2.8 Monopoles in General Gauge Theories

Home , Andrew Strominger, Cumrun Vafa, Igor Klebanov, Parity anomaly

Strings and Monopoles in Strongly Interacting Gauge Theories ~ARCHWE

by OF TECHNOLOGY Ethan Stanley Dyer JUL 0 1 2014 Submitted to the Department of Physics I LIBRARIES in partial fulfillment of the requirements for the degree of

Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY

June 2014

Signature redacted A uthorg ...... Department of Physics May 8, 2014 Signature redacted Certified by.... Allan Wilfred Adams III Associate Professor Thesis Supervisor -/ /I / Signature redacted Accepted by...... Krishna Rajagopal Associate Department Head for Education Room 14-0551 77 Massachusetts Avenue Cambridge, MA 02139 Ph: 617.253.2800 MTLibraries Email: [email protected] Document Services http://libraries.mit.edu/docs

DISCLAIMER OF QUALITY

Due to the condition of the original material, there are unavoidable flaws in this reproduction. We have made every effort possible to provide you with the best copy available. If you are dissatisfied with this product and find it unusable, please contact Document Services as soon as possible.

Thank you. 2 Strings and Monopoles in Strongly Interacting Gauge Theories by Ethan Stanley Dyer

Submitted to the Department of Physics on May 8, 2014, in partial fulfillment of the requirements for the degree of Doctor of Philosophy

Abstract

In this thesis we discuss aspects of strongly coupled gauge theories in two and three dimensions. In three dimensions, we present results for the scaling dimension and transformation properties of monopole operators in gauge theories with large numbers of fermions. In two dimensions, we study (0,2) gauge theories as a tool for constructing string backgrounds with non trivial H-flux. We demonstrate how chiral matter content in the gauge theory allows the construction of infrared fixed points outside of the usual Calabi-Yau framework, and further derive consistency relations for a special class of torsional models.

Thesis Supervisor: Allan Wilfred Adams III Title: Associate Professor

3 4 Acknowledgments

There are many people without whom this thesis would not be possible. I would like to thank my family, especially my parents, Barbara and Sam Dyer; grandpar- ents, Betty and Ira Dyer; and girlfriend Gabrielle Lurie for their continual support of my interest in physics, and tolerance of long work hours. I would also like to thank my advisor, collaborator, and thesis committee member, Allan Wilfred Adams III, for his encouragement, critiques, and undying enthusiasm for all things physics, as well as for his help in navigating the world of academia. I am deeply indebted to my collaborators, Jaehoon Lee, Mark Mezei, Silviu Pufu, and Sho Yaida, who have been instrumental in shaping my graduate experience and research, and a pleasure to interact with. I am also grateful to my fellow center for theoretical physics(CTP) classmates who have made the past few years a joy both academically and socially, and the CTP faculty who have provided answers to countless questions, especially

John McGreevy and Jesse Thaler who helped guide my research on numerous occa- sions. I would like to express my appreciation to the CTP administrative staff, Joyce

Berggren, Scott Morley, and Charles Suggs who have helped me in many ways, and without whom I would most likely still be locked out of my office. I would like to give a special thanks to my committee members, Allan Adams, Hong Liu, and Michael

Williams for their willingness to read this thesis. Lastly, I would like to acknowledge the United States taxpayers and private donors, without whose support physics could not go on.

Thank you

5 6 Contents

1 Introduction 19 1.1 Monopoles and Confinement in Three Dimensions ...... 21 1.1.1 M onopoles ...... 23 1.1.2 Fate of the IR ...... 25 1.2 Strongly Coupled Gauge Theory for Chiral Strings ...... 27 1.2.1 Consistancy conditions from world-sheet and space-time .. . 31 1.2.2 Gauge Linear Sigma Models: The basic idea ...... 33 1.2.3 Lorentz Symmetry, Supersymmetry, and Anomalies in Two Di- m ensions ...... 38

2 Monopole Operators in Strongly Coupled Gauge Theories 43 2.1 Introduction ...... 43

2.2 Monopole operators via the state-operator correspondence ...... 47

2.2.1 Classical Monopole Backgrounds ...... 48

2.2.2 Three Dimensional Gauge Theories with Fermions ...... 50

2.2.3 Quantum Monopole Operators ...... 52

2.3 Free energy on S 2 x R ...... 54

2.3.1 Setup ...... 56 2.3.2 Gauge Field Effective Action ...... 58

2.4 Functional determinants ...... 63

2.4.1 The fermion determinant ...... 63

2.4.2 The Faddeev-Popov determinant ...... 67

2.4.3 The gauge fluctuations determinant ...... 69

7 2.4.4 Combining the subleading terms in the free energy ...... 81 2.4.5 Summary and an example ...... 84 2.5 Stability ...... 86 2.5.1 A systematic study of monopole stability in QCD 3 ...... 88 2.6 Monopole operator dimensions ...... 92 2.6.1 Monopole operator dimensions in QED ...... 93 2.6.2 Monopole operator dimensions in U(Nc) QCD ...... 93 2.7 Other quantum numbers of monopole operators ...... 95 2.7.1 Quantum numbers of monopole operators in QED ...... 96 2.7.2 Quantum numbers of monopole operators in U(Nc) QCD ... 106 2.8 Monopoles in general gauge theories ...... 109 2.8.1 Anomalous dimensions for general groups ...... 110 2.8.2 Exam ples ...... 116 2.9 D iscussion ...... 131 2.9.1 Summ ary ...... 131 2.9.2 Confinement and chiral symmetry breaking . .... 133 2.9.3 QED and and algebraic spin liquids ...... 136

3 Chiral Gauge Theory for Stringy Backgrounds 139 3.1 Introduction ...... 139 3.2 Generating dH in a (0,2) GLSM ...... 142 3.2.1 Torsion in (0, 2) NLSMs ...... 142 3.2.2 Adding dH to a (0,2) GLSM by hand: the Green Schwarz m echanism ...... 144 3.2.3 On the geometry of GS GLSMs ...... 147

3.2.4 Generating dH in a garden-variety (0, 2) GLSM ...... 149 3.3 Verifying Quantum Consistency in a Special Class of Models . .... 154 3.3.1 The M odels ...... 154 3.3.2 Methodology ...... 158 3.3.3 Gauge Invariant Model ...... 160

8 3.3.4 Anomalous Model with Green-Schwarz Mechanism ...... 171 3.3.5 Multiple U(1)s ...... 175 3.4 Conclusions ...... 179

A Monopole Harmonics 183 A.1 Definition and Properties of Monopole Harmonics ...... 183 A.1.1 Scalar Harmonics ...... 183 A.1.2 Spin s Harmonics ...... 185 A.1.3 Spin 1/2 Harmonics ...... 186 A.1.4 Spin 1 Harmonics ...... 187

B (0,2) Details 189 B.1 Conventions ...... 189 B.1.1 Coordinates ...... 189 B.1.2 Superfields ...... 190 B .2 A ction ...... 192 B.3 OPEs ...... 194 B.3.1 Operator Product Expansion with single anomalous U(1) . . 194 B.3.2 Operator Product Expansion with multiple U(1)s ...... 196 B.4 Quantum Chirality ...... 197

9 10 List of Figures

1-1 Anomalous diagrams in four and two dimensions ...... 40

2-1 We plot the terms in the infinite sum over j (2.74) that give the matrix

element [Kq ,(a)] u for q = -1, q' = 1/2, Q = 1, and J = 35/2.

We show the stage of the calculation where all the finite sums (over

6q, ', 1, and j') in (2.74) have been done and only the infinite sum

over j remains. The dots represent the actual terms in the sum, while

the solid line is the asymptotic expansion of the summand to 9(1/jl 8 )

that we determined analytically. We perform the infinite sum by zeta-

function regularization of the asymptotic form for j > jc, where j, is

the value below which we use the numerical values of the terms in the

sum. We check the numerical precision by changing jc and we reach our

goal of 10-1' precision by choosing j, ~ 40. This precision is needed

to get the free energy with 10- 3 precision...... 77

2-2 The eigenvalues of K ,(Q) for some example q, q' and low J values as

a function of Q. Zero eigenvalues corresponding to pure gauge modes

are omitted. Note that the eigenvalues are monotonic in J and Q,

hence it suffices to examine the Q = 0 behavior of the lowest J mode

for stability. Also note that in both examples IQI ;> 1 and the two

lowest lying J modes have one non-zero eigenvalue, while higher J

modes come with two eigenvalues. (The smaller number of eigenvalues

corresponds to the reduced size of the matrix K ,(Q).) ...... 81

11 2-3 We plot the ratio of the non-zero eigenvalues Ajgauge(Q) of the gauge

kernel divided by their asymptotic behavior Aaymp(Q). We chose q -1, q' = 1/2 for this example. Because IQI = 3/2 the J = 1/2, 3/2

modes contribute one eigenvalue, while for higher J eigenvalues come in

pairs. We used the same colors to plot the pair of eigenvalues for these

higher J modes. Because the ghosts give a contribution proportional

to Aaymp(Q) this plot shows that the low energy modes are the most

important in determining the free energy...... 83

2-4 We plot the subleading term in the free energy, 6F(q, q') for q =

-1, q' = 1/2 as a function of the cutoff A. We extrapolate to 1/A -+ 0

by fitting the data points by a second order polynomial. Our results

are reliable to 10- 3 precision...... 86

2-5 The lowest eigenvalue A = Kqaq 1 (0) of the aAb component of the

gauge field fluctuations around the GNO monopole background (2.18).

We have marked explicitly the plane z = 0. The region where this

eigenvalue dips below zero corresponds to an instability of aAb. If this

eigenvalue is positive, then the action for a b is positive-definite. . . 90

2-6 A summary plot of the stability of GNO monopoles. A GNO monopole

with charges {qi,. . . qNpe is stable provided that all pairs (qa, qb) cor-

respond to (open or filled) black circles, and it is unstable otherwise.

