`(∂ X)2 2 0 + 2π α −` n=+∞ 1 X = (∂ X)2 . (2.62) 2 + n=−∞ They obey {Lm,Ln}P.B. = −i(m − n)Lm+n . (2.63) Now we turn to the DD strings. For them, the endpoints are ﬁxed by the requirement that ˙ µ µ X (τ, σ = {0, `}) = 0, which must hold for all τ. Integrating this gives X (τ, σ = 0) = xA µ while X (τ, σ = `) = xB, where xA,B are constants. This implies the mode expansion

1 √ X 1 πnσ (DD) Xµ(τ, σ) = xµ + (xµ − xµ )σ + 2α0 αµe−iπnτ/` sin( ) (2.64) A ` B A n n ` n=06

This time, 1 αµ = √ (xµ − xµ ) (2.65) 0 π 2α0 B A and of course the centre of mass is at (xA + xB)/2. The only other material diﬀerence with NN open strings is that

1 π X H = (x − x )2 + α · α , (2.66) 4πα0` B A 2` −n n n=06

19 i.e. that the Hamiltonian must contain a term describing the potential energy of the string stretched between xA and xB. Next, we examine the mixed ND strings. These beasts have different half-integer modings, because the first end (traditionally chosen at σ = 0) is Neumann while the other (at σ = `) is Dirichlet. Specifically, √ X 1 πrσ (ND) Xµ(τ, σ) = xµ + 2α0 αµe−iπrτ/` cos( ) , (2.67) B r r ` r∈Z+1/2 µ where xB is the position of the second (D) end. The DN strings are also half-integer moded, √ X 1 πrσ (DN) Xµ(τ, σ) = xµ + 2α0 αµe−iπrτ/` sin( ) , (2.68) A r r ` r∈Z+1/2 The reality condition for all four kinds of boundary conditions is

µ ∗ µ (αn) = α−n . (2.69) µ We can in fact extend the doubling trick to combine the light-cone derivatives ∂±X (τ, σ) into one field, traditionally defined as left-moving, defined on the doubled interval σ ∈ [0, 2`],

µ µ ∂+X (τ, σ) σ ∈ [0, `] ∂+X = µ (2.70) ±∂−X (τ, 2` − σ) σ ∈ [`, 2`] where the ± sign is + for (NN) and (DD) strings and is - for (ND) and (DN) strings. Then r π α0 X ∂ Xµ(τ, σ) = αµe−iπn(τ+σ)/` (2.71) + ` 2 n n where the n is summed over integers for (NN) and (DD) strings and over half-integer modes for (ND) and (DN) strings. For use with the doubling trick, periodicity conditions can be expressed as

µ µ ∂+X (τ, σ + 2`) = +∂+X (τ, σ) (NN), (DD) µ µ ∂+X (τ, σ + 2`) = −∂+X (τ, σ) (ND), (DN) . (2.72) Why do people talk about Dp-branes with different dimensions p? Because one may choose to select Dirichlet boundary conditions on different numbers of the Xµ maps for various µ. If there are no Dirichlet directions at all, this is referred to as either ‘open string endpoints moving freely at the speed of light’ or ‘a space-filling D-brane’. In D = 26 this is would be a D25-brane. If by contrast there are no Neumann directions, this is referred to as a D0-brane, a pointlike object. (In Euclidean spacetime signature we can even put Dirichlet boundary conditions on the time direction, making what is called a D-instanton.) All the options in between, D1-branes through D24-branes, are obtained by choosing (D − 1 − p) of the Xµ to have Dirichlet BCs and the other p directions to be Neumann. The D1-brane is sometimes called a D-string, which is physically very distinct from a fundamental string because it has a nonperturbative tension of order 1/gs in string units while the fundamental string tension is of order unity. All D-branes have a tension of order√ 1/gs in string units; the greater the worldvolume dimensionality p, the more powers of 1/ α0 we need to soak up the dimensions of length.

20 3 Quantum Bosonic String Theory

3.1 Canonical quantization and negative norm states The standard prescription in canonical quantization is to replace Poisson Brackets by quantum mechanical commutators4 1 { , } −→ [ , ] (3.1) P.B. i Using what we derived in the section on classical string theory, the equal time commutators then become [Xµ(τ, σ),Xν(τ, σ)] = 0 = [X˙ µ(τ, σ), X˙ ν(τ, σ)] . (3.2) and [Xµ(τ, σ), X˙ ν(τ, σ)] = 2πiα0ηµνδ(σ − σ0) , (3.3) In our mode expansion for the closed string, when we quantize, the Fourier mode coeﬃ- cients are promoted to operators5. The operator commutation relations they obey are

[xµ, pν] = iηµν , µ ν µν µ ν [αm, αn] = mδm+nη = [¯αm, α¯n] , µ ν [αm, α¯n] = 0 . (3.4)

(For the open string, the barred sector is absent.) The operator Xµ(τ, σ) must be hermitian, so µ † µ µ † µ (αm) = α−m , (¯αm) =α ¯−m . (3.5) These guys are quite familiar to you already: they just describe a simple harmonic oscillator µ √ µ where αm = mam. Notice that the number operator for the mth mode Nm =: αm · α−m : contains a factor of m compared to what you are used to from undergraduate QM. µ These commutators imply that the positive modes αm, m > 0 correspond to the annihilation operators while the negative modes with m < 0 correspond to the creation operators. The vacuum can have a CoM momentum pµ. Therefore the vacuum state obeys

µ µ αm|0; p i = 0 , m > 0 , pµ|0; pµi = pµ|0; pµi . (3.6)

In the second equation, the [first] pµ on the LHS is to be understood as an operator, while the [first] pµ on the RHS as its eigenvalue. µ ν Notice that the constant on the RHS of the [αm, αn] commutator involves the Minkowski metric ηµν which has Lorentzian, not Euclidean signature. The minus sign in the time- time component is a harbinger of a serious physical problem known as negative norm states 0 or ghosts. To see this, consider states of the form α−m|0i with m > 0 involving the 0 0 troublesome time component. These satisfy h0|αmα−m|0i = −mh0|0i < 0; in other words, they have negative norm. This is the key reason why canonical quantization of string theory is nontrivial: we have to figure out what to do about negative norm states.

4Recall that we are working throughout in grownup units in which ~ = c = 1. 5Following BLT, will only put hats on operators when it is necessary to avoid physical confusion.

21 There is a way forward. Remember the constraint equations that followed from symmetry under reparametrizations? We can impose them as operator equations constraining physical states. The hope is that the nasty ghosts decouple from the physical Hilbert space. The algebra is sufficiently long and unilluminating compared to more modern approaches to be discussed soon that we will not show any of the details here; if you are curious you can look it up in Green, Schwarz and Witten volume 1. It is possible to prove a no-ghost theorem, provided that two conditions are met in quantum bosonic string theory. First, the spacetime dimension has to be exactly 26, so that the number of physically transverse modes is (D − 2) = 24. Second, the normal ordering constant a in the mode expansion for the Virasoro generator L0 has to be exactly a = −1. It is very interesting that we have to make these specific choices in order to get quantum string theory to work in canonical quantization – and they are very physically relevant choices. Note in particular that we did not obtain D = 4 for the critical dimension! When we study light-cone quantization and modern BRST quantization, we will see how to derive these results in a very different way. To give you a sneak peek: in light-cone gauge, D = 26 and a = −1 will be demanded by the closure of the Virasoro algebra. In the most modern approach, the result will be obtained with the least amount of drudgery. Our next order of business is to discuss the propagators for our physical fields the Xµ(τ, σ). In order to begin solving a 2D wave equation on a worldsheet, we need the Green’s function. As you should already know, for 2D electrostatics, a point charge q located at z gives a potential V (w) = −q ln |z − w|2, and the method of images is fabulously helpful when we have boundaries. Another handy fact to know from complex analysis is that 1 1 ∂∂¯ln |z|2 = ∂ = ∂¯ = 2πδ2(z, z¯) . (3.7) z¯ z So we will be looking for logs. As usual, the propagators are defined as the difference between the time-ordered product and the normal-ordered product (c.f. Wick’s Theorem),

hXµ(τ, σ)Xν(τ 0, σ0)i = T [Xµ(τ, σ)Xν(τ, σ)]− :[Xµ(τ, σ)Xν(τ, σ)] : . (3.8)

Zero modes have to be handled with care. We need to handle the closed and open string cases differently because they have physically different boundary conditions. As usual, we do the closed string case first. We work on the cylinder with coordinates

(z, z¯) = e2πi(τ−σ)/`, e2πi(τ+σ)/` (3.9)

Notice thatz ¯ is not the complex conjugate of z, because we are currently still working in Lorentzian signature. If we were to Wick rotate, thenz ¯ would indeed be the complex conjugate. (Technically, this is why we chose z, z¯ this way.) Suppose that we re-use our classical mode expansions from the previous section but with the mode coeﬃcients promoted to operators. However, we cannot honestly write such an expression without specifying exactly what to do with the zero modes at the quantum operator level. If they are not chosen correctly, then the propagators for left- and right- handed ﬁeld modes fail to separate, which is a necessary condition for constructing the

22 heterotic string among other things. The correct choice for zero modes is

µ ν µν µ ν µ ν µ ν [xR, pR] = iη = [xL, pL] , [xL, pR] = 0 = [xR, pL] . (3.10) Then the propagators become α0 hXµ(z)Xν (w)i = − ηµν ln(z − w) , R R 2 α0 hXµ(z)Xν (w)i = − ηµν ln(¯z − w¯) . (3.11) L L 2 The expressions for the open string are more intricate. The worldsheet is now not a cylinder but a strip of width `, and so the relevant worldsheet coordinates are

(z, z¯) = eiπ(τ−σ)/`, eiπ(τ+σ)/` . (3.12)

The propagators are

0 µ ν α µν 2 2 hX (z, z¯)X (w, w¯)iNN,DD = − η ln |z − w| ± ln |z − w¯| , 2 √ 0 " √ √ 2 √ 2# µ ν α µν z − w z − w¯ hX (z, z¯)X (w, w¯)iND,DN = − η ln √ √ ± ln √ √ . (3.13) 2 z + w z + w¯

If you expand out the |...|2 pieces, you will be able to see that these expressions are both manifestly symmetric under interchange of (z, z¯) ↔ (w, w¯). The ± signs are delicately ar- ranged in order to ensure satisfaction of the boundary conditions, e.g. that the DD propagator vanishes at the endpoints where z =z ¯. So, how about those famous reparametrization constraints? What good will they do us here? Well, we know that classically they enforce T±± = 0, or in Fourier language, ¯ Ln = Ln = 0. But quantum mechanically we cannot be careless about operator ordering in defining composite operators like the Virasoro generators. By inspection you can see that the only Virasoro operator which has such an ambiguity is the zeroth mode L0. We will see this represented in equations as L0 → L0 + a , (3.14) where a is the normal ordering constant. This seemingly unimportant constant has physical ramifications: as we will see, it affects the spectrum of excitations of the string. (It does not, however, meddle with the angular momentum operators involved in the Poincaréalgebra.) After a bit of drudgery (see BLT §3.5), it can be proven that the Virasoro algebra for the quantum closed bosonic string has what is called a central extension c [L ,L ] = (m − n)L + m(m2 − 1)δ , (3.15) m n m+n 12 m+n where c is called the central charge. For one free boson, c = 1. Translating back to T±±, this implies iπc [T (σ),T (σ0)] = −2πi [∂ δ(σ − σ0)] {T (σ) + T (σ0)} − ∂3δ(σ − σ0) , ++ ++ σ ++ ++ 6 σ 23 iπc [T (σ),T (σ0)] = +2πi [∂ δ(σ − σ0)] {T (σ) + T (σ0)} + ∂3δ(σ − σ0) , −− −− σ −− −− 6 σ 0 [T++(σ),T−−(σ )] = 0 . (3.16)

Now we can ﬁnally address the question of whether we can get rid of the negative norm states. We want to work by analogy with electromagnetism, where one can only impose the positive frequency part of the gauge condition (e.g. ∂ · A = 0) on physical states, which ensures that longitudinal/scalar photons decouple. Here, the analogy would be to consider imposing only that the positive-n Virasoro modes annihilate physical states

Ln|physi = 0 , n > 0 . (3.17)

This is not a silly option to consider because, by the centrally extended Virasoro algebra, the Ln with n positive (i.e. not including n = 0) form a closed subalgebra. It also avoids the problem that c hφ|[L ,L ]|φi = 2nhφ|L |φi + n(n2 − 1)hφ|φi , (3.18) n −n 0 12 i.e., that we cannot impose Ln|φi = 0 ∀n and stay consistent with the symmetry algebra. Motivated by the above equation, we can try

Ln|physi = 0 , (L0 + a) |physi = 0 . (3.19)

Similar equations hold for the barred (left-moving) sector. For strings propagating on ﬂat Minkowski spacetime (no winding), the momentum constraint that we ﬁrst discussed for the classical bosonic string imposes ¯ L0 = L0 , a =a ¯ . (3.20)

The last thing to do in this section is to work out the mass formula for open and closed strings. This comes about from inspecting the deﬁnition of the number operator closely. For the open string,

∞ X µ i X a Nopen = α−nαµ,n + α−nαi,n + α−rαa,r (3.21) n=1 r∈N0+1/2 where the indices µ enumerate NN directions, the i do DD directions, and the a (not to be confused with the normal ordering constant!) cover the DN and ND directions. Using the pair of physical constraint conditions above, the mass formula is found to be

0 2 0 2 α mopen = Nopen + α T (∆x) + a , (3.22)

2 i where (∆x) = ∆x ∆xi is the distance between the two Dirichlet endpoints and T is the string tension. The important physics in this equation has two parts: (1) the zero of energy is determined by the normal ordering constant; (2) the |∆x| dependent piece is just describing the energy cost of extending the body of a straight (unexcited) open DD string a distance ∆x.

24 P∞ 2 0 2 For the closed string, since L0 = n=1 α−n · αn + α0/2 = N + α p /4 and similarly for the barred sector, 0 2 ¯ α mclosed = 2(N + N + 2a) , (3.23) while momentum balance requires N = N.¯ (3.24) A spurious state |spuri is one that obeys the mass shell condition and is orthogonal to all physical states: hspur|physi = 0. How can we ﬁnd the normal-ordering constant from insisting that unphysical states decouple? Consider a class of zero-norm spurious states of the form

∞ X |ψi = Ln|χni (3.25) n=1 where

Ln|χni = 0 (L0 + n + a)|χni = 0 . (3.26) By the Virasoro algebra, any such state can actually be rewritten as a linear combination

|ψi = L−1|χ1i + L−2|χ2i (3.27) because we have relations like [L−1,L−2] = +1L−3. Now, suppose that |ψi = L−1|χ1i. If this is a physical state, then by the Virasoro algebra L1L−1 = 2L0 + L−1L1, so that

L1|ψi = L1L−1|χ1i = (2L0 + L−1L1)|χ1i = −2(a + 1)|χ1i = 0 if a = −1 . (3.28) This is why we insist that the normal ordering constant appearing in the string mass spectrum formula takes the value a = −1. How about finding D? One way to see this is to consider zero-norm spurious states of the form 2 |ψi = (L−2 + γL−1)|χ˜i (3.29) where |χ˜i satisifes (L0 + 1)|χ˜i = 0 = Lm|χ˜i, for positive m. Since |ψi is physical, Ln|ψi = 0 for positive n. By the Virasoro algebra, requiring it to be annihilated by just L1 and L2 is enough, because you can build all the higher Ln, n ≥ 3 from those two. Now let us find L1|ψi, which we want to be zero. From the Virasoro algebra, we can find L1 in terms of other generators in a useful way. Specifically,

2 [L1,L−2 + γL−1] = 3L−1 + 2γL0L−1 + 2γL−1L0 = 3L−1 + 2γL−1(L0 + 1) + 2γ(L0 − 1)L−1 = (3 − 2γ)L−1 + 2γL−1 + 2γ(L−1L0 + L0L−1) = (3 − 2γ)L−1 + 4γL0L−1 . (3.30)

25 Now, L0L−1|χ˜i = L−1(L0 + 1)|χ˜i = 0 . (3.32) 3 So we have that L1 annihilates |ψi when γ = 2 . Next, we need to check whether or not |ψi is also annihilated by L2. By the Virasoro algebra,

3 2 D 3 [L2,L−2 + 2 L−1] = 4L0 + 2 + 2 [(3L1)L−1 + 3L−1L1] (3.33) 9 = 4L0 + 2 (2L0 + 2L−1L1) (3.34) D = 13L0 + 9L−1L1 + 2 . (3.35) Using the same technique as above, this gives

3.2 Light cone quantization

Light-cone coordinates select out one spatial direction as special: X± = √1 (X0 ± X1). The 2 Minkowski metric becomes off-diagonal: a · b = −a+b− − a−b+ + aibi, where i = 2 ...D − 1. The relativistic mass shell relation becomes p− = (m2 + pipi)/(2p+), which makes it look vaguely non-relativistic. Note that these light-cone indices refer to spacetime directions, not the worldsheet z = τ − σ andz ¯ = τ + σ which give rise to unbarred and barred worldsheet derivatives and will turn into complex conjugates once we Wick rotate. Analyzing QFT in light-cone makes counting the degrees of freedom quite straightforward. It also makes interrogating the string theory spectrum a lot easier for beginners. Light-cone gauge (more properly called light-front gauge) is a reparametrization gauge fixing in which the X+ direction is tied to be proportional to worldsheet time τ. Both the X0 and X1 directions must be Neumann if we have open strings around. In BLT conventions, 2πα0 X+ = p+τ . (3.40) ` This gauge choice permits the Virasoro constraints to be solved explicitly and the theory can be formulated in terms of physical (transverse) degrees of freedom only. The price of fixing the light-cone gauge is that the theory does not automatically have D-dimensional Poincaré symmetry. It is only in D = 26 that the light-cone gauge Poincaréalgebra closes. Light cone gauge clearly requires that X+ does not oscillate. We can solve for X− by using the constraints (X˙ µ ± X0µ)2 = 0,

` ∂ X− = (∂ Xi)2 . (3.41) ± 2πα0p+ ±

26 − This implies that the αn oscillators are proportional to transverse Virasoro generators

⊥ X I I Ln = αn−pαp (3.42) p∈Z where the I are summed over transverse directions I = 2,..., (D − 1). The mass-squared formula becomes a sum over oscillators that point in transverse directions only. No negative- norm states appear anywhere. The all-important mass formula takes the same form as for canonical quantization, with aLC being the zero-point energy (ZPE) X 1 a = p (D − 2) (3.43) LC 2 p∈N obtained by using the canonical commutation relations for the XI fields. How are we to regularize this infinite sum? There are two relatively fast ways to address this. The first is to take the mathematicians seriously when they tell us about Riemann zeta functions P∞ −s 1 ζ(s) = n=1 n . We have s = −1, and ζ(−1) = − 12 , which implies that 1 −1 (D − 2) a = (D − 2) = − . (3.44) LC 2 12 24 Note: for the case of half integrally moded fields, we would need to use instead the generalized P∞ −s ζ-function ζ(s, q) = n=0(n + q) . For the zero-point energy, the answer appearing is 1 2 1 1 1 proportional to ζ(1, q) = − 12 (6q − 6q + 1). When q = 2 , the ZPE is + 24 rather than − 12 . This drives home the fact that the ZPE is not just about summing up an infinite number of modes: it is also about how the fields in question are moded. The second way is to regularize by putting in a Gaussian exponential tail with UV cutoff Λ,

∞ ∞ π X π X π n n → n exp − (3.45) ` ` ` Λ n=1 n=1 π e−π/`Λ = (3.46) ` (1 − e−π/`Λ)2 ` 1 π = Λ2 − + O(1/Λ) . (3.47) π 12 `

2 The Λ divergence√ is proportional to the length of the string and can be cancelled by in- 2 R 2 1 troducing Λ d σ h to the Polyakov action. The remaining − 12 per integrally moded worldsheet boson XI is what sums up the vacuum energies of the oscillators. You will be working out some details of the low-lying spectrum in HW1.

3.3 Path integral quantization and Fadeev-Popov ghosts The Feynman path integral (FPI) writes the generating functional for correlation functions as a sum over all possible paths, with a well-deﬁned measure, weighted by eiS in Lorentzian signature where S is the action. It is straightforward to write a suitable measure for the FPI whenever there is no gauge symmetry, like for the ordinary QFT of a spin-0 or spin-1/2

27 field. The presence of a gauge symmetry makes defining the measure considerably more subtle, because one must ensure that the measure does not overcount physically equivalent configurations of the quantum fields. The fancy-pants word for the technology to handle this is Fadeev-Popov ghosts. They come in a variety of incarnations, each set relevant to the particular gauge symmetry and the particular gauge choice being used. They make use of a basic fact about functional determinants, a topic for which we now give a lightning review. Consider a Gaussian integral involving commuting scalars of the form Y Z Ib = dξk exp (−ξiBijξj) , (3.48) k where B is a symmetric matrix with eigenvalues bi. Diagonalize B via orthogonal matrix O, and switch to xi variables defined by ξi = Oijxj. Then Z Z Z Y Y X 2 Y X 2 Ib = dξk exp(−ξiBijξj) = dξk exp(− bixi ) = dxi exp(− bixi ) k k i i i Y r π 1 = = const. × b p i i det(B) In other words, Z Y −ξiBij ξj −1/2 Ib = dξke = (const.)(det(B)) (3.49) k 1 R D 2 2 For a spin-0 scalar field, we can write the free action as S0[φ] = 2 d x [φ (−∂ − m ) φ]. 2 2 R Formally we can write B = (−∂ − m I) as a matrix. Therefore Zφ = Dφ exp (iS0[φ]) 2 2 −1/2 becomes Zφ = (const) · [det(m + ∂ )] . Now let us do something similar for fermion fields. In path integrals, these are represented via Grassmann variables, which anticommute. For any two anticommuting Grassmann variables θ, η we have {θ, η} = 0. In other words, θη = −ηθ. Notice that a fermion bilinear is again a boson. Since the above equation holds for any θ, η, it holds in particular when θ = η, i.e. θ2 ≡ 0. This mathematizes the Pauli principle. It also makes Taylor expansions splendidly easy, because each such Taylor series terminates after the linear piece: f(θ) = A + Bθ, where A, B are constants. The definition of integration for Grassmann fields is given by R dθ = 0 and R dθθ = 1, and the sign convention we use is R dθ R dη · η · θ = +1. For complex Grassmann variables, (θη)∗ = η∗θ∗. Also, R dθ∗dθ θ θ∗ = +1. Now, because Taylor expansions truncate so early owing to the anticommutation property of Grassmann fields, we have exp (−θ∗bθ) = 1 − θ∗bθ + 0 (3.50) Therefore, Z Z dθ∗dθe−θ∗bθ = dθ∗dθ (1 − θ∗bθ) = b (3.51) by the rules of Grassmann integration and the anticommuting property. In doing Feynman path integrals to find correlation functions of physical fields, we would insert fields into the integrand. So let us consider for instance Z Z Z dθ∗dθ θ θ∗e−θ∗bθ = dθ∗dθ θ θ∗ (1 − θ∗bθ) = dθ∗dθ θ θ∗ = 1 (3.52)

28 Compare this to the result we had obtained just above. Looking carefully, we notice that the Gaussian integral with θθ∗ in the integrand brings down an extra factor of (1/b) compared to the case without. So let us deﬁne the following integral for an invertible matrix Bij, Z Y ∗ ∗ If = dθi dθi exp (−θi Bijθj) . (3.53) i

In the diagonal basis, Bij has eigenvalues {bi}. Then Z Y ∗ X ∗ Y If = dθi dθi exp(− θi biθi) = bi = det (B) (3.54) i i i We will make use of a functional version of this shortly to convert a determinant that will arise in the FPI measure into a covariant path integral over (scalar) Grassmann fields called Fadeev-Popov ghosts. The ghost action will enforce the rule that the measure does not overcount fields related by the symmetries of the Polyakov action. Note: for our first look at ghosts and measures in the FPI, we will ignore the issue of ghost zero modes. We will have more to say about this subtle and important physics question later when we analyze the superstring. What gauge choice do we want to enforce? We use the conformal gauge which puts the conformal symmetry on the worldsheet front and centre. Earlier we mentioned that µ ν on-shell, the induced metric gµν∂αX ∂βX is equal to the intrinsic worldsheet metric hαβ. This is true because of the classical equation of motion. Quantum mechanically, we have to µ do better – we need to integrate over all paths and over different hαβ and embeddings X . The important feature we need in the measure is that it respects reparametrizations and Weyl rescalings. Classically this symmetry was enough to gauge away the three independent components of hαβ. But will this still be true quantum mechanically? The answer will turn out to be: only for specific string theories referred to as critical string theories which have total central charge zero, counting the Xµ as well as the (b, c) ghosts. It is possible to define noncritical string theories as well, at the price of including an extra field in the action called the Liouville field. For our first try, we will write the string theory path integral as Z Z = DhDXeiSP [h,X] , (3.55) where the norms are defined via Z √ 2 µ ||δX|| = d σ −h δX δXµ (3.56) Z √ 2 αβ γδ ||δh|| = d σ −h h h δhαγδhβδ . (3.57)

We can use the reparametrization invariance of the Polyakov action to go to a gauge where the intrinsic worldsheet metric is 2φˆ hαβ = e hαβ . (3.58)

29 Further, under reparametrizations and Weyl rescalings the changes of the metric can be decomposed into a traceless symmetric tensor part and a trace part (orthogonal w.r.t. the norms deﬁned above), ˜ δhαβ = −(Pξ)αβ + 2Λhαβ , (3.59) where P maps vectors to traceless symmetric tensors. To see this, note that reparametrization ensures that we can pick conformal gauge locally, with the conformal factor being e2φ(τ,σ). But can it be done globally? Under a reparametrization,

δhαβ = −(∇αξβ + ∇βξα) + 2Λhαβ (3.60) ˜ ≡ −(Pξ)αβ + 2Λhαβ (3.61) ˜ 1 γ γ where Λ = Λ − 2 ∇γξ , and where (Pξ)αβ = ∇αξβ + ∇βξα − (∇γξ )hαβ. The trace part can always be cancelled by a suitable choice of Λ. So for conformally gauge to be allowed α globally, there must exist a globally defined vector field ξ such that (Pξ)αβ = tαβ for † arbitrary symmetric traceless tαβ. The adjoint P maps symmetric traceless tensors to † β vectors via (P ξ)α = −2∇ tαβ. Whether or not the zero mode subtleties allow a consistent choice of conformal gauge will be mentioned further when we get to superstrings. For now, it suffices to suppress this detail. Getting back to our path integral measure, we have then

∂(Pξ, Λ˜ ˜ Dh = D(Pξ)D(Λ) = DξDΛ . (3.62) ∂(ξ, Λ) This gives a Jacobian of the form P 0 † / det = | det P| = (det P P)1 2 . (3.63) ∗ 1 The integral over reparametrizations gives the volume of the part of the diﬀeomorphism group connected to the identity. This volume depends on the Weyl degree of freedom as does Dξ. But in the critical dimension this all drops out. We assume critical dimension, and write for our second try Z 2φˆ µ Z = DXµ (det P † P) eiSP [e hαβ ,X ] (3.64)

This is almost suitable for use, but at present it has a nasty functional determinant in the integrand. Now we can bring in our Grassmann-functional-determinants trick and use Fadeev-Popov ghost ﬁelds c(τ, σ), b(τ, σ) to write Z 1 Z √ (det P † P) = Dc Db exp d2σ −hhαβb ∇ cγ , (3.65) 2π βγ α

2φˆ α where hαβ = e hαβ. In physical language, the ghost c corresponds to inﬁnitesimal reparametrizations, and the antighost bαβ corresponds to variations perpendicular to the gauge slice. Then our ﬁnal expression for the Feynman Path Integral for strings is Z µ ˆ Z = DXµ Dc Db eiSP [X ,h,b,c] , (3.66)

30 where 1 Z q S [Xµ, h,ˆ b, c] = − d2σ −hˆhˆαβ ∂ Xµ∂ X + 2iα0b ∇ˆ cγ . (3.67) P 4πα0 α β µ βγ α Note that the Weyl factor is gone because we are working in the critical dimension. In ˆ conformal gauge, hαβ = ηαβ, i Z S = d2σ c+∂ b + c−∂ b , (3.68) gh π − ++ + −− which is real when b, c are Hermitean by dint of the anticommuting property. The (b, c) system has stress-energy tensor

γ γ γ γ δ Tαβ = −i bαγ∇βc + bβγ∇αc − c ∇γbαβ − hαβbγδ∇ c . (3.69)

It is important to recall that bαβ is symmetric and traceless while deriving this. Then, the ghost stress-energy tensor becomes in conformal gauge

± ± T±± = −i 2b±±∂±c + (∂±b±±)c . (3.70)

The equations of motion for the ghosts are

− + ∂−b++ = 0 = ∂+b−− , ∂+c = 0 = ∂−c , (3.71) and both ﬁelds are taken to have periodic boundary conditions for the closed string, e.g. + − b(σ + `) = b(σ). For the open string, at the endpoints we obtain b++ = b−− and c = c . These relations are generically not true anywhere else on the string. Physically, since ghosts are Grassmann, they obey canonical anticommutation relations,

± 0 0 {b±±(τ, σ), c (τ, σ )} = 2πδ(σ − σ ) . (3.72)

The b and c mode expansions show integrally moded ﬁelds, and Virasoro operators become

+∞ X ◦ ◦ Lm = (m − n) ◦ bm+nc−n ◦ (3.73) m=−∞ The weird-looking normal ordering symbols are designed to stand out: they tell you that in putting annihilation operators to the right the Grassmann property is taken into account properly. Hermiticity of the ghosts implies that

† † † cn = c−n , bn = b−n ,Ln = L−n . (3.74)