We denote Qab = qa - q, as in the main text. The orange dots corre-

spond to values of (qa, qg) for which Kq$a- 1 (0) < 0, i.e. the effective

action for aab has a negative mode with J = jQabI - 1. The open and

filled black circles correspond to values of (qa, qg) for which there is no

such negative mode. The difference between the open and filled black

circles is that for the filled ones the lowest angular momentum mode

has J = lQab| - 1, while for the open ones the lowest value of J is IQabI. 91

12 2-7 The weight lattice of SU(2) (blue dots) as well as the lattice of all

possible monopole charges (dotted circles). The monopole charges are

defined modulo the action of the Weyl group, which in this case is Z2 and acts as reflections about the origin...... 118

2-8 The SU(2) monopoles appearing as black dotted circles in Figure 2-

7. In the presence of N1 fundamental fermions these backgrounds are all stable, and we list the scaling dimensions A of the corresponding

monopole operators...... 119

2-9 The SO(3) weight lattice (blue dots) and its dual lattice (dashed cir-

cles). The weight lattice is a sublattice of the SU(2) weight lattice in

Figure 2-7. The dual lattice contains more monopole charges q than

the dual lattice of SU(2). As in the SU(2) case, the Weyl acts by

reflections about the origin, so it provides the identification q - -q on

the set of monopole charges...... 119

2-10 The SO(3) monopoles appearing as black dotted circles in Figure 2-9.

Here, we consider these backgrounds in the presence of N1 fermions transforming in the three-dimensional fundamental representation of

SO(3). The orange circles correspond to unstable backgrounds. For

the stable backgrounds (represented in black by various shapes), we

list the scaling dimensions A of the corresponding monopole operators. 120

2-11 The weight lattice of SU(3) (blue dots) as well as the lattice of all

possible monopole charges (dotted circles). The monopole charges are

defined modulo the action of the Weyl group, which in this case is S 3

and is generated by 120 degree rotations as well as reflections about

the q2 axis...... 121

13 2-12 The SU(3) monopoles appearing as black dotted circles in Figure 2-11.

Here, we consider these backgrounds in the presence of Nf fermions

transforming in the three-dimensional fundamental representation of

SU(3). The orange circles correspond to unstable backgrounds. For

the stable backgrounds (represented in black by various shapes), we

list the scaling dimensions A of the corresponding monopole operators. 122

2-13 The weight lattice of Sp( 4 ) (blue dots) as well as the lattice of all

possible monopole charges (dotted circles). The monopole charges are

3 defined modulo the action of the Weyl group, which in this case is (Z 2 )

and is generated by reflections about the q1 axis, q2 axis, and the line that makes a 45 degree angle with the qi axis...... 123

2-14 The Sp( 4 ) monopoles appearing as black dotted circles in Figure 2-13.

Here, we consider these backgrounds in the presence of N fermions

transforming in the four-dimensional fundamental representation of

Sp( 4 ). The orange circles correspond to unstable backgrounds. For

the stable backgrounds (represented in black by various shapes), we

list the scaling dimensions A of the corresponding monopole operators. 124

2-15 The weight lattice of SO(5) (blue dots) as well as the lattice of all

possible monopole charges (dotted circles). The monopole charges are

defined modulo the action of the Weyl group, which, as in the Sp( 4 ) 3 case, can be identified with the (Z 2 ) generated by reflections about

the qi axis, q2 axis, and the line that makes a 45 degree angle with the

qi axis...... 125

2-16 The SO(5) monopoles appearing as black dotted circles in Figure 2-

15. Here, we consider these backgrounds in the presence of N fermions

transforming in the five-dimensional fundamental representation of SO(5).

The orange circles correspond to unstable backgrounds. For the sta-

ble backgrounds (represented in black by various shapes), we list the

scaling dimensions A of the corresponding monopole operators. . .. 126

14 2-17 The weight lattice of G2 (blue dots) as well as the lattice of all possible monopole charges (dotted circles). The monopole charges are defined

modulo the action of the Weyl group, which in this case is D6 (the dihedral group of order 12) and is generated by 60 degree rotations as well as reflections about the line that makes a 45 degree angle with the qi axis...... 127

2-18 The G2 monopoles appearing as black dotted circles in Figure 2-17. Here, we consider these backgrounds in the presence of Nf fermions transforming in the seven-dimensional fundamental representation of

G2. The orange circles correspond to unstable backgrounds. For the stable backgrounds (represented in black by various shapes), we list the scaling dimensions A of the corresponding monopole operators. . 128 2-19 In the top right corner we show the number of (inequivalent) stable monopoles for SU(3) gauge theory with nrfundNf fundamental fermions and nadjNf adjoint fermions as a function of the ratio nrfuna/nadj. The solid line is divided into five regions that correspond to the diagrams on the left and bottom, where we show explicitly which monopoles are stable in each region. The dashed line is a continuation of the solid line for larger values of nfund/naj, but in this region we do not show explicitly which monopoles are stable...... 130

15 16 List of Tables

1.1 Monopole operator dimension Aq for monopole charge q in QED 3 . 26

1.2 Estimates of the smallest number of fermions, Nd*nf for which the IR

of QCD3 is in a deconfined quantum critical point ...... 27 1.3 Standard model gauge charges for a fermion generation ...... 41

2.1 Monopole operator dimension Aq for monopole charge q in U(1) gauge

theory...... 93

2.2 6F(q, q') for various values of q and q'. The orange dots mean that the

corresponding W boson is unstable...... 95

2.3 The transformation properties of the first few (bare) monopole op-

erators under the flavor SU(N) global symmetry of QED 3 with Nf flavors. The dimensions of the irreps were calculated using (2.131). All

these monopole operators are singlets under spatial rotations. .... 105

2.4 Estimates of the smallest number of fermions, Nd*nf for which the IR

of QCD 3 with gauge group G is in a deconfined quantum critical point. Results are listed for various rank one and two gauge groups. .... 134

17 18 Chapter 1

Introduction

Strongly interacting gauge theories are ubiquitous in physics, perhaps the most familiar example of a strongly interacting gauge theory is quantum chromo dynamics in 1 four space-time dimensions, QCD 4 . At high energies, this theory describes how the six flavors of quarks interact with gluons. At low energies, it describes the spectrum and interaction of hadrons, including the proton and neutron, which make up the nuclei of atoms. The theory can be described by the Lagrangian:

1 6

LQCD 4 - F F" + (iq., - maqaqa). (1.1) Q=1

At first blush, it is not clear why this theory describes strong interactions in the infrared (IR). The coupling that dictates the strength of interactions, g, is classically dimensionless, and so one might expect that its strength is insensitive to energy scale.

Furhtermore, very similar theories, such as QED 4 become weekly interacting. This puzzle was resolved many years ago [1-3]. Though classically dimensionless, quantum effects conspire to drive g to large values in the IR. In particular, the # function which describes how the coupling g changes with energy is given at one loop by:

'We will use the notation QEDD and QCDD throughout the introduction to denote abelian and non-abelian gauge theories with matter in D space-time dimensions

19 ,3QCD4, W1d - _ 11 - -Nf , (1.2) dlog(A) 3 167r2

indicating that g becomes large at low energies. This strong coupling leads to a

very rich structure for the IR physics, the perturbation theory in g breaks down, and

we must appeal to other techniques to describe the zoo of interacting baryons and mesons that result from the confinement of quarks.

Viewing this story from the other direction, it is an amazing simplification that the

messy low energy physics of hadronic resonances can be described by the relatively

simple theory of quarks at high energies. This ultraviolet simplification allows for relatively easy computation of some aspects of the strong interactions.

The focus of this thesis is not with the strong interactions in four dimensions, but

rather with strongly interacting gauge theories in two and three space-time dimensions. As we will see, these strongly interacting gauge theories play important roles in string theory, condensed matter physics, and mathematics, as well as serving as toy models to help understand features of gauge theories in four dimensions, such as confinement.

A particularly important feature of gauge theories in lower space-time dimensions, is that strong coupling at low-energies is almost unavoidable. In particular, the classical scaling dimension of the gauge coupling is:

[g 2] = 4 - D, (1.3) from which we can create the dimensionless coupling, Ag = g/E 2 D/ 2 , where E is an appropriate energy scale. As E becomes small, A9 becomes large, indicating that the theory runs strong in the infrared and interesting low energy phenomena are possible.

This thesis addresses two such classes of strongly coupled gauge theories. The first such class consists of three dimensional gauge theories with fermionic matter. These gauge theories, denoted by QED 3 in the abelian case and QCD 3 in the non-abelian,

20 display interesting strong-coupling dynamics, such as confinement or chiral symmetry breaking for certain choices of matter content and gauge group. The second class of theories are two dimensional gauge theories with a particular amount of supersymmetry, suited to describing heterotic string theory. In these, an asymptotically free gauge theory provides a UV completion of a 2d strongly interacting fixed point.

In the remainder of the introduction, we will provide background on the two classes of theories described, beginning with gauge theories in three dimensions and then moving down to two, setting the stage for the results in chapterss 2 and 3 which make up the body of the thesis. 2

1.1 Monopoles and Confinement in Three Dimen- sions

A major focus of this thesis will be on three dimensional gauge theories with fermionic matter. As mentioned above, these models are often strongly interacting at low energies, and perturbative analysis in the gauge coupling is typically not feasible. Nev- ertheless, three dimensional gauge theories provide a useful setting in which to try and understand field theoretic phenomena such as confinement [6], chiral symmetry breaking [7,8], and duality [9]. Furthermore, three dimensional gauge theories have recently emerged as candidate descriptions of quantum phase transitions in two spatial dimensions [10]. The aim of chapter 2 is to examine the possible low energy descriptions of gauge theories with large amounts of fermionic matter.

There is a natural guess for this low energy physics based on classical dimensional analysis. To understand this guess, let's discuss QED 3. We will introduce the tools to address general gauge groups when appropriate. QED 3 with NJ fermions can be described by a Lagrangian,

2The work appearing below has appeared previously in [4] and [5].

21 I Nf LQED 3 = - FvFM" + Z P@ ba. (1.4) a=1

Here, F,, = iAv-avA, is the usual electro-magnetic field strength, and a=1 ,...,Nf are a collection of complex two component fermions, with the covariant derivative given by:

= D = - i (1.5)

We would like to understand the behavior of such a theory at low energies. As mentioned above, e2 has dimensions of mass in three dimensions, and so we might guess that the low energy limit is the same as the e -+ oo limit. Said another way, the kinetic term for the gauge field, FF"" has scaling dimension four, and so is a classically irrelevant operator. We therefore might expect that the low energy description of the theory can be understood by dropping the gauge kinetic term from (1.4).

Nf LIR = i ( 00c'. (1.6) a=1

We will refer to this guess for the IR theory as a deconfined critical theory. In addition to their interest as toy models for four dimensional physics, such deconfined theories have been suggested as describing critical points in quantum phase transitions of spin systems [10,11]. For large amounts of matter, our work in chapter 2, as well as previous work by [12-15], indicate that the deconfined critical theory is the correct description of the low energy physics. For small numbers of fermions, however, the fate of the infrared is far less clear.

As mentioned above, three dimensional gauge theories can exhibit diverse IR phenomena not described by the deconfined fixed point, (1.6). The seminal work of Polyakov [6] showed that the classical argument leading to (1.6) can miss crucial

22 quantum effects. In particular, even though the operator F,,Fi" is irrelevant there are additional operators, called monopole operators, which can become relevant, caus- ing the theory to confine. More generally, understanding whether these additional monopole operators are relevant plays a crucial role in understanding the low energy

physics of three dimensional gauge theories, giving insight into confinement in quan-

tum field theory, as well as constraining whether the deconfined critical theories are

viable descriptions of phase transitions. The bulk of chapter 2 is devoted to calculat-

ing the dimension of monopole operators, as well as their other quantum numbers,

as a way to understand the low energy physics of the gauge theories in which they

occur.

In the next few subsections we introduce monopole operators in more detail and

explain the techniques used in chapter 2 to compute the scaling dimension and trans-

formation properties of these operators. In the last subsection, we give a preview of

the results of chapter 2 and comment on the implications for the IR of the deconfined

critical theories.

1.1.1 Monopoles

As alluded to above, monopole operators play in important role in understanding the

structure of many gauge theories. We will be interested in them because they can

become relevant, in the RG sense, and destabilize the deconfined fixed point described

by (1.6), as such, they provide a valuable probe of low energy physics.

In four space-time dimensions, monopoles are a localized sources of magnetic flux.

A particularly nice choice for the gauge potential describing a localized source of flux

was given by Wu and Yang [16].

Au(l) JA(N) q(1 - cosO)do if6$ir, (1.7) A(s) =q(-1 - cos )dO if 0 ,0,

23 which gives the field strength,

FU(1) = q sin Od9 A do. (1.8)

As was originally realized by Dirac, [17], the charge of a monopole, 47rq = fS2 T, is not arbitrary, but is quantized. With the conventions we have chosen, q E Z/2.

We wrote down, (1.7), thinking of it as a static source of magnetic flux in four

dimensions, but as it is time independent, it can also be thought of as a field config-

uration in three Euclidean dimensions. In this thesis, the term monopole will always refer to an object localized in three Euclidean dimensions.

We will not be so concerned with this classical gauge configuration, but rather with

the quantum monopole operator associated to this background. More explicitly, given

a classical field configuration such as (1.7) we can construct a local gauge invariant

operator Mq, often referred to as a disorder operator, [18], by performing the path

integral subject to the boundary conditions that near the operator, fields approach a

given classical configuration. In chapter 2 we give a more careful explanation of this construction.

For non-abelian gauge theories we can consider more general monopoles known as

Goddard-Nuyts-Olive (GNO) monopoles [19]. For gauge group G, these monopoles are given by:

AGNO = HAu(l), (1.9) where H is an element of the Lie-Algebra of G. We can use a gauge transformation to rotate H into a diagonal form. For example, for G = U(Nc) we can rotate H into the form:

H = diag(qconditin , q2, )- e.

In this case, the Dirac quantization condition becomes, qi E Z/2, more generally,

24 we can write H = qah', where {h } form a basis for the Cartan subalgebra of the

4 Lie algebra of G. The charges, qa must be chosen such that e 1riqaha = 1. This is the non-abelian generalization of the Dirac quantization condition. The collection

{qi, q2 , ... , qr} is gauge invariant and characterizes the GNO monopole. There is a much coarser characterization of monopoles that is often important.

The collection {qi, q2 ,... , qr} are known as GNO charges. Monopoles may also carry a

topological charge, Qtp. For abelian monopoles this is precisely the charge, QtP = q, and the conserved current is given by:

1 JA = "PF.(1.11) 47r

For U(N) monopoles, the charge is the sum of the GNO charges. QtiO = E qj, more

generally the topological charge is an element of -r,(G).

For a nice introduction to non-abelian monopoles, see [20].

1.1.2 Fate of the IR

Given a particular gauge theory, we would like to understand what effects monopole

operators have on the renormalization group (RG) trajectory. The strategy that we

will take in chapter 2 is to assume that the low energy physics is described by the

deconfined critical theory, (1.6), and check the self consistency of this assumption.

As explained below, we are able to identify and compute the dimensions of monopole

operators at the deconfined fixed points. For some choices of gauge group and matter

content we find that these fixed points are self consistent, that is the dimension of all

monopole operators is greater than 3 and so they are all irrelevant, however for small

amounts of matter, and small gauge groups, we find that the monopole operators are

relevant and represent an instability of the IR fixed point.

The technique for calculating the dimension of monopole operators to leading

order was pioneered in [14] and extended to calculate the dimension of the lowest

charge monopole in QED 3 in [21]. The idea, as we explain in chapter 2, is to map the calculation of the scaling dimension, Am on R' to the computation of an energy

25 2 on S x R, which can in turn be evaluated by performing the Euclidean path integral on S 2 x R.

1 AM = lim -- log(ZE)- (1.12)

Here:

ZE = D[A]D[Vt]D[V]e-sE[AVb,'t] = D[A]e-NSEff[A], (113)

is the Euclidean partition function on S 2 x R.

N1 appears in this expression multiplying the whole action, we can thus perform

a saddle point expansion in 1/N where the saddles are just the classical monopole configurations described above.

One distinguishing feature of non-abelian gauge theory is that not all classical

saddles are minima. For some classical saddles, the quadratic fluctuations of gauge

fields are unstable, and so these saddles do not correspond to monopole operators in the low energy theory.

For QED 3 the monopole dimensions are given in Table 1.1.

|Iq|l Aq 0 0 1/2 0.265 Nf - 0.0383 + 0(1/N) 1 0.673 N1 - 0.194 + 0(1/N) 3/2 1.186 N1 - 0.422 + 0(1/N) 2 1.786 N - 0.706 + 0(1/N) 5/2 2.462 N - 1.04 + 0(1/N)

Table 1.1: Monopole operator dimension Aq for monopole charge q in QED3

The most relevant (smallest dimension) monopole is the charge q = 1/2 monopole. With:

A1 / 2 = 0.265 N1 - 0.0383 + 0(1/N 1 ). (1.14)

26 The assumption that monopoles are irrelevant is only valid if, A1/ 2 3, which requires Nf > 12 and we predict an instability for smaller N1 . We can run similar arguments for other gauge groups. The predictions are given in Table 1.2, where

Nd"""f is the number of fermions for which the assumption of the deconfined theory becomes self consistent. We present the complete results in chapter 2.

Gauge group N onf U(1) 12 U(2) 14 SU(2) 8 SO(3) 8 SU(3) 10 Sp(4) 10 SO(5) 10

G 2 6

Table 1.2: Estimates of the smallest number of fermions, N fd,*f for which the IR of QCD 3 is in a deconfined quantum critical point.

In the next subsection we move away from gauge theories in three dimensions and begin discussing the two dimensional gauge theories which make up the second half of this thesis. We return to three dimensions in chapter 2.

1.2 Strongly Coupled Gauge Theory for Chiral Strings

In this section we will lay the groundwork for the models introduced in chapter 3.

In the previous section we used an asymptotically free gauge theory to understand interacting conformal field theories in three-dimensions, here we will use a class of gauge theories in two dimensions to study novel two-dimensional CFTs. The conformal field theories we will discuss are particularly relevant for string theory, where they describe potential backgrounds for the heterotic string.

Perturbative string theory describes the propagation of one-dimensional strings in a (typically) higher dimensional target space in a similar fashion to how non- relativistic quantum mechanics describes the motion of point particles in space.

27 Non-relativistic point particles moving in d spatial dimensions can be described by the action:

SNR Jdt ( m4ii - V(qi)). (1.15)

Here i = 1, ... , d, and V describes some potential for the point particles. This

action can be thought of as defining a 0 + 1 dimensional field theory, where the coordinates of the particle, qg(t) are the fields. This is a one dimensional quantum field theory describing maps from the one dimensional world line of the particle into the d dimensional target space.

Similarly, the gauge fixed form of the Polyakov action describing the motion of strings in target space takes the form:

Sp = - 1 d2x,#kia#qi. (1.16)

Here, the two dimensional coordinates, (xO, x 1 ) parameterize the two dimensional world sheet traced out by a string moving in time. The fields, #i(xO, x 1 ), i = 1, ... , D

give the coordinates in the D dimensional space-time, of the point (x 0, 1).3

The action, (1.16), describes strings propagating in D flat space-time dimensions.

Consistency of the theory puts constraints on the allowed values of D. The simplest string backgrounds describe the propagation of strings in either 26 flat dimensions, for the bosonic string, or 10 for the superstring. As our universe appears to be

four dimensional this has lead to the search for more general backgrounds of string theory. A particularly geometric approach to solving this problem is to consider strings propagating in four flat non-compact directions, with all remaining directions

3 We have been slightly careless here, for the sake of a simple analogy. For the non-relativistic point particle the indices i only run over the spatial coordinates, and are thus contracted with the Euclidean metric, while in the case of the string, i runs over all of space-time, and is contracted with the Minkowski metric. We have also brushed all of the gauge fixing required to arrive at (1.16) under the rug. There is a much closer parallel between the relativistic string and the relativistic point particle

28 compactified. Such a theory can be encoded in a non-linear sigma model describing propagation of strings in this geometry,

SNLSM ~- 1 d2x (Gjj(0)am#'a1 + Bij(O)E M± + . . .) . (1.17)

The background field, Gj(#) is the target space metric, while Bij(#) is an an-

tisymmetric two form. The dots represent other possible background fields coupled

to world sheet fermions. One coupling of particular significance for the models considered below, is the possibility of a background gauge field, Ai(#). The detailed expansion for the models we will consider can be found in chapter 3. As the non-

compact directions describe Minkowski space, we often use (1.17) to refer to purely

the compact directions.