Using zeta function regularization as before, it is straightforward to obtain

cgh = −26 . (3.75)

31 4 Conformal Field Theory

4.1 Radial quantization To start with we will discuss several deﬁnitions. The ﬁrst is how we got from our world sheet coordinates to the complex conformal coordinates we were using. First recall our world sheet coordinates can be parametrized as σ ∈ [0, 2π] and τ ∈ [−∞, ∞]. Now Wick rotate to a complex space (i.e. τ → −iτ). To get to the complex plane, we simply conformal transform by z = eτ−iσ z¯ = eτ+iσ (4.1) Pictorially,

Now we can see how the worldsheet time is mapped. For τ = −∞, we see that z = 0 and for τ = ∞, we see that z = ∞. Furthermore, lines of constant τ are circles around the origin. So to define a time ordered product, we now do a radially ordered product. This will be important in our definitions of operator product expansions (OPEs), since we will naturally want to time order them for physics reasons. σ-translations on the cylinder become rotations on the complex plane, while τ-translations on the cylinder become dilatations. Recall that holomorphic functions are functions which are complex and infinitely differ- entiable everywhere. We use the same definition in CFT. Functions that are holomorphic in z are called such, but holomorphic functions inz ¯ are called anti-holomorphic. There are several different types of fields, the most basic of which is called a primary field. Primary field transform as tensors under conformal transformations: ¯ ∂z0 h∂z¯0 h φ(z, z¯) → φ0(z, z¯) = φ(z0, z¯0). (4.2) ∂z ∂z¯

Note we call h and h¯ conformal weights under analytic and anti-analytic transforms, respectively. The scaling dimension of a field is given by h + h¯ and the conformal spin is given by h − h¯. These indicate how field behave under dilations or rotations. Fields that are single valued will have h = h¯ ∈ Z. Holomorphic fields will have h¯ = 0 and anti-holomorphic fields will have h = 0. If we introduce a infinitesimal conformal transformation

z0 = z + ξ(z)z ¯0 =z ¯ + ξ¯(¯z), (4.3) then the ﬁeld will transform as 0 φ = φ + δξ,ξ¯φ, (4.4) where ¯ ¯ ¯ δξ,ξ¯φ = (h∂zξ + h∂z¯ξ + ξ∂z + ξ∂z¯)φ (4.5)

32 Using the relation between the cylinder and plane coordinates, we see quickly that

X −n−h φplane = φnz . (4.6) n∈Z One final note is about the Hermitian conjugate of a field. The general definition for a field of weight h is 1 1 [φ(z)]† = φ†( ) . (4.7) z¯ z¯2h and this implies that the modes satisfy

† † φ −n = φn (4.8) How does T act on primary fields? As we saw, the Virasoro algebra leads to an infinite number of conserved charges I dz T = ξ(z)T (z) . (4.9) ξ 2πi These can be used to generate the infinitesimal conformal transformations

z → z0 = z + ξ(z). (4.10)

To generate the conformal transforms on primary ﬁelds, we use the commutator,

δξφ(w) = [Tξ, φ(w)]. (4.11)

Now we wish to map the two definitions of the transformations. Suppose we have a field with h¯ = 0. Then the above definition gives us (recalling radial ordering)

I dz δξφ(w) = ξ(z)T (z)φ(w). (4.12) Cw 2πi The contour is

On the other hand, including (4.5), we see that

I dz ξ(z)T (z)φ(w) = (h∂wξ + ξ∂w)φ. (4.13) Cw 2πi Now let us propose that the time ordered product takes the form

hφ(w) ∂ φ(w) T (z)φ(w) = + w + regular , (4.14) (z − w)2 z − w

33 then we can see that I dz I dz hφ(w) ∂ φ(w) ξ(z)T (z)φ(w) = ξ(z) + w + regular (4.15) 2 Cw 2πi Cw 2πi (z − w) z − w d ξ(z) ∂ φ(w) = lim hφ(w) (z − w)2 + (z − w) w (4.16) z→w dz (z − w)2 z − w

= (h∂wξ + ξ∂w)φ (4.17)

So our proposition was correct. Goodie! Now we might expect to be able to expand T similarly. In order to determine the correct expansion, we need to have some constraint for the product T (z)T (w) to satisfy. To start with consider the commutator of two inﬁnitesimal conformal transformations,

[δξ1 , δξ2 ]φ = δξ1 δξ2 φ − δξ1 δξ2 φ. (4.18) Using our deﬁnition for the transformations,

[δξ1 , δξ2 ]φ = δξ1 δξ2 φ − δξ1 δξ2 φ (4.19)

= (h∂zξ1 + ξ1∂z)(h∂zξ2 + ξ2∂z)φ − (h∂zξ2 + ξ2∂z)(h∂zξ1 + ξ1∂z)φ (4.20) 2 2 = (h(ξ1∂ ξ2 − ξ2∂ ξ1) + (ξ1∂ξ2 − ξ2∂ξ1)∂)φ (4.21)

= (h∂(ξ1∂ξ2 − ξ2∂ξ1) + (ξ1∂ξ2 − ξ2∂ξ1)∂)φ (4.22)

= δ(ξ1∂ξ2−ξ2∂ξ1)φ. (4.23) This implies that

[δξ1 , δξ2 ] = δ(ξ1∂ξ2−ξ2∂ξ1). (4.24) So now we can consider our other deﬁnition,

[δξ1 , δξ2 ]φ = δξ1 δξ2 φ − δξ1 δξ2 φ (4.25)

= [Tξ1 , [Tξ2 , φ]] − [Tξ2 , [Tξ1 , φ]] (4.26)

= Tξ1 Tξ2 φ − Tξ2 φTξ1 − Tξ1 φTξ2 + φTξ2 Tξ1 − 1 ↔ 2 (4.27)

= Tξ1 Tξ2 φ + φTξ2 Tξ1 − Tξ2 Tξ1 φ − φTξ1 Tξ2 (4.28)

= [[Tξ1 ,Tξ2 ], φ]. (4.29) Additionally, we can see that

δ(ξ1∂ξ2−ξ2∂ξ1)φ = [T(ξ1∂ξ2−ξ2∂ξ1), φ]. (4.30) Now we apply our condition, equation 4.24, and we will get

[Tξ1 ,Tξ2 ] = T(ξ1∂ξ2−ξ2∂ξ1). (4.31) This is the condition that T (z) must satisfy. Now we propose the following form for the T − T operator product expansion, c/2 2T (w) ∂ T (w) T (z)T (w) = + + w + regular. (4.32) (z − w)4 (z − w)2 z − w

34 We will work on the right side of our condition, I dw I dz [Tξ1 ,Tξ2 ] = ξ1(z)T (z)ξ2(w)T (w) (4.33) 2πi Cw 2πi I dw I dz c/2 2T (w) ∂ T (w) = ξ (z)ξ (w) + + w + regular (4.34) 1 2 4 2 2πi Cw 2πi (z − w) (z − w) z − w I dw c = ξ (w) ∂3 ξ (w) + 2∂ ξ (w)T (w) + ξ (w)∂ T (w) . (4.35) 2πi 2 12 w 1 w 1 1 w

Note that the ﬁrst term has no singularities (we said both ξ1,ξ2 were both analytical). So the contour vanishes. Also note that we can integrate the last term by parts. Since it is a contour integral, the surface term vanishes. Doing both theses things we will ﬁnd

I dw c [T ,T ] = ξ (w) ∂3 ξ (w) + 2∂ ξ (w)T (w) + ξ (w)∂ T (w) (4.36) ξ1 ξ2 2πi 2 12 w 1 w 1 1 w I dw = 2ξ (w)∂ ξ (w)T (w) − ∂ (ξ (w)ξ (w))T (w) (4.37) 2πi 2 w 1 w 2 1 I dw = ξ (w)∂ ξ (w) − ξ (w)∂ ξ (w)T (w) (4.38) 2πi 2 w 1 1 w 2

= T(ξ1∂ξ2−ξ2∂ξ1). (4.39)

So we have shown that our proposal for the OPE was correct. Note that we did not need to have the term proportional to c. However, it does have the correct scaling dimension and Bose symmetry. We also recall that this is the term that comes from the Weyl anomaly, which we needed to include. Let us look at the operator product expansion of Tzz (and by extension Tz¯z¯). We will expand the Laurent series, ∞ X −n−2 Tzz = z Ln (4.40) −∞ Like before, we will cut down on the pain of presentation by proposing a form for this T − T OPE and show that it works. It is

λ 1 2T (0) ∂ T (0) T T = + ww + w ww + regular , (4.41) zz ww 2 (z − w)4 (z − w)2 z − w where by regular, we mean no singular behaviour as z → w. In order to infer this deﬁning equation, let us recall that I dz z−k = δ . (4.42) 2πi k,1 So then we can see that to ﬁnd the operators we can perform a contour integral, I dz L = zm+1T = (4.43) n 2πi zz

35 So we can turn our expression for the product of two energy momentum operators into an equation for the commutator that these Ln operators satisfy. Doing so we ﬁnd that I dz I dw [L ,L ] = zn+1wm+1T T . (4.44) n m 2πi 2πi zz ww We can now plug in our equation for the energy momentum components. We can drop out the regular terms since they will integrate away. Then we have ! I dw I dz λ 1 2T (0) ∂ T (0) [L ,L ] = zn+1wm+1 + ww + w ww . (4.45) n m 4 2 2πi Cw 2πi 2 (z − w) (z − w) z − w

Examining the ﬁrst term alone,

I dw I dz λ 1 I dw 1 d3 λ zn+1 zn+1wm+1 = wm+1 lim (z − w)4 (4.46) 4 3 4 2πi Cw 2πi 2 (z − w) 2πi z→w 3! dz 2 (z − w) λ I dw = (n + 1)n(n − 1) wm+1wn−2 (4.47) 12 2πi λ = n(n2 − 1)δ . (4.48) 12 n+m,0 For the second two terms, we can use the expansion we had earlier. Putting that in, it becomes I dw I dz 2T ∂ T zn+1wm+1( ww + w ww ) (4.49) 2 2πi Cw 2πi (z − w) z − w I dw I dz X 2w−p−2 w−p−3 = zn+1wm+1 − (p + 2) L (4.50) 2 p 2πi Cw 2πi (z − w) z − w p∈Z I n+1 n+1 X dw m+1 −p−2 d 2 z −p−3 z = Lp w lim 2w (z − w) − (p + 2)w (z − w) 2πi z→w dz (z − w)2 z − w p∈Z (4.51) X I dw = L wm+n−p−1(2(n + 1) − (p + 2)) (4.52) p 2πi p∈Z X = Lp(2n − p)δm+n,p (4.53) p∈Z

= (n − m)Ln+m. (4.54)

So putting it all together, the operators satisfy the commutator, λ [L ,L ] = (n − m)L + n(n2 − 1)δ . (4.55) n m n+m 12 n+m,0 We can now recognize that this is the Virasoro algebra! Everything works in an exactly analo- ¯ gous way for the barred sector; the two Virasoro algebras are independent (i.e. [Ln,Lm] = 0).

36 4.2 The operator product expansion So this brings us to our more general idea of a product expansion of two operators. Suppose that {Oi} are a complete set of local operators with deﬁnite scaling dimensions. We can then expand a product of these two operators in terms of the others. Since should have scaling dimensions (since the product must have a deﬁnite scaling dimension), we expand as

X hk−hi−hk Oi(z)Oj(w) = Cijk(z − w) Ok(w). (4.56) k It is important to note that the (anti)commumators of the operators are determined by the singularities in the operator product expansion. Of course, there are still other properties of the Virasoro algebra to learn. Consider the commutator of Ln with a ﬁeld φ I dw [L , φ(z)] = wn+1[T (w), φ(z)] (4.57) n 2πi I dw = wn+1T (w)φ(z) (4.58) Cz 2πi

= [Tzn+1 , φ] (4.59)

= δzn+1 φ (4.60) = zn[z∂ + (n + 1)h]φ. (4.61)

To see what happens for the modes, we simply use our contours, I dz [L , φ ] = zm+h−1[L , φ(z)] (4.62) n m 2πi n I dz = zm+h−1zn[z∂ + (n + 1)h]φ(z) (4.63) 2πi I dz = zm+n+h−1[z∂ + (n + 1)h]φ(z) (4.64) 2πi I dz = (n + 1)hφ − ∂zm+n+hφ(z) (4.65) m+n 2πi

= ((n + 1)h − (m + n + h))φm+n (4.66)

= (n(h − 1) − m)φm+n. (4.67)

n+1 So we see that Ln are generators of conformal transformations with ξ = z . A important note is that L−1,L0,L1 are the generators of SL(2, R), the maximal closed subalgebra of the conformal group. They have finite transformations, called fractional linear or Möbius transformations, of the form az + b z → z0 = (4.68) cz + d with the coefficients satisfying a b ∈ SL(2, ). (4.69) c d R

37 ¯ ¯ ¯ If we add L−1,L0,L1, then we get SL(2, C). We could also consider the effect of a finite conformal transformation, z → f(z), on T (z). We can use our OPE for T (z)T (w) to find the infinitesimal transformation of T (z) and then integrating the result. We obtain c T (z) → T 0(z) = (∂f(z))2T (f(z)) + D(f) . (4.70) 12 z Note that D(f) is the Schwarzian derivative, which is given by ∂f(z)∂3f(z) − (3/2)(∂2f(z))2 D(f) = . (4.71) z (∂f(z))2 The Schwarzian derivative is unique in that it is the only weight two object that satisfies the properties below az + b D(f) = 0 ⇔ f(z) = (4.72) z cz + d af + b D( ) = D(f) (4.73) cf + d z z 2 D(f)z = (∂zg) D(f)g + D(g)z. (4.74) To get the energy momentum tensor from the cylinder to the plane, we can use equation 4.70 on z0 → ez0 = z and get c T (z0) = z2T (z) − . (4.75) cylinder plane 24 4.3 Action of Virasoro generators on states: highest weights and descendants

Now let us discuss the action of the Ln on states in the Hilbert space. As usual, |0i, h0| will be the in and out vacua respectively. Now we wish for the energy momentum tensor to be regular at z = 0 and z = ∞. For z = 0, then we just need to make sure that the Virasoro algebras that are generate regular conformal transformations at the origin annihilate the vacuum. This is done by

Ln|0i = 0 for n ≥ −1 (4.76) For z = ∞, we need to bring in the point at inﬁnity to the origin. We do this using the map w = −1/z. Using our transformation law form equation 4.70,

T 0(w) = (1/w2)2T (−1/w). (4.77)

So then we can expand out our energy momentum tensor in these coordinates, ﬁnding 0 P n−2 T (w) = (−w) Ln. So the Virasoro operators that generate regular conformal transformations at the origin are diﬀerent than in the other coordinates. To properly regularize the energy momentum, we need to enforce

h0|Ln = 0 for n ≤ 1 (4.78)

38 Another way of coming to the same result is to notice that these are the Hermitian conjugates of one another. We can argue similarly for primary ﬁelds of weight h. Here the conditions become

φn|0i = 0 for n ≥ 1 − h (4.79)

h0|φn = 0 for n ≤ h − 1. (4.80) One might be afraid that if h ≤ 0, then there might be non-zero vacuum expectation values. However, unitarity restricts us to states that are h ≥ 0. Note that h = 0 corresponds to the identity from unitarity and so its mode expansion is itself. However, ghost fields can avoid unitarity restrictions since on-shell ghosts are always countered by non-physical polarizations. Now we can construct some other states. To define an in state, we wish for it to be defined at the origin (i.e. τ = −∞). So then

|φji = lim φj(z)|0i (4.81) z→0

X −n−hj = lim z φn|0i (4.82) z→0 n∈Z

X −n−hj = lim z φn|0i (4.83) z→0 n≤−hj X = δn,−hj φn|0i (4.84)

n≤−hj

= φ−hj |0i. (4.85) This might have seemed trivial, but we have used our ideas of expanding and our requirements of regularity. The out state can be deﬁned as we did out states earlier, or we can just take the Hermitian conjugate, † hφj| = h0|(φ )hj . (4.86) Now we wish to know how our Virasoro operators act on our states. For the moment, we will consider only n ≥ 0. We make use of equation 4.67 and can see that

Ln|φji = Lnφ−hj |0i (4.87)

= φ−hj Ln|0i + (n(hj − 1) + hj)φn−hj |0i. (4.88) Note for n ≥ 0, the ﬁrst term is annihilated. The second term is annihilated for any n > 0 but not if n = 0. So the result is

L0|φji = hj|φji (4.89)

39 So we see that the Virasoro operators L−n act as raising operators for n > 0. Note that states that satisfy equations 4.89 and 4.90 are called highest weight states. (Even though one can raise their weight higher using L−n: it is a bit weird.) Highest weight states with diﬀerent eigenvalues of L0 (hj) are orthogonal. We can show this noticing that

hφi|L0|φji = hjhφi|φji (4.93)

† but if we act on the out state (since Ln = L−n, L0 is Hermitian), then

hφi|L0|φji = hihφi|φji (4.94) and for both equations to hold true for different h, the states must be orthogonal. We expect that we can now complete the Hilbert space by using our raising operators. The resulting states are called descendant states (which are distinct from the primary fields we discussed earlier). For every highest weight state |φji, we can find a representation of the Virasoro algebra labelled by hj. It is called a Verma module and it consists of all the states that have the form of k1...km |φj i = L−k1 ...L−km |φji. (4.95) P Note this only makes sense if ki > 0. These states will have L0 eigenvalues of hj + i ki. Note that these states are secondary operators, but are contained in the operator product of the primary field with the energy momentum tensor:

∞ X −2+k (−k) T (z)φ(w) = (z − w) φi (w). (4.96) k=0 In other words, I −k dz 1−k ˆ φi (w) = (z − w) T (z)φi(w) ≡ L−kφi(w). (4.97) Cw 2πi

(−k1) (−k1,−k2) We could then consider T (z)φi (w) and get ﬁelds like φi (w) and so on. So we generalize to the descendant ﬁeld that creates the state in equation 4.95,

k1...km ˆ ˆ φj (z) = L−k1 ...L−km φj(z). (4.98)

These fields constitute the conformal family of [φj]. Note that from this definition we can see that T is not a primary field, rather it is in the conformal family of the identity operator [I]. To see this, consider I dw I(−2)(z) = Lˆ I(z) = (w − z)−1T (w)I(z) = T (z). (4.99) −2 2πi One final note about this. We can define the character of a conformal family (partition function) as ∞ c hj − Y n −1 Chj(τ) = q 24 (1 − q ) , (4.100) n=1 where q = e2πiτ . More details can be found in BLT.

40 4.4 Correlation functions Now that we have our definitions of fields and operators sorted, we would like to investigate how the correlation functions behave. To do this we will restrict ourselves to fields called quasi-primary fields. These are fields that transform as tensors under SL(2, C), but not necessarily under the full conformal group (i.e. all primary fields are quasi-primary but not vice versa). There are three important transformations under this group, all given in the table below. Transformation Generator Coordinate Field 0 0 Translations L−1 z = z + b φi(z) = φi(z + b) 0 0 hi Dilation and rotations L0 z = az φi(z) = a φi(az) 0 z 0 1 2hi z Special conformal L1 z = cz+1 φi(z) = ( cz+1 ) φi( cz+1 ) Demanding invariance under these restricts the forms that are allowed for the correlation functions. See BLT for details. The results are as follows,

hφi(z)i = 0 (4.101)

Cij hφi(z )φj(z )i = δh ,h (4.102) 1 2 2hi i j z12 Cijk hφi(z1)φj(z2)φk(z3)i = (4.103) hi+hj −hk hi+hk−hj hk+hj −hi z12 z13 z23 h2+h4 h1+h3 z13 z24 z12z34 hφ1(z1)φ2(z2)φ3(z3)φ4(z4)i = f( ) (4.104) h1+h2 h2+h3 h3+h4 h1+h4 z12 z23 z34 z14 z13z24 Y −γ zijzkl hφ (z )...φ (z )i = z ij f( ) (4.105) 1 1 n n ij z z i

Note that the four-point amplitudes can help us put constraints on Cijk by a method of the conformal bootstrap.

41 5 Ghosts, BRST invariance, and scattering amplitudes

5.1 Properties of (b,c) ghosts and the critical dimension The equations of motion are simple first order differential equations which one could just read off. They imply that

z c = c(z), bzz = b(z), (5.1) z¯ ¯ c =c ¯(¯z), bz¯z¯ = b(¯z). (5.2) ¯ Now the tensor character of bαβ implies that b transforms with conformal weight (h, h) = (2, 0) and ¯b with (h, h¯) = (0, 2). For the action to remain invariant under conformal transformations, we need c to transform with conformal weight (−1, 0) andc ¯ with (0, −1). This negative conformal dimension is necessary to make the ghost sector of the critical bosonic string CFT work correctly. We will see this can be done another way straightforwardly using CFT technology. For the moment, we will just focus on the analytic ﬁelds; everything goes through similarly for the barred sector. Continuing our analysis, the propagator should satisfy,

(2) ∂z¯hb(z)c(w)i = 2πδ (z − w). (5.3)

So we suggest that the propagator takes the form of 1 hb(z)c(w)i = . (5.4) z − w This in turn gives us the operator product expansion, 1 b(z)c(w) = + regular = c(z)b(w), (5.5) z − w where we used the fact that they are Grassman ﬁelds to anticommute them in the second equality. We can now expand them into their modes,

X −n+1 c(z) = cnz (5.6) n X −n−2 b(z) = bnz . (5.7) n We can now use our OPE to ﬁnd the anticommutation relations these modes will satisfy. Since c2 and b2 are zero due to being Grassman, we can immediately see that

{cn, cm} = 0, {bn, bm} = 0. (5.8)

Now we will use contours to ﬁnd out the ﬁnal anticommutators. Now since we instead have

42 anticommuting ﬁelds, the contours will give us an anticommutator instead. We can see that I I dw m−2 dz n+1 {bn, cm} = w z b(z)c(w) (5.9) 2πi Cw 2πi I dw I dz zn+1 = wm−2 (5.10) 2πi Cw 2πi z − w I dw = wm+n−1 (5.11) 2πi

= δn+m,0 (5.12)

Now we should examine the energy momentum tensor,

T (z) = −∂b(z)c(z) − 2b(z)∂c(z) (5.13) which should be normal ordered in the quantum theory. Expanding T into modes using b and c, X −n−m−2 T (z) = (2m − 2 + p + 2) : bpcm : z (5.14) m,p But we could expand it into Virasoro generators by

X b,c −n−2 T (z) = Ln z . (5.15) n To find the Virasoro generators in terms of the ghost modes, we use contour integration. I dz Lb,c = zn+1T (z) (5.16) n 2πi X = (n − m): bn+mc−m : . (5.17) m Finding the T −T OPE for the ghost sector is a tad tedious, and only involves techniques that you already know how to use (e.g. Wick’s theorem, Taylor expansion, the Grassmann property of the ghosts), so I will suppress the boring steps. I plan to set working out the details as a homework problem. Very similar types of calculations are needed to work out the operator product expansions of T (z)c(w) and T (z)b(w), which provide the answer that b and c are primary conformal fields of the weights we stated above. The final result for the ghost T −T OPE is

−13 2T (w) ∂T (w) T (z)T (w) = + + + regular (5.18) (z − w)4 (z − w)2 (z − w)

Recall then that half of the coeﬃcient of the ﬁrst term gives us the central charge. So our ghost system has a central charge of c = −26. By itself it does not make much sense, but for us it always comes along for the ride with a CFT of D bosons Xµ. It is therefore possible to write down a sensible path integral for critical string theory provided that D = 26. This completes the ghost-style derivation of the critical dimension of bosonic string theory.

43 5.2 BRST invariance and unitarity Becchi, Rouet, Stora, and Tyutin noticed that if we choose the symmetry transformation parameter ∆ωA(x) to be ∆ωA(x) = −cA(x), where cA(x) and are both Grassmann and is a constant, then the action is invariant – provided that cA(x) transforms with a particular form. The BRST transformation equations involve the original fields you were trying to quantize and their symmetries, as well as ghost, antighost and Lagrange multiplier fields. Us- ing identities satisfied by the symmetry algebra generators, it is possible – although typically a tad arduous algebraically – to show that the BRST transformation squares to zero:

2 δBRST = 0 (5.19) Since the BRST operator generating these BRST symmetry transformations is nilpotent, any term added to the action which is the BRST-variation of something will automatically be BRST-invariant. Formally, we can write this as δ L = L + BRST Ξ (5.20) tot cl δ where Ξ is known as the gauge fermion. The structure of BRST transformations implies very nontrivial things about unitarity of scattering amplitudes. These all originate from having a nilpotent operator QBRST that commutes with the Hamiltonian H:[H,QBRST] = 0. To see this, let H1 be the subspace of states |ψ1i which are not annihilated by the BRST operator QBRST, and let H2 be the subspace of states |ψ2i of the form |ψ2i = QBRST|ψ1i for some |ψ1i ∈ H1. Also let H0 be the subspace of states |ψ0i satisfying QBRST|ψ1i = 0 but that cannot be written as the QBRST of any state (the states which are BRST-closed but not BRST-exact). Having these three subspaces of the space of states H0, H1, H2 follows directly from having a nilpotent operator which commutes with the Hamiltonian. Incidentally, note that the subspace H2 is a rather weird subspace, as any two states in it have zero inner product. This fact also follows directly from the nilpotency of QBRST. Let |A; ⊥i denote an external state containing no ghosts or antighosts and only physical (transverse) polarizations of X ﬁelds. We would like to show that the S-matrix for these guys is unitary, i.e. that

X † hA; ⊥ |S |C; ⊥ihC; ⊥ |S|B; ⊥i = hA; ⊥ |I|B; ⊥i (5.21) C

Recall that H0 is the space of states |ψ0i such that QBRST|ψ0i = 0 but which cannot be written as QBRST|λi for any λ (i.e., the states which are BRST-closed but not BRST-exact). Also, we know that [QBRST,H] = 0 so that any time-evolved state S|ψ0i is also killed by the BRST operator QBRST: it takes the form

QBRST · S|A; ⊥i = 0 (5.22)

This implies that S|A; ⊥i must be linear combinations of states in H0 (physical ones) and states in H2 (BRST-exact ones). But any two states in H2 have zero inner product with

44 one another, and also hψ2|ψ0i = 0 (by deﬁnition of H2). So the inner product of any two states of that form must arise solely from the overlap of the components in H0. Therefore, X hA; ⊥ |S†S|B; ⊥i = hA; ⊥ |S†|C; ⊥ihC; ⊥ |S|B; ⊥i (5.23) C and so not only is the full S-matrix unitary but its restriction to the subspace H0 is also unitary. The BRST method is really powerful! So far in the discussion of BRST invariance I have suppressed the details of the symmetry. It is time to now lift the veil from your eyes and show how this part works.

5.3 Applying BRST to the bosonic string We can deﬁne an operator called the U(1) ghost number current j(z) by j(z) = − : b(z)c(z): . (5.24) We expect that j(z) should be a conformal ﬁeld of weight 1. So we want,

X n−1 j(z) = jnz . (5.25) n To find the coefficients, we use contour integration. I dz j = znj(z) (5.26) n 2πi X = : cn−mbm : . (5.27) m Classically, this number is conserved. However, there is a quantum anomaly, which will soon turn out to be proportional to D − 26. This will be how the critical dimension will show up in the context of BRST technology. The ghost charge is given by the contour integral I dz X N = j(z) = j = : c b : . (5.28) gh 2πi 0 −m m m To see the ghost charge of a conformal field φ(z), we can just look at the operator product expansion, N φ(w) j(z)φ(w) = gh + regular. (5.29) z − w

Note, if were we to do this OPE, we would see that c(z) has Ngh = 1 and b(z) has Ngh = −1 as we would expect. The general BRST method goes as follows. Suppose that you have a Lie algebra

k [Ki,Kj] = fij Kk (5.30) where i, j, k run over 1,..., dimG. Deﬁne the BRST operator by

i 1 j QBRST = c Ki − 2 c bk (5.31) i 1 gh = c Ki + 2 Ki (5.32)

45 i i where {c , bj} = δj. Using the symmetry algebra, the canonical anticommutation relations and the antisymmetry of the structure constants, it follows that

2 1 k m j i l QBRST = 4 f[ij fl]k (c c c bm) = 0 (5.33) Without much work you can see that with ˜ gh Ki := Ki + Ki (5.34) ˜ the Ki also satisfy the same symmetry algebra with identical structure constants. Under BRST transformations,

i i 1 i k l δBRSTc = {QBRST, c } = − 2 fkl c c (5.35) k j gh ˜ δBRSTbi = {QBRST, bi} = Ki − fij c bk = Ki + Ki = Ki . (5.36) Physical states obey QBRST|ψi = 0 (5.37) which ensures that they have zero ghostiness. Now, how about doing this kind of malarkey when we have an inﬁnite number of symmetry generators, as is the case for our Virasoro algebra? In that case we have to worry about normal ordering. We have

+∞ +∞ ! X X 1 X QBRST = c−mLm − 2 (m − n): c−mc−nbm+n : (5.38) m=−∞ n=−∞ or, as a contour integral, I dz X 1 b,c QBRST = : c(z) T (z) + 2 T (z) : (5.39) C0 2πi Then we can specify the BRST current as

X 1 b,c 3 2 jBRST = cT + 2 cT + 2 ∂ c (5.40) where the 3/2 is required in order that the BRST current has conformal weight unity. 2 How bad is it to check that QBRST = 0? It is perhaps easiest to do as the jBRSTjBRST OPE, which is equivalent. If you go through the exercise, you will ﬁnd that (D − 26) 1 j (z)j (w) = ... + [(∂3c)c](w) + ... (5.41) BRST BRST 12 (z − w)

tot From this it also follows straightforwardly that in the critical dimension both jBRST and T are both BRST-exact. If you would like more detail you can consult BLT §5.2. Some useful relations are [Q, Xµ] = c∂Xµ (5.42) (D − 26) [Q, T tot] = ∂3c (5.43) 12 {Q, c} = c∂c (5.44) {Q, b} = T tot (5.45)

46 We can transform these into operator relations by contour integrals. X [Q, αn] = − ncmαn−m (5.46) m 1 [Q, L ] = − (D − 26)n(n2 − 1)c (5.47) n 12 n X {Q, cn} = − (2n + m)c−mcm+n (5.48) m tot {Q, bn} = Ln . (5.49) Then, via mode-ology, the expression for the nth total Virasoro generator is

+∞ tot X 1 Ln = 2 : αn−mαm : +(n + m): bn−mcm : (5.50) m=−∞ so that α0 Ltot = N tot + p2 − 1 (5.51) 0 4 where N tot counts the oscillator energy in all the modes. In CFT there is a state-operator correspondence emanating from the fact that specifying the boundary condition on the path integral at τ = −∞ inserts a local operator at the origin (in the plane; see radial quantization). For a more detailed exposition I recommend Polchinski Volume 1 §2.8 on Vertex Operators. Consider a vertex operator φ. Its commutator with the BRST operator involves a contour integral of jBRST against φ. By BRST invariance, the jBRST − φ OPE must not have a pole of order one, or if it does, the residue is a total derivative so that, integrated over the insertion point, it vanishes. Now specialize to a physical state with no ghost excitations. Then I dw I dw j (w)φ(z) = c(w)T X (w)φ(z) (5.52) 2πi BRST 2πi I dw h φ(z) ∂φ(z) = c(w) φ + + ... (5.53) 2πi (w − z)2 (w − z)

= hφ(∂c)φ(z) + c(∂φ)(z) (5.54) and this is a total derivative iﬀ hφ = 1 i.e. we are dealing with a physical vertex operator. So overall: primaries of dimension one create asymptotic BRST-invariant states. How about the ghost sector? Our main complication here is that there are ghost zero 2 2 modes. These satisfy b0 = 0 = c0 by the canonical anticommutation relations, and they both commute with the Hamiltonian. So there are two degenerate states for the ghost vacuum. Let us denote them as |↑i and |↓i, where

c0|↑i = 0 b0|↑i = |↓i (5.55)

b0|↓i = 0 c0|↓i = |↑i (5.56)

We can add a constant to j0 to symmetrize these as follows, 1 X N = (c b − b c ) + (c b − b c ) (5.57) gh 2 0 0 0 0 −n n −n n n>0

47 or 3 j(z) = − : b(z)c(z): − (5.58) 2z Now, for the b, c modes the SL(2, C) invariant vacuum is deﬁned by

bn|0ib,c = 0 , n ≥= −1 (5.59)

cn|0ib,c = 0 , n ≥= +2 . (5.60)

Unfortunately, the vacuum so constructed is not annihilated by c1, which means that it is not a highest-weight state of the b, c algebra, ergo is not the groundstate of the ghost system. Note that c1|0ib,c and c0c1|0ib,c both have L0 eigenvalue −1. Identify them as | ↓i and | ↑i respectively. Pick the normalization such that

b,ch0|c−1c0c1|0ib,c = +1 . (5.61)

In other words, the SL(2, C)-invariant vacuum carries 3 units of ghost number, corresponding to the 3 independent diﬀeomorphisms of the sphere generated by {L0,L−1,L1}. Using the above, we will build physical states of the form

|ψi = |φiX ⊗ |↓ib,c . (5.62)

Then requiring BRST invariance gives ! X X X Q|ψi = 0 = c0(L0 − 1) + c−nLn |ψi (5.63) n>0 which requires both

(L0 − 1)|ψi = 0 (5.64)

Ln|ψi = 0 , n > 0 , (5.65) exactly as we would have expected from our last lecture. Note: this trick would not have worked if we had built states on the up vacuum instead of the down vacuum, because c0|↑i = 0. Requiring the following will ensure those options are eliminated: ¯ b0|ψi = 0 , b0|ψi = 0 . (5.66) ¯ ¯ Then {QBRST, b0 − b0}|ψi = (L0 − L0)|ψi = 0, which is the level matching constraint. tot tot Now, how about identifying the physical states for all N ? Since {QBRST, b0} = L0 , any µ tot 0 2 tot state of deﬁnite p and N satisfying b0|ψi = 0 and Q|ψi = 0 satisﬁes α m = −4(N −1). Vertex operators of physical states are of the form

ψ(z) = c(z)φ(z) . (5.67)

Then [Q, ψ(z)] = (hφ − 1) : ∂ccφ(z) := 0 , for hφ = 1 . (5.68)

48 This is called the unintegrated form of the vertex operator. The integrated form is Z d2zφ(z, z¯) (5.69) where ψ = ccφ¯ such that [Q, φ] = ∂(cφ) and similarly for the barred sector. This ensures that [Q, ψ] = 0 and similarly for the barred sector. What are the vertex operators for the groundstates of closed bosonic strings? We are about to find out that there is not only a graviton in spacetime but also an antisymmetric tensor field Bµν (also with its own gauge symmetry) and a scalar field known as the dilaton Φ. To do so, we need only remind ourselves of the salient physical LEGOs, α0/2 h∂X(z)∂X(w)i = − (5.70) (z − w)2 and 1 T = − : ∂X∂X : (5.71) α0 which have OPEs ∂X(w) ∂(∂X(w)) T (z)∂X(w) = + + ... (5.72) (z − w)2 (z − w) α0k2/2 ∂ T (z): eik·X : = + w : eik·X : (5.73) (z − w)2 (z − w) Therefore, ∂X has conformal dimension 1 and the normal-ordered exponential has conformal dimension h = α0k2/4 (remember, |z|2 = zz¯). The field X itself does not have a well-defined conformal dimension. Two other identities worth knowing are

: eip·X(z) :: eiq·X(w) : = |z − w|α0p·q : ei(p·X(z)+q·X(w)) : = |z − w|α0p·q : ei(p+q)·X : +i|z − w|α0p·qp · (z − w): ∂X(w)ei(p+q)·X(w) : +barred (5.74) α0 ip ∂X(z): eip·X(w) : = − : eip·X(w) : + ... (5.75) 2 (z − w)

Physical string states obey (L0 −1)|ψi = 0, Ln|ψi = 0, n > 0 and similarly for the barred ¯ sector; they also obey the momentum constraint (L0 − L0)|ψi = 0. Therefore, they are just the primaries of the CFT |φi = φ(0)|0i = lim φ(z, z¯)|0i . (5.76) z,z¯→0 The really cool thing that follows from this state-operator correspondence is that string scattering amplitudes then become integrals over the worldsheet (in a sense that we will become technically exact about shortly) of correlation functions of vertex operators associated to the physical states. While this is morally simple, the precise details can get very messy indeed (see e.g. §16 of BLT). For now, we just focus on what is necessary to build vertex operators for closed string tachyons and massless modes. Now, the condition (L0 − 1)|ψi = 0 requires that R d2xφ(z, z¯) has h = (1, 1) in order to be well-deﬁned. So then a [closed string] tachyon

49 has α0k2/4 = −1, i.e., m2 = −k2 = −4/α0. It is easy to check from the definition of pµ as an operator written as a contour integral of ∂X that this vertex operator thingie does carry momentum kµ. That was for the tachyons. How about for the first excited states, which are massless? We define the vertex operator for these as

2 µ ¯ ν ik·X(z,z¯) |k; i = − µν(k) lim : ∂X (z)∂X (¯z)e : |0i (5.77) α0 z,z¯→0 µ ν = α−1α¯−1µν|ki (5.78) We need to check whether this has (h, h¯) = (1, 1) so that it can represent a physical state. Taking the OPE with the energy-momentum tensor gives

µ ¯ ν ik·X(w,w¯) T (z): µν∂X (w)∂X (w ¯)e : (5.79) iα0 kµ = − µν : ∂X¯ ν(w ¯)eik·X(w,w¯) : + (5.80) 2 (z − w)3 α0k2/4 + 1 ∂ + + w : ∂Xµ∂X¯ νeik·X : (5.81) (z − w)2 (z − w) µν So for physicality we require µ k µν = 0 (5.82) from the holomorphic sector. From the barred sector we obtain an analogous equation

ν k µν = 0 (5.83) These two conditions requires the polarization vectors to be transverse. What fields do we obtain? The tensorial property of a general two-index tensor can be split up as follows: (1) trace, (2) symmetric traceless and (3) antisymmetric. The resulting bulk spacetime fields are known as the dilaton Φ, graviton Gµν (with its well-known diffeomorphism symmetry) and the Kalb-Ramond field Bµν which also has a gauge symmetry analogous to that of electromagnetism only with one more spacetime index. As for higher excited states, it is possible to prove a formula known as the Cardy formula describing the density of states at large level number, r ! cL ρ(L ) ∼ exp 2π 0 . (5.84) 0 6

This might look unassuming but please notice that it becomes exponentially large at large L0 eigenvalue. This means that the entropy S grows like the energy E at high-energy, meaning that there is a limiting temperature at which the canonical ensemble still makes sense - delineated by the point at which the Cardy exponential overwhelms the Boltzmann weight. This limiting temperature is called the Hagedorn temperature.

5.4 The Virasoro-Shapiro amplitude In this subsection, we will only look at (1) tree amplitudes and (2) n-point amplitudes where n ≤ 4. This is done in order to keep the bit rate lower than that of a ﬁrehose. For > 4-point

50 amplitudes the integration over insertion points is a nontrivial technical challenge, and for loop amplitudes the conformal Killing vectors (CKVs) have to be analyzed in full careful detail – parametrizing the moduli space of Riemann surfaces of higher genera is a bitch. Conformal symmetry on the worldsheet (in critical dimension) means that diagrams shaped like (say) two closed strings joining and then splitting again, can be conformally mapped into the sphere with four marked points. The diagrams are traditionally drawn with × denoting insertions of the vertex operators representing physical states (that are killed by the BRST operator QBRST) at particular points on the worldsheet in the z, z¯ coordinates. Generally, our n-particle amplitude takes the form

Z 2 2 n−2 d z1 . . . d zn A ∼ g hV (z ) ...V (z )i 2 (5.85) tree s (vol(CKV) 1 1 n n S where gs is the string coupling. The n − 2 in front comes about because this is the Euler 2 2 number of an n-punctured S . Alternatively, you can count it as 1/gs for the sphere and gs for each vertex operator. At tree level, the holomorphic and anti-holomorphic parts just factorize simply. This simple factorization does not hold at loop level and it is important to dig into the full technical detail to see this. We do not have time in this survey-level course to get this technical. BLT §6 is where they develop in detail the nasty details; we will just cherry pick a few key facts here to enable us to build up to a four-point tree amplitude before we move on to the superstring and heterotic constructions next week. First, the CKVs correspond to zero modes of the c ghost. The zero modes of the b antighost correspond to animals known as Beltrami differentials, and the number of them is given by the Riemann-Roch theorem Nb − Nc = 3(g − 1). Luckily for us, at genus zero (the sphere, worldsheet tree level) there are no b zero modes as there are no moduli. There are 2 however 3 c zero modes corresponding to the generators ∂z, z∂z, z ∂z. We met these buggers already when figuring out the ghost vacuum business. Fix 3 vertex operators at positions z1, z2, z3. Vector fields generating the 3 CKV sym- 2 R 2 R 2 metries, (α + βz + γz )∂z, mean that we can trade d {z1, z2, z3} for d {α, β, γ} modulo the Jacobian ∂(z1, z2, z3) 2 = |(z1 − z2)(z2 − z3)(z3 − z1)| , (5.86) ∂(α, β, γ) R 2 i 2 i and d {α, β, γ} =vol(CKG). This can also be found as | det(V (zj))| where {V } = {1, z, z2} are the CKVs and

  2 1 1 1 2 2 2 2 2 |h0|c(z1)c(z2)c(z3)|0i| = |h0|c−1c0c1|0i| det z3 z2 z1 = z12z23z31 (5.87) 2 2 2 z3 z2 z1 So, formally, in the path integral what we do to divide out by the volume of the conformal Killing vectors is to insert a ghost ﬁeld at 3 chosen points and drop the integrations over their positions. Usually, the 3 marked points are chosen at {0, 1, ∞}. SL(2, C) invariance actually makes the zero-, one-, and two-point functions vanish. The ﬁrst nontrivial amplitude is the three-point amplitude, and we will only consider the simplest

51 situation of the bosonic string involving three tachyons. Using the above rules, we have (3) 3 3 Abos(T ) = CS2 gc hccV¯ (1)ccV¯ (2)ccV¯ (3)i = gc CS2 (5.88) where CS2 is a physical normalization constant that can be determined by unitarity and known physics of gauge and graviton particle theory amplitudes (see BLT and/or Polchinski Vol.1 for details). Here, we used the form of the vertex operators V (k) =: eik·X : and the fact 2 2 0 0 that, on-shell, k = −m = 4/α , as well as α ki · kj/2 = −2 for i 6= j. The only other input we have used is the usual trick of ignoring the overall momentum conservation delta-function proportional to the volume of spacetime familiar from particle theory. One last step you will need to reproduce the result is n D D X Y 2α0ki·kj /2 hV (1)V (2)V (3)i = (2π) δ ( ki) |zij| (5.89) i i

2 Y 2α0ki·kj /2 |z12z14z24| |zi − zj| (5.91) i

(4) 4 Γ(α(s))Γ(α(t))Γ(α(u)) A (k , k , k , k ) = 2πg C 2 (5.94) bos 1 2 3 4 c S Γ(α(t) + α(u))Γ(α(s) + α(u))Γ(α(s) + α(t)) where α0 α(s) = −1 + s (5.95) 4 and similarly for t, u, where {s, t, u} are the standard Mandelstam kinematic invariants 2 2 2 P4 2 s = −(k1 +k2) , t = −(k1 +k3) , u = −(k1 +k4) which obey the relation s+t+u = i=1 mi . For our closed bosonic string tachyons, m2 = −4/α0. The above formula is called the Viraroso-Shapiro amplitude, the closed string cousin of the open-string Veneziano amplitude. It has two properties which you can immediately see are physically important. First, it has crossing symmetry. This is why it was called a dual model when string theory was originally invented in the late 1960s, in an attempt to explain the strong nuclear force. Second, it has simple poles in each channel (and an inﬁnite number of them), e.g. in the s-channel you get poles at s = −4/α0, 0, +4/α0, +8/α0, etc. These correspond to exchange of a closed excited string state. The Virasoro-Shapiro amplitude does not have any higher order poles.

52 6 Superstrings

The famous Coleman-Mandula theorem states that spacetime symmetries cannot be com- bined with internal symmetries (such as colour or hypercharge) in a physically nontrivial fashion. That is why we treat Standard Model gauge group indices as completely independent of spacetime indices when we write down quantum fields for quarks and leptons: the structure is a trivial tensor product. Now, the Coleman-Mandula theorem assumed that the generators of symmetry transformations were bosonic. If you allow them to be fermionic then Poincarécan actually be nontrivially extended, and the result is called supersymmetry. SUSY transformations have fermionic parameters and fermionic generators; the group elements formed via the exponential map are still bosonic. So in addition to the momen- i tum Pµ and angular momentum/boost generators Mµν, there are also SUSY generators Qα, where the i index is used only for extended SUSY, which obey very specific anticommutation relations that we began to introduce in the first lecture. In particular, we saw that if you do two SUSY transformations in a row you get back a translation (etc.), as we saw when we discussed the role of BPS states. In order to be able to say anything about the physics of supersymmetry, we first need to learn how to build spinors in diverse dimensions. We pattern our discussion here largely on Appendix B1 of Polchinski Volume 2, referred to in the following as Pol.vol2.appB1.

6.1 Spinors in diverse dimensions Let spacetime have signature (D − 1, 1). The Dirac algebra is deﬁned by

{Γµ, Γν} = 2gµν1 . (6.1)

First consider the case of even dimensions; the odd dimensional case is a relatively simple gen- eralization. The fundamental algebra above implies that different Γµ anticommute with each other, while each one squares to ±1. Therefore, the product of a finite number of them can always be reduced to the following set of 2D matrices {ΓA} = {1, Γµ, Γ[µΓν],..., Γ0Γ1 ··· ΓD−1} which are all linearly independent. Notice that (ΓA)2 = ±1, for any A, regardless of whether A µ µ −1 A a b h the Γ is a product of 1 or more Γ s. Note also that (Γµ) = (Γ ) ; also, if Γ = Γ Γ ... Γ A A for some a, b, . . . , h then ΓA = Γh ... ΓbΓa and these satisfy Γ ΓA1 = ΓAΓ . In even dimensions only, we can define a ΓD+1 by ΓD+1 = i−kΓ0Γ1 ··· ΓD−1, where D = 2k+2, which squares 1 µ µν i µ ν to and commutes with the Γ as well as with the Lorentz generators Σ = − 4 [Γ , Γ ]. The only nonzero trace of a ΓA occurs for the identity which produces D; otherwise, tr(ΓA) = 0 if ΓA 6= 1. We also have the Fundamental Theorem: if Γµ and Γ˜µ are two different irreducible representations of the Dirac algebra in the gamma matrix space, then there exists an invertible matrix S defined up to a constant such that Γµ = SΓ˜µS−1 for all µ ∈ {0, 1,...,D − 1}. What changes when D is odd? Here, we select D − 1 of the Γs (i.e., Γ0, Γ1,..., ΓD−2) and apply the previous analysis. Then, obviously, ΓD−1 anticommutes with the other ones by the Dirac algebra. Since these D − 1 generators fully span the space of matrices, ΓD−1 = (const.) × Γ0Γ1 ... ΓD−2, because only that product of all the Γs in a row can anticommute with the first D − 2 of them. Note also that ΓD+1 has to be proportional to the identity in odd dimensions.

53 Now we get to the interesting bit: building the Fock space representation for spinors. We do it by selecting pairs of the original set of Γ matrices,

0 ± 1 0 1 (Γ ) := 2 ±Γ + Γ (6.2) a ± 1 2a 2a+1 (Γ ) := 2 ±Γ ± iΓ (6.3) where a = 1, . . . , k. These satisfy {Γa+, Γb−} = δab and {Γa+, Γb+} = 0 = {Γa−, Γb−} and so can be thought of as anticommuting creation and annihilation operators. This is very reminiscent of Pauli exclusion. Indeed, for each pair of Γ±, the Γ+ behaves just like a b† and Γ− like a b. We start by ﬁnding the ζ annihilated by all the Γa−. Then acting with Γa+ at ~s k s 1 s 1 1 + k+ 2 0+ 0+ 2 most once each gives ~s = (s0, s1, . . . , sk) where si = ± 2 ∀i: ζ = (Γ ) ··· (Γ ) ζ. Overall, we see that spinors must have dimension 2[D/2]. Note that this is 2 to the power [D/2], where [...] denotes the integer part. This is because if you are in odd dimensions and have an extra Γ to play with you cannot enlarge the Fock space because the extra Γ is not paired up. So spinor dimensionality is not mysterious: it is pre-ordained through the Dirac algebra from which all of the above follows. This is why spinors in 4D are four dimensional. There are three subclasses of spinors available: Weyl, Majorana, and Majorana- Weyl. Weyl spinors exist in even dimensions and are chiral; the Weyl projectors P± := 1 1 D+1 2 ± Γ split the spinor into left- and right-chiral pieces. Majorana spinors are real; Majorana-Weyl are both chiral and real. Which types of spinors can occur in which dimensions can be looked up in any decent textbook; we summarize the reults from Pol.vol2.appB1 in Figure 7.3. We will indicate how the Majorana analysis works for even dimensions here; see Pol.vol2.appB1 for more details on the odd-dimensional case and further generalities. After the dust settles, one noteworthy example will be that Majorana-Weyl spinors are available both for the worldsheet case D = 1 + 1, where they are one-dimensional, and the critical dimension for superstrings D = 9 + 1, where they are sixteen-dimensional.

54 In the basis ~s, matrix elements of Γa± are real. Using the deﬁnitions for Γa± we can easily see that Γ3, Γ5,..., ΓD−1 are imaginary while all the other Γµ are real. Deﬁne

3 5 D−1 D+1 B1 = Γ Γ ··· Γ ,B2 = Γ B1 . (6.4)

µ −1 k µ∗ µ −1 k+1 µ∗ Then by the Dirac algebra, B1Γ B1 = (−1) Γ and B2Γ B2 = (−1) Γ Also, by µν −1 µν∗ −1 ∗ virtue of the form of the Lorentz generators, B1,2Σ B1,2 = −Σ . Therefore, ζ and B1,2 ζ transform the same way under Lorentz transformations so that the Dirac representation is D+1 −1 k D+1∗ its own conjugate. Also, B1,2Γ B1,2 = (−1) Γ , so that either B1 or B2 will change the eigenvalue of ΓD+1 when k is odd and not when k is even. Further, for k even (D = 2 mod 4) each Weyl representation is its own conjugate. For k odd (D = 0 mod 4) each Weyl representation is conjugate to the other. A reality condition ζ∗ = Bζ is consistent only if ∗ ∗ k(k+1)/2 ∗ k(k−1)/2 B B = 1. In fact, B1 B1 = (−1) and B1 B1 = (−1) . This means that you can use B1 for a Majorana condition if k = 0 mod 4 or k = 3 mod 4, and you can use B2 for one if k = 0 mod 4 or k = 1 mod 4. For k = 0 mod 4, B1 and B2 are related by a similarity transformation and hence physically equivalent. Finally, we note that Majorana- Weyl spinors require (a) a Majorana condition and (b) that the spinor be self-conjugate. This requires k = 0 mod 4 or equivalently D − 2 = 0 mod 8, which is relevant to both the D = 1 + 1 worldsheet and the bulk D = 9 + 1 target spacetime.

6.2 Green-Schwarz formalism There are several ways to handle the issue of SUSY for superstrings. The two which you will see the most often in standard string theory textbooks like Polchinski, BBS, BLT, and Green-Schwarz-Witten are the Neveu-Schwarz-Ramond formalism, which insists on worldsheet supersymmetry, and the Green-Schwarz formalism, which insists on spacetime supersymmetry. Both preserve Poincarésymmetry. The Berkovits pure spinor formalism gives a very powerful (and very technical!) third way forwards; we will not have time to say anything about it here. I already have several pages of notes online describing the basics of the Green-Schwarz approach and kappa symmetry at http://ap.io/2406s/notes/st20080305.pdf, first for point particles and then for superstrings, using light-cone gauge when necessary to illustrate the physics. Accordingly, I will not repeat the material here in LATEX: I cannot afford the typing pain! I will just give a brief summary of the main points. Those interested in further details should consult Green-Schwarz-Witten. We already know that the action principle we start from has reparametrization invariance. Introduce fermionic superpartners for the bosonic degrees of freedom Xµ, which are θa, where a is the spinor dimension appropriate to D dimensions. The action constructed has SUSY, but it also possesses a qualitatively different symmetry involving the bosonic and fermionic worldsheet fields called kappa symmetry. Kappa symmetry serves to provide a projection operator that reduces the rank of a spinor by a factor of two, and it is neither SUSY on the worldsheet nor SUSY in the target spacetime. It is possible to write down a kappa symmetric worldsheet action for the superstring, known as the Green-Schwarz action, which preserves worldsheet reparametrization invariance as well as target spacetime super- Poincaréinvariance. But it requires a gnarly self-consistency condition that arises out of

55 ¯ µ making the most of Fierz identities: ¯Γµψ[1ψ2Γ ψ3] ≡ 0, with (ψ1, ψ2, ψ3) = (θ, ∂σθ, ∂τ θ). This puts a very strict condition on the dimension of spacetime for classical superstrings. It can be satisﬁed only when D = 3, 4, 6, 10 (6.5) when we have one timelike dimension of spacetime, which is the physical number. No other dimension permits the correct matching of bosonic and fermionic degrees of freedom. Those are also the dimensions in which local kappa symmetry can be implemented. Quantum consistency of the superstring will select out the case D = 10 as the unique allowed spacetime dimension. A useful mathematical property known as triality, an automorphism of SO(8), allows you to relate the eight-dimensional vector spinor and the conjugate spinor representations of SO(8). In light-cone gauge, the fermion analogue of the light-cone gauge condition + + − welding X to τ lets you set ψα = 0; in turn, this allows solving for the ψα oscillators in I terms of the transverse ψα. Working through the kappa symmetric details you end up with equations of motion on the worldsheet which are the Weyl equations of motion for chiral spinors recognizable from QFT1. Type I superstring theory eventuates when you set to zero one or other of the two spinors of N = 2 worldsheet supersymmetry allowed by kappa symmetry. Type IIB is a chiral theory which eventuates when both spinors have the same chirality, and Type IIA is a nonchiral theory which eventuates when they have the opposite chirality. Heterotic theories arise via a hybrid construction where the left side is bosonic i while the right side is supersymmetric; the remaining 16 leftover XL end up being bundled up into a current algebra. Requiring that all anomalies cancel gives SO(32) or E8 × E8.

6.3 Neveu-Schwarz-Ramond formalism With worldsheet SUSY in the NSR formalism, we partner every worldsheet boson ﬁeld with a worldsheet fermion ﬁeld. In particular, this means that our worldsheet fermions ψµ carry a spacetime vector index just like the Xµ do. The following superstring action on the worldsheet in conformal gauge (see Polchinski eq.10.1.5) has a global worldsheet supersymmetry,

1 Z 2 S = d2z ∂Xµ∂X¯ + ψµ∂ψ¯ + ψ˜µ∂ψ˜ , (6.6) 4π α0 µ µ µ where ψµ are anticommuting worldsheet ﬁelds (Majorana-Weyl spinors. Look for a minute at the fermionic term. Its variation must vanish for the initial value problem to be well deﬁned; a simple integration by parts shows that Z 1 ¯ ¯ σ=π δSf = dτ ψ+δψ+ − ψ−δψ− . (6.7) 4π σ=0 where we have temporarily restored the + subscript on the right-movers and the − subscript on the left-movers for clarity. In order to make the variation vanish for open strings, the endpoint terms have to cancel separately at σ = 0 and σ = π, implying that at those two points (and only at those two µ µ points) ψ+ = ±ψ−. The overall relative sign is conventional and taken to be positive at

56 σ = 0. Then

µ µ R: ψ+(τ, σ = π) = +ψ−(τ, σ = π) , (6.8) µ µ NS : ψ+(τ, σ = π) = −ψ−(τ, σ = π) . (6.9) These R ﬁelds are integrally moded on the cylinder while the NS ﬁelds are half-integrally moded. In order to make the variation vanish for closed strings, we need

ψ±(τ, σ) = ±ψ±(τ, σ + π) (6.10) so we impose either NS or R boundary conditions in both the holomorphic and anti- holomorphic sectors. This gives rise to NS-NS, R-R, NS-R and R-NS states.

6.4 ZPEs, physical state conditions and the GSO projection Old covariant quantization yields decoupling of spurious states for

1 D = 10 , aNS = − 2 , aR = 0 . (6.11) 1 Recall that for the bosonic string we found a = − 24 for each transverse boson; this came 1 from 2 ζ(−1) in zeta function regularization. If you rerun the calculation for the fermion 1 contributions in the NS sector you find − 48 for each fermion. So the ZPE for our superstring 1 1 1 in the NS sector arises from (a) 8 × − 24 + 8 × − 48 = − 2 . In the R sector, which preserves 1 1 unbroken worldsheet supersymmetry, you find − 24 × 8 + 24 × 8 = 0. The vacuum in the NS sector is unique, but the Ramond vacuum carries a representation of the Dirac algebra via µ the zero modes ψ0 and is a spacetime spinor. The physical state conditions are as for the bosonic string, except that (a) there are fermionic as well as bosonic contributions to the mode sums and (b) there is a new condition that the fermionic partners of Ln – the Gr – also annihilate the physical states for positive mode number r. Light cone quantization welds X+ to τ, disallowing it any oscillators. The fermionic analogue is ψ+ = 0, which can be used to solve for the + direction oscillators in terms of our friends the transverse oscillators. The result for the mass formula is the same as previously found for the bosonic string except that the zero-point energies are shifted according to (6.11). First consider the open string. In the NS sector the groundstate |0NS; ki is again a 0 2 1 i scalar, a tachyon with α m = − 2 ; the first excited state b−1/2|0NS; ki is a massless vector of SO(8) which is the transverse Lorentz group. In the R sector the groundstate has unbroken supersymmetry and its groundstate is a SO(8) spinor; all the excited states are also spinors. Define G-parity by

P bi bi +1 FNS +1 r + 1 r r GNS = (−1) = (−1) ∈N 2 − (6.12) P i i \ n d ndn GR = Γ (−1) ∈N − (6.13) where Γ\ = Γ0Γ1 ··· ΓD−1 (note that we use \ to denote the number that follows 9 as a single digit). The GSO projection sets

(−1)FNS = +1 (6.14)

57 which projects out the tachyon and every second oscillator level above it. This is consistent not only in free string theory but to all orders in perturbation theory. Note that in the R sector the G-parity is conventional depending on the chirality of the spinor groundstate. The second major advantage of the GSO projection (apart from killing the unphysical tachyons) is that it ensures spacetime supersymmetry with all the nice implications thereof. It sounds simple in principle, and it is, but the technical details are quite involved. See e.g. BLT. The spectrum of massless states for the closed string is straightforward to work out; see e.g.BBS §4.6 for further details.

NS − NS : Gµν(35) ⊕ Bµν(28) + Φ(1) (6.15) α α NS − R, R − NS : ψµ (56) ⊕ λ (8) (6.16) + R − R : IIA : Cµ(8) ⊕ Cµνλ(56); IIB : C(1) ⊕ Cµν(28) ⊕ Cµνλσ(35) . (6.17) The names of the NS-R and R-NS fermions are the gravitino and the dilatino. The Cs are called R-R p-form antisymmetric tensors. The NS-NS sector is common to all string theories, except for the unoriented ones which project out the Kalb-Ramond ﬁeld B.