The theory as written in (1.17) does not describe a consistent quantum theory

of strings for every choice of background fields. Rather, quantum consistency of the

string theory imposes constraints on G, B, A, and any other interactions included.

The consistency conditions are equivalent to demanding that (1.17) describe a mod-

ular invariant conformal field theory with a particular central charge.

This is an example, of a much more general statement, that any two dimensional

conformal field theory with the right central charge (a parameter in the conformal

algebra) defines a consistent background of string theory. In this way, the study of

perturbative string theory is intimately tied to the study of two dimensional conformal

field theories.

One rich application of string theory has been the interplay of two dimensional

CFT, on the one hand, with the target space geometry on the other. The additional

structure provided by two dimensional super-conformal field theories, has provided

tools for understanding and classifying manifolds. This has been most fruitful in

the case of conformal field theories with P2 = (2, 2)4 supersymmetry, where confor-

mal field theory, and string theory more generally have lead to many new tools for

4 The notation K2 = (M, N) is described in subsection 1.2.3.

29 understanding a class of geometries known as Calabi-Yau manifolds. However these Calabi-Yau compactifications, with (2, 2) supersymmetry only represent a small patch of the string landscape. We will be interested in exploring two dimensional theories with N2 = (0, 2) supersymmetry. These models are appropriate for compactifications of the heterotic string to four dimensions which preserve at least A4 = 1 target space supersymmetry [22]. One hope for these models is that they will shed light on geometric structures beyond the Calabi-Yau framework.

Though we can write down non-linear sigma models for these (0, 2) theories, in general, an NLSM described by (1.17) can be difficult to study, as the metric and other background fields my induce complicated interactions. The main tool we will use to study these non-linear sigma models is the gauge-linear sigma model (GLSM). In the context of string theory, the GLSM was introduced by Witten in [23], however the basic idea behind the GLSM is much older. As we illustrate in subsection 1.2.2 the purpose of the GLSM is to realize the non-linear sigma model as an effective low energy description of a simpler, linear gauge theory. In this way, we will again be concerned with gauge theories, in this case as a convenient UV description of NLSMs appropriate for the heterotic string. 5

The main focus of chapter 3 will not be with phenomenology of the heterotic string, or even with a detailed analysis of specific (0, 2) geometries, but rather with understanding the implications and constraints of chirality in two dimensional, supersymmetric, abelian gauge theories. As we explain, just as fermions can have a handedness in four dimensions, fermions in two dimensions can be labeled by their chirality. One of the interesting, and under-explored aspects of (0, 2) compactifications is taking full advantage of this chirality. It is this feature, and especially anomalies resulting from chiral fermions, that allows for the construction of models that have the potential to expand the space of geometries string theory is able to help understand.

In the remainder of this section we will lay out some of the basic tools necessary to

5 These same gauge theories have also featured prominently in recent developments attempting to use two dimensional field theory to understand four-manifolds [24].

30 understand these models. We begin by reviewing the geometric constraints imposed by target space and world sheet supersymmetry on the compact internal geometry.

We will then give a simplistic introduction to the GLSM, demonstrating how a non- linear sigma model can arise as the low energy description of an asymptotically free gauge theory. We then set up the language of two dimensional supersymmetry as well as discussing gauge anomalies in two dimensions. We close by discussing how the interplay of gauge anomalies with classical gauge non-invariance allows the models we introduce in chapter 3 to go beyond the typical non-chiral geometric framework

and exhibit properties generic in chiral theories that have only begun to be explored.

1.2.1 Consistancy conditions from world-sheet and space- time

Before launching into the discussion of the gauge theories we will use to explore

conformal field theory, let's discuss what constraints conformal invariance puts on

the non-linear sigma model itself. The most well known example of this is when we

look for consistent, supersymmetric theories of the form (1.17) with H = dB = 0. In

this case, the leading order consistency conditions for the six dimensional manifold, is

that it be a compact Kiihler manifold 6 that satisfies the vacuum Einstein equations.

R%3 = 0. (1.18)

Such manifolds are known as Calabi-Yau manifolds. 7

More generally, the lowest order consistency conditions on (1.17) will reduce to

Einstein's equation for the metric, with the other background fields sourcing the

energy momentum tensor, as well as additional constraints on the background fields

themselves.

6A Kahler manifold is a special class of complex manifold, where a particular two form (the Kshler form) is closed. 7The bars on the indices indicate the fact that supersymmetry requires the compact manifold to be complex, and thus we have a splitting into holomorphic indices and anti-holomorphic indices. This is analogous to the coordinates z = x + iy and = x - iy on the plane.

31 There are typically two approaches to finding these conditions. One approach is to study the two dimensional field theory described by (1.17) directly. Here, both

the cancellation of sigma model anomalies, and the condition that the theory be conformal impose conditions on the background fields [25-31].

There is a complimentary approach to understanding the NLSM consistency con-

ditions, which relies on studying the ten dimensional target space description of the

string. Requiring that a given geometry solve the ten dimensional supergravity equa-

tions of motion also serves to constrain the form of the geometry [32].

Unfortunately, even if a solution to the perturbative equations is found to some

order, it may not satisfy the higher order equations. Both the sigma model beta

function computations and the supergravity computations are perturbative in a', and receive modifications up to arbitrarily high order in perturbation theory [33,33].

As such, it is useful to identify topological properties of the low energy manifolds which will be unaffected by perturbative corrections.

For the case of Calabi-Yau manifolds, (1.18), implies the apparently weaker topological condition,8

c1(TM) = 0. (1.19)

It turns out that allowing for more general backgrounds does not alter this con-

dition, but the background fields must satisfy additional constraints.

c1(V) = 0 mod 2, (1.20)

8 writing 1n the topological conditions that compact geometries must satisfy, we use the notation of characteristic classes. These classes represent equivalence classes of closed forms. We have already run into one such class in the discussion of monopoles c1 (Au(1)) = - [F]. The brackets represent the fact that c1 (V) is only defined up to the addition of a closed form, but the integral, fs2 c1 (AU(l)) = A fs2 FU(1) = 2q is well defined, and is just twice the monopole charge. ch 2 also has an expresion in terms of the field strength. These equivalence classes are invariants of the manifold, in the case of TM, and the gauge configuration in the case of V.

32 and,

ch 2(TM) = ch 2 (V) , (1.21) which follow from more restrictive geometric equations. Here, V represents the vector bundle (gauge configuration) specified by the background gauge field A'(0). Whether or not there is guaranteed to be a solution to the geometric equations given a manifold that satisfies the topological restrictions is a difficult question. For the special case of H = dB = 0, and thus a compact Kihler manifold with vanishing first Chern class (c1 (TM) = 0), the answer was given by Yau [34] as a proof of the Calabi conjecture [35]. For this case there is guaranteed to be a solution to (1.18).9 Outside of this class of models it is difficult to proove whether a manifold which satisfies all of the topological requirements, (1.19 - 1.21), admits field configurations that satisfy the string consistency conditions. As such, it is important to develop other techniques to study the flow of two dimensional sigma models, ultimately identifying and describing candidate IR fixed points.

The gauge linear sigma model, which we introduce in the next section, provides just such a technique, by realizing the conformal theory as the low energy fixed point of a gauge theory, we are often able to address the existence of a conformal theory in the infrared by an analysis of consistency in the gauge theory. The next section is an introduction to the GLSM, which we use in chapter 3 to understand candidate conformal theories which are not described by Calabi-Yau geometry.

1.2.2 Gauge Linear Sigma Models: The basic idea

As emphasized above, in string theory, we are often concerned with the physics of a string propagating in some potentially complicated geometry. This can be described in terms of a two dimensional non-linear sigma model (NLSM).

9 Yau, together with Uhlenbeck, also showed that given some assumptions about the vector bundle V, any vector bundle satisfying [J] A [J] Ac 1 (V) = 0 admits a connection, that satisfies the supergravity equation of motion, Fj = F3 = giF3 = 0 [36].

33 1 LNLSM = +... (1.22) 2

Here 0', i = 1, ... , D can be thought of as two dimensional scalar fields and as maps from the two dimensional world-sheet into the D dimensional target space. Gij is the target space metric, while the derivatives, 0 , are two dimensional derivatives contracted with the flat metric, . The dots represent other potential couplings of the #b to background fields, as well as additional two dimensional fields.

The idea behind the gauge linear sigma model is to realize this potentially complicated non-linear sigma model as an effective low energy description of a simpler, linear gauge theory at high energies. This basic idea is quite old, however in the context of string geometry it was introduced by Witten in [23]. Before introducing the full machinery of the supersymmetric GLSM, it is nice to consider a few simple examples which illustrate the basic idea.

O(N) Model:

The O(N) model is a theory of N scalar fields, #, constrained to be on a sphere,

2 010I = r .

This can be described by the Lagrangian:

LO(N) -1 - A (101 _ , (1.23)

in the limit A -+ oo.

Alternatively, as A has dimension 2, we can think of the O(N) model as being given by the low energy theory of the above Lagrangian.

Note that we can also solve the constraint 010I = r 2 explicitly. For simplicity lets consider the case of N = 3, and take the fields q1 = X, 02 = y, q3 = z. The constraint is just:

x2 + y2 + z2 = r2, (1.24)

34 describing a two sphere, S 2 in R'. Taking spherical coordinates to solve this constraint,

x = r sin () cos(6)

y = r sin(#) sin(9) (1.25)

z = r cos(#) ,

and plugging into 1.23, yields:

1 3 LNLSM = -- Gj (0,0),a5 1 90 . (1.26)

Here, 60 = (0, #) and G = diag(r 2 , r 2 sin 2 (0)) is the usual round metric on S2.

In this way, we see that (1.23) reduces to the non-linear sigma model (1.26) at low energies. In contrast, at high energies, we can neglect the interaction term in (1.23) as [A] = 2, so we are left with just a free, linear theory.

2 As a final point on the O(N) model, note that Gij has an overall factor of r , similarly, if we rescale red we can pull out a pre-factor of r 2 in (1.23). In the path integral, Z = f D[<]ew[1], and so r2 appears in front of the action, just as .. This means that r controls the saddle point approximation, and perturbation theory is

only valid when r is large.

The above example didn't incorporate gauge symmetry, in order to get a feel for

the types of models that will be discussed in chapter 3 it is good to look at a relatively

simple example of a geometry that can be realized through a gauge theory at high

energies.

pN Model:

As apposed to the two dimensional O(N) model, which described maps from two

dimensional space to the sphere, SN1, the two dimensional PN model describes maps

from two dimensions to complex projective space, pN (also referred to as CpN).

The space, pN is the space of lines in CN+1-{O}. More explicitly, if (Z', . .. , ZN+1)

35 are coordinates for CN+1, and we introduce an equivalence relation,

(Z,... , ZN+1 ) (AZ 1 ... AZN+ 1) V A E C*, (1.27) then pNis given by:

pN = (oN+1 _ {o}) (1.28)

Here, {0} denotes the origin in CN+1, and C* = C - 0.

1 2 We can use the N + 1 numbers, (Z : Z : ... : ZN+1) to represent a point in pN, keeping in mind that the overall scale has no meaning (thus the ":" notation). These are known as homogeneous coordinates. Alternatively we can work locally, defining inhomogenous coordinates in say the ZN+1 5 0 region by z = Zi/ZN+1. This defines a set of N coordinates on a patch of pN.

We can write down an NLSM describing maps into this geometry by using the natural metric on 1pN called the Fubini-Study metric.

2 G.(FS) -. = + k(1k Izk )J - ZZ j '1 (1 + E Izk 12)2 -- (1.29)

The NLSM is just,

LNLSM - 213G ~,2D~ 43 (1.30)

Here, the NLSM is written in terms of N complex fields z'.

There is another useful way to view the space pN. Given the homogenous coor- 1 dinates, (Z : ... : ZN+1), we can use the equivalence, 2 (1.27), to fix EN 1 jZ 1 - 1, this condition defines an S2N+1 inside of CN+1. This is not enough to totally fix the equivalence, as multiplying by a complex phase doesn't effect E 1 2 = 1. This means that pN can be thought of as S2N+1 subject to the equivalence of multiplying the coordinates by a phase.

36 PN = S2N+1/U(1). (1.31)

This perspective on pN allows us to write down a relatively simple theory, which flows to (1.30) at low energies. Using the example of the O(N) model, we know how to write down a theory which describes fields living on a 2N + 1 dimensional sphere,

LO(2N+2) = ~0pZI 2O - A ( I ZI12 - r2). (1.32)

Here we have written the 2N + 2 real fields of the 0(2N + 2) model in terms of N + 1 complex fields ZI.

This theory has a global symmetry, ZI -+ e"ZI, but this is exactly the equivalence we would like to impose to get pN, therefore gauging the symmetry, by introducing a U(1) gauge field,

LpN = -- FF" - DIIZIDA2 - A 7ZI|2 - r2, (1.33)

gives a gauge theory which flows to a sigma model on PN at low energies, and a free theory at high energies.

These two examples illustrate the idea behind the GLSM, however the discussion here has been purely classical, and there are subtleties in these models quantum mechanically that destroy this simple picture. The types of models we will be interested in in chapter 3 are supersymmetric, and as a result, the quantum effects which are crucial for understanding the O(N) and pN models are under better control. Before introducing these models, let's briefly review fermionic physics in two dimensions.

37 1.2.3 Lorentz Symmetry, Supersymmetry, and Anomalies in Two Dimensions

The focus of the second half of this thesis, will be on strongly interacting, supersymmetric gauge theories in two space-time dimensions. As such, it is useful to get a little familiarity with Lorentz invariance and supersymmetry in two dimensions, we record the detailed conventions in appendix B and here remind the reader of a few basic facts about two dimensional physics.

Recall, that all massless relativistic fields, D satisfy the Klein-Gordan equation:

E14) = (-O + a2)& = 0 . (1.34)

Just as in higher dimensions this equation admits plain-wave solutions. An interesting feature of two dimensions, is that a general solution to (1.34) can be written as a superposition of left moving and right moving waves. This can be seen by noting that in the light cone coordinates,

X* - I(XO i X1) . (1.35) 2

The wave equation becomes:

0EI = -4a+O-_ = 0, (1.36)

and so admits solutions of the form 1(x) = D+(x+) + D-(x-).

This is a result of the fact that the defining representation of the Lorentz group in two dimensions is reducible. Explicitly, Lorentz transformations act on the coordinates, x± as multiplication.

38 X- e±o+X. (1.37)

Similarly, fermions in two Minkowski dimensions can be written in terms of real one component fermions, I±(x), which transform as:

,0(x) = eg2±(A-lx). (1.38)

For the models under consideration in chapter 3, we will be interested in super-

symmetric theories in two dimensions. Supersymmetry is an extension of the Poincare

group that includes fermionic symmetry generators, Q' and Q' , in addition to the

usual bosonic generators, PA and M"'. As fermions come in two chiralities, t, we

can consider two dimensional theories with some number, M, of negative chirality

supersymmetry generators, Q_', and sum number N or positive, QS. These theories are said to have 2 = (M, N) supersymmetry.

We will be interested in the case 2 = (0, 2), relevant for describing the heterotic string propagating in a supersymmetric target. In this case we can package the two real supersymmetry generators into a single complex generator. At supersymmetric conformal fixed points, this is enhanced to the (0, 2) superconformal algebra.

As (0,2) supersymmetry is chiral, the matter content of these theories is generally chiral as well. One aspect of theories with chiral fermions is the possibility of anomalies. An anomaly refers to a symmetry of the classical theory which is not preserved quantum mechanically. Anomalies in global symmetries have important implications for the content of the theory. For instance in four dimensions the decay rate for the process 7ro -+ 7 7 crucially depends on an anomalous global U(1) axial symmetry [37, 38]. In two dimensions, the U(1) axial anomaly results in a gauge invariant mass for the photon [39].

In four dimensions anomalies appear in correlators involving three currents, or

39 (a) Triangle anomaly in four dimensions

(b) Diangle anomaly in two dimensions

Figure 1-1: Anomalous diagrams in four and two dimensions

gauge fields, and show up diagrammatically in evaluation of triangle diagrams, such as shown in Figure 1 - la. In two dimensions, they show up as diangle diagrams, such as Figure 1 - 1b.

As apposed to global anomalies, which alter the phsyics of a theory, anomalies in gauge symmetries are typically said to render a theory inconsistent. As we will see below, and more explicitly in chapter 3, care should be made in making this statement.

It has been known since the work of Green and Schwarz that some theories with apparently anomalous matter content are actually rendered non-anomalous by the cancelation of quatum anomalies with a classical gauge non-invariance [40].We will see that for chiral gauge theories in two dimensions, such cancellations are typical and facilitate the construction of gauge theories for geometries outside of the Calabi-Yau framework.

In chapter 3 we present the two dimensional story, but the basic pronciple can be understood in the more familiar four dimensional context. To see this explicitly let's consider the standard model. The standard model describes an SU(3) x SU(2) x U(1)y gauge theory at high energies, which at low energies breaks to SU(3) x U(1)EM. The fermionic content of this theory is three generations of quarks and leptons, where each generation has the charges listed in Table 1.3.

It is not difficult to check that all of the gauge anomalies, which are proportional to Tr(T1 {T 2 T3}), vanish. Here T is the generator of the ith gauge group. For example,

40 I SU(3) SU(2) U(1)y U(1)EM

UL 3 2

dL 3 2 1

UR 12 dR 33

VL 1 2 1 0

eL 1 2 1 -1

eR 1 1 1 1

Table 1.3: Standard model gauge charges for a fermion generation the cancellation of the U(1)3 anomaly is given by:

0,j" oc Tr(Q3) = 3 (2(1/6)3 - (2/3)3 + (1/3)3) + (-2(1/2)3 + (1)3) = 0. (1.39)

quarks = -3/4 leptons = 3/4

One thing to note about the cancellation, is that it relies crucially on having a full generation. If one drops the contribution from the top quark, for instance, the anomaly no longer cancels. This presents a puzzle. The top quark is about forty times heaver than the bottom quark, and so we should be able to consider an effective theory where we integrate out the top quark. Even worse, if we consider a theory like the standard model, but where we are free to change the Higgs-quark Yukawa couplings, we can take the mass of the top and bottom to infinity without changing the mass of any other particles.

For energies above the scale of spontaneous symmetry breaking, but below the mass of the arbitrarily heavy quarks, the theory is naively described by a gauge theory with anomalous matter content which appears inconsistent. On the other hand, it should be consistent to integrate out the heavy quarks.

This puzzle was resolved by D'Hoker and Farhi [41,42], who showed that integrating out the massive fermions induces self interaction terms of the Higgs field which conspire to cancel the gauge anomaly in the low energy theory.

41 This same phenomena is common in the two dimensional chiral gauge theories we study, and as we explain in chapter 3 it is precisely this cancellation between quantum anomaly and classical gauge non-invariance which allows these models to describe low energy fixed points outside of the usual Calabi-Yau framework.

The rest of this thesis goes into more detail trying to understand the strongly interacting gauge theories introduced above. In section two we explore monopoles in three dimensional gauge theories with large amounts of matter. Using the 1/Nf expansion we compute the scaling dimension and quantum numbers of monopoles for a variety of gauge groups ultimately making predictions for the infrared behavior of these gauge theories. In section three we move on to discuss chiral gauge theories in two dimensions exploring the consistency conditions and geometric implications for the low energy conformal field theories.

42 Chapter 2

Monopole Operators in Strongly Coupled Gauge Theories

In this chapter, we cary out the investigation described above, identifying monopole operators and studying their scaling dimension and quantum numbers in strongly coupled gauge theories in three dimensions.