6.5 SCFT, (β, γ) ghosts, and BRST Armed with the NSR worldsheet action in conformal gauge, we can work out some important basic things. First is equations of motion give us that

∂∂X¯ µ = 0 ⇒ Xµ(z, z¯) = Xµ(z) + Xµ(¯z) (6.18) ∂ψ¯ (z, z¯) = 0 ⇒ ψ(z, z¯) = ψ(z) (6.19) ∂ψ˜(z, z¯) = 0 ⇒ ψ˜(z, z¯) = ψ˜(¯z) (6.20) and the propagators are given by

α0 ηµν α0 ηµν h∂Xµ(z)∂Xν(w)i = − h∂X¯ µ(¯z)∂X¯ ν(w ¯)i = − (6.21) 2 (z − w)2 2 (¯z − w¯)2 ηµν ηµν hψ(z)ψ(w)i = hψ˜(¯z)ψ˜(¯z)i = (6.22) z − w z¯ − w¯ We can then determine that the energy momentum tensors are given by

1 2 T (z) = − : ∂Xµ(z)∂X (z) : + : ψµ(z)∂ψ (z): (6.23) B 2 α0 µ µ 1 2 T˜ (¯z) = − : ∂X¯ µ(¯z)∂X¯ (¯z) : + : ψ˜µ(¯z)∂¯ψ˜ (¯z): . (6.24) B 2 α0 µ µ

We can work out the world-sheet supercurrents associated with supersymmetry transformations, which gives us

r 2 r 2 T (z) = i ψµ(z)∂X (z) T˜ (¯z) = i ψ˜µ(¯z)∂X¯ (¯z). (6.25) F α0 µ F α0 µ

58 To find the supersymmetry transformations, we can find the OPE with these generators with µ µ ˜ ˜µ the fields. So when we calculate TF (z)X (w), TF (z)ψ (w), and TF (¯z)ψ (w ¯), the singular terms will give us

rα0 δXµ(z, z¯) = − ((z)ψµ(z) + ˜(¯z)ψ˜µ(¯z)) (6.26) 2 r 2 δψµ(z) = (z)∂Xµ(z) (6.27) α0 r 2 δψ˜µ(¯z) = ˜(¯z)∂X¯ µ(¯z). (6.28) α0 Just for example, we can work out

2 1/2 T (z)Xµ(w) = i : ψν(z)X (z): Xµ(w) (6.29) F α0 ν 2 1/2 = i ∂ hX (z)X (w)iψν(z) (6.30) α0 z ν µ α0 1/2 ψµ(w) = −i + regular (6.31) 2 z − w 1 = iδXµ(w) . (6.32) z − w So, we get the transformation above as stated. Now we could go back and show that the action is invariant under this transformation. From here, we could work out a variety of diﬀerent OPEs. They are given by

3D 2T (w) ∂T (w) T (z)T (w) = + B + B (6.33) B B 4(z − w)4 (z − w)2 z − w 3T (w) ∂T (w) T (z)T (w) = F + F (6.34) B F 2(z − w)2 z − w D 2T (w) T (z)T (w) = + B . (6.35) F F (z − w)3 z − w

We can glean important information out of these. The ﬁrst OPE tells us that the central 3 charge for this SCFT is given by c = 2 D. This comes from the fact that each scalar ﬁeld 1 adds 1 to the central charge and each fermion adds 2 (as you are proving for yourself in 3 HW2). The second OPE tells us that the supercurrent is a tensor of weight ( 2 , 0). How about the ghosts? In addition to the bc ghosts from the bosonic part, we must add a commuting (β, γ) ghost system to handle the Fadeev-Popov determinant. Whereas the 3 1 b ghost has weight (2, 0) and c has (−1, 0), the new ghosts have weight ( 2 , 0) and (− 2 , 0) for β and γ respectively. We also have their antiholomorphic counterparts. i Z S = d2z(b∂c¯ + ¯b∂c¯ + β∂γ¯ + β∂¯ γ¯) (6.36) gh π

59 so that

gh 3 1 TB = −2b∂c + c∂b − 2 β∂γ − 2 γ∂β (6.37) gh 3 TF = −2bγ + c∂β + 2 β∂c . (6.38) For this superstring case, 1 I Q = d2z cT matter + γT matter + bc∂c − 1 cγ∂β − 3 cβ∂γ − bγ2 (6.39) BRST 2πi B F 2 2

How about the ghost vacuum? In the NS sector, the (β, γ) system has half-integer moding on the cylinder so we have a two-fold degeneracy like we did for the (b, c) system 1 and the physical states have ghost number − 2 . In the R sector the Fock space built from zero modes β0, γ0 gives rise to an infinite degeneracy. These are referred to as different pictures which are physically equivalent for most purposes and are labelled by an integer. Picture-changing operators allow you to transform from one picture to another. In the story of vertex operators, some restrictions apply on which pictures you can choose. You also have to be a lot less cavalier about the conformal Killing vectors, moduli space, etc. How about the critical dimension? If we worked out the boring details using SCFT techniques we would find that the total central charge for the ghost system is −26+11 = −15. Here the 11 comes from the (β, γ) portion. This means that if we include all the matter fields – bosons and fermions – and the ghosts, the total central charge is

1 ctot = D(1 + 2 ) + (−26 + 11) . (6.40) If we want the Weyl anomaly to vanish, we require that D = 10, which gives us the critical dimension for superstring theories. How about modings on the plane? The Ramond and Neveu-Schwarz boundary conditions are, on the z plane,

Ramond (R): ψµ(w + 2π) = ψµ(w) (6.41) Neveu-Schwarz (NS): ψµ(w + 2π) = −ψµ(w). (6.42)

Note that since we wish for our theories to have maximum Poincar`einvariance, we require Xµ to be periodic (in other words, antiperiodicity breaks translation invariance). So we can expand into modes on the plane by,

µ X µ −r−1/2 ˜µ X ˜µ −r−1/2 ψ = ψr z ψ = ψr z¯ (6.43) r∈Z+v r∈Z+˜v

1 where v,v ˜ are 0 or 2 for R or NS BCs respectively (i.e. sum over integers or half-integers). Following this expansion, we could expand our worldsheet supercurrent, TF out in terms of its modes by X −r−3/2 TF (z) = Grz . (6.44) r∈Z+v

60 q α0 P −m−1 P −m−2 We can also expand ∂X = −i 2 m αz and TB = m Lmz as usual. Then we could use our OPEs that we discovered before and contour integrals to ﬁnd the commutation relations and anticommutation relations for the operators. These are given by c [L ,L ] = (m − n)L + (m3 − m)δ (6.45) m n m+n 12 m+n,0 c {G ,G } = 2L + (4r2 − 1)δ (6.46) r s r+s 12 r+s,0 m − 2r [L ,G ] = G . (6.47) m r 2 m+r For r,s integer, this is the Ramond algebra and for r,s half-integer, this is the Neveu-Schwarz algebra. We also get their anti-holomorphic copies. We can work out an example, I I dw dz r+1/2 s+1/2 {Gr,Gs} = z w TF (z)TF (w) (6.48) 2πi Cw 2πi I dw I dz D 2T (w) = zr+1/2ws+1/2 + B (6.49) 3 2πi Cw 2πi (z − w) z − w I I dw dz D r+s−1 r+s+1 = (r + 1/2)(r − 1/2)w + 2w TB(w) (6.50) 2πi Cw 2πi 2 D = (r + 1/2)(r − 1/2)δ + 2L (6.51) 2 r+s,0 r+s c = (4r2 − 1)δ + 2L , (6.52) 12 r+s,0 r+s

3 where we have used c = 2 D. Finally, we could expand these operators in terms of the mode operators of the matter ﬁelds. We would ﬁnd that

1 X 1 X L = : αµ α : + (2r − m): ψµ ψ : +amδ (6.53) m 2 m−n µn 4 m−n µn m,0 n∈Z r∈Z+v X µ Gr = αnψµ r−n. (6.54) n∈Z Finally, in this section, let us say something about the states in the superstring SCFT. Recall the commutation and anticommutation relations for Lm and Gr. If we look at what the commutators are with L0, we will ﬁnd [L0,Lm] = −mLm and [L0,Gr] = −rGr. So we see that the Gr operators will act as lowering operators as well (for m, r > 0). We can look at our highest weight states as we did in the bosonic CFT. These should satisfy

L0|hi = h|hi (6.55)

Lm|hi = 0 for m > 0 (6.56)

Gr|hi = 0 for r > 0. (6.57)

We will discuss the interpretation of G0 (which is in the R sector) later. In the bosonic case, we found that unitarity forced non-ghost ﬁelds to have h, c ≥ 0. We can now use the Gr

61 operators to further constrict these inequalities. We consider the condition,

hh|GrG−r|hi = hh|{Gr,G−r}|hi (6.58) c = hh|(2L + (4r2 − 1)|hi (6.59) 0 12 c = 2h + (4r2 − 1)hh|hi ≥ 0. (6.60) 12 For the NS sector, the lowest non-negative value for r is r = 1/2. In this case, we only require h ≥ 0 as before. However, in the R sector the lowest is r = 0, so we see that h ≥ c/24. To continue, let us focus on the NS sector. So we wish for our two energy momen- P −n−2 tum tensors to be regular on our vacuum. Recall that TB(z) = n Lnz and TF (z) = P G z−r−3/2. So then we require r∈Z+1/2 r

Ln|0i = 0 n ≥ −1 h0|Ln = 0 n ≤ 1 (6.61)

Gr|0i = 0 r ≥ −1/2 h0|Gr = 0 r ≤ 1/2. (6.62)

Suppose we have a primary conformal field of weight h, given by φ and its superpartner field of weight h + 1/2, given by ψ. We can define the states

φ(0)|0i , ψ(0)|0i. (6.63)

We can recall that for our a primary ﬁelds that

1 [Lm, φn] = [m(h − 1) − n] φn+m [Lm, ψn] = m(h − 2 ) − n ψn+m. (6.64) However, with the supersymmetry generators, we have a different set of commutators. To do this formally, we would define the supersymmetric conformal fields, show their supersymmetry transformations, find OPEs for TF (z)φ(w) and TF (z)ψ(w), and then use these in the commutator relationship using contour integration. The details are very similar in spirit to previous computations shown. The results for the commutators are given by [G , φ ] = ψ [G , ψ ] = m h − 1 − 1 n φ . (6.65) r n 2 n+m r n 2 2 n+m where is a constant Grassman parameter. Obviously we have seen that the state φ(0)|0i = φ−h|0i = |hi is a highest weight state with conformal weight h. However, we want to see 1 how the new states given by ψ(0)|0i = ψ−h−1/2|0i go. Note that [G−1/2, φ−h] = 2 ψ−h−1/2. So we see that

ψ(0)|0i = ψ−h−1/2|0i (6.66)

= 2[G−1/2, φ−h]|0i (6.67)

= G−1/2|hi. (6.68)

From here, we could use our commutators to work out further properties if desired. Now we can look at the R sector. Things are a bit different here. We wish to find a vacuum state, as we did before. However, we have forgotten to take into account our supersymmetry. In the NS sector, supersymmetry is not broken by our definition of the

62 vacuum. However, we must be more careful here. We note that from our commutation 2 c relations, we can see that G0 = L0 − 24 . G0 is a global supersymmetry charge. Therefore, for our vacuum to have unbroken supersymmetry, we require that G0|0iR = 0. However, c this implies that L0|0iR = 24 |0i. This is a good thing however, as our unitarity condition meant that even the vacuum had to have some non-zero weight and this vacuum satisﬁes this condition. The other operators acting on the vacuum remains unchanged from the NS conditions (other than the fact that Gr have r ∈ Z). This leads to an interesting point. Since [G0,L0] = 0, we can deﬁne two heighest wieght states,

+ − + |h i, |h i = G0|h i. (6.69)

This may look bad as there might be two diﬀerent vacua for our theory. However, we note that if |h+i is the vacuum, then

− − + 2 + hh |h i = hh |G0|h i (6.70) c = hh+|L − |h+i (6.71) 0 24 = 0. (6.72)

So then the |h−i simply decouples from the physics and ignored (of course, this is only true if the supersymmetry is unbroken). One final note. In order to introduce desired field operators, we need to introduce the spin fields,

|h+i = S+(0)|0i, |h−i = S−(0)|0i. (6.73)

These spin ﬁelds satisfy the OPEs 1 T (z)S+(w) = S−(w) + less singular (6.74) F 2(z − w)3/2 h − c/24 T (z)S−(w) = S+(w) + less singular. (6.75) F 2(z − w)3/2

It is these fields that bring in our anti-symmetric boundary conditions ψ(ze2πi) = −ψ(z). As one can see, they have branch cuts, which are designed very carefully to ensure that the mathematics on the complex plane gives the right physics. As the spin fields interpolate between our NS vacuum and the R vacuum, we can define the action of the spin fields on our fermionic operators as, I dw ψRS±(w) = ψNS(w)(w − z)n+h−1S±(z) (6.76) n 2πi There is much MUCH more to say about the physical and mathematical delights of the superstring, but we have run out of time. I hope this aperitif was at least sufficient to pique your interest. Next week, we will get to discussing D-branes and T-duality for open and closed strings, and superstring duality with its connection to eleven dimensional M Theory.

63 7 T-duality, D-branes and superstring duality

In essence, T-duality is a symmetry that relies on the ability of strings to wind around a circular dimension, something which particles obviously cannot do. The rough intuition goes as follows. Strings, like particles, must fit an integer number of de Broglie wavelengths around the circular dimension z in order to have a nonzero wavefunction. This requires the 1 KK momentum pz to be quantized in units of 1/R, where R is the radius of the S . Winding of a string around the circular dimension is also quantized, with the winding number labelling topologically different sectors, and it literally represents the body of the string winding around z an integer number of times before joining back up with itself. The ability of strings to have KK momentum and winding simultaneously is what makes T-duality, the first string duality, possible.

7.1 KK reduction in QFT Before we dive in to the physics of putting one coordinate on a circle for string theory, let us do a lightning review of the Kaluza-Klein procedure in regular old quantum field theory. Let us split up D spacetime coordinates {xM } into d = D − 1 coordinates xµ and the Dth coordinate z around the S1. The SO(D − 1, 1) symmetry is broken down to SO(d − 1, 1) which we can use to classify the tensors. Then the D-dimensional metric tensor can be decomposed into a d-dimensional metric tensor gµν, a vector gµz, and a scalar gzz. The vector is known as the Kaluza-Klein gauge field and objects in the theory can carry electric or magnetic charge. Now, what happens to the action principle when we have circle compactification? If you work out the Ricci scalar in D dimensions, you find that it is a sum of three terms written in the language of d dimensional tensors: (1) the Ricci scalar in d dimensions, (2) a scalar field kinetic term, and (3) a KK gauge field strength term with an exponential coupling to the scalar, which has a gauge symmetry descended from D dimensional diffeomorphism invariance. (The presence of the scalar is mandatory, and it is what threw the monkey wrench into the idea of unifying gravity and electromagnetism by recruiting a fifth dimension.) The scalar φ, representing the size of the compactified dimension, has no potential energy function. It is massless and called a modulus: the different values of φ label the degenerate vacua of the theory. Furthermore, by just chasing the constants, we can see quickly that 1 2πR = z (7.1) Gd GD which provides the link between the Planck length `P,D in D dimensions and the Planck D−3 D−2 length `P,d in d dimensions. Note that we use the convention 16πGD = (2π) `P,D . Now suppose that we inspect the physics of a scalar field, which we take to be massless in D dimensions for simplicity. Assume that it couples minimally to gravity so obeys a simple Klein-Gordon equation. If we simplify further to the case where the spacetime metric is flat in the z dimension, then we can expand the scalar in plane waves easy peasy,

µ X µ inz/R φ(x , z) = φn(x )e (7.2) n∈Z

64 to find n2 ∂µ∂ − φ = 0 . (7.3) µ R2 n In order to describe the full D-dimensional dynamics for φ, we need to include the entire infinite tower of KK modes. Were we to chop off the tower at some finite level, we would lose information and we would also find that gauge invariances interrelating different KK modes would break. If we work at such low energies that the ambient energy budget is much lower than the gap energy 1/R, then all we will be left with for fluctuating degrees of freedom is the zero modes, i.e., we can treat the physics as d dimensional instead of D dimensional. This is a one-dimensional prototype of the idea of string compactification: start in the native ten dimensions, find the low-energy Lagrangian using the beta-function story we will describe in the next lecture, and then compactify dimensions to find the physics of the four-dimensional fields appropriate to low energy scales. One final physical remark: notice that we do not need z ∈ (−∞, +∞) to produce 1 i`x0 momentum eigenstates on S . The operators O` = e for l ∈ Z are well-defined, and using † the formal canonical commutation relation [x0, p] = i it follows that O` p (O`) = p + `/R i.e. (O`|pi) has momentum (p + `/R).

7.2 T-duality for closed bosonic strings Now let us get more serious and ask about compactifying the 25th direction of bosonic string theory X25 on a circle of radius R. Putting it on a circle does not change the wave equation satisﬁed but it does change the boundary conditions. This means that our old friend the mode expansion will change when we do physics on compact spaces rather than non-compact ones. Let us see how this process works. We have rα0 rα0 X25(z, z¯) = 1 (xµ +x ˜µ) + −i (αµ +α ˜µ) τ + (αµ − α˜µ) σ + oscillators . (7.4) 2 2 0 0 2 0 0 We also know that for closed strings rα0 pµ = 1 (α +α ˜µ) . (7.5) 2 2 0 0 Further, we have that the Xµ are periodic under σ → σ + 2π: Xµ(z, z¯) → Xµ(z, z¯) + q α0 µ µ µ 2π 2 (α0 − α˜0 ). If we demand that X is single-valued like for a noncompact direction in µ µ µ target space, then we get back familiar relations between α0 , α˜0 , and p . But here we have a qualitatively and quantitatively diﬀerent situation: p25 has values n/R for n ∈ Z, and X25(z, z¯)(τ, σ + 2π) = X25(τ, σ) + 2πRw . (7.6)

25 This gives us enough information to solve for the α0 and its barred cousin, n wR rα0 rα0 α25 = + ≡ p (7.7) 0 R α0 2 L 2 n wR rα0 rα0 α˜25 = − ≡ p (7.8) 0 R α0 2 R 2

65 Then 2 4 2 4 m2 ≡ −pµp = (α25)2 + (N − 1) = (˜α25)2 + N˜ − 1 (7.9) 25 µ α0 0 α0 α0 0 α0 which can be rearranged to give

n2 w2R2 2 m2 = + + N + N˜ − 2 (7.10) 25 R2 (α0)2 α0 0 = n · w − (N − N˜) . (7.11)

The second equation here is known as the momentum constraint and it allows N and N˜ to differ as long as there is a nonzero dot product between momentum and winding numbers. The first equation is again known as the mass formula. Notice that by having integer quantized momentum we lost some states in the theory, but by having integer quantized winding we gained some states in the theory. Clearly, as R → ∞, momentum modes are cheap and winding modes are expensive. In the opposite limit R → 0, winding modes are cheap and momentum modes are expensive. In fact, the entire spectrum of our circle-compactified X25 has a symmetry under

n ↔ w (7.12) √ R α0 √ ↔ (7.13) α0 R This is known as T-duality. Note that this is stated as a symmetry of the free spectrum of closed string theory, but it actually can be proven to hold at the level of the string path integral. √ Notice further that at R → α0 something very special happens. There, and only there, at the self-dual radius, new massless modes appear! They have the quantum numbers n = −w = +1 and N = 1, N˜ = 0 and n = +w = +1 and N˜ = 1,N = 0. No other choices satisfy the momentum constraint. These extra massless vectors (vectors because they have one oscillator) are the W-bosons of an enhanced√SU(2) symmetry from wrapped fundamental strings at the self-dual radius. When |R| > α0, this SU(2) is Higgsed. The vertex operators for these extra gauge bosons are √ 25 ¯ µ 25 ¯ µ ±2iX (z)/ α0 SU(2)L : ∂X ∂X , ∂X e ; (7.14) √ 25 µ ¯ 25 µ ±2iX (¯z)/ α0 SU(2)R : ∂X ∂X , ∂X e . (7.15)

I have a few notes online at http://ap.io/2406s/notes/Hagedorn.pdf describing the Hagedorn transition in string theory which arises because string states at high mass level become exponentially numerous, and this density of states can at a high enough temperature overwhelm the Boltzmann weight and render the statistical mechanical partition function undeﬁned. I am not willing to re-type this information in LATEXhere so please click on the above link to see more. For open strings, there is no longer a conserved winding number labelling topologically distinct sectors. We will still be able to ﬁgure out the physics, by using the following really cool fact about T-duality.

66 7.3 T-duality for open bosonic strings T-duality corresponds to a worldsheet symmetry transformation

µ β µ ∂αX ↔ αβ∂ X . (7.16)

Notice that this worldsheet Hodge duality transformation switches N and D boundary conditions for open strings. This is what we will use in a second to ﬁgure out how T-duality works for open strings. The previous equation can be recast as

X25 = X25(z) + X25(¯z) ↔ X250 = X25(z) − X25(¯z) . (7.17)

You should explicitly check for yourself that the X250 also satisﬁes the worldsheet Klein- Gordon equation, by antisymmetry of the Levi-Civita pseudotensor αβ. For the open string we write

Xµ(z, z¯) = Xµ(z) + Xµ(¯z) (7.18) where r α0 X 1 Xµ(z) = 1 (xµ + x0µ) − iα0pµ ln(z) + i αµz−n , (7.19) 2 2 n n n=06 r α0 X 1 Xµ(¯z) = 1 (xµ − x0µ) − iαpµ ln(¯z) + i α¯µz¯−n , (7.20) 2 2 n n n=06

Now put X25 on a circle S1 of radius R. We can ﬁnd the T-dual X 025 by using (7.17),

X 025(z, z¯) = X25(z) − X25(¯z) (7.21) √ X 1 = x025 − iα0pµ ln(z/z¯) + 2 2α0 α25e−inτ sin(nσ) (7.22) n n n=06 n √ X 1 = x025 + 2α0 σ + i 2α0 α25e−inτ sin(nσ) (7.23) R n n n=06

Look what just happened! There is no τ dependence in the zero mode part of the oscillator expansion. Physically this means that there is no momentum in the T-dual X 025. Also, note that the oscillator terms do not contribute at σ = 0, π, in other words, there are no wiggles at the endpoints either! This is the origin of the famous fact about D-branes that for open strings their endpoints must be stuck to a codimension one surface, known as a D24-brane. It is codimension one because we only compactiﬁed one coordinate on a circle. Looking a bit more closely, we see that 2παn X 025(τ, σ = π) − X 025(τ, σ = 0) = = 2πnR0 . (7.24) R

67 Now let us turn to the topic of Chan-Paton factors, which are gauge labels carried by open string endpoints without energy cost, and Wilson lines. What is a Wilson line? For a U(1) H µ gauge field, you can think of it as a little bit like an Aharonov-Bohm phase: W ≡ eiq Aµdx . (If we had a non-Abelian gauge field, then we would need to path order the exponential.) Here, let us consider the following configuration of a U(1) gauge field A25 which has zero field strength, θ A ≡ − (7.25) 25 2πR where θ is the angle on the S1 direction X25. The resulting Wilson line is W = e−iqθ. Now suppose that we have something a bit more interesting, diag{θ , . . . , θ } A = 1 N (7.26) 25 2πR which breaks U(N), the gauge symmetry appropriate to N D24-branes in unoriented string theory, down to U(1)N generically. Under X25 → X25 + 2πR, we will pick up a phase in our fields of diag{e−iθ1 , . . . , e−iθN }, so that open string momenta are now fractional. This has a simple meaning: the endpoints are no longer on the same D24-brane. More generally, if you have |iji Chan-Paton factors, the phase you get is ei(θi−θj ) giving momenta (2πn + θ − θ ) i j . (7.27) 2πR Then 025 025 0 X (τ, σ = π) − X (τ, σ = 0) = (2πn + θi − θj)R , (7.28) so that the endpoint for state i is

025 0 0 X = θiR = 2πα A25 ii (no sum) . (7.29) So our U(N) Chan-Paton factors distinguish which D24-brane our oriented open strings end on. There are N 2 degrees of freedom because there are N 2 ways of arranging the D24-branes. For example, with N = 3 D24-branes, there are i¯j strings for i, j = 1, 2, 3.

68 The key thing about D-branes discovered by Polchinski in 1995 is that they carry target spacetime Ramond-Ramond charge. This provided the missing link in the story of the spectrum of charge-carrying states in superstring theory.

7.4 Unit conventions and dimensional reduction The following ﬁgure from Polchinski is our focus for the rest of today’s lecture.

For units, we will be using the conventions of the Polchinski textbook because they are by far the most widespread ones used. The fundamental string tension is 1 1 τ = ≡ . (7.30) F1 0 2 2πα 2π`s while the Dp-brane tension (mass per unit p-volume) in superstring theory is 1 τ p = , (7.31) D p p+1 gs(2π) `s and the NS5-brane tension is 1 τ = . (7.32) NS5 2 5 6 gs (2π) `s

In ten dimensions the Newton constant G is related to the gravitational coupling κ and gs, `s by 2 7 2 8 16πG10 ≡ 2κ10 = (2π) gs `s . (7.33) To get units convenient for T-duality, we deﬁne any volume V to have implicit 2π’s in it. If the ﬁelds of the theory are independent of (10 − d) coordinates, then the integration

69 R 10 10−d R d measure factorizes as d x = (2π) V10−d d x. We can use this directly to ﬁnd any lower-dimensional Newton constant from the ten-dimensional one, as follows: G G = 10 , (7.34) d 10−d (2π) V10−d

The Planck length in d dimensions, `d, is deﬁned by

d−3 d−2 16πGd ≡ (2π) `d . (7.35) From these facts we can see that there is a neat interdimensional consistency in the expression for the Bekenstein-Hawking entropy. Let us take a black p-brane and wrap it on T p to make d = 10 − p black hole. Translational symmetry along the p-brane means that the horizon has a product structure, and so the entropy is

p Ad+p Ad(2π) Vp SBH = = 4Gd+p 4Gd+p (7.36) A = d , 4Gd which is the same as the black hole entropy. As a reminder, we mention that the event horizon area in the Bekenstein-Hawking formula must always be computed in the Einstein frame, which is the frame where the kinetic term for the metric is canonically normalized, 1 Z √ Sgrav = −gR[g] . (7.37) 16πGd The relation between the Einstein and string metrics is

−4Φ/(D−2) gµν = e Gµν . (7.38)

Figuring out the constants is only one small part of the mechanics of dimensional reduction. We now move to a simple example of Kaluza-Klein reduction of ﬁelds in string frame, by reducing on a circle of radius R. Label the d dimensional system with no hats and the (d − 1) system with hats. Split the indices as {xµ} = {xµˆ, z}. The vielbeins decompose as

Eâˆ eχÂˆ Gˆ + e2ˆχAˆ Aˆ e2ˆχAˆ Ea = µˆ µˆ ⇒ (G ) = µˆνˆ µˆ νˆ µˆ , (7.39) µ χˆ µν ˆ 2ˆχ 2ˆχ 0 e Aνê e and 1 Φ = Φˆ + χˆ ; (7.40) 2 which yield Z √ 1 d −2Φ d x −Ge RG = 16πGd Z q (7.41) 1 d−1 ˆ −2Φˆ ˆ 2 2 1 2ˆχ ˆ 2 d x −Ge RGˆ + 4(∂Φ) − (∂χˆ) − e |dA| . 16πGd−1 4

70 The Kaluza-Klein procedure can also be done in Einstein frame. Taking the metric 2 2 2αχˆ 2 2βχˆ ˆ µˆ ds = e dsˆ + e dz + Aµˆdx , (7.42) with β = (2 − D)α and α2 = 1/[2(D − 1)(D − 2)] gives √ p 1 2 1 −2(D−1)αχˆ 2 −gRg = −gˆ Rgˆ − 2 (∂χˆ) − 4 e F , (7.43) where F is the field strength of Aˆ. What actions do you get at low energy for superstrings? There are too many cases to discuss in detail here but we will show IIA for purposes of illustration. The 11D M theory R √ 1 2 1 R action is even simpler, S11 = −g{R(g) + 48 |dA3 | + (gravitino)} + 96 A3 ∧ dA3 ∧ dA3 . For Type IIA, the independent R-R potentials are C1 ,C3 . The low-energy effective action of IIA string theory is d = 10 IIA supergravity, 1 Z √ e−2Φ 1 S = d10x −G R + 4 (∂Φ)2 − |dB |2 + (fermions) A 7 8 2 G 2 (2π) ls gs 12 1 1 1 Z 1 − |dC |2 − |dC − dB ∧ C |2 + dC ∧ dC ∧ B . 1 3 2 1 7 8 3 3 2 4 12 (2π) ls 64 We have shifted the dilaton field so that it is zero at infinity. Note that the dilaton kinetic energy looks like it has the wrong sign, but this is fixed once you recognize that an integration by parts needs to be done in order to write it in canonical form; in Einstein frame it has the correct sign. Note also that in the action we could have used the Hodge dual ‘magnetic’ (8−n)-form potentials instead of the ‘electric’ ones, e.g. a 6-form NS-NS potential instead of the 2-form. However, we cannot allow both the electric and magnetic potentials in the same Lagrangian, as it would result in propagating ghosts. The funny cross terms, such as dC3 ∧ dC3 ∧ B2 , are required by supersymmetry. In many cases there is a consistent truncation to an action without the cross terms, but compatibility with the field equations has to be checked in every case. For Type IIB string theory, the R-R 5-form field strength + F5 ≡ dC4 is self-dual, and so there is no covariant action from which the field equations can be derived. It is the equations of motion that are more fundamental: they are derived from the beta function equations for target spacetime fields that we will describe next week. Now recall that in d = 4 electromagnetism, an electrically charged particle couples to A1 ∗ (or its field strength F2 ), while the dual field strength F2 gives rise to a magnetic coupling to point particles. By analogy, a p-brane in d=10 couples to Cn=p+1 electrically, or C7 −p magnetically. As a result, we find 1-branes “F1” and 5-branes “NS5” coupling to the NS-NS potential B2 , and p-branes “Dp” coupling to the R-R potentials Cp+1 (or their Hodge duals). What happens when we probe a spacetime, using a Dp-brane? The kappa symmetric action of a probe brane in a supergravity background has two pieces,

Sprobe = SDBI + SWZ , (7.44) which are, to lowest order in derivatives, Z 1 p+1 −Φp S = − d σe − det (Gαβ + [2πFαβ + Bαβ]) , DBI p p+1 P g (2π) ls s (7.45) 1 Z S = − exp (2πF2 + B2 ) ∧ ⊕nCn . WZ p p+1 P (2π) ls

71 where the σ are the worldvolume coordinates and P denotes pullback to the worldvolume of bulk ﬁelds. The brane action encodes both kinetic and potential information, such as which branes can end on other branes. The WZ term, in particular, encodes the fact that Dp-branes can carry charge of smaller D-branes by having worldvolume ﬁeld strength F2 turned on.