2.1 Introduction

In three-dimensional gauge theories, one can define local disorder operators by requiring the gauge field to have a certain singular profile close to the point where the operator is inserted [13,14]. These operators are commonly referred to as monopole operators, because in Euclidean signature the gauge field singularity looks like that of a Dirac monopole [17] or a non-Abelian generalization thereof [19,43,44], as will be the case in this chapter.

Monopole operators are of interest for many reasons.' As explained by Polyakov in 3d Maxwell theory without matter, the proliferation of monopoles provides a mechanism for confinement [6,46]. If one adds enough fermionic or bosonic matter, how-

'We restrict ourselves to the study of monopole operators in three-dimensional gauge theory. In four-dimensional gauge theories one can define line operators by requiring the gauge field to asymptote to that of a monopole close to the line singularity [18]. These operators play an important role in the geometric Langlands program; see, for instance, [45].

43 ever, the monopole operators become irrelevant in the renormalization group (RG) sense [12-15], and in the deep infrared one finds a deconfined quantum critical theory. As stressed in [47] (see also [48]), the existence of these deconfined quantum critical theories relies on not having any monopole operators with scaling dimensions smaller than three. Another reason why monopole operators are of interest comes from certain spin systems whose low energy physics is described by an emergent gauge theory, such as the CPN model [10,11]. In these gauge theories, monopole operators can act as order parameters [49,50] for second-order quantum phase transitions that cannot be described within the Landau-Ginzburg-Wilson paradigm [10,11]. The scaling dimensions of these monopole operators constitute interesting critical exponents that can also be computed directly from quantum Monte Carlo simulations of the spin systems [51-53].

Monopole operators play a prominent role in supersymmetric theories as well. For instance, in the model introduced by Aharony, Bergman, Jafferis, and Maldacena (ABJM) [54] (see also the related model in [55]), it was shown that when the Chern- Simons level is k = 1 or 2, there exist BPS monopole operators that are Lorentz vectors and have scaling dimension precisely equal to two [55-57]. In other words, these operators are conserved currents. The existence of these conserved currents is what makes possible an enhancement in the amount of supersymmetry from K = 6, which is the manifest supersymmetry of the ABJM Lagrangian, to K = 8, which is the expected amount of supersymmetry that follows from M-theory. In the same ABJM model, as well as in many other superconformal field theories with gravity duals [58-67], monopole operators are also needed to match the spectrum of supergravity fluctuations in the bulk, and indeed certain holographic RG flows are triggered by operators with non-vanishing monopole charge [68]. Monopole operators also play important roles in various supersymmetric dualities (see for example [69-71]) and mirror symmetry (see for example [14, 72]), where the duality transformations map them to more conventional operators. It is important to know the quantum numbers of these monopole operators if one wishes to check these dualities.

The goal of this chapter is to study monopole operators in (non-supersymmetric)

44 three-dimensional QCD with gauge group G (which includes QED in the case G =

U(1)) and N1 flavors of fermions transforming in some representation of G. We study these operators perturbatively to next-to-leading order in 1/Nf. While in the absence of matter fields, 3d gauge theory with any compact gauge group is believed to confine [6,46,73-76], in the presence of a sufficiently large number of matter fields the theory flows to an interacting conformal field theory (CFT) in the infrared (IR) [77]. We are interested in studying monopole operators at this interacting IR fixed point. We want to answer the questions: How many independent monopole operators are there, and what are their quantum numbers?

Of course, starting with any monopole operator, we can take its product with various gauge-invariant local operators built out of the fermions, and construct new monopole operators this way. Throughout this work, however, we will focus only on the "bare" monopole operators, namely those that cannot be written as such composites.

Like in any CFT, one can use the state-operator correspondence to identify the space of local operators that can be inserted at a given point on R3 with the Hilbert space of states on S2 x R. In general, if one defines a monopole operator by requiring the gauge field to have a fixed behavior close to the insertion point, the resulting operator will not have a well-defined scaling dimension. It is quite subtle, in general, to associate a certain monopole profile to an operator with well-defined scaling dimension, or, equivalently, to a certain energy eigenstate on S2 x R. We will discuss this subtlety in Section 2.2. As we will explain, in the large N limit that we study, the subtlety is ameliorated by the fact that the gauge field fluctuations are suppressed, and one can indeed say that certain energy eigenstates on S 2 x R correspond to monopole operators. However, not all possible non-Abelian generalizations of the Dirac monopole can be associated with linearly-independent monopole operators of well-defined scaling dimension.

As we will review in Section 2.2.1, the non-Abelian generalization of a Dirac monopole involves several discrete parameters referred to as Goddard-Nuyts-Olive (GNO) charges [19]; the monopoles are also classified topologically by iri(G) [78], and

45 unless G = U(1), there are infinitely many GNO monopoles in the same topological class. One of our main results is that, at least in the limit of large Nf, only certain sets of GNO charges yield independent bare monopole operators. These sets are the ones for which the corresponding monopole background is stable, in the sense that it is a local minimum of the gauge effective action on S 2 x R. A surprising result is that we find more than one independent monopole operator per topological class.

The monopole operators must transform as representations of the global symmetry group, which includes the conformal group and the flavor group. The quantum numbers under the conformal group are the spin and the scaling dimension. We devote a significant part of our work to computing the scaling dimensions of the monopole operators to second order in 1/Nf. For clarity, we first present our computations in the case where the gauge group is G = U(Nc) and where the fermions are two-component complex spinors transforming in the fundamental representation of U(Nc). We later generalize our computations to other gauge groups and/or other representations of the fermion flavors. Our work improves on existing results in the literature: in the QED case, N, = 1, the leading large N1 behavior of the monopole operator dimensions was found in [13]; for the monopole with lowest charge, the first subleading correction was computed in [21]; lastly, in U(Nc) QCD with N1 fundamental fermions, the dimensions of the monopole operators at leading order in N1 were found in [72]. Re- lated computations can be found in [12,15, 79] in non-supersymmetric theories, and in [14, 55,56,65] in a supersymmetric context.

We also calculate the representations of the monopole operators under the flavor

symmetry group, but only in the case where the gauge group is G = U(Nc). In this

case, the flavor group is SU(Nf). We find that the monopole operators transform in

irreducible representations of SU(Nf) whose Young diagrams are rectangles with a

number of rows equal to N1 /2 and a number of columns that depends on the GNO charges. Our results apply, of course, in particular to the QED case, N, = 1, and agree

with the results of [13] for the monopole of smallest charge, but disagree with [13] on

the monopole with two units of charge. (Our computation for monopole operators

with greater than two units of charge is a new result.) We also find disagreement

46 with the results of [72] in the case of the simplest GNO monopole in U(Nc) QCD.

The rest of this chapter is organized as follows. Section 2.2 is a rather non-technical and highly recommended read that includes a definition of monopole operators in 3d

gauge theory (in particular in QED3 and QCD 3), as well as a discussion of how these operators can be studied via the state-operator correspondence. In Section 2.3 we set up the computation of the scaling dimensions of the monopole operators in U(Nc) QCD with Nf fundamental fermion flavors as an expansion in 1/Nf. To evaluate these scaling dimensions through order O(NO), we need to compute three functional determinants corresponding to the fluctuations of the fermions, of the ghosts, and of the gauge field. We study the effective actions of these fields in Section 2.4. The gauge field effective action is not positive-definite for all sets of GNO charges, thus making certain GNO monopole backgrounds unstable and the corresponding monopole operators poorly defined. We discuss this stability issue in Section 2.5. For the monopoles that are stable, we collect the results on their scaling dimensions in Section 2.6. Our results include the QED case N, = 1 as a particular case. In Section 2.7, we find how the monopole operators transform under the SU(N) flavor symmetry group. In Section 2.8 we generalize the results of Sections 2.3-2.6 to other gauge groups and/or representations of the fermions. Lastly, we end with a discussion of our results in Section 2.9. The reader interested only in the results can skip Sections 2.3 and 2.4.

2.2 Monopole operators via the state-operator correspondence

We now start by addressing some of the preliminaries necessary for studying properties of monopole operators. In Section 2.2.1, we introduce classical monopole backgrounds in both Abelian and non-Abelian gauge theories. In Section 2.2.2, we then review the gauge theories of interest for this chapter and highlight the role played by large Nf in studying them. Lastly, in Section 2.2.3, we introduce carefully the monopole operators that we will study in the rest of the chapter, and discuss two ways of

47 defining them that become equivalent in the limit of large Nf.

2.2.1 Classical Monopole Backgrounds

To begin discussing monopole operators more explicitly, it is convenient to first think about classical backgrounds. The simplest and perhaps most familiar such backgrounds can be constructed in Abelian gauge theory as follows. A monopole of charge q in a U(1) gauge theory in three dimensions is a rotationally-invariant background

A for the gauge field A, where the field strength F = dA integrates to 47rq over any two-sphere surrounding the center of the monopole. For a monopole at the origin, we can write the gauge field and its field strength in spherical coordinates as

cosO)do if 09# 7r, F = q sinOdO A do A(N) q(1 A(S) q(-1-cos)d if 6O0,

where the expressions A(N) and A(S) satisfy dA(N) = dA(S) - F and are defined everywhere away from 0 = 7r (the North chart) and away from 0 = 0 (the South chart), respectively.2 In the overlap region, these two expressions differ by a gauge transformation, A(N) - A(S) = dA, with gauge parameter A = 2q. The condition that this gauge transformation is well-defined, namely that the same U(1) group element eiA is associated both with 4 and 0 + 27r (assuming that the U(1) gauge group is a circle of circumference 21r), implies the quantization condition q E Z/2.

For a gauge theory with gauge group G, one can define similar monopole backgrounds by simply considering a U(1) subgroup of G for which one can construct a monopole just like (2.1) [19]. For instance, if the gauge group is G = U(Nc), we can write

A = H(±1 - cos 9)d#, (2.2)

2The expression for the monopole background is given in the dual coordinate basis {dr, dO, d}. It is also common to present this background in the frame basis {r, 0, $}, where it takes the form A(N) _ q soS and - qrsin in the North and South charts, respectively.