7.5 Spacetime ﬁelds exerted by BPS objects in string/M theory The BPS M2-brane spacetime has worldvolume symmetry group SO(1, 2), and the transverse symmetry group is SO(8). Let us deﬁne the coordinates parallel and perpendicular to the brane to be (t, xk) , x⊥, respectively. Then, using these symmetries and a no-hair theorem, the spacetime metric turns out to depend only on |x⊥| ≡ r, and has the form

2 −2/3 2 1/3 2 −1 ds11 = H2 dxk + H2 dx⊥ ,A012 = −H2 . (7.46) The fact that the same function appears in the metric and gauge field is a consequence of supersymmetry. Note that the metric is automatically in Einstein frame because there is no string frame in d=11. It turns out that supersymmetry alone is not enough to give the equation that the function H must satisfy; rather, the supergravity equations of motion must be used. One finds that H2 must be harmonic as it satisfies a Laplace equation in x⊥. The solution is r6 H = 1 + 2 , where r6 = 32π2N `6 , (7.47) 2 r6 2 2 11 where we remind the reader that `11 is the eleven-dimensional Planck length. The BPS M5-brane has symmetry group SO(1, 5) × SO(5), and the metric is

2 −1/3 2 2/3 2 ds11 = H2 dxk + H2 dx⊥ , (7.48) and the harmonic function is this time r3 H = 1 + 2 , where r3 = πN `3 . (7.49) 5 r3 5 5 11

In this case, the gauge ﬁeld is magnetically coupled, F4 is proportional to the volume element on the S4 transverse to the M5-brane. For the M2, the origin of coordinates r = 0 is singular and so there must be a δ-function source there, to wit the fundamental M2-brane. This happens essentially because the M2- brane is electrically coupled. The magnetically coupled BPS M5-brane is solitonic and nonsingular, in that the geometry admits a maximal analytic extension without singularities. However, the nonextremal version of the M5 has a singularity and does need a source. Near- 7 4 horizon, the M2 spacetime is AdS4 × S and the M5 is AdS7 × S . Since the M2 and M5 are asymptotically ﬂat, again we have interpolation between 2 highly supersymmetric vacua. Let us now move down to ten dimensions. The symmetry for BPS Dp-branes is SO(1, p)× SO(9 − p). In the string frame, the solutions are

2 − 1 2 2 + 1 2 dS = Hp(r) 2 −dt + dxk + Hp(r) 2 dx⊥ ,

Φ 1 (3−p) (7.50) e = Hp(r) 4 , −1 −1 C01···p = gs [1 − Hp(r) ] .

72 2 The function Hp(r) is harmonic; it satisﬁes ∂⊥Hp(r) = 0,

7−p cpgsNp` √ H = 1 + s , c ≡ (2 π)(5−p)Γ[ 1 (7 − p)] . (7.51) p r7−p p 2

Note that the function Hp would still be harmonic if the constant piece, namely the 1, were missing. The asymptotically flat part of the geometry would be absent for this solution. The (double) horizon of the Dp-brane geometry occurs at r = 0, and in every case except the D3-branes the singularity is located there as well. Hence, for the Dp-branes with p 6= 3, the singularity is null. Since the singularity and the horizons coincide for these cases, we may worry that the singularity is not properly hidden behind an event horizon, and so perhaps it should be classified as naked. We therefore demand that a null or timelike geodesic coming from infinity should not be able to bang into the singularity in finite affine parameter. Interestingly, this condition separates out the D6-brane from the others as being the only one possessing a naked singularity. For the D3-brane the dilaton is constant, and the spacetime turns out to be totally nonsingular: all curvature invariants are finite everywhere. This allows a smooth analytic extension inside the horizon, like the case of the M5-brane. 5 The near-horizon D3-brane spacetime is AdS5 × S . The Penrose diagram for the D3 is like that of the M5. The F1 and NS5 spacetimes may be found by using the T- and S-duality formulæ that we gave earlier. We can also show two purely gravitational objects√ in the BPS spectrum of string theory. Defining light-cone coordinates dz± ≡ (t ± z)/ 2,

2 + − 2 2 2 dSd = −2dz dz + dρ + ρ dΩd−3 . (7.52) This is the gravitational wave W, which has zero ADM mass in d dimensions. In d − 1 dimensional language, Q2 M 2 = 0 = M 2 − . (7.53) d d−1 R2 The d − 1-dimensional charge is the z-component of the d-dimensional momentum. The other purely gravitational object is the KK monopole. Labelling the ﬁve longitudinal directions y1···5, and the four transverse directions xi, i = 1, 2, 3, and z; the metric is

2 2 2 −1 i 2 2 ds = −dt + dy + H (x)(dz + Aidx ) + H(x)dx , 1···5 1···3 (7.54) 2∂[iAj](x) = ijk∂kH(x) .

The Ai can be found via the curl equation, given that H = 1 + k/|x|. The periodicity of the azimuthal angle (an isometry direction) must be 4π to avoid conical singularities. We now turn to a lightning review of some common and useful dualities.

7.6 Superstring dualities and the deerskin diagram Type IIA ↔ M-theory The 11th coordinate x\ is compactiﬁed on a circle of radius

R\ = gs`s . (7.55)

73 The supergravity ﬁelds decompose as

ds2 = e−2Φ/3dS2 + e4Φ/3 dx\ + C dxµ2 11 10 1 µ (7.56) \ A3 = C3 + B2 ∧ dx .

4Φ/3 Φ/3 \ where F4 = dA3 = e (dC3 + C1 ∧ H3 ) + e H3 ∧ dx , and H3 = dB2 . We can turn M-theory objects into Type IIA objects by pointing them in the 11th direction (.) or not (↓). W M2 M5 KK . ↓ . ↓ . ↓ . ↓ . (7.57) D0 W F1 D2 D4 NS5 D6 KK S-duality of IIB The low-energy limit of IIB string theory, IIB supergravity, possesses a SL(2,R) symmetry (it is broken to SL(2,Z) in the full string theory). Deﬁne −Φ dB2 λ ≡ C0 + ie and H ≡ . (7.58) dC2

Under an SL(2,R) transformation represented by the matrix a b U = ∈ SL(2, ) , (7.59) c d R the fields transform as aλ + b H → U H λ → . (7.60) cλ + d The d = 10 Einstein metric and the self-dual five-form field strength are invariant. A commonly considered Z2 subgroup obtains when C0 = 0. The Z2 flips the sign of Φ, and exchanges B2 and C2 . The result is

D1 ↔ F 1 ,D5 ↔ NS5 ; (7.61) all others such as W and KK are unaffected, and the D3 goes into itself. The effect of this Z2 on units is 1 1 ˜ 1 g˜s = , g˜s 4 `s = gs 4 `s . (7.62) gs From this one can easily check that the tensions of e.g. F1 and D1’s transform into each other under the Z2 flip. T-duality The operation of T-duality on a circle switches winding and momentum modes of fundamental strings (F1) and exchanges Type IIA and IIB. The effect on units is to invert the radius in string units, and leave the string coupling in one lower dimension unchanged:

R˜ ` g˜ g = s , s = s , `˜ = ` . (7.63) ˜ q p s s `s R ˜ ˜ R/`s R/`s

74 T-duality does not leave all branes invariant; it changes the dimension of a D-brane depending on whether the transformation is performed on a circle parallel (k) or perpendicular (⊥) to the worldvolume. It also changes the character of a KK or NS5; doing T-duality along the isometry direction (isom) of the KK gives an NS5. Summarising, we have:

Dp ↔ Dp − 1(k) or Dp + 1(⊥) , KK (isom) ↔ NS5 ; (7.64)

Everything else is unaﬀected. Let z be the isometry direction. Then T-duality acts on NS-NS ﬁelds as follows:

2Φ˜ 2Φ ˜ ˜ ˜ e = e /Gzz , Gzz = 1/Gzz , Gµz = Bµz/Gzz , Bµz = Gµz/Gzz , ˜ Gµν = Gµν − (GµzGνz − BµzBνz) /Gzz , (7.65) ˜ Bµν = Bµν − (BµzGνz − GµzBνz) /Gzz . T-duality also acts on R-R fields, and the formulæ are a little more involved. For simple situations involving no NS-NS B-field and no off-diagonal metric components, we have either ˜ ˜ Cn+1=Cn ∧ dz (⊥) or Cn ∧ dz=Cn+1 (k), as appropriate. Note that if we do T-duality on a supergravity Dp-brane in a direction perpendicular to its worldvolume, we are dualising in a direction which is not an isometry, because the metric and other fields depend on the coordinates transverse to the brane. But the T-duality formulæ for supergravity fields apply only when the direction along which the T-duality is done is an isometry direction. If it is not, then we should first “smear” the Dp-brane in that direction to create an isometry and then do T-duality. Note also that in the presence of some branes, string momentum or winding number may not be conserved, e.g. string winding number in a KK background. However, the conserved quantity transforms as expected under T-duality.

8 Sigma Models and String Theory

GR teaches us that “matter tells spacetime how to curve, and spacetime tells matter how to move”. Well, in string theory the graviton and the matter are included under the same umbrella, so the equations for gravity in string theory are one giant self-consistency equation. Morally speaking, the string does not like to propagate on any old spacetime. Instead, it forces conditions on the background fields in order to be bona fide solutions of string theory. The main question we want to answer in this chapter is: what happens when you do not assume that the spacetime metric is the Minkowski metric? What if there is also a nontrival Kalb-Ramond field turned on, and a dilaton? Our guiding principle will be to demand conformal invariance of the action principle in the quantum theory. We will find a conformal anomaly in a general background, which will give β-functions for our couplings telling us how the couplings run with energy scale. These β-function equations will then be re-interpreted as equations of motion for the spacetime fields like the string metric. In other words, this chapter is about giving you an outline of how the [NS-NS part of the] 10D SUGRA action mentioned in the previous chapter is derived from first principles. It will take a bit of work, but the physics principles involved are not actually that complicated. The procedure is just a little long-winded for a first-year graduate student!

75 That is why I will lead you through the details. The main ingredients that we have not yet recruited are the background field method from QFT and the Riemann normal coordinate expansion from GR. You can find a good review of the background field method pitched towards gauge field theories like the Standard Model in Peskin and Schroeder, and a discussion of Riemann normal coordinates in any decent book on GR. Sometimes, critics of string theory harp on about how this is not a background independent formalism. Most of them misunderstand the degree to which the σ-model worldsheet approach is background independent. The key thing is that the fluctuations of the background are included in the analysis. So the worldsheet formulation of string theory does have a degree of background dependence, but it is a lot less na¨ıve than most dilettantes believe. I also like to ask the question: what would background dependence even look like when there is no classical background around which to expand, i.e. if you are in a truly quantum phase of quantum gravity? Using GR intuition is unlikely to be a good guide to that physics.

8.1 Sigma Model couplings The worldsheet theory of strings propagating in nontrivial backgrounds is described by a σ-model. Sigma models are used in many other contexts in both relativistic quantum ﬁeld theory and and nonrelativistic many-body theory. For now, we stick with the string theory application. Recall our original Polyakov action, 1 Z √ S = d2ξ γγab∂ Xµ∂ XνG (X). (8.1) P 4πα0 a b µν It is this term that gives us our graviton. For the purposes of the string worldsheet, the λ spacetime background in which the string propagates, Gµν(X ) is a coupling function of the primary dynamical variables Xλ(τ, σ). Suppose that we also had an antisymmetric coupling on the worldsheet as well. It would need to be reparameterization invariant. The only term that matches all the criteria is 1 Z S = d2ξab∂ Xµ∂ XνB (X). (8.2) AS 4πα0 a b µν Here note that ab is the two-dimensional antisymmetric Levi-Civita symbol, which is a tensor √ density (i.e. has a factor of γ in it). We recognize that this will be the Neveu-Schwarz (NS) two-form that we love. Note that it indeed is invariant under our spacetime ”gauge transformations”, Bµν → Bµν + ∂µΓν − ∂µΓν. (8.3) Of course, renormalization will bring in terms that are of lower dimension. Another two-dimensional term is given by 1 Z √ S = d2ξ γR(2)Φ(X). (8.4) D 4π We can see that this is the dilaton. There is a slight problem here. This is not Weyl invariant! The important thing to note is that we will be expanding in powers of α0, which means that

76 this term will only be relevant at one loop level and so our overall theory will be invariant at the tree level. This is fine, since we already have an anomaly at one loop, so we can just include it to be a term than needs to cancel. One final term of dimension zero is given by 1 Z √ S = d2ξ γT (x). (8.5) T 4π This is the tachyon. It is reparameterization invariant and not Weyl invariant as above. The tachyon is needed to remove quadratic divergences in the vacuum. However, we will not need to use them for our discussion. If we were to look at open strings, we could include a vector gauge field that would couple to the end of the strings. The action for such a coupling would take the form of I dXµ S = i dsA (X) . (8.6) A µ ds Then this would modify boundary conditions that are need to keep conformal invariance there. To avoid this discussion, we will just work with closed strings. So our full action will be of the form,

A[X, γ] = SP + SAS + SD. (8.7) The general idea is to do the analysis loop by loop, and the loop counting parameter is α0, i.e. the inverse string tension. Our analysis will only go to one loop in α0. There is also the string loop expansion in gs, which we do not cover here.

8.2 Background field method and Riemann normal coordinates There are two pieces of QFT technology that are extremely useful for figuring out how the sigma model action can give rise to equations of motion for spacetime fields. The first is the background field method. Basically, it is done by expanding out the fields into a classical component and a quantum component and shifting the integration to the quantum component. Let us first see how this works for Yang-Mills gauge fields, which should be familiar to those of you taking QFT2. Suppose we have some gauge field A which we separate into background plus fluctuations: A = A¯ + Q. We can look at how it transforms under infinitesimal tranformations. In Peskin and Schroeder conventions, 1 δAa = ∂ αa + f abcAb αc. (8.8) µ g µ µ Then we see that we can define, 1 δA¯a = ∂ αa + f abcA¯b αc (8.9) µ g µ µ a abc b c δQµ = f Qµα (8.10) which are consistant with the original gauge transformations. Notice that the ‘background’ part involving A¯ transforms in an entire familiar way. The really cool part is that the

77 ‘quantum’ part involving Q transforms tensorially, not like a connection. This simpliﬁes the process of handling the gauge invariance. To do the same sort of thing for the string, deﬁne

µ µ µ X = X0 + π (8.11) µ µ where X0 will be our classical component and π will be the quantum. Our path integral will then become

R 2 δA[X,γ] µ Z −(A[X +π,γ]−A[X ,γ]− d ξ µ π ) 0 0 δX Ω[X0, γ] = [Dπ]e 0 . (8.12)

Note that we simply expanded the action out into quadratic or higher terms in πµ. This makes sense, since the order zero terms would just be a constant and the linear terms are just proportional to the classical equation of motion, which the classical part satisfies. The second method is the idea of Riemann normal coordinates. Consider our quantum µ µ µ field π = X −X0 . This is the difference of two coordinates, which is generally not a vector under Lorentz transformations (in the Xµ, it is a scalar under Lorentz tranformations for ξ). To make sure we do this all covariantly, we should use a field that is a spacetime vector. We will do this by choosing a tangent vector to the spacetime geodesic that runs through µ µ X and X0 . To set up the geodesic, have the geodesic λ. Choose an affine parameter t normalized so µ µ ˙ ˙ df that λ(0) = X0 and λ(1) = X . Define the tangent vector η = λ(0), where f = dt for any f. The geodesic equation then reads, ä a ˙ b ˙ c λ + Γbcλ λ = 0. (8.13) We can build up a Taylor series expansion around t = 0 using this equation. Consider the geodesic equation at t = 0, ä a b c ä a b c λ (0) + Γbc(0)η η = 0 ⇒ λ (0) = −Γbc(0)η η . (8.14) Next, we take a single derivative with respect to t on the geodesic equation and then set t = 0. Doing this we find, ... a a ¨b c ˙ a b c λ (0) + 2Γbc(0)λ (0)η + Γbc(0)η η = 0 (8.15) ...a a b e d c a d b c ⇒ λ (0) = 2Γbc(0)Γde(0)η η η − ∂dΓbc(0)η η η . (8.16) We can continue to do this to find the nth derivative as we like. Putting down up to the order we have, 1 1 µ µ µ µ σ1 σ2 2 µ ν µ σ1 σ2 σ3 3 λ (t) = X + η t − Γ η η t − (∂σ Γ − 2Γ Γ )η η η t + ... (8.17) 0 2 σ1σ2 6 1 σ2σ3 σ1σ2 σ3ν Generalizing to higher orders, we can define a symbol Γµ = ∇0 ...∇0 Γµ . This is σ1...σn σ1 σn−2 σn 1σn a bit of an abuse of notation, since covariant derivatives are not defined on symbols− that are 0 not tensors. In this case, we mean that ∇σ acts as if the symbol to the right of it was a true tensor in only the lower indices. Now, if we set t = 1 we will find, 1 πµ = ηµ − Γµ ησ1 ησ2 + ... (8.18) 2 σ1σ2 78 In order to get that πµ = ηµ, we need for the Christoffel symbols and the symmetric part of Γµ to vanish. Luckily, we can do this, but only at a point (which we will choose to be σ1...σn X0). The coordinates that this occur in are called the Riemann normal coordinates. These coordinates have some great properties. A few examples include,

¯µ Γνσ = 0 (8.19) Γ¯µ = 0 (8.20) (σ1...σn) ¯µ ¯µ ¯µ Rνσρ = ∂σΓνρ − ∂ρΓνσ (8.21) 1 ∂ Γ¯µ = (R¯µ − R¯µ ). (8.22) ν σρ 3 σνρ ρνσ Keep in mind these only hold at the origin. Of course there are far more, but we won’t put them all here. We can of course expand all quantities out into normal coordinates through their Taylor series. For a general covariant tensor,

∞ X 1 T¯ = (∂ ...∂ T¯ )ην1 ...ηνn (8.23) µ1..µn m! ν1 νm µ1..µn m=0 We want to work in a manifesitly covariant formulation. To transform the partial derivatives to covariant, we also make use of the properties of normal coordinates. For example, a two index covariant tensor, 1 1 1 T¯ (X) = T¯ + ∇ T¯ ηλ + (∇ ∇ T¯ − R¯ρ T¯ − R¯ρ T¯ )ηλησ + ... (8.24) µν µν λ µν 2 λ σ µν 3 λµσ ρν 3 λνσ µρ ¯ where all of the Tµν on the right side are evaluated at X0.

8.3 Graviton beta function So now we will start expanding out our actions using these two methods, which will allow us to pick oﬀ the propagators and vertices for our Feynman diagrams. Then we can start looking at loops and calculate the beta functions for each of our propagators. First, consider our spacetime metric, 1 1 1 G (X) = G + ∇ G ηλ + (∇ ∇ G − Rρ G − Rρ G )ηλησ + ... (8.25) µν µν λ µν 2 λ σ µν 3 λµσ ρν 3 λνσ µρ 1 = G − (R + R )ηλησ + ... (8.26) µν 6 νλµσ µλνσ 1 = G − R ηλησ + ... (8.27) µν 3 νλµσ where we have kept in mind that the covariant derivative of the metric vanishes. Next, we expand the derivatives,

µ µ µ ∂aX = ∂a(X0 + π ) (8.28) 1 µ µ µ σ1 σ2 = ∂a(X + η − Γ η η + ...) (8.29) 0 2 σ1σ2

79 where we have used equation 8.18. Expanding this out, we find 1 µ µ µ µ σ1 σ2 µ σ1 σ2 ∂aX = ∂aX + ∂aη − ∂aΓ η η − Γ ∂aη η + ... (8.30) 0 2 σ1σ2 σ1σ2 1 µ µ µ ρ σ1 σ2 µ σ1 σ2 = ∂aX + ∂aη − ∂ρΓ ∂aX η η − Γ ∂aη η + ... (8.31) 0 2 σ1σ2 σ1σ2 1 µ µ µ ρ σ1 σ2 µ σ1 σ2 = ∂aX + ∂aη − ∂ρΓ ∂aX η η − Γ ∂aη η + ... (8.32) 0 2 σ1σ2 0 σ1σ2 We recall equations 8.19 and 8.22, which will kill the final term and simplify the derivative. So the final equation reads, 1 µ µ µ µ ρ σ1 σ2 ∂aX = ∂aX + ∇aη − R ∂aX η η + ... (8.33) 0 3 σ1ρσ2 0 With these two expantions in play, we can expand the Polyakov action out to quadratic order in ηµ. Doing this we can see that 1 Z √ S [X] = d2ξ γγab∂ Xµ∂ XνG (X) (8.34) P 4πα0 a b µν 1 Z √ 1 2 ab µ µ µ ρ σ1 σ2 = d ξ γγ (∂aX + ∇aη − R ∂aX η η + ...) (8.35) 4πα0 0 3 σ1ρσ2 0 1 1 ν ν ν ρ σ1 σ2 λ σ ∗ (∂bX + ∇bη − R ∂bX η η + ...)(Gµν − Rνλµση η + ...) 0 3 σ1ρσ2 0 3 1 Z √ = d2ξ γγab(∂ Xµ∂ XνG (X) + 2∂ Xµ∇ ηνG (8.36) 4πα0 a 0 b 0 µν a 0 b µν µ ν µ ν λ σ + ∇aη ∇bη Gµν − Rµλνσ∂aX0 ∂bX0 η η + ...) where we have used our index symmetry to collect a few terms. The first and second terms will not contribute to our quantum system because they do not depend on ηµ. We argued why earlier with the idea that we only need quadratic term or higher in this expansion. The two quadratic terms are interesting. The first is clearly a kinetic term with the second a µ vertex term between two X0 background fields and two η fields. We can clear the kinetic i up a bit using vielbeins, eµ(x0). We introduce them by defining,

i i µ η = eµ(X0)η . (8.37)

These satisfy the immensely useful property of

i j eµ(X0)eν(X0)δij = Gµν(X0). (8.38)

Considering the kinetic term, we see that

µ ν i ∇aη ∇bη Gµν = (∇aη) (∇bη)i, (8.39)

i i ij j ij with (∇aη) = ∂aη + ωµ ∂aX0η . Here, ωµ is the spin connection in spacetime, which you need in order to take covariant derivatives of spinors. It obeys mathematical equations known as the Cartan structure equations. If you have a spacetime which is torsion-free, then using

80 ij ij µ the language of differential forms we can write ω ≡ ωµ dx and the torsion-free equations i i j become de +dω j ∧e = 0 and the relationship between the spin connection and the Riemann ij ij i jk tensor two-form is R = dω + ω k ∧ ω . For a simple diagonal spacetime metric, it is often faster to find the vielbeins, then get the spin connection by using the torsion-free equation, then compute the Riemann two-form, than to compute using the Christoffel connection. Just saying! :D

8.4 Kalb-Ramond field beta function Anyways, we move onto expanding out the other two terms. If we expand the antisymmetric term, 1 1 1 B (X) = B + ∇ B ηλ + (∇ ∇ B − Rρ B − Rρ B )ηλησ + ... (8.40) µν µν λ µν 2 λ σ µν 3 λµσ ρν 3 λνσ µρ Unlike our previous case, the terms with the covariant derivatives do not vanish. So we can exand as before, 1 Z S = d2ξab∂ Xµ∂ XνB (X) (8.41) AS 4πα0 a b µν 1 Z 1 2 ab µ µ µ ρ σ1 σ2 = d ξ (∂aX + ∇aη − R ∂aX η η + ...) (8.42) 4πα0 0 3 σ1ρσ2 0 1 ν ν ν ρ σ1 σ2 λ ∗ (∂bX + ∇bη − R ∂bX η η + ...)(Bµν + ∇λBµνη 0 3 σ1ρσ2 0 1 1 1 + (∇ ∇ B − Rρ B − Rρ B )ηλησ + ...) 2 λ σ µν 3 λµσ ρν 3 λνσ µρ 1 Z = d2ξab ∂ Xµ∂ XνB + 2∂ XµB ∇ ην + ∇ B ∂ Xµ∂ Xνηλ (8.43) 4πα0 a 0 b 0 µν a 0 µν b λ µν a 0 b 0 1 + B ∇ ηµ∇ ην + 2∂ Xµ∇ B ∇ ηνηλ + ∂ Xµ∂ Xν(∇ ∇ B µν a b a 0 λ µν b 2 a 0 b 0 λ σ µν ρ ρ λ σ − RλµσBρν − RλνσBµρ)η η where we have used our antisymmetry properties to collect a few terms. As before, the first two terms do not contribute. However, the quadratic terms look quite a mess. Furthermore, they are not in a nice gauge invariant form, as we would expect given that we have not broken the gauge symmetry here. We wish to write it in terms of the field strength, Hµνλ = ∇µBνλ + ∇νBλµ + ∇λBµν, which is gauge invariant. To do this, we need to integrate by parts. The quadratic term comes out to 1 Z 1 d2ξab H ∂ Xµ∇ ηνηλ + ∇ H ∂ Xν∂ Xληµησ (8.44) 4πα0 µνλ a 0 b 2 µ νλσ a 0 b 0 Of course we can introduce our vielbeins again and get 1 Z 1 d2ξab H ∂ Xµ∇ ηiηj + ∇ H ∂ Xν∂ Xληiηj . (8.45) 4πα0 µij a 0 b 2 i νλj a 0 b 0

81 We may end up needing a cubic term too. 1 Z d2ξabH ∇ ηj∇ ηkηi. (8.46) 12πα0 ijk a b

8.5 Dilaton beta function With that done, we can expand our dilaton term. It is a scalar term, so the expansion is straight forward, 1 Φ(X) = Φ + ∇ Φηµ + ∇ ∇ Φηµην + ... (8.47) µ 2 µ ν So the dilaton action expands out like, 1 Z √ S = d2ξ γR(2)Φ(X) (8.48) D 4π 1 Z √ 1 = d2ξ γR(2)(Φ + ∇ Φηµ + ∇ ∇ Φηµην + ...) (8.49) 4π µ 2 µ ν 1 Z √ 1 Z √ = S [X ] + d2ξ γR(2)∇ Φηi + d2ξ γR(2)∇ ∇ Φηiηj + ... (8.50) D 0 4π i 8π i j Now we have all the terms that we will need for our loop calculations.

8.6 Weyl anomaly at one loop Now we turn to focusing on working through the Feynman diagrams that contribute to the 0 0 Weyl anomaly at the zeroth order in α (classical level is 1/α ) and to first order in Rµνσλ. Recall that the conservation of the energy momentum that arises from variations of the worldsheet momentum, z¯ z ∇ hTzz¯i + ∇ hTzzi = 0. (8.51) z We will work out our loops and find that at one loop, the ∇ hTzzi term contributes a non-zero value, which implies that the trace must have a non zero value to satisfy the conservation of energy-momentum. This is an interesting dualism between the reparameterization invariance (which gives us our conservation equation) and the Weyl invariance (which would give us a non-vanishing trace). We can have one or the other but not both. Since we demand reparameterization invariance, we will have our Weyl anomaly which we will have to set to zero to remove. To begin with, we will consider a flat, Minkovski worldsheet to start and consider curvature later (hint, it will involve the dilaton). We will also work in momentum space. We can translate our conservation equation into,

q+hT−+i + q−hT++i = 0 (8.52) Note that this is a bit confusing. The important things to remember are that

γ++ = γ−− = 0 (8.53)

γ+− = −1 (8.54)

82 To determine the one loop contribution to hT−+i (which is proportional to the trace) from the Polyakov action. To do this, we will calculate the hT++i diagram and use our conservation equation. The diagram we will consider is

This diagram has a contribution of

Z d2l l (l + q ) + + + {R ∂ Xµ∂aXν}(q), (8.55) 2π l2(l + q)2 µν a 0 0 where q is the momentum carried away and l is the loop momentum. At this point, we should slow down and explain where these terms came from. First off, i i note that this has an insertion of ∂+η ∂+η . This comes from the fact that we are looking at hT++i. The term hT++i also has a contribution proportional to Rµνσλ. But this would make a diagram with two factors of the Riemann tensor, whereas we are only working to first order. Note that the integral comes in with a 1/4π2 factor, the two propagators give a 4π2α02, the vertex with background fields give a 1/4πα0, and the insertion comes with a factor of 2/α0. This gives us the appropriate prefactor. Furthermore, we note that the propagators and the energy momentum insertion are both a sum over η, and so our vertex will be summed over as well, which is where the Ricci tensor (rather than Riemann) comes in. 2 ab The bottom factors of momentum, q = γ qaqb, obviously come from the two propaga- i tors. The momentum on the top come from the vertex factor (i.e. ∂+ηi∂+η → l+(l+ + q+). With these things sorted out, we then we turn to evaluating the momentum integral. Luckily we can use our loop integral calculation techniques. Let’s introduce a Feynman parameter,

1 Z d2l l (l + q ) Z Z d2l l (l + q ) + + + = dx + + + (8.56) 2π l2(l + q)2 2π (xl2 + (1 − x)(l + q)2)2 0 1 Z Z d2l l (l + q ) = dx + + + (8.57) 2π (l2 + 2(1 − x)l · q + (1 − x)q2)2 0 1 Z Z d2l l (l + q ) = dx + + + . (8.58) 2π ((l + (1 − x)q)2 − (1 − x)2q2 + (1 − x)q2)2 0 So we shift the momentum integral, l → l0 = l + (1 − x)q and relabel it back to l. This gives

83 us,

1 Z d2l l (l + q ) Z Z d2l (l − (1 − x)q )(l + xq ) + + + = dx + + + + (8.59) 2π l2(l + q)2 2π (l2 + x(1 − x)q2)2 0

Note that the integral with two powers of the momentum is proportional to the metric, γ++, so it will vanish. Obviously the portion proportional to one loop momentum vanishes as well. This leaves,

1 Z d2l l (l + q ) Z Z d2l x(x − 1)q2 + + + = dx + (8.60) 2π l2(l + q)2 2π (l2 + x(1 − x)q2)2 0 1 Z x(x − 1)q2 q q = dx + = + = − + (8.61) 2 2 2x(x − 1)q 2q 4q− 0 So now we use our equation 8.52 and ﬁnd that 1 hT i = R (X )∂ Xµ(ξ)∂aXν(ξ). (8.62) −+ 4 µν 0 a 0 0

84 We can now look at the contribution from the antisymmetric tensor coupling action. The easiest diagram to examine is

This is almost exactly as the one we did before, except with a slightly diﬀerent vertex factor. One can easily replace it and ﬁnd 1 hT i = ∇λH (X )∂ Xµ(ξ)∂ Xν(ξ)ab. (8.63) −+ 8 λµν 0 a 0 b 0 The other diagram to that we can get is from the other vertex,

This is slightly diﬀerent from our previous diagram, but it is almost the same. The only major diﬀerence is the vertex factor. As before, we sum over all the indices belonging to contractions to η. Consider the following,

ab cd ij µ ν γbdHµijHν ∂aX0 ∂cX0 . (8.64) Note that we know the Levi-Civita pseudotensor satisﬁes

ab cd ac γbd = γ , (8.65)

So we can see that the vertex factor will be proportional to

ac ij µ ν ij µ a ν γ HµijHν ∂aX0 ∂cX0 = HµijHν ∂aX0 ∂ X0 . (8.66) The contribution is 1 hT i = H (X )H ij(X )∂ Xµ(ξ)∂aXν(ξ). (8.67) −+ 16 µij 0 ν 0 a 0 0 Now we turn to the contribution for the dilaton. The dilaton will add additional corrections similar to the ones above but it will also add its own unqiue corrections. This is due to the fact that the above work was done on a flat worldsheet and the dilaton still give a contribution even if it is a flat worldsheet. (i.e. the action may be zero, but the variation with respect to the metric does not need to be.) However, to find the additional dilaton contributions, we will have to go beyond the flat background.

85 To start, we will examine the energy momentum tensor that arrises from the dilaton in ﬂat space. This is given by dil Tab = (∂a∂b − δab)Φ(X). (8.68) The is a d’Alembertian with respect to the worldsheet. We can do the trace and ﬁnd,

a a a Ta = (∂ ∂a − δa)Φ(X) (8.69) = −Φ(X) (8.70) So then we see that,

dil T+− = ξΦ(X(ξ)). (8.71)

We can expand this at X0 to see what terms are there.

a ξΦ(X0) = ∂ ∂aΦ(X0) (8.72) a µ = ∂ (∂aX0 ∂µΦ(X0)) (8.73) a µ µ a = ∂ ∂aX0 ∂µΦ(X0) + ∂aX0 ∂ ∂µΦ(X0) (8.74) µ µ a ν = ξX0 ∂µΦ(X0) + ∂aX0 ∂ X0 ∂µ∂νΦ(X0). (8.75) This is not Lorentz invariant from the spacetime point to view. We could just try to put it in a covariant form right now, but it will be more useful to use the classical equation of motion for X0, since we know that it is satisfied. Since we are working in a flat worldsheet so far, we can ignore the dilaton action in deriving it. The equation is given by 1 Xµ = Γµ ∂ Xλ∂aXσ − Hµ ∂ Xλ∂ Xσab. (8.76) ξ 0 λσ a 0 0 2 λσ a 0 b 0 We can just throw this into our equation above and easily see that 1 Φ(X ) = ∂ Xµ∂aXν∇ ∇ Φ(X ) − ∇λΦ(X )H (X )∂ Xµ∂ Xνab (8.77) ξ 0 a 0 0 µ ν 0 2 0 λµν 0 a 0 b 0 These two terms should look very familar from the terms we did above. So to include this, we just add them to get the total contribution from the flat worldsheet parts to the zeroth 0 power in α and to one power in Rµνσλ. The contribution is 1 √ 1 hT i = βG ∂ Xµ∂ Xν γγab + βB ∂ Xµ∂ Xνab (8.78) −+ 4 µν a 0 b 0 8 µν a 0 b 0 where 1 βG = R − H H λσ + 2∇ ∇ Φ (8.79) µν µν 4 µλσ ν µ ν 1 βB = ∇λH − ∇λΦH . (8.80) µν 2 λµν λµν We are almost there! Our next task is to look at the contributions form the dilaton with non flat worldsheet geometry and see the beta function that comes from that.

86 8.7 Nailing the dilaton This part is focused on ﬁguring out the ﬁnal, and arguably the most important, contribution to the Weyl anomaly. This contribution is from the dilaton, and it is a bit more involved. To start, we will backtrack a bit and look at the energy momentum for the dilaton. The action for the dilaton is given by 1 Z √ S = d2ξ γR(2)Φ(X). (8.81) D 4π So if we to a variation with respect to the metric to get the engery momentum tensor, 1 √ √ δL = (δ( γ)R(2)Φ(X) + γδ(R(2))Φ(X)) (8.82) 4π We know from GR that the variations take the form of √ 1√ δ( γ) = − γγ δγab, δ(R(2)) = R δγab + δ(R )γab (8.83) 2 ab ab ab In the Einstein Hilbert action, the term proportional to the variation of the Ricci tensor produced a total derivative that we dropped. Here, we do not have that luxury. We found that the variation took the form of

ab a bc ab δ(Rab)γ = ∇a(γbc∇ δγ − ∇bδγ ) (8.84) a bc ab = ∇a∇ γbcδγ − ∇a∇bδγ (8.85)

Since this is now multiplied by Φ(X) it is no longer a total derivative! Putting this together, 1 √ 1 δL = γ(− γ R(2)Φ(X)δγab +R Φ(X)δγab +(∇ ∇aγ δγbc −∇ ∇ δγab)Φ(X)) (8.86) 4π 2 ab ab a bc a b We need to remove the covariant derivatives off of the variations of the metric. So we integrate by parts twice and find, 1 √ 1 δL = γ(− γ R(2)Φ(X) + R Φ(X) + (γ ∇ ∇c − ∇ ∇ )Φ(X))δγab (8.87) 8π 2 ab ab ab c b a We can simplify this. Note that the first two terms together are proportional to the Einstein tensor, which vanishes identically for any 2D worldsheet. This implies that, δL 1 √ = γ(∇ ∇ − γ ∇ ∇c)Φ(X). (8.88) δγab 4π a b ab c The energy momentum tensor is then 4π δL T dil = √ (8.89) ab γ δγab c = (∇a∇b − γab∇c∇ )Φ(X) (8.90)

87 Now we can return to where we left off. We wish to see what the effect of a curved worldsheet on these beta functions. We can get some of the pieces by using two point functions on a flat worldsheet. φ Recall that the conformal metric can be written as γab = e δab. Now we wish for the conformal anomaly to vanish for any generic worldsheet. However, we will soften this result to approximate and only demand that the first variation vanishes. We can see that

δ 1 φ hT−+(0)ie δab = − hT−+(ξ)T−+(0)iδab . (8.91) δφ(ξ) φ=0 4π To find out this contribution, there will be three different types of terms. The first type is the one loop corrections, which will be of order zero in α0. The second type will be two loop diagrams corrections, which will be first order. The third type will be tree and one loop corrections using the dilaton energy momentum instead of just the Polyakov and antisymmetric tensor, which will be of first order. As before, we will use the conservation of energy momentum to allow us to calucated tensor diagrams of the + + ++ and transform them to the + − +− character.

88 There is only one diagram at one loop with no dilaton terms to consider.

It is similar to what we have seen before. The contribution here is Z l2 (l + q )2 πD q3 hT (q)T (−q)i = 2D d2l + + + = − + . (8.92) ++ ++ 2 2 l (l + q) 6 q− There are few things to note. Since there is no momentum carried oﬀ, the energy momentum insertions must have opposite momentum contributions. The factor of D is there because i there are D η , each of which have a diagram like this. Note that the two T++ insertions bring 4/α02, the momentum integral brings 1/4π2, the two propagators give 4π2α02, and we have a 1/2 as a symmetry factor. This is what gives us the 2 out front and why there this contribution is zeroth order in α0. We can now use the fact that q+T−+ + q−T++ = 0 twice and we can see that, πD hT (q)T (−q)i = − q q . (8.93) −+ −+ 6 + −

2 We would like return to coordinate space. We note that q = −2q+q−. So then we can easily see that πD hT (ξ)T (0)i = δ(2)(ξ). (8.94) −+ −+ 12 With this in hand, we can use equation 8.91 and by inspection see that D hT (ξ)i φ = − φ(ξ). (8.95) −+ e δab 48 We recall our earlier equation for the curavture and ﬁnd that our ﬁnal expression is

D √ (2) hT (ξ)i φ = γR . (8.96) −+ e δab 24

89 We will now discuss the two loop contributions. Getting the correct forms for these would be tricky, as it would involve calculating two loop diagrams. The ones that are relavent for the dilaton correction at ﬁrst order in α0 are

0 α 1 µνσ √ (2) hT (ξ)i φ = (−R + H H ) γR (8.97) −+ e δab 8 12 µνσ Finally, we have to calculate two point functions with insertions of the dilaton energy momentum. The easier diagram is actually a tree diagram, since it involves two insertions of the dilaton energy momentum. This is simply a connection between two vertices

Here the X0 ﬁeld are allowed to be internal lines, since tree level is the classical level. This gives

0 dil dil 0 µ (2) α µ √ (2) hT (ξ)T (0)i = πα ∇ Φ∇ Φ δ (ξ) ⇒ hT (ξ)i φ = ∇ Φ∇ Φ γR (8.98) −+ −+ µ −+ e δab 2 µ The other diagram with an insertion of the energy momentum is a one loop with only one insertion of the dilaton ﬁeld. The diagram is of the form

90 This gives

0 dil 0 µ (2) α µ √ (2) hT (ξ)T (0)i = −πα ∇ ∇ Φ δ (ξ) ⇒ hT (ξ)i φ = − ∇ ∇ Φ γR (8.99) −+ −+ µ −+ e δab 2 µ Now that we have all the pieces, we can take a look at the form of the Weyl anomaly. We can ﬁnally see that putting everything together, 1 √ 1 1 √ hT i = βG ∂ Xµ∂ Xν γγab + βB ∂ Xµ∂ Xνab + βΦ γR(2) (8.100) −+ 4 µν a 0 b 0 8 µν a 0 b 0 4 where 1 βG = R − H H λσ + 2∇ ∇ Φ (8.101) µν µν 4 µλσ ν µ ν 1 βB = ∇λH − ∇λΦH (8.102) µν 2 λµν λµν D − 26 α0 1 βφ = + −R + H Hµνσ + 4∇ Φ∇µΦ − 4∇ ∇µΦ (8.103) 6 2 12 µνσ µ µ

26 Note that we have added and extra term that is of the form − 6 . This comes from the ghost system. In order to ensure that the Weyl anomaly is not present in our quantum system, we need that the anomaly is free at all orders. We will set these beta functions to zero (to remove the anomaly) and see what equations result. G B To start, consider a ﬂat, free worldsheet. Then we see both βµν and βµν are zero. However, this is not true for βΦ. In order to remove the anomaly, we need to set D = 26. This is called a critical string theory, since the central charge is zero. It is possible to allow D 6= 26, which are non-critcal string theories, but we would need to introduce Liouville terms in order to compensate. These are signifcantly more complicated than critical string theories, and we will not get into detail here. Of course, there is a slight trouble that we might be overlooking. If we were to look back at our central charge, we would note that it is proportional to βΦ. So the central charge is in actuality an operator (even though we sometimes call the eigenvalue and the operator the same name). So what if the central charge operator has eigenvalues that depend on G B position? The theory manages to safely avoid this by noticing that if both βµν and βµν are zero, then βφ is a constant. So we are indeed safe to return to consider the central charge to be a c-number. Furthermore, it is possible to prove that this conclusion is safe to all orders. Now that we have set D = 26, we can go ahead and consider a curved worldsheet with

91 our ﬁelds. Then we have three conditions, 1 0 = R − H H λσ + 2∇ ∇ Φ (8.104) µν 4 µλσ ν µ ν 1 0 = ∇λH − ∇λΦH (8.105) 2 λµν λµν 1 0 = −R + H Hµνσ + 4∇ Φ∇µΦ − 4∇ ∇µΦ. (8.106) 12 µνσ µ µ Now suppose I take the trace of equation 8.104. I would ﬁnd, 1 0 = R − H Hµλσ + 2∇ ∇µΦ. (8.107) 4 µλσ µ I now use this equation to eliminate the R from the the bottom equation. We then have the conditions 1 0 = R − H H λσ + 2∇ ∇ Φ (8.108) µν 4 µλσ ν µ ν λ λ ∇ Hλµν = 2∇ ΦHλµν (8.109) 1 ∇µ∇ Φ − 2∇µΦ∇ Φ = − H Hµνσ (8.110) µ µ 12 µνσ

1 We could also then reintroduce the curvature scalar using equation 8.107 times 2 Gµν. Finally, we then get 1 1 1 1 R − G R = (H H λσ − G H Hλρσ) − 2(∇ ∇ Φ − G ∇ρ∇ Φ) (8.111) µν 2 µν 4 µλσ ν 2 µν λρσ µ ν 2 µν ρ λ λ ∇ Hλµν = 2∇ ΦHλµν (8.112) 1 ∇µ∇ Φ − 2∇µΦ∇ Φ = − H Hµνσ (8.113) µ µ 12 µνσ These are the equations we were looking for the whole time. They are indeed equivalent G B Φ to the conditions to the vanishing of the Weyl conditions, βµν = βµν = β = 0. However they are in a more suggestive from. Note that on the left hand side of equation 8.111, we have the Einstein tensor. If the right hand side side was the energy momentum tensor of some action, and the following two conditions were equations of motion, then we know we could satisfy these conditions simulanteously. The eﬀective action that reproduces these equations is

Z √ 1 S = dDX Ge−2Φ R + 4(∇Φ)2 − H2 . (8.114) 12

This is the NS sector of the action in the string frame we were looking at in a previous report! In the Einstein frame, this action is given by Z D √ 4 2 1 − 8Φ 2 S = d X g R − (∇Φ) − e D 2 H . (8.115) D − 2 12 −

92 This should look familiar now. The first term is the Einstein-Hilbert action that gives us our regular GR and the other terms are just fields that live in the spacetime (although they are coupled quite strangely). So string theory really is a theory of gravity! So we finally arrive at the main point. We started out with a reparameterization invariant and Weyl invariant (to 1/α0) string action on the worldsheet. We looked at the quantum picture and found that if we demand reparameterization invariance holds, there is an Weyl anomaly. If we demand that the Weyl invariance holds to all orders of α0, then not only do we get a dimensionality for our spacetime, but we also have a low energy effective action that reproduces gravity and other fields. So we can see that string theory is a theory of gravity and other fields.

9 Compactiﬁcation

9.1 Hiding extra dimensions Essential problem we face in building real-world models: taking our string theory action principle defining the UV physics and flowing down to the phenomenologically relevant IR. This is generically messy and hard. β-functions tell us that our 10D spacetime M10 must obey the string theory equations of motion. String theorists commonly take a direct product (KK) ansatz M10 = M4 × K6, where K6 is some compact six-manifold. The details of what theory emerges on M4 depends on the details of the manifold K6 – not every 6-manifold is a solution. The low-energy theory in 4D that you get also depends on which superstring theory you chose: the precise structure of the 10D SUGRA field equations (the antisymmetric tensor fields especially) depends delicately on whether you have IIA, IIB, I, HE, or HO theory. One challenge in model-building is getting the right spectrum of gauge forces and quark/lepton matter. Before the advent of D-branes, a no-go theorem prevented compactifications of IIA or IIB producing chiral fermions in 4D – this is why heterotic string theory was so popular in the first superstring revolution. One of the most difficult aspects of building a credible compactification scenario is stabilizing all of the moduli, scalar fields which do not develop a potential to any order in perturbation theory but which we know must be absent from the low-energy massless spectrum. Recruit nonperturbative physics to fix.

9.2 SUSY

Both the geometry and topology of K6 play an important role in what low-energy physics we end up getting on M4. This originates in the fact that the 10D string theory worldsheet β-function equations are very picky about the spacetimes on which strings can propagate – if we make a product space ansatz it must be compatible with the field equations. In particular, the number of light generations of fermion fields depends sensitively on the holonomy of the Calabi-Yau in KK compactifications. In the first superstring revolution, we discovered how to explain 3 generations in terms of the mathematics of one special type of Calabi-Yau.

93 Why Calabi-Yaus? These are manifolds with special holonomy which support the existence of Killing spinors, allowing SUSY to be present in the 4D theory. SUSY is not an observed low-energy symmetry in Nature (so far), but it is technically important in controlling UV physics. SUSY should be at most N = 1; models with N ≥ 2 have unrealistic spectra. SUSY introduces at least one extra scale in the problem (masses of superpartners), and this can actually permit Grand Uniﬁcation. Just extrapolating Standard Model gauge couplings up to higher energy scales does not yield a GUT; this was proved experimentally via precision electroweak measurements at LEP in the tunnels now occupied by LHC.

9.3 Brane world models Alternatives to Calabi-Yau compactifications of heterotic string theories? Use Type IIA/IIB and add D-branes, fluxes, and orientifolds, which are objects which possess both a negative charge and a negative tension. N.B.: orientifolds do not destabilize the vacuum because they are fixed planes of a symmetry: they cannot fluctuate physically. So their negative tension is harmless. The physically crucial thing is that these negative tension objects which are fundamentally string theoretic let you evade previous no-go theorems which prevented building de Sitter compactifications in string theory. Brane world idea # 1: we ‘effectively’ compactify the physics using a brane world (e.g. Randall-Sundrum) type model. These have a warped product space structure, in which the overall scale of the 4D geometry depends on the coordinate in the compact dimension. For example, in the RS models the bulk has Λ < 0 while the brane is Minkowski, and the radius of curvature of the AdS space provides an effective compactification radius. Brane world idea # 2: build models where Standard Model gauge and matter fields are restricted to the worldvolume of an intersecting D-brane configuration. We are made of open strings stretched between various pairs of D-branes. Only closed strings (gravitons) can move off-world.

9.4 Heterotic string theory on CY3s − + The 10D gravity multiplet comes from the NS-NS sector and has the ﬁelds {GMN ,BMN , Φ, ψM , λ } − − where ψM is the gravitino and λ is the dilatino. (c.f. Jesse’s Final Project presentation.) Since the heterotic string theory already has a big gauge symmetry in 10D, we also have the − − vector multiplet {AM , χ }, where χ is the gaugino, and we have suppressed the Yang-Mills indices. The direct product ansatz M10 = M4 × CY3 is not necessarily enough to ensure that the β-function equations are satisﬁed. For heterotic string theory on a CY3, the Bianchi identity like equation for B2 yields the important condition α0 dH = [Tr (R ∧ R) − Tr (F ∧ F )] . 3 4

α0 This implies that H3 6= dB2 , but rather H3 = dB2 + 4 (ΩL − ΩYM ), where ωL and ωYM are the Lorentz and Yang-Mills Chern-Simons terms which play a central role in the analysis of [gauge and gravitational] anomalies.

94 Since Tr (R ∧ R) is nontrivial in cohomology (it is the 2nd Chern character of the tangent bundle), it requires turning on a nontrivial background ﬁeld strength in order to satisfy Tr (R ∧ R) = Tr (F ∧ F ). We ‘embed the spin connection in the gauge group’, using the fact of SU(3) holonomy group.

9.5 Calabi-Yaus What is a Calabi-Yau manifold? It is a Kählermanifold with n complex dimensions and vanishing first Chern class 1 c = [R] = 0 . 1 2π To unpack this definition, let us start with the definition of a complex structure. It is a rank (1,1) tensor field J such that J 2 = −1 when regarded as an isomorphism on the tangent bundle. In physicist’s terms, J acts like i. A Kählermanifold has three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. You have already met Riemannian manifolds in GR. You have also met symplectic structures in classical mechanics in the part about Hamiltonian dynamics on phase space. We also need Chern classes, which are characteristic classes. These are topological invariants associated to vector bundles on a smooth manifold. A theorem conjectured by Calabi and proven by Yau says these requirements for Calabi-Yaus imply SU(n) holonomy. (For non-compact case, care with BCs at ∞ is required to make this fly.) This implies that CYs admit a covariantly constant spinor. In turn, this implies compactification on a CY leaves some SUSY preserved. Also, Ricci-flat, in SUGRA approx.

9.6 Geometry µ Consider the exterior derivative operator d = dx ∂µ. This object can be used to act on differential forms, which are antisymmetric tensors of rank p. For example, in EM we µ write F = dA where A = dx Aµ is the gauge potential and F the field strength tensor. The Bianchi identity is extremely simple: dF = 0, and it expresses the fact that d2 = 0. Maxwell equation also becomes v.simple: ∗d ∗ F = j, where j is the current. For higher p-forms, such 1 µ1 µp as NS-NS B2 and R-R Cp+1, Ap = p! dx ∧ ... ∧ dx Aµ1...µp where ∧ is the wedge product. This operator that squares to zero should ring a bell vs. BRST cohomology. Indeed, we can ask: are there differential forms that are closed (killed by d) but are not exact (the d of something)? The pth de Rham cohomology group Hp(K) is defined to be the quotient of closed p-forms by exact p-forms and it tells us important topological information for compactification. Next, define the formal adjoint of d, known as d†; the Laplacian on forms is then defined by ∆ = dd† + d†d. This can be used to show that for each de Rham cohomology class on K, there is a unique harmonic representative.

Calabi-Yau metric can be written locally in the form ga¯b = ∂za ∂z¯b K where K is the a b Kählerpotential. Then the Ricci form R = dz dz¯ iRa¯b is closed because Kählermanifolds have no torsion. This is where the first Chern class arises from.

95 For more details, see BBS Appendix to §9 starting on p.440, or §14 of BLT.

9.7 Hodge numbers

Betti numbers bp give fundamental topological information about a manifold. bp is dimension of pth de Rham cohomology Hp(K) of manifold K. When K has a metric, it counts Pk p,k−p number of linearly independent harmonic p-forms on K. When K is Kähler, bk = p=0 h where the hp,q are the Hodge numbers counting the number of harmonic (p, q)-forms on K. These beasts are very useful for helping figure out the spectrum of dimensionally reduced fields. A Calabi-Yau is [partially] characterized by its Hodge numbers. The properties of p,0 n−p,0 p,q q,p p,q n−q,n−p CYs relate h = h (use c.c. of Ω, ga¯b), h = h (c.c.), and h = h (Poincaré duality). Any compact connected Kählermanifold has h0,0 = 1 (constant fns). Also, a simply connected manifold has vanishing first homotopy group, and therefore has vanishing first homology group. (Homology and homotopy are both about defining and categorizing holes in a shape but capture different information.) This gives h1,0 = h0,1 = 0. So for n = 3, the dimension of interest for us, we only need to specify h1,1 and h2,1.

9.8 Mirror symmetry and the conifold Calabi-Yaus with n = 1 are either C (non-compact) or T 2 (compact). For n = 2, you get (a) products of n = 1 CYs, for non-compact cases, or for the compact cases either (b) T 4 or (c) K3. For n = 3 the options become much more numerous, with examples including weighted projective spaces. For most CYs the metric is not explicitly known, except in special limits, which makes compactification life more interesting! CYs are not completely characterized by their Hodge numbers. Indeed, there are pairs of distinct CYs that have the same Hodge numbers and which obey mirror symmetry, which is like a more powerful and complicated version of T-duality. It can be shown at the level of the path integral that the string theories are physically identical for mirror pairs. This involves some pretty math. Physically, CYs can have deformations. These correspond to changing parameters characterizing their shapes and sizes, described by moduli fields. Branes wrapped on supersymmetric cycles can become very light under certain conditions analogous to approaching the self-dual radius in circle compactified string theory. It is critical to include these light modes in your low-energy effective action principle or you will miss important nonperturbative effects. A signature example of this is the conifold transition which can, unlike flop transitions, change Hodge numbers. Need NP string theory, not

96 just SUGRA.

9.9 Dp-brane probes Consider probing a big fat Dp-brane spacetime with a single ‘test’ Dp-brane. Its (kappa symmetric) action in a SUGRA background has two pieces,

Sprobe = SDBI + SWZ , which are, to lowest order in derivatives, for the U(1) part, Z 1 p+1 −Φp S = − d σ e − det (Gαβ + [2πFαβ + Bαβ]) , DBI p p+1 P gs(2π) ls 1 Z S = − exp (2πF2 + B2 ) ∧ ⊕nCn . WZ p p+1 P (2π) ls where the σ are the worldvolume coordinates and P denotes pullback to the worldvolume of bulk fields. The brane action encodes both kinetic and potential information, such as which branes can end on other branes. The WZ term, in particular, encodes the fact that Dp-branes can carry charge of smaller D-branes by having worldvolume field strength F2 (or B2 -fields) turned on. Works for a brane that is topologically R1,p or wrapped on tori. If the D-brane is wrapped on a manifold which is not flat, extra terms arise in the probe action (e.g. ‘A-roof genus’ for K3). Does not capture non-Abelian physics.

9.10 Example: 1 Dp probing N Dps For our supergravity background exerted by N Dp-branes, we had

− 1 1 2 2 2 2 + 2 2 dS = Hp (−dt + dxk) + Hp dx⊥ , 1 −p Φ 4 (3 ) e = Hp , −1 −1 C01···p = gs 1 − Hp . The physics is easiest to interpret in the static gauge, where we ﬁx the worldvolume reparametrisation invariance by setting Xα = σαˆ, α = 0, . . . p. We also have the 9 − p transverse scalar ﬁelds Xi, which for simplicity we take to be functions of time only, Xi = Xi(t), i dXi i = p + 1 ... 9. Denote the transverse velocities as v ≡ dt . Now we can compute the pullback of the metric to the brane. α β i i P (G00) = (∂0X )(∂0X )Gαβ + (∂0X )(∂0X )Gij − 1 1 i j 2 + 2 2 = G00 + Gijv v = −Hp + Hp ~v ; − 1 2 P (Gαβ) = Hp . The last ingredient we need is the determinant of the metric.

q − 1 (p+1) 4 p 2 − det P(Gαβ) = Hp 1 − ~v Hp .

97 9.11 Forces between D-branes Putting this all together yields 1 Z h i p+1 −1p 2 −1 SDBI + SWZ = d σ −H 1 − ~v Hp + H − 1 . p+1 p+1 p p (2π) gs`s From this action we learn that the position-dependent part of the static potential vanishes, as it must for a supersymmetric system such as we have here. The constant piece is of course just the Dp-brane tension. In addition, we can expand out this action in powers of the transverse velocity. To lowest order, 1 Z p+1 1 2 4 Sprobe = d σ −1 + ~v + O(~v ) , p+1 p+1 2 (2π) gs`s so the metric on moduli space, the coefficient of vivj, is flat. This is a consequence of having 16 supercharges preserved by the static system. In SUGRA, SUSY field transformations have a spinorial parameter . For preserved SUSYs, you find a projection condition for Dp-branes. Schematically,

1 + [sgn(Z)] Γ01···p = 0 .

So generically, Dp-branes break half SUSY. Whether or not Dp and Dq can be in equilibrium is determined by calculation. Find: Dp k D(p + 4) is 1/4 SUSYic.

9.12 Open string BCFT The specific form of the action we presented is valid for the U(1) part of the gauge field only. For a stack of Dp-branes, generally in an oriented string theory we get U(N) gauge group, not U(1). The story of how the non-Abelian information is encoded in the DBI+WZ action has been worked out in quite some beautiful detail, e.g. in some classic papers of Rob Myers from the 1990s. We have suppressed it here in an attempt to keep the number of details flying around more manageable. The DBI+WZ action is actually far more than a kappa-symmetric action suitable for D-branes in SUGRA backgrounds. It can be derived in a completely different way as a partial resummation of open string corrections to the SYMp+1 action for a stack of N Dp-branes. This was done in the mid-1980s using worldsheet β-function techniques and starts with a worldsheet coupling of the form I dXµ ds Aµ(X) . ∂Σ ds There is also an open string tachyon, but the GSO projection gets rid of it too. Boundary CFT methods were exploited nicely in this context. You can find a great deal 0 more detail about how to calculate α corrections to the lowest-energy SYMp+1 Lagrangian using BCFT methods in BLT.