48 where H is a constant N, x N, Hermitian matrix, and the two possible signs correspond to the North and South charts, as in (2.1). Requiring that on the overlap region between the two charts, the expressions for the gauge field in (2.2) should differ by a

4 U(Nc) gauge transformation, implies e riH = 1. Making use of the gauge symmetry, we can always rotate H to the diagonal form

H=diag{qi,q 2, ... ,qNc, (2-3)

4 with qi q2 --- The condition e riH = 1 implies qa E Z/2 for all a.

In general, for a gauge group G there exist monopoles of the form (2.2), with H an element of the Lie algebra of G. Using the gauge symmetry, H can always be rotated into the Cartan of the gauge group [19]. More explicitly, if ha (with a = 1,... ,r, where r is the rank of the gauge group) is a basis for the Cartan subalgebra, then

H can be written as H = EIl qaha for some set of numbers qa. This rotation does not completely exhaust the gauge symmetry, as the Weyl group acts non-trivially on the ha, and consequently on the qa as well. One should therefore regard as equivalent any two sets of qa that are related by a Weyl group transformation. The collection of numbers {qa} are called GNO charges after the authors of [19]. The GNO charges must satisfy the quantization condition exp [47ri >E qaha] = 1, where "exp" is the usual exponential map defined on the Lie algebra and valued in the gauge group. It is equivalent to say that exp [47ri Er=1 qaha = 1 in any representation of the gauge group, where the ha are now matrices, and "exp" is the matrix exponential.

It is worth noting that the GNO charges qa are not all conserved, or equivalently, they do not all provide a topological characterization of the singular gauge configurations (2.2). Indeed, there exists a much coarser classification of monopoles by elements of the fundamental group 7ri(G) [78]. As we discussed, in order to make sure that we have a well-defined monopole background (2.2), we need to specify a gauge transformation (i.e. an element of the gauge group G) in the overlap region between the North and South charts. Since the overlap region has the topology of a circle, these gauge transformations are classified topologically by maps from a circle

49 into the gauge group, or in other words by elements of 7ri(G). In the case of U(1) this topological charge is the same as the GNO charge q. When G = U(Nc), the topological charge can be derived from the current

JItop = 1 , trF"P, (2.4) which is conserved provided that the field strength F,, satisfies the (non-Abelian)

Bianchi identity. It follows that in this case it is only the sum qtop = .=, q, that is a conserved topological charge, as opposed to all the individual q,. In other words, in a non-Abelian gauge theory there are several GNO monopoles (in fact, infinitely many) belonging to the same topological class.

2.2.2 Three Dimensional Gauge Theories with Fermions

To make the discussion of monopoles more concrete, let us focus our attention on a specific class of three-dimensional gauge theories. The class of theories whose monopole operators we want to study is QCD3 with gauge group G and N fermion flavors transforming in some representation of G. These theories have a parity anomaly if N1 is odd [80-82], so we will restrict the following discussion to the case where N is even. When the gauge group is G = U(Nc), the Lagrangian in Euclidean signature is

Nc Nf N. L=42 (Fab)2 + N N (#Q,(ioabO, + Aab)b,c] (2.5) gYM a,b=1 a,b=101

Here, the indices a, b are fundamental color indices, and a is a flavor index, while the two-component spinor indices on the fermions and gamma matrices are suppressed.

When N1 is sufficiently large, this theory flows to a CFT in the infrared [77]. This CFT can be studied by simply erasing the Yang-Mills term from the action, which by dimensional analysis is an irrelevant operator. We can therefore write the Lagrangian

50 for this CFT as

Nf Nc CFT , + A3ab) pb,] - (2.6) a=1 a,b=1

In the infrared theory, in the absence of the Yang-Mills term, the only role played by the gauge field is that of a Lagrange multiplier that imposes the constraint that the non-Abelian current vanishes, jab(X) = 0.3 Being a conformal field theory, the fixed point (2.6) can be studied on any conformally flat space. The Lagrangian on such a space differs from (2.6) only in that the partial derivative ,, should be replaced by the covariant derivative V,. The gauge field components Aajb, considered as components of a one-form in the coordinate basis, remain invariant under the Weyl transformation used to map the theory on R' to that on a different conformally flat space. For large numbers of fermion flavors, this theory simplifies. Indeed, integrating out the matter fields, the effective action for the gauge field takes the form

Seff[A = -Nf log det(i ab yV, + .yIAab) (2.7)

The factor of Nf in front of the action means that we can perform a semiclassical expansion about any saddle point of (2.7) that is also a local minimum of the effective action, with Nf playing the role of 1/h. As in any such expansion, the typical size of fluctuations about the saddle is of order vh, or 1/VNI in our case, as can be seen from expanding (2.7) around the saddle point configuration and examining the term quadratic in the fluctuations. 4 The monopole backgrounds introduced in Section 2.2.1 are rotationally symmetric about their center and invariant under conformal inversions, as one can easily check. These properties guarantee that they are saddle points of the effective action (2.7) on any conformally flat space. It is not guaranteed, however, that they are all local 3In the quantum theory, this constraint translates into the condition that (xIjablx) = 0 for all physical states IX). 4Note that for a saddle point of the effective action that is not a local minimum, the fluctuations would grow with time and eventually become large.

51 minima of the effective action, which is a fact that will become important in studying monopoles in the quantum theory.

2.2.3 Quantum Monopole Operators

In the previous two subsections we established the existence of monopole saddles in a class of three-dimensional gauge theories. We now explain how to define local operators with well-defined scaling dimensions that are associated with these saddles in the infrared CFT. As we explain below, for small Nf there is a tension between defining an operator that corresponds to a classical monopole background, and defining an operator with definite scaling dimension. In particular, the operator most easily identified with a monopole background does not have definite scaling dimension, but rather corresponds to a sum of such operators. At large Nf, however, this tension is alleviated, and we can indeed associate an operator of fixed scaling dimension to a monopole background. Before delving into monopole operators, let us briefly review what we know about local operators in the IR theories of interest. The most familiar local operators are those that can be written as gauge-invariant combinations of the fundamental fields, such as 0 = Z Y 4 Pa. These operators are sometimes referred to as order operators [18]. In addition to order operators, one can also define local disorder operators, which cannot be written simply in terms of the fundamental fields. Rather than being defined as local products of fields, disorder operators can be thought of as creating singularities for the fundamental fields. In the context of the path integral, we can define a disorder operator inserted at a point p by integrating only over field configurations that asymptotically approach a prescribed singular configuration in a neighborhood of p.5 For the classical monopole backgrounds described in Section 2.2.1, the gauge field has such a localized singularity. We can thus define a local disorder operator asso-

'Concretely, this procedure can be realized by cutting out a ball of radius e about the point p, fixing the boundary conditions for fields on the surface of this ball, and only integrating over fluctuations outside the radius E. Away from the insertion point, this regularized disorder operator acts just like a local operator. A similar prescription in four dimensions was used in [18].

52 ciated to a monopole background by requiring that the gauge field asymptotically

approach that of the classical background near the insertion point of the operator. In

this way we can indeed associate a quantum operator with a classical monopole back-

ground. Unfortunately, the disorder operator so defined does not transform nicely

under the conformal symmetries present at the infrared fixed point. In a CFT it is

convenient to work in a basis of local operators with definite spin and scaling dimen-

sion. As we will see, the disorder operator defined above does not have a definite

scaling dimension, but rather can be written as a sum of operators with definite

scaling dimension.

To see that the disorder operator cannot in general have a well-defined scaling

dimension, it is convenient to change perspectives from operators on R 3 to states on

S2 x R by using the state-operator correspondence. In a CFT, the state-operator

correspondence maps operators inserted at the origin of R 3 to normalizable states on

S2 x R. The R coordinate r is interpreted as Euclidean time and is related to the radial coordinate r on R 3 through r = er. The scaling dimension of an operator on R3

is identified with the energy of the corresponding state on S2. Restricting to disorder

operators, the correspondence identifies the disorder operator defined by boundary

conditions at a point in R3 to the state on S2 x R given by a wave-functional on field

space with delta-function support on the classical field configuration at r = -oo.6

This state is not an energy eigenstate, but rather a superposition of energy eigenstates. The wave function, which is localized about the classical configuration at early times, spreads out at late times. The corresponding operator on R3 is therefore

a sum of operators with definite scaling dimension. In a generic theory, there is no principle that singles out any one operator in this sum, and correspondingly there is a significant distinction between a disorder operator defined by boundary conditions and an operator of definite scaling dimension.

At large Nf, however, the situation is better. The monopole background is a classical saddle, and for large Nf the gauge fluctuations are suppressed. If the sad-

6 If we regularize the disorder operator by smearing it over a sphere of radius E, as in footnote 5, the wave-functional would have delta-function support on the classical field configuration at r = log e.

53 dle is stable, the state corresponding to the disorder operator is close to an energy eigenstate. It is the operator of definite scaling dimension corresponding to this energy eigenstate that we refer to as the monopole operator for the remainder of this chapter. 7 In the next section we explain how to use the path integral on S2 x R to calculate the energy of eigenstates associated with stable saddles, and in Section 2.5 we determine which saddles are stable.

2.3 Free energy on S 2 x R

In the previous two sections we discussed at some length the precise definition of monopole operators at the infrared conformal fixed point of QCD 3 with many flavors of fermions. In summary, for the GNO backgrounds (2.2) on S2 x R that are local minima of the gauge field effective action, and only for those backgrounds, there exist several degenerate lowest-energy states whose wavefunctions are highly peaked around the saddle (2.2); it is these states that, via the state-operator correspondence, get mapped to the bare monopole operators on R3 whose properties we want to study. We will refer to these states on S2 x R as "ground states" in the presence of the monopole flux (2.2). The use of the term "ground states" can be justified only in large NJ perturbation theory, where one can define a Fock space of states for every stable GNO configuration, and these Fock spaces mix only non-perturbatively in 1/Nf. One aspect of the state-operator correspondence is that the scaling dimensions of operators on R3 are equal to the energies of the corresponding states on S2 x R. In particular, the scaling dimension A of a bare monopole operator equals the ground

71n the U(1) case each operator in the decomposition of the disorder operator must carry the same topological charge. As such, even away from large Nf, there is a natural operator with definite scaling dimension to associate with a monopole background, namely the operator corresponding to the lowest energy state with the given topological charge. As discussed in Section 2.2.1, for non-Abelian theories there are many classical backgrounds with the same topological charge, and so one would be able to identify only one monopole operator per topological class this way. In supersymmetric theories, however, it may be possible to identify BPS monopole operators with certain GNO backgrounds after performing a Q-exact deformation of the theory to weak coupling (see, for example, [55,65]).