98 9.13 What is an orbifold? Suppose that X is a smooth manifold with a discrete isometry group G. Then we can construct the quotient space X/G. A point in that quotient space corresponds to an entire orbit of points in X, i.e., a point and all its images under the isometry group. The quotient space has singularities if nontrivial group elements leave points of X invariant (called fixed points). Locally, the orbifold is indistinguishable from the original manifold. GR is ill-defined on singular spaces. But strings can actually propagate consistently on manifolds with [spatial*] orbifold singularities! The essential physical reason for this is that strings are extended objects, and so they have significantly softer behaviour at short distance than particles do. When you mod out by a symmetry, you lose some sectors of the theory, but for orbifolds you also gain back some sectors known as twisted sectors, roughly speaking where the fields only come back to themselves modulo a symmetry transformation. (This is morally similar to what we saw in circle compactified string theory where KK momentum got quantized, losing states, but we also developed a whole new sector of winding modes.) * If you try to quotient timelike directions, you generically end up with closed timelike curves. These are very bad for your credibility. Interestingly, people have managed to make some sense of some null string theory orbifolds.

9.14 Simple compact and non-compact orbifolds

1 The simplest example of a compact orbifold is S /Z2, where the coordinate x is identiﬁed with −x. This case is relevant to how the E8 ×E8 heterotic string is obtained from M theory 1 – by compactifying on the interval S /Z2 with end-of-the-world branes carrying the E8 × E8 gauge symmetry.

Alternatively, take the complex plane C and identify z with −z. This produces the orbifold C/Z2. What does this space look like? The orbifolding identifies the upper half plane’s positive real axis with its negative real axis under a group transformation, and it is a cone. The conical deficit angle is π. The point (0, 0) is a fixed point of the group action. We could also consider C/ZN , where the group is generated by a 2π/N rotation. This time, the singularity at the origin signifies a deficit angle of 2π(N − 1)/N and it is of AN type (part of the ADE classification).

99 9.15 Spectra of states for orbifolds Untwisted sector states are those which exist on X and are invariant under the symmetry group G. You just take states Ψ such that gΨ = Ψ for all g ∈ G. If your group is finite, then you can make a G-invariant state by starting with any representative Ψ0 and superposing all the images gΨ0. Twisted sector states arise in the following way. For a closed string propagating on an ordinary manifold, we know that translating σ by 2π brings the embedding map field back to itself. But when you have an orbifold, you only need to produce the same map up to a group transformation: Xµ(τ, σ + 2π) = g Xµ(τ, σ) . For orbifolds there are various distinct twisted sectors labelled by the group element used to make the identification. In more fancy mathy language, they are labelled by the conjugacy classes of G. For the C/Z2 example, it is clear that twisted sector string states enclose the singular point of the orbifold. In the quantum spectrum, individual twisted sector quantum states are localized at the orbifold singularities that the classical configurations enclose. It is easy to see this approximately for low-lying string states; harder for full-on oscillator content. Orbifolding enables breaking of some SUSYs of the original manifold.

9.16 What is an orientifold? The one superstring theory we have hardly talked about is Type I, open superstring theory. It can be understood as arising from a projection of Type IIB. For IIB theory, the two worldsheet superpartners of Xµ have the same chirality. World- sheet parity Ω : σ → −σ is therefore a Z2 symmetry of the theory, as it exchanges left- and right-moving modes. The bright idea people had was to gauge this worldsheet parity symmetry, and this is what produces Type I from Type IIB. Recall that T-duality switches the sign of the right-movers only. Then in the T-dual picture, the symmetry transformation above becomes a product of world-sheet parity and a spacetime reflection in directions that were T-dualized. 1 The closed-string spectrum is obtained by keeping states that are even under 2 (1+Ω). The projection condition kills the Kalb-Ramond field and leaves the string metric and dilaton fields preserved. For the gravitini, only the sum of the two survives the projection. Similarly, only one of the IIB dilatini survives, so there are overall 56 + 8 = 64 massless fermionic d.o.f. How about the R-R sector? Requiring SUSY implies that there are 64 bosonic d.o.f. A simple light-cone counting shows that C0 and C4 are projected out, leaving only the R-R 2-form C2. This time, the counting is 35 + 1 + 28 = 64.

9.17 What are Op-planes good for? One of the main applications orientifold planes have found in string compactiﬁcations is to provide sinks for sources of ﬂux. These arise because of tadpole cancellation conditions stating that the total number of sources and sinks must be balanced out to zero overall.

100 Turning on fluxes is one way to help address the perennial problem of stabilization of moduli, so it comes up often. One application this found was to creating string theoretic models of Randall-Sundrum compactifications involving warped product spaces. H.Verlinde showed how if you concen- trated D-branes in a Calabi-Yau nearby one another, such as for D3-branes of the AdS/CFT correspondence, you would develop a long AdS throat with a warp factor that scales exponentially in the radial coordinate away from the D3-brane stack. One may also ask whether Op-planes have other physical roles. They have a very important property other than negative charge (see above): negative tension. If you are a half-decent theoretical physicist, the idea of negative-tension objects should jolt you out of your seat, because they will reliably destabilize the vacuum of your theory if they are allowed to fluctuate. But since the Op-planes are actually fixed planes, they do not have a physical interpretation familiar from classical physics. They never fluctuate. Ever.

9.18 Flux compactifications Cycles in e.g. a Calabi-Yau manifold can have fluxes threading through them. What dimension of cycle you need is obtained by looking at how to integrate up your field strength or its Hodge dual: Z Z Fp+2 or ?Fp+2 . γp+2 γD p 2 − − Fluxes are actually quantized. If they are sourced by D-branes, then it is clear why fluxes are integers: this corresponds to having an integer number of branes. For manifolds of nontrivial homology, under special conditions, integer quantized fluxes can also be turned on even when there are no brane sources. Flux quantization arises from the generalized Dirac quantization conditions (electric charge e and magnetic charge g obey e · g ∈ 2πZ, via holonomy). What kinds of fluxes can be turned on, and how they can be mutually compatible, depends sensitively on the type of superstring theory you start with and the compactification you choose. Their options are highly constrained by the ultraviolet physics of the worldsheet. One of the most difficult tasks in building a relatively realistic string compactification is stabilizing the moduli. Two ubiquitous types of moduli are the dilaton field and the radial modulus describing the overall scale of the CY.

9.19 Flux-ology How do you decide what fluxes to turn on? First of all, if you want low-energy SUSY, you have to ensure that their presence is consistent with the existence of covariantly constant spinors. The fluxes end up appearing in covariant derivatives of spinors, contracted with an antisymmetric combination of gamma matrices. (For example, in IIB compactifications you find that the 3-form field strength needs to be imaginary self-dual to preserve SUSY properly.) Tadpole cancellation conditions also provide an important constraint. Flux compactifications typically produce a non-trivial scalar potential for the [would- be] moduli fields. A handwaving description of this goes as follows. We may have a modulus

101 field ϕ that couples differently to two different fluxes, Z −ϕ 2 +ϕ 2 S ⊃ − e |F1| + e |F2| .

The value of ϕ can get dynamically fixed by the ratio of fluxes via the equations of motion. Very few models have been constructed in which all moduli are stabilized without nonperturbative effects. Noone knew how to do nonperturbative string theory with any confidence until the second superstring revolution of the mid-1990s.

9.20 Inflation in string theory BBS §10.7 has a nice quick introduction to early universe cosmology and inflation. Unfor- tunately I have no time to give a quick review here so must refer you there. I can also recommend the gigantic review on string theory modelling of early universe cosmology available from Baumann and McAllister, arxiv:1404.2601 for anyone who has an appetite for more. It will be published soon in proper textbook form by Cambridge University Press. String theory compactifications have many moving parts. It is possible to obtain inflation, or a phenomenon producing many of the same effects in our uinverse today, using quite a number of different string theoretic mechanisms. Examples of classes of models that have been studied include the following (borrowed from the B-McA table of contents):-

• Inflating with unwarped or warped branes, such as D3-D7. • Inflating with relativistic branes - Dirac-Born-Infeld style. • Inflating with axions. Compactified Kalb-Ramond components routinely end up giving rise to an axion field from string theory. • Inflating with Kählermoduli. • Inflating with dissipation.

You can also build ekpyrotic universe type models as well, although those suﬀer from their own controversies.

9.21 KKLT and the Landscape Can we build string models with small positive cosmological constant? H.Verlinde late 90s: Consider a CY3 with a certain number of O3-planes. Tadpole cancellation allows you to have a limited number of D3-branes living at various points in the CY3. Idea: group enough D3-branes together to make a long AdS5 throat. This introduces a way of understanding developing large hierarchies via exponentially large redshifting, `ala Randall-Sundrum. [K]KLT early 2000s: put in an anti-D3 to break SUSY spontaneously and uplift AdS vacua slightly to deS. Vacua obtained in this way are only metastable. But if their lifetime is extremely long then it does not bother us. The big question is: how controlled is the approximation of adding the anti-D3?

102 The embarrassing thing about the second superstring revolution is that it eventually yielded the stark realization that there is an extremely large number of possible Standard- Model-like vacua in the theory. It looks very unlikely that our universe would be the only solution to the fundamental equations of string theory. It now seems much more likely that we are a statistical accident. In this context, the value of the cosmological constant is seen to be a fortunate envi- ronmental accident. (Hot tip: believe Susskind, not Smolin!) String theorists who felt bereft after these revelations have satisﬁed themselves with investigating the statistical properties of superstring vacua. 9.22 Ur potential StringString TheoryTheory LandscapeLandscape

Perhaps 101000 different minima

Lerche, Lust, Schellekens 1987

Bousso, Polchinski; Susskind; Douglas, Denef,…

10 AdS/CFT

10.1 Origin of AdS/CFT AdS/CFT holography is an equivalence between gravitational (string) theories and non- gravitational ﬁeld theories in one lower dimension. It gave the ﬁrst concrete working models of the concept of holography for quantum gravity and grew out of studying nonperturbative

103 D-branes in string theory. Consider a stack of Np Dp-branes. Our key idea will be to have large Np.

To see why, consider how the gravitational warping of a single Dp-brane scales. Correc- tions to the ﬂat Minkowski metric scale as (G )(τ ) N 1 g N δg ∼ N Dp ∼ g2`8 p ∼ s p µν 7−p s s p+1 7−p 7−p r gs`s r (r/`s)

This shows that the variable controlling gravitational warping is gsNp. So we can obtain a seriously large warping of ﬂat spacetime by having Np become parametrically large while keeping gs small.

10.2 Maldacena’s decoupling limit Spacetime warping can be thought of as physics of closed strings. After all, the lowest state of excitation of a closed superstring includes the graviton mode. We can also look at the open string picture of Dp-brane physics. Lowest mode of open superstring has spin-one and mass zero. Theory living on world- 0 volume of N Dp-branes is SYMp+1 + α corrections + gs corrections. The variable controlling open string corrections when D-branes are present is actually gsNp, which is exactly what we found also controls spacetime warping. In general, dynamics of the full closed + open string sectors is unknown (c.f. string ﬁeld theory). Maldacena’s key insight was to take a very special ‘decoupling limit’ of this stack of Np Dp-branes, which turns oﬀ interactions between closed strings in the bulk and open strings whose endpoints lie on the D-branes.

* Bulk

Brane

10.3 AdS=CFT In the decoupling limit, on the open string side you get just d = p + 1 SUSY Yang-Mills living on the Dp-brane worldvolume. On the closed string side, the decoupling limit zooms in on the near-horizon region of the geometry.

104 Maldacena boldly realized that this decoupling limit meant that you could think about the stack of N3 D3-branes either in terms of the near-horizon geometry or the worldvolume Yang-Mills theory. But because it is the same stack of D3-branes either way, those representations must be equivalent! D−p−2 The near-core geometries of D3, M2, M5, and D1-D5 systems produce AdSp+2×S spacetimes, where D is the spacetime dimension, 10 in superstring theory and 11 in M theory. You will see how this works in HW4. In brief: the S5 factor arises for D3 because the r2 in 2 +1/2 the transverse piece of the metric proportional to dΩ5 is cancelled off by the H3 factor, because for D3-branes the harmonic function scales as 1/r4 in the near-core region. Yang-Mills theory on flat D3-branes has maximal SUSY, i.e. 16 supercharges. In d = 4 language this translates into N = 4 SUSY. The supermultiplet has adjoint fermions and scalars partnering up with the adjoint vector field. The field content of this theory is so special that it ends up having zero β-function. It is actually a conformal field theory. Hence, we say AdS=CFT.

10.4 Probes and IR/UV relations AdS/CFT gives a definition of quantum gravity in asymptotically AdS space in Lorentzian signature, via the duality and Wick rotation in the CFT. AdS/CFT sheds light on the question of background independence in quantum gravity. It is independent of bulk background except for its asymptotics which are locked down by boundary physics. Not yet known how to do a worldsheet analysis for string theory in AdS, because a Ramond-Ramond flux is turned on. (c.f. v.technical Pure Spinor formalism of N.Berkovits, which is v.different to the NSR and GS formalisms.) One of the key ideas string theorists use every day is the concept of probes. Probes are objects that exist in the theory and are used to interrogate the physics. For fundamental string probes stretched between Dp-branes, field theory energy E is proportional to distance r between D-branes in bulk, r E = . 2πα0 This is known as an IR/UV relation and is central to AdS/CFT. It says that the UV in the field theory is welded to the IR in the bulk and vice versa. Polchinski and I showed that gravitons have different IR/UV relations. Still, how deep it penetrates into the bulk scales inversely with energy on the brane.

10.5 IMSY and applications of AdS/CFT For more general Dp-branes, you ﬁnd that the harmonic functions conspire to not give D−p−2 you AdSp+2 × S spacetimes. Instead [for p < 5], you get a warped version where there is an r- (and p-) dependent conformal factor. What does this mean for the possibility of a correspondence like AdS/CFT? The 10D bulk spacetime background is only reliable for certain ranges of r, because either α0R or eΦ is monotonically growing as r → 0 or r → ∞. By dimensional analysis, the ﬁeld theory coupling also runs with energy. 10D SUGRA

105 2 p−3 geometry becomes unreliable when geff (E) := gsN(`sE) becomes of order unity, which is where the SYM field theory description takes over. This gives rise to the idea of a phase diagram for a D-brane/string system, which turns out to be a very generic and physically rich feature in holographic setups. There can be more than one description of the physics, but only one of them at a time can be weakly coupled. AdS/CFT has been applied to modelling quark-gluon plasma and cond-mat, with limited successes. Can calculate transport properties; for cond-mat the physics is quite different in spirit to the story of quasiparticles. Also inspired the study of dS/CFT correspondence where the CFT is non-unitary. H.Nastase review of AdS/CFT basics for beginners: 0712.0689.

10.6 Holographic dictionary AdS/CFT is a strong/weak duality,

L4 1 AdS ↔ g2 N , g ↔ . 4 YM s `s N This is why it is so powerful: to learn about strongly coupled CFT we use weakly coupled string theory in AdS and vice versa. Isometries of the bulk match up with symmetries of the CFT.

1. The S5 factor has a natural SO(6) isometry. This is is also the R-symmetry of the N = 4 SUSY SYM theory.

2. The AdS5 factor has isometry group SO(4, 2). This is the conformal symmetry group of N = 4 d = 3 + 1 SYM. SO(3, 1), the Lorentz group, is a subgroup.

One of the reasons why AdS/CFT works is that near infinity, the area element grows like the volume element. The dimensionful constant of proportionality is furnished by the radius of curvature of AdS. Another very useful physics fact to know about AdS is that higher-` partial waves do not fall off with higher powers of 1/r like they do in asymptotically Minkowski space. Influences deep in the AdS bulk can be easily seen from infinity. This is related to the fact that AdS has a timelike boundary.

10.7 Fefferman-Graham Maldacena derived AdS/CFT for the gauge theory on the worldvolume living on Rp. Technically, the spacetime obtained was the Poincarépatch of AdS. Witten taught us that the correspondence actually extends to global AdS, which is dual to field theory living on Sp. Much beautiful physics derived. In asymptotically AdS, Fefferman-Graham coordinates are very handy,

L2 ds2 = dz2 + g (z, ~x)dxadxb , z2 ab

106 where L is the AdS scale, and z = 1/r ∈ [0, ∞). gab(z, ~x) is the boundary metric which obeys an expansion in powers of z/L. It is only fixed up to conformal transformation. Other bulk fields obey a similar near-boundary expansion. In general, get an overall power of z/L times a power law expansion with some logarithmic terms. Bulk EOMs are 2nd order PDEs, which yield two independent solutions. Working through the Fefferman-Graham details shows that coefficients of power law and log terms are all determined in terms of the first nontrivial coefficient function of ~x. The key fact is that this boundary value of the bulk field, up to an overall power of z/L, is interpreted as a source term for a dual operator in the boundary field theory.

10.8 BF bound and GKP/W Example: scalar of mass m. Get

d−∆ z φ(r, ~x) = (z/L) {φ(0)(~x) + L φ(1)(~x) + ···} 2∆−d +(z/L) {φ(2∆−d)(~x) + ···} , where ∆ solves m2L2 − ∆(∆ − d) = 0, giving p ∆ = d/2 ± (d/2)2 + m2L2 .

∆ will turn out to be the scaling dimension of the dual operator in the CFT. Requiring ∆ ∈ R gives m2 ≥ −(d/2L)2 . This is the Breitenlohner-Freedman bound famous from SUGRA. Scalar ﬁelds in AdS can be a little bit tachyonic, but not too much, or AdS goes unstable. Gubser-Klebanov-Polyakov + Witten took this further, conjecturing

R d ∆−d −S+ d xφ(0)O(x) Zstring z φ(x, z)|z=0 = φ(0)(x) = he iCFT

Prime example of operator dictionary: gµν(bulk) ↔ Tµν(boundary). Non-normalizable modes: ↔ turn on ∆LCFT = αφOφ in boundary (irrelevant ops). Normal- izable modes: ↔ turn on VEV hOφi = βφ (relevant ops).

10.9 Holography and black holes Nonlocal probes: correlation functions, Wilson loops, entanglement entropy. Witten diagrams: bulk-boundary propagators and bulk vertices.

107 UV/IR relations taught us that high-energy in CFT corresponds to near-boundary in bulk. Holographic RG: running understood via bulk Hamilton-Jacobi, including countert- erms. Flows change interior of AdS. A very deep fact about AdS/CFT is that it is not just a zero-temperature equivalence. Asymptotically AdS black holes correspond to turning on ﬁnite temperature in the CFT. The Hawking-Page transition from BH to hot AdS is dual to the deconﬁnement transition in the boundary theory. AdS/CFT shows us how one extra bulk spacelike dimension emerges, but generically without a path integral proof as yet. It also does not show us how time emerges. That may require going beyond QM as we currently know it.

10.10 Maldacena’s eternal AdS BH Maldacena also proposed in hep-th/0106112 that the eternal AdS black hole is dual to a direct product of two uncoupled conformal ﬁeld theories, CFTL × CFTR, with thermal entanglement between the L and R CFTs. Key feature: if you trace over either the left or right set of degrees of freedom, then you obtain a thermal density matrix at the Hawking temperature. In the holography literature this is known as the ‘two-sided’ black hole. Black holes formed in gravitational collapse are fundamentally diﬀerent: they do not have a white hole singularity or a past event horizon and are called ‘one-sided’.

Maldacena’s idea rests on an older construction by Werner Israel, a very famous Cana- dian gravity theorist, called the thermoﬁeld double or TFD state, √ X −βE/2 |TFDi = (1/ Z) e |ψiL × |ψiR . i

10.11 Less symmetric holography To use holography to model real-world systems, need to break increasing degrees of SUSY and other symmetries. Big literature on this, divided into quark-gluon plasma modelling, AdS/condensed matter; also, dS/CFT. Fixing the asymptotics does not prevent you from having interesting phase transitions originating in interesting hair on the bulk solutions. There are much bigger, wilder classes of geometries available than previously imagined.

108 Holography is applicable to systems other than N = 4 SYM. Sometimes in a bottom- up setup we do not know what the dual QFT is, but we can still use holography to discern universal aspects of strongly coupled systems. Breaking boost: can get residual Schrödingeror Lifshitz symmetry. Breaking anisotropy and homogeneity. Modelling superconductors, glasses, strange metals, Fermi surfaces, hyperscaling violation, disorder. Holographic lattices. My interest in holography is more bottom-up style than top-down. Recent(ish) papers were on Lifshitz black holes [Bertoldi-Burrington-Peet] and on hyperscaling violating crossovers [O’Keeffe-Peet]. Current preprint on holographic modelling of disorder almost ready for posting [O’Keeffe-Peet]. Neat part about holography from the point of view of someone interested in gravity: geometrizing phases of the dual QFT.

10.12 Higher-spin/vector holography Higher-spin theory is a lab that may provide a bridge between classical gravity and full quantum ST. Tower of modes. (c.f. ABJM, multi-M2, localization.) Old theorems had put stringent constraints on low-E scattering in flat spacetime that forbid m = 0 particles with spins s > 2 from participating in any interacting QFT. But in AdS, Λ < 0 provides dimensional coupling and IR cutoff, which can reconcile HS gauge symmetry w equivalence principle, giving M.Vasiliev’s nonlinear unfolded equations of motion. Review: 1404.1948. J.Maldacena-A.Zhibodaev 1112.1016 showed why for 3D CFT with higher-spin symmetry you get a free theory, found for the dual of Vasiliev higher-spin theory in AdS4. S.Giombi-X.Yin AdS4 VH review: 1208.4036. R.Gokapumar-M.Gaberdiel 1207.6697 found the dual for AdS3: a minimal model coset CFT2 with WN symmetry at large-N. (Scalar accompanying graviton and HS fields in 2D GG duality is massive; for AdS4 it is massless.) GG actually got even further: they found that higher-spin in AdS3 is a subsector of string theory, in the tensionless limit. Technically important. Higher-spin theory in 3D has BHs 1208.5182. Existence of horizons and singularities is not invariant under HS gauge transformations, but can define via holonomy if use Chern- Simons formulation of 3D gravity, e.g. 1302.0816.

10.13 Bulk locality AdS/CFT is a fascinating laboratory for studying process of thermalization. Study quenches, try to extract universalities. Review: 1103.2683. (Strong time dependence is harder than weak or none.) See also ﬂuid/gravity correspondence; review by Hubeny: 1501.00007. (A.Hamilton-)D.Kabat-G.Lifschytz-D.Lowe showed in hep-th/0506118, hep-th/0606141, 1102.2910 how local operators in the AdS bulk can be represented via smeared operators in the CFT. Only regions in the causally relevant zone contribute. Their construction can be obstructed if there are bulk normal modes with exponentially small near-boundary imprint, such as for the AdS black hole 1304.6821. Is bulk locality emergent?

109 M.vanRaamsdonk conjectured 0907.2939, 1005.3035 that smooth connected patches of geometry emerge from entanglement of regions on the boundary. Entanglement may not be enough to fully probe bulk geometry, esp. if BH. 1406.5859 by BCCdB discussed entanglement shadows and entwinement. Key Q: how much information can you ever reconstruct from the boundary? A.Almheiri-X.Dong-D.Harlow 1411.7041 argued that localization of bulk information should be understood in terms of quantum error correction. E.Mintun-J.Polchinski- V.Rosenhaus 1501.06577 connected this to boundary gauge invariance, suggesting it is closely connected to spacetime emergence.

10.14 Geometrization of entanglement

S.Ryu-T.Takayanagi conjecture hep-th/0603001 relates the entanglement entropy SEnt associated to a region R in the field theory to the area of the minimal surface in the bulk whose boundary is R. The holographic RT formula is important because it connects a geo- metrical bulk computation with an information theoretic field theory computation. Reviews: T.Nishioka-S.Ryu-T.Takayanagi 0905.0932, Headrick 1312.6717. An explanation of the RT formula was provided by A.Lewkowycz-J.Maldacena in 1304.4926, using a bulk version of the replica trick. ‘Hole-ography’ method computes entanglement for a hole in AdS spacetime 1310.4204 V.Balasubramanian-B.Chowdhury-B.Czech-J.deBoer-M.Heller. Uses differencing of RT formula, residual entropy. N.Lashkari-J.Simon in 1402.4829 argued that emergence of an effective notion of spacetime locality originates in restricting to a subset of observables unable to resolve black hole microstates from the maximally entangled state. People e.g. 1312.7856 also found that the first law for SEnt – for small perturbations about CFT vacuum states, for ball-shaped regions – translates in the bulk to satisfaction of equations of motion linearized about AdS! Constraining the nonlinear story: 1405.3743. Also, entanglement inequalities can be used to derive conditions on bulk Tµν: 1412.3514.

11 Black hole entropy and the information paradox

11.1 Cooking up D1 k D5 Recipe for making BPS black holes is considerably simpler than recipe for making nonextremal ones. Today, make BPS, qualitative comments only regarding nonextremal. First part of recipe is how to combine diﬀerent ingredients. In other words, rules for intersecting branes. We know two clumps of parallel BPS p-branes can be in static equilibrium. Also, BPS p-branes and q-branes for some choices of p, q can be in equilibrium with each other under certain conditions. One way to ﬁnd many rules is to start with the fundamental string intersecting a Dp-brane at a point, F 1 ⊥ Dp, and use S- and T-duality.

F 1 ⊥ D3 −→ D1 ⊥ D3 −→ D0 k D4 −→ D1 k D5

110 F1 D1

D0 D1

D3 D3 D4 D5

11.2 Cooking up W k D5 and W k D1 and also F 1 ⊥ D3 −→ D1 ⊥ D3 −→ D0 k D4 −→ MW k M5 −→ W k D5

F1 D1 W D0 MW D5 D3 D3 D4 M5

and by T-duality, W k D1.

W W D5 D1

Therefore, W k D1 k D5 can all be in neutral equilibrium in a mutually consistent fashion.

11.3 Problems with too few ingredients BPS black holes in dimensions d = 4 ... 9 may be constructed from BPS building blocks. Typically, however, they have zero horizon area and therefore non-macroscopic entropy. Example: consider D1-brane 2 −1/2 2 2 +1/2 2 2 2 ds = H1 −dt + dx + H1 dr + r dΩ7 where 32π2g N `6 H = 1 + s p s 1 r6 Now compactify the x direction on a circle of radius R at inﬁnity. At the horizon, 1 Vol(S ) p − 1 3 = G = (H ) 2 ∼ r → 0 (2π)R xx 1 How about Bekenstein-Hawking entropy? Transform to Einstein frame: −1/2 1/2 −1/4 gµν = H1 Gµν = H1 Gµν so that 2 −3/4 2 2 +1/4 2 2 2 ds = H1 −dt + dx + H1 dr + r dΩ7

111 11.4 Why we started with D = 5 BH Hence (entropy same if evaluate in d = 10 or d = 9 !)

1 16π3 7/2 S = r2H1/4 BH 6 2 8 1 4[8π gs `s] 15 horizon

−6 and since H1 ∼ r near horizon,

SBH (BPS D1) = 0. More generally, study SUGRA field equations to find what BHs can have macroscopic entropy. Sizes of internal manifolds, plus dilaton, are scalar fields in lower-d. Horizon area depends on these scalars, which are ratios of functions of charges like Hp’s. But in any given d, have only a few independent charges on a black hole – fewer gauge fields than scalars. Too few independent charges to give all scalar fields well-behaved vevs everywhere in spacetime. E.g. for stringy black holes made by compactifying on tori, only asymptotically flat BPS black holes with macroscopic finite-area occur with 3 charges in d = 5 and 4 charges in d = 4. The d = 4 case where all 4 charges are equal is Reissner-Nordstrøm. Woohoo!

11.5 The harmonic function rule A systematic Ansatz is available for construction of SUGRA solutions corresponding to pairwise intersections of BPS branes. Known as “harmonic function rule”. Ansatz: metric factorizes as product structure: simply “superpose” harmonic functions. This ansatz works for both parallel and perpendicular intersections. Important restriction: harmonic functions can depend only on overall transverse coordinates. In this way, get only “smeared” intersecting brane solutions. Representation convention: − means brane is extended in that dimension, · means it is pointlike, and ∼ says although brane is not extended in that direction a priori, its dependence on those coordinates has been smeared away. E.g. for D5 with D1 smeared over its worldvolume: 0 1 2 3 4 5 6 7 8 9 D1 − − ∼ ∼ ∼ ∼ · · · · D5 − − − − − − · · · ·

11.6 Cooking the D1-D5 system 2 2 P9 i 2 For D1-D5 system, let us deﬁne r ≡ x⊥ = i=6(x ) to be overall transverse coordinate. Then string frame metric is, using harmonic function rule,

2 − 1 − 1 2 2 + 1 − 1 2 dS10 = H1(r) 2 H5(r) 2 (−dt + dx1) + H1(r) 2 H5(r) 2 dx2···5 + 1 + 1 2 2 2 +H1(r) 2 H5(r) 2 (dr + r dΩ3) and dilaton is Φ + 1 − 1 e = H1(r) 2 H5(r) 2

112 while R-R gauge ﬁelds are as before,

−1 −1 −1 −1 C01 = gs H1(r) C01...5 = gs H5(r) Independent D1 and D5 harmonic functions both go like r−2, g N `2 g N `6/V H (r) = 1 + s 5 s H (r) = 1 + s 1 s 4 5 r2 1 r2 Wrap x2 ··· x5 on T 4 to make d = 6 black string with two charges. Internal T 4 is ﬁnite-size at event horizon r = 0:

1 4 4 4 p H1 N1(`s/V4) G22 ··· G55 = → H5 N5

11.7 Adding the gravitational wave Next step is to roll up direction of black string, to make black hole in d = 5. Behaviour of radius of x1 direction near horizon? p r G = (H H )−1/4 ∼ → 0 11 1 5 1/4 (N1N5) Oops! Still need another quantum number to stabilize this S1 as well as our T4. We can use knowledge from solution-generating to puﬀ up this horizon to a macroscopic size by using ∞ boost in longitudinal direction x1. Ingredients for building this black hole are then previous branes with addition of a gravitational wave W: 0 1 2 3 4 5 6 7 8 9 D1 − − ∼ ∼ ∼ ∼ · · · · D5 − − − − − − · · · · W − → ∼ ∼ ∼ ∼ · · · ·

→ denotes direction in which gravitational Wave moves (at speed of light).

11.8 The D1-D5-W metric in 5D BPS metric for this system is obtained from simpler metric for plain D1-D5 system by boosting and taking extremal limit. To get rid of ﬁve dimensions to make a d = 5 black hole, compactify D5-brane on the T 4 of volume (2π)4V , and then D1 and remaining extended dimension of D5 on S1, volume 2πR. d = 5 Einstein frame metric becomes

2 −2/3 2 ds5 = − (H1(r)H5(r) (1 + K(r))) dt 1/3 2 2 2 + (H1(r)H5(r) (1 + K(r))) [dr + r dΩ3] where harmonic functions are r2 r2 r2 H (r) = 1 + 1 H (r) = 1 + 5 K(r) = m 1 r2 5 r2 r2 113 r2 `2 r2 r2 `8 1 = (g N ) s 5 = (g N ) m = (g2N ) s 2 s 1 2 s 5 2 s m 2 `s V `s `s R V This SUGRA solution has limits to its validity. For e.g. curvature, ﬁnd e.g. R(d = 2 2 2 1/3 µν 2 2 5) → −2/(r1r5rm) at small r; or R Rµν(d = 10) → −24/(r1r5). So if stringy α0 corrections to geometry are to be small, need large radius parameters. Dila- 2Φ ton? E.g. d = 10 e → N1/N5.

11.9 Bekenstein-Hawking entropy Suppose we keep volumes V,R ﬁxed in string units. Therefore, need

2 gsN1 1 gsN5 1 gs Np 1

Can also control closed-string loop corrections if gs 1. These two conditions are compatible if we have large numbers of branes and large momentum number for gravitational wave W. Also note that Np needs to be hierarchically larger than N1,N5. Next properties of this spacetime to compute are thermodynamic quantities. BPS black hole is extremal and it has TH = 0. For Bekenstein-Hawking entropy,

A 1 2 n 3 3/6o SBH = = 2π r [H1(r)H5(r) (1 + K(r))] (11.1) 4G5 4G5 r=0 2 2π 1/2 = (r r rm) (11.2) 2 8 1 5 4 (π/4)gs `s /(VR) 1 2πV R g N ` 6 g 2N ` 8 2 = s 1 s g N ` 2 s m s (11.3) 2 8 s 5 s 2 gs `s V R V p = 2π N1N5Nm (11.4)

11.10 Properties of SBH This entropy p SBH = 2π N1N5Nm is macroscopically large. Notice that it is also independent of R and of V . More generally, SBH for BPS guys is independent of all moduli. This is to be contrasted with ADM mass

Nm N1R N5RV M = + 2 + 6 R gs`s gs`s which depends on R,V explicitly. √ For entropy of black hole just constructed out of D1 D5 and W, we had SBH = 2π N1N5Nm. More generally, for a more general black hole solution of maximal supergravity arising from compactifying Type II on T 5, it is r ∆ S = 2π BH 48

114 where quantity ∆ in surd is cubic invariant of the E6,6 duality group,

4 X 3 ∆ = 2 λi i=1 and λi are eigenvalues of central charge matrix Z.

11.11 Yes, extreme BH can have ﬁnite SBH A few years ago a claim was made that all extremal black holes have zero entropy. Arguments were in Euclidean spacetime signature, and made the point that adding in surface terms at horizon was necessary to make sure Euler number of horizon was not fractional. This result is not trustworthy in the context of string theory. 1. There is actually no good physical reason why zero-temperature black holes should have zero entropy. Standard statements of the Third Law make unnecessary assumptions about the equation of state of physical matter. 2. Faulty nature of classical reasoning in string theory context was pointed out in a G.Horowitz review article from mid-1990s. In Euclidean geometry, for any periodicity in Euclidean time β at r = ∞, presence of extremal horizon results in a redshift which forces that periodicity to be substringy very close to horizon. Since light strings wound around this tiny circle can condense, a Hagedorn transition can occur. Classical approximation is not reliable there; in particular, arguments based on classical topology are not believable. 3. This entropy would be hugely smaller than entropy of very-nearly-extremal BH! Where would all the entropy go?

11.12 Four charges in four dimensions Extremal Reissner-Nordströmblack hole can be embedded in string theory using D-branes. For the extremal Reissner-Nordströmspacetime metric in isotropic coordinates we find H±2(r)s appearing in metric:

2 −2 2 2 2 2 2 ds = H (−dt ) + H (dr + r dΩ ) H = 1 + r0/r

1 This is to be contrasted with the H 2 ’s to be found in a generic p-brane metric:

2 −1/2 2 2 +1/2 2 2 2 ds = H (−dt + dx1...p) + H (dr + r dΩ ) From this we can guess (correctly) that, in order to embed extremal Reissner-Nordstr¨om black hole in string theory, we will need 4 independent brane constituents. Restrictions must be obeyed, however, in order for that black hole to be Reissner- Nordstr¨om. To make more general d = 4 black holes with four independent charges, we simply lift these restrictions and allow charges to be anything - so long as they are large enough to permit a supergravity description.

115 11.13 The D2-D6-W-NS5 duality frame For making d = 4 black hole, one set of ingredients would be

0 1 2 3 4 5 6 7 8 9 D2 − − − ∼ ∼ ∼ ∼ · · · D6 − − − − − − − · · · NS5 − − − − − − ∼ · · · W − → ∼ ∼ ∼ ∼ ∼ · · ·

By U-duality, we could consider instead 4 mutually orthogonal D3-branes, or indeed many other more complicated arrangements. In ten dimensions we can construct BPS solution by using the harmonic function rule. So far we have not exhibited metric for NS5-branes but that can be easily obtained using D5 metric and using fact that Einstein metric is invariant under S-duality. We then have

− 1 − 1 − 1 − 1 2 2 2 2 2 2 2 2 2 dS10 = H2 H6 −dt + dx1 + K(dt + dx1) + H5H2 H6 (dx2) 1 − 1 1 1 + 2 2 2 + 2 + 2 2 2 2 +H2 H6 H5(dx3···6) + H5H2 H6 (dr + r dΩ2) (11.5) and 1 1 − 1 Φ + 2 + 4 4 (3) e = H5 H2 H6

11.14 SBH for the 4D 4-charge BH Smearing and Newton’s constant formulæ give

g N ` 5 g N ` N ` 2 g 2N ` 8 r = s 2 s r = s 6 s r = 5 s r = s m s 2 6 5 m 2 2V 2 2Rb 2VRaRb Kaluza-Klein reduction formulæ give ﬁrst a d = 5 black string and then ﬁnally the d = 4 black hole. Final Einstein metric in d = 4 is

−1 2 2 hp i ds = −dt (1 + K(r))H2(r)H6(r)H5(r) 2 2 2 hp i +(dr + r dΩ2) (1 + K(r))H2(r)H6(r)H5(r)

Reissner-Nordstr¨omblack hole is obtained by setting all four gravitational radii to be identical: r2 = r6 = r5 = rm. Bekenstein-Hawking entropy is p SBH = 2π N2N6N5Nm

More generally, in surd is quantity ♦/256, where ♦ is quartic invariant of E7,7

4 4 X 2 X 2 2 ♦ = |λi| − 2 |λi| |λj| + 4 λ1λ2λ3λ4 + λ1λ2λ3λ4 i=1 i

116 11.15 The D-brane picture Our setup of branes for d = 5 BPS BH with 3 charges was

0 1 2 3 4 5 6 7 8 9 D1 − − ∼ ∼ ∼ ∼ · · · · D5 − − − − − − · · · · W − → ∼ ∼ ∼ ∼ · · · ·

This system preserves 4 real supercharges, or N = 1 in d = 5. Each constituent breaks half of SUSYs. Necessary for SUSY to orient branes in a relatively supersymmetric way. If not, e.g. if an orientation is reversed, D-brane system corresponds to a black hole that is extremal (double horizon) but has no SUSY. Beginning ingredients: D1 branes and D5 branes. What are degrees of freedom carrying momentum quantum number? D5 branes and smeared D1 branes have a symmetry group

SO(1, 1)×SO(4)k ×SO(4)⊥ .

This symmetry forbids (rigid) branes from carrying linear or angular momentum, so we need something else.

11.16 Open string dynamics

T4 S1

Obvious modes in the system to try are massless 1-1, 5-5 and 1-5 strings, which come in both bosonic and fermionic varieties. • Momentum Nm/R carried by bosonic and fermionic strings, 1/R each. 1 • Angular momentum is carried only by fermionic strings, 2 ~ each. Both linear and angular momenta can be built up to macroscopic levels. Next step: identify degeneracy of states of this system. Simpliﬁcation made by [Strominger-Vafa] is to choose the four-volume small by comparison to circle radius,

1 V 4 R

Makes theory on D-branes a d = 1 + 1 theory. This theory has (4, 4) SUSY in d = 1 + 1 language.

117 11.17 String partition function d = 1 + 1 partition function of a number n of boson fields and an equal number of fermion fields is " ∞ #n Nm Y 1 + w X Nm Z = ≡ Ω(Nm)w 1 − wNm Nm=1 where Ω(Nm) is degeneracy of states at d = 1 + 1 energy E = Nm/R. At large-degeneracies, which happen with big quantum numbers like we have here, we can use Cardy formula r π c E (2πR) r c Ω(N )∼exp = exp 2π ER m 3 6 (Technical note: This formula assumes that lowest eigenvalue of energy operator is zero, as it is in our system. Otherwise must use instead ceff = c − 24∆0, where ∆0 is ground state energy.) We know R, radius of circular dimension. Need c and E. Central charge 1 c = nbose + 2 nfermi How many bosons (and fermions) do we have?

11.18 Degeneracy of states Boson and fermion count in system of D1, D5 and open strings? Can be done rigorously; here is the basic physics: • N1N5 1-5 strings that can move in 4 directions of torus, hence c = 6N1N5. • Alternatively, we can use neat fact that D1-branes are instantons in D5-brane theory. Have N1 instantons in U(N5) gauge theory, and N5 orientations to point them in. Etc... Now, how about energy E? System is supersymmetric, and since no Z’s down here in µ this d = 1 + 1 story, need P Pµ = 0. So E = |P |. In d = 1 + 1 things can move only to R or L. Our sign conventions make us have R-moving groundstate, and put all the action in L-movers. Momentum was P = ±Nm/R, so E = Nm/R. Cardy said r π c E (2πR) r c Ω(N )∼exp = exp 2π ER m 3 6 Therefore p Smicro = 2π N1 N5 Nm OMG: this agrees exactly with the black hole result!

11.19 Adding rotation [BMPV] In d = 5 there are two independent angular momentum parameters, because rotation group transverse to D1’s and D5’s splits up as

SO(4)⊥ ' SU(2) ⊗ SU(2)

118 Angular momentum is consistent with d = 5 superalgebra. Metrics for general rotating black holes are algebraically rather messy, we will not write them here. We will simply quote result for BPS entropy:

p 2 SBH = 2π N1N5Nm − J

BPS black holes have a nonextremal generalisation, in which the two angular momenta are independent. However, in extremal limit something interesting happens: two angular momenta are forced to be equal and opposite, Jφ = −Jψ ≡ J. There is also a bound on angular momentum, p |Jmax| = N1N5Nm

Beyond Jmax, closed timelike curves develop, and entropy walks oﬀ into complex plane.

11.20 Rotating entropy agreement Another notable feature of this BPS black hole: those funny Chern-Simons terms in the R-R sector of the SUGRA Lagrangian are turned on. So this black hole is not a solution of d = 5 Einstein-Maxwell theory! Note that gauge charges are unmodified by the funny Chern-Simons terms because they fall off too quickly to contribute to surface integrals. Reduced entropy can be understood rigorously in D-brane field theory. 1 But basic physics is simple: aligning 2 ~’s all in a row to build up macroscopic angular momentum costs oscillator degeneracy. Energy is reduced as

2 Nm 1 J −→ Nm − R R N1N5 So entropy reduced to p 2 Smicro = 2π N1N5Nm − J Agrees with black hole calculation again. Also, ﬁnd Jφ = −Jψ from SUSY.

11.21 d = 4 entropy counting A canonical set of ingredients for building d = 4 system is what we had previously in building black hole: 0 1 2 3 4 5 6 7 8 9 D2 − − − ∼ ∼ ∼ ∼ · · · D6 − − − − − − − · · · NS5 − − − − − − ∼ · · · W − → ∼ ∼ ∼ ∼ ∼ · · · First three ingredients are simply T-dual to our (D1, D5, W) system. New feature: NS5-branes. New physics: D2-branes can end on NS5-branes. It costs zero energy to break up a D2-brane like so:

119 D2

NS5

11.22 d = 4 microscopic entropy and non-extremality These extra massless degrees of freedom in system lead to an extra label on 2-6 strings, giving rise to an extra factor of NNS5 in degeneracy. Entropy counting proceeds just as before, and yields p Smicro = 2π N2N6NNS5Nm which again agrees exactly with Bekenstein-Hawking black hole entropy. A major diﬀerence between this and d = 5 case is that the single rotation rotation parameter is incompatible with supersymmetry. How about nonextremality? No SUSY nonrenormalization theorem here. New ingredient: add extra energy (but no other charges) to system of D-branes (and NS-branes) and open strings carrying linear and angular momenta. SUGRA: nonextremal branes cannot be in static equilibrium with each other – they want to fall towards each other, and they do not satisfy simple harmonic function superposition rule. Least confusing way to construct nonextremal multi-charge solutions is to start with appropriate higher-d neutral Schwarzschild or Kerr type solution, and to use multiple boost- ings and duality transformations to generate required charges.

11.23 Nonextremal entropy and greybody factors For nearly BPS systems, D-brane pictures for (D1,D5,W ) and (D2,D6,NS5,W ) stay in d = 1 + 1. Physics: new energy adds a small number of R-movers as well as L-movers. (Breaks BPS condition.) Think of R-movers and L-movers as dilute gases, interacting only very infrequently. Energy and momentum are additive, and so is entropy. Amazingly, entropy agrees with near-extremal black brane entropy. Why? - no theorem protecting degeneracy of non-BPS states. What is going on physically is that conformal symmetry possessed by the d = 1 + 1 theory is sufficiently restrictive, even when it is broken by finite temperature, for black hole entropy to be reproduced by field theory. Multi-parameter agreement. ↑↓

Also greybody factors can be computed. Mindbogglingly, D-brane story gives same answer!

120 11.24 String theory, D-branes, and SBH S.Hawking 1974: quanta emitted by BH do not carry info about anything behind the horizon, other than what can be measured at infinity: M,Ja,Qi. S.Mathur proved a 2009 theorem 0909.1038 (more on that soon) that subleading quantum gravity corrections cannot resolve the BH information paradox. Only order one corrections to semiclassical BH expectations around the horizon can rescue unitarity. So we need lots of hair. But is there any? No-hair folk theorems for higher-D built on D ≤ 4 intuition turned out to be quite wrong. In D ≥ 5, there is a much wider variety of solutions available as ingredients for building BH. See e.g. I.Bena-N.Warner review 1311.4538. D-branes arise as loci where open strings end; this is enough to determine their kine- matics and dynamics. Nonperturbative: tension τp ∝ 1/gs. Key fact about a stack of N D-branes: for large-N, distance scales you might think are natively `s or `P can get parametrically enhanced to be as large as a BH horizon. Why? 2 Open (closed) string corrections scale as gsN (gs N). A.Strominger-C.Vafa rocked the world in 1996 by computing SBH for special D=5 BPS black holes from string statistical mechanics. This was the first computation of the Bekenstein-Hawking entropy from first principles. Similar methods correctly account for entropy even for rotating and near-BPS BHs in 5D, 4D. But a microscopic model of 4D Kerr BH remains elusive.

11.25 Emission rates and the fuzzball programme Morally, we need to know the wavefunction behind the horizon as well as in front of it to be able to solve the BHIP as well as compute entropy. String theorists got further than computing SBH. Microscopic calculations of open/closed string scattering yielded gorgeous agreement w Hawking emission from classes of near-BPS BHs, including multi-parameter greybody factors. From ST POV, ‘4D’ BHs are hiding higher-D physics near the singularity. Motivated partly by new solutions, and by string CFT emission rate successes, S.Mathur conjectured in 2001 that conventional BH geometry emerges as a coarse-graining over microstates: non-singular, horizonless, non spheroidally symmetric geometries with same asymptotics as BH but diﬀer inside region of order horizon size. Exponentially large density of states. Top-down POV. For limited classes of less-complicated fuzzballs, it is possible to check Mathur’s conjecture with some rigour. Nice fuzzball FAQ by Mathur: physics.ohio-state.edu/∼mathur/faq2.pdf. Mathur’s 2009 theorem on BHIP used only two assumptions: (1) Hawking pairs created fresh from vacuum independently of other pairs; (2) quantum gravity obeys strong subadditivity, like any other reasonable quantum theory. S.Mathur also clariﬁed in 0909.1038, 1108.0302 that just having AdS/CFT duality does not resolve the BHIP in principle.

121 What is the dual of two entangled CFTs?

Samir D. Mathur∗ Department of Physics, The Ohio State University, Columbus,OH43210,USA

It has been conjectured that the dual of the eternal black holeinAdSistwoentangledbut disconnected CFTs. We show that the entanglement created by the process of Hawking radiation creates several challenges for this conjecture. The nature of fuzzball states suggests a diﬀerent picture, where the dual to two entangled CFTs is two entangledbutdisconnectedspacetimes.We argue for a process of ‘quick tunneling’ where the Einstein-RosenWhat bridge of is the the eternal dual hole of tunnels two entangled CFTs? rapidly into fuzzball states, preventing the existence of the eternal hole as a semiclassical11.26 spacetime. D1-D5 CFT

The regions behind the horizon then emerge only in the approximation of fuzzball complementarity, 1 SamirPrototype D. Mathur microscopic∗ model: N1 D1-branes wrapped on S + N5 D5-branes wrapped on where one considers the impact of probes with energy E T . 1 Department≫ of Physics, The OhioS State× M University,4. This system Columbus has a moduli,OH43210,USA space. At one point it is best described in terms of BH geometry; at another, by a D = 1 + 1 SCFT. It has been conjectured that the dual of the eternal black holeinAdSistwoentangledbut1 p4 In the low-energy limit with R(S ) Vol(M4), the SCFT is a symmetric product I. INTRODUCTION disconnectedno-hair CFTs. theorems We [8]show and that gives the entanglementthe required created modificationN by the process of Hawking radiation 1 orbifold (M4) /SN . Related physics: strings wrapped around S fractionate: lowest mode createsof theseveral hole. challenges for this conjecture. The nature of fuzzball states suggests a different has energy 1/(N1N5R) rather than naive 1/R. Hawking’s discovery of black hole evaporation led to apicture, where the dual to two entangled CFTs isEasy two to entangle calculatedbutdisconnectedspacetimes.We in microscopic SCFT at orbifold point where it is free. And for BPS argue for a process of ‘quick tunneling’ where the Einstein-Rosen bridge of the eternal hole tunnels deep puzzle [1]. Particle pairs are created by the gravita- states, SUSY non-renormalization theorem ensures entropy agrees. rapidly into fuzzball states, preventing the existenceBut of to t connecthe eternal honestly hole as with a semiclassical macroscopic spacetime. BH physics and solve information paradox, tional field around the horizon. One member of the pair, F The regions behind the horizon then emergeneed only to in deform the approx SCFTimation away from of fuzzball orbifold complementarity,point towards black hole. Top-down framing. This b, escapes to infinity as radiation, while the other mem-where oneCFT considers the impactCFT of probes with energy E T . 1 2 is one focusL of our≫ research.R ber c falls into the hole to reduce its mass. These two Recent projects: computing anomalous dimensions of low-lying string states in confor- P particles are in an entangled state, so the entanglement of mal perturbation theory [Burrington-Peet-Zadeh] and analyzing aspects of squeezed states I. INTRODUCTION no-hair theorems [8] and gives the required modification the radiation with the remaining hole keeps rising. This generated by twist deformations [Burrington-Mathur-Peet-Zadeh]. + [Burrington-Jardine- (a) of the(b) hole. leads to a puzzle near the endpoint of evaporation: how Peet-Zadeh in progress] can the small residual hole have the huge degeneracyHawking’s re- discovery of black hole evaporation led to a How exactly will we see emergence of effective BH geometry? e.g.:- quired to carry this entanglement? deep puzzle [1]. Particle pairs are created by the gravita- Many aspects of string theory suggest thattional the evapo- field around the horizon. One member of the pair, F b, escapes to infinity as radiation, while the other mem- ration of the hole should be no different from the burn- CFT CFT iiCFT CFT vs L R 2 Σi 1 2 ing away of a piece of paper; thus we shouldber notc fallshave into the hole1 to reduce its mass. These two particles are in an entangled state, so the entanglement of P such a monotonically growing entanglement. In [2] it the radiation with the remaining hole keeps rising. This was shown, using strong subadditivity, that small correc- (b) leads to a puzzle near the endpoint(c) of evaporation: how (d) (a) tions to the physics around the horizon cannotcan the resolve small residual hole have the huge degeneracy11.27 re- Firewalls the problem; one needs corrections of order unity.1 Then quired to carry thisFIG. entanglement? 1. The conjecture of [10] says thatHawking two entangled pairs straddling CFTs horizon are max entangled: their S is ln 2. we have, a priori, two possibilities: Ent Many aspects of(a) string gives theorythe connected suggest spacetime that the evapo-(b). ThePage’s nature theorem of fuzzbal on quantumls subsystem entropy: SEnt between BH and Hrad grows as ration of the holesuggests should bethat no two diff entanglederent from CFTs the burn-(c)BH give radiates, two entangled butCFT must go but back toCFT zero again by time BH evaporatesii away. So new Hrad just 2 Σi (P1) The black hole has a traditional horizon,ing away where of a piecedisconnected of paper; spacetimes thus we should (d). not haveoutside BH should1 be max entangled with old Hrad. the spacetime around the horizon is in the localsuch vacuum a monotonically growing entanglement. In [2] it But monogamy of entanglement rules out max entanglement with two others. Old state. Then entangled pairs will be producedwas at the shown, hori- using strong subadditivity, that small correc-BH complementarity of L.Susskind et al finessed this by arguing that BH blueshift prevents A significant role in this debateexperimenters has been from played seeing(c) by violation of no-xerox theorem. (d) zon, but one can conjecture that some newtions (nonlocal) to the physicsconsideration around the of horizon the eternal cannot hole resolve in AdS space. We are the problem; one needs corrections of order unity.1 Then AMPS 1207.3123 pointed out new flaws in old BH complementarity, ignited firewall effect solves the problem of growing entanglement. In interested in the notion of AdS/CFTdebate duality aboutFIG. validity in 1. the The of con- conjecture GR as an ofeffective [10] says field that theory two around entangled BHs. CFTs Consider 4 postulates: we have, a priori, two possibilities: discussing this possibility, we will focus on the recent pro- text of this eternal hole. The eternal(1) unitary hole(a) S-matrix. gives in AdS the (2) connected has EFT works spacetime outside BH (b). horizon. The nature (3) BH of fuzzbalappearsls to distant observer posal of Maldacena and Susskind [6] where it is conjec- as quantumsuggests system that with two discrete entangled energy CFTs levels. (c) (4) give Nothing two entangled bad happens but at the horizon. arXiv:1402.6378v1 [hep-th] 25 Feb 2014 two asymptotically AdS boundaries, so the usual notion disconnected spacetimes (d). tured that entangled particles are connected by(P1) a ‘worm- The blackof hole AdS/CFT has a traditional duality [9] horizon, suggests whereThe that main the result eternal of AMPS: hole one of (1,2,4) has to be false. They believe in (2) so yelled “Fire!”. hole’, regardless of how far apart they are. the spacetime aroundspacetime the horizon is dual is to in two the CFTs. local vacuum TheTechnical two AdS argument boundaries was about excitation of field modes, for infaller vs Hrad. state. Then entangled pairs will be produced at the hori- T.Banks had previously warned that energy may not be the only variable deciding (P2) The black hole does not have a traditional hori- are not connected, so we have two disconnectedA significantCFTs. role in this debate has been played by zon, but one can conjecture that some new (nonlocal)effectiveness of GR as an EFT. Must also look at entropy. zon, so the radiation is not emitted through the Hawking Corresponding to the possibilities (P1), (P2)consideration above, we of the eternal hole in AdS space. We are effect solves the problem of growing entanglement. In S.Hawkinginterested hated in firewalls the notion so much of AdS/CFT he wrote a duality paper basically in the con- saying that he would process of pair creation. In this case we are not forced have the following two possibilities:rather giving up on event horizons entirely! [CBC article] into Hawking’s problem of rising entanglement.discussing But the this possibility, we will focus on the recent pro- text of this eternal hole. The eternal hole in AdS has posal of Maldacena and Susskind [6] where it is conjec- Recent substantial review article on BHIP ⊃ FW by D.Harlow: 1409.1231. nontrivial task is to find the alterationarXiv:1402.6378v1 [hep-th] 25 Feb 2014 of the state at (P1’) In [10], it was conjectured that whentwo asymptotically these dis- AdS boundaries, so the usual notion tured that entangled particles are connected by a ‘worm- of AdS/CFT duality [9] suggests that the eternal hole the horizon, since the ‘no-hair’ theorems suggest that the connected CFTs are placed in a particular entangled hole’, regardless of how far apart they are. spacetime is dual to two CFTs. The two AdS boundaries hole always settles down to its unique metric which has state, the CFT dual is the eternal hole. Thus when dis- 122 (P2) The black hole does not have a traditional hori- are not connected, so we have two disconnected CFTs. the vacuum state around the horizon. In stringzon, so theory the radiationconnected is not emitted CFTs throughare placed the in Hawking a state that is entangled, we find the fuzzball construction [7], which evades the Corresponding to the possibilities (P1), (P2) above, we process of pair creation.the dual In spacetime this case can we arebe connected not forced(fig.1(a,have b)). the following two possibilities: into Hawking’s problem(P2’) of In rising the fuzzball entanglement. proposal, But each the state of a CFT is nontrivial task isdual to find to a the spacetime alteration that of ends the before state at reaching(P1’) the horizon. In [10], it was conjectured that when these dis- the horizon, sinceThis the ‘no-hair’ suggests theorems that the suggest dual to that the entangled the connected CFTs is CFTs just are placed in a particular entangled ∗ [email protected] hole always settlesa pairdown of todisconnected its unique metricspacetimes, which has with wavefunctionalsstate, the CFT dual is the eternal hole. Thus when dis- 1 See [3–5] for furthur comments in this direction. the vacuum statein around the corresponding the horizon. entangled In string theory state (fig.1(c),(d)).connected CFTs The are placed in a state that is entangled, we find the fuzzball construction [7], which evades the the dual spacetime can be connected (fig.1(a, b)). (P2’) In the fuzzball proposal, each state of a CFT is dual to a spacetime that ends before reaching the horizon. This suggests that the dual to the entangled CFTs is just ∗ [email protected] a pair of disconnected spacetimes, with wavefunctionals 1 See [3–5] for furthur comments in this direction. in the corresponding entangled state (fig.1(c),(d)). The 11.28 Avoiding firewalls Lots of papers have been written about how firewalls might be avoided. (They all differ drastically from the LQG community focus on remnants.) D.Harlow-P.Hayden 1301.4504: quantum information theory constraints on getting 1 info out of a BH prevent firewalls. It takes the Page time (when SBH drops to 2 its initial value) to be able to do experiments detecting a firewall. Aspects of this were explained more intuitively by L.Susskind, 1301.4505. S.Giddings 1211.7070: a small ‘nonviolent’ nonlocality hidden to large scale observers may save you from firewalls. Challenge: it is generally very difficult to introduce only a ‘small’ amount of nonlocality theoretically. S.Mathur-D.Turton in 1306.5488 clarified a number of issues surrounding black hole complementarity, and explained the advantages the fuzzball approach provides in evading firewalls. The essential technical point is that a fuzzball has collective modes, and infalling quanta with E kBT interact with these differently than Hawking radiation does. K.Papadodimas-S.Raju conjectured in 1310.6335 that the mapping of CFT operators to local bulk operators in AdS/CFT depends on the state of the CFT. Mirror operators needed for 1-sided BH, to describe behind-horizon physics in a holographic setup and avoid firewalls. So far only describes small fluctuations about a given reference state. Status: murky at best.

11.29 ‘ER=EPR’ J.Maldacena-L.Susskind 1306.0533 proposed an intriguing new take on wormholes to address firewalls that has become known as ’ER=EPR’. It is built on Maldacena’s proposal hep-th/0106112 that the AdS eternal BH can be constructed via CFTL × CFTR with thermal entanglement between L and R, built on Israel’s |TFDi = √1 P e−βE/2|ψi × |ψi . Z i L R They propose entanglements are encoded by having ER bridges, but note that these wormholes are far from classical. For good explanations of the proposal, see series of papers by Susskind, e.g. 1311.3335, 1411.0690. L.Susskind advocated in 1311.7379, 1402.5674 for connection with computational complexity: length of ER bridge ∝ 1/entanglement. ‘Precursor’ in boundary CFT: nonlocal object set up in boundary theory to create desired thing in the bulk in the causal future. These have played an important role in questions about avoiding firewalls. Precursors that cause firewalls are ‘hard’, and have exponentially large computational complexity. V.Balasubramanian-M.Berkooz-S.Ross-J.Simon provided some interesting caveats in 1404.6198, arguing that spectral information is also needed to diagnose spacetime connect- edness in the AdS/CFT context. Perhaps, as Mathur has suggested, the non-classical Einstein-Rosen bridges of ER=EPR rapidly tunnel into fuzzball states?

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