54 state energy in the presence of some constant GNO flux through the two-sphere.

The goal of this section is to exploit this equality in order to calculate the scaling

dimensions A. In later sections, we will calculate the other quantum numbers of the bare monopole operators.

The ground state energy in the presence of some constant GNO flux can in turn be calculated by performing the path integral on S 2 x R. More explicitly, we have

A = - log Z[A] = F[A], (2.8) where Z[A] is the Euclidean partition function on S2 x R in the presence of the background A, and F[A] is the corresponding ground state energy (or free energy). 8

In principle, the quantity log Z[A] should be understood as the limit

log Z[A] = lim - log Z6[A], (2.9) where Z[A] is a similar partition function on S2 x S 1 calculated after first compact- ifying the R direction into a circle of circumference 3. In practice, as we will see, it is easy to isolate the leading term in the large 3 expansion of log Z,3[A] while working directly on S2 x R.

Our procedure for calculating F[A] consists of three steps:

1. We perform the path integral over the matter fields. Integrating out the matter

fields generates a gauge-invariant effective action for the gauge fluctuations and leads to a sensible 1/Nf expansion.

2. We fix the gauge by introducing Faddeev-Popov ghosts.

3. We evaluate the path integral over the gauge fluctuations and the ghosts at next-to-leading order in 1/Nf.

We now provide an explanation of this procedure, while the next section is devoted to the details of the calculation. 8 The expression (2.8) should be taken to include only perturbative contributions in the 1/N expansion. All non-perturbative contributions in 1/Nf should not be taken into account.

55 2.3.1 Setup

As discussed previously, the IR conformally-invariant action for QCD with U(Nc) gauge group and N1 flavors of complex two-component fermions is

Nf N. S d VfH- [OT',yp abV + A al+ al,)4b,a] , (2.10) a=1 a,b=1 where, in anticipation of having to study this theory in the presence of a background monopole flux, we split the non-Abelian gauge field Aab into a sum between a background Aab and fluctuations aab. Here, the indices a, b are color indices, and a is a flavor index. The spinor indices on the fermions and gamma matrices are suppressed.

The action (2.10) can be more compactly written as

S[A; a,44 , 0] = So[A; 0t,4] + d3 X V Z ab (211 a,b where So[A; 4t,40] is the action (2.11) with gauge fluctuations set to zero, and

3a= Z: ?4,c74,,a: (2.12) is the non-Abelian covariantly conserved current. Double colons stand for normal ordering. As in any gauge theory, we can write the partition function as

Z [A] = V 1 Da Dt DV) exp [-S [A; a, Vt, V], (2.13) Vol (G) with Vol(G) being the volume of the group of gauge transformations, which we need to divide by because we do not want to count gauge-equivalent configurations multiple times.

Let us set up our conventions for this calculation. We write the standard line

56 element on R' in spherical coordinates as

s3= d e2 [d 2 + d92 + sin 2 2] = e (dqsin 9 cos # sin 0sin 4 cos 0)

(2.14)

We want to calculate the partition function on S2 x R. The metric on S2 x R is obtained by rescaling the R3 metric (2.14) by e- 2r:

dsS2xR - dO2 + sin2 6 dq2 + dr 2 . (2.15)

Recall that the dynamics of a CFT is insensitive to such a rescaling. We will be doing calculations with spinors on the curved space S2 x R, hence we need to specify a frame ei. We obtain the frame by the conformal transformation of the standard frame ei = dxi on R3

e = e-dx (2.16)

We choose the set of gamma matrices iy= o-&, where the a are the Pauli matrices. All subsequent formulae for spinors are understood to follow these conventions. A point on S 2 x R will be denoted by x = (r,9, #). Sometimes we will also use the decomposition x = (r,ft), where fi is a unit vector pointing to a point on S 2 . The covariant derivative on S2 x R will be denoted by V,. The gauge covariant derivative for a fundamental fermion 0' and current jab (which transforms in the adjoint representation of U(Nc)) is

N. [D,)] a = (v, jab - iab b D Ab= 1 i ) O b 1(2 1 7)

(DA)] ab = Vjab - i [A,, j]ab where [A,, j=]a - Zl 1 (A/ JcbV - c AA) is the matrix commutator.

As explained in the previous sections, for a U(Nc) gauge group, the most general

57 monopole background can be taken to be

A"a = diag{qi, q2 ,... qNJ}U(1) AU(1) (1 cos 0)do if 9 7r, (2.18) (-I - cos 0)do if 0 4 0,

with q q2 -- qN, and qa E Z/2.

2.3.2 Gauge Field Effective Action

The first step in our general procedure for evaluating the ground state energy F[A] is to integrate out the matter fields. Doing so yields a gauge-invariant effective action for the gauge field fluctuations. The gauge effective action is defined in such a way that the partition function is simply

Z[A] = Vol(G) Da exp -Seff[a]]. (2.19)

Comparing with (2.13), using the decomposition of the action in (2.11), and expanding in powers of a, one can write

Seff[a] = -log Zo[A] + . K(Inzd /7VYagb(x) ()j(x) , (2.20) n=1 ni ( a COnnI where the correlators on the right-hand side are evaluated using the action of free fermions in the background A, namely the action So[A; V), Of] introduced above. In other words, the effective action Seff [a] is the generating functional of connected correlators of the current operator jP in this theory of free fermions. The quantity Zo [A] appearing in (2.20) is the partition function associated with So[A; 4P, Ot]; it is just a Gaussian integral, which evaluates to

ZO[A] = D'tDoexp [-So[A; , t]] = (det(iO")))N" (2.21)

58 Here, the subscript (A) denotes a background gauge covariant derivative, as in (2.17) with A replaced by A.

For us, the GNO monopole background (2.18) is static as well as invariant under rotations and time reversal, and therefore the one-point function of the current operator must vanish, (j"(x)) = 0 (see also the last paragraph of Section 2.2.2). Therefore, the term linear in a in (2.20) vanishes. In general, the term quadratic in the a does not vanish, and its coefficient is given by the current-current correlator

Kp'cd(xx') ' (2.22)

This current-current correlator should be thought of as an integration kernel that defines an operator on the space of square-integrable one-forms on S 2 x R.

The kernel Kf"c(x, x') can be written more explicitly in terms of a quantity

Gq(x, x'), which can be identified with the Green's function of a single fermion in an Abelian gauge theory in the presence of a monopole background (2.1), namely

G,(xx') = (*(x)pt(x')) = x U . (2.23)

(This theory would be described by the action (2.10) with Nc = Nf = 1 and a = 0.) Indeed, substituting the normal-ordered expression (2.12) into (2.22), and notic- ing that the contractions between the fermions take the form (/,a(X)>4,(X')) =

6 a,56 abGqa(X, x'), one obtains

K11c(x, X') = N 6&c6adKq(x, x), (2.24) with

With t(x, e x')s in tr (handwec(ax 'Gi. (x, x'))a. (2.25)

With the expression (2.24) in hand, we can write the effective action for the gauge

59 field fluctuations as

Seff [a] = Nf tr log(i; A)) + Jdxd d3x' g(X) g(x')aabt(x)Kq x,.)a.b(X/)

(2.26)

The ellipses denote terms with higher powers of a, which one can easily show are also proportional to Nf. We are now in business. That the quadratic part of the action is proportional to Nf means that the typical gauge field fluctuations are a oc 1/V/N-, and we can calculate Z[A] (see (2.19)) approximately at large Nf using a saddle point approximation.

We can now try to perform the integral over the gauge fluctuations by keeping only the terms up to quadratic order in a in Seff[a], and write down the free energy on S2 x R as:

F[A] = -Nf tr log(i(A) 1 " tr log K" + O(l1/N). (2.27) 2

The quotation marks are meant to emphasize that the O(NO) term is only rough, because we ignored the issue of gauge invariance when we performed the integral over the gauge fluctuations. To obtain a more explicit answer, we now proceed to a more careful analysis of gauge invariance.

Just like the original action (2.10), the effective action Seff[a] is invariant under gauge transformations that in the gauge sector act as

(Al + a,) -+ i U&iIU + U(A + a,,)Ut. (2.28)

Correspondingly, the integrand in (2.19) has flat directions corresponding to these gauge transformations. Therefore, one cannot simply identify the functional integral f Da exp[-aKa] with the determinant of the kernel K, as this kernel has many eigenvalues that vanish.

60 It is most convenient to work in background field gauge by imposing the condition:

D(A)all = 0. (2.29)

This condition distinguishes one gauge configuration in every gauge-equivalence class, so if we restrict our integral over a to configurations that satisfy (2.29) then the integrand e-sff[a] will no longer have any flat directions. The condition (2.29) does not exhaust, however, the group of all possible gauge transformations, because there are residual gauge transformations that leave a completely untouched. These residual gauge transformations form the isotropy group H(A). The Faddeev-Popov trick is to insert

1 = Det' (-D(A)D(A+a)p) X V DU 6 [D(A)aU,1] , (2.30) A ~ Vol (H(A)) jAJ into the path integral, where Det' denotes the functional determinant with zero modes omitted, and af = iUaUt + U(A, + a)Ut - A, is the gauge transformed a,.

Changing variables9 a -+ au in the partition function (2.19), inserting (2.30), and then renaming au -4 a gives:

[A] Vol (H(A)) Da e-sff [a]-sFPJI [DA)a] Det' (-D )D(A)A ,

(2.31) with

SFp[a] = - Tr'log (-D A+a)D(A+a),) , (2.32) where Tr' is a trace over the non-zero modes. The factor in the parenthesis in (2.31) multiplying the delta-function is precisely the inverse of the Jacobian factor that one obtains when taking D (A) outside of the delta-function. We can therefore perform the path integral (2.31) by integrating only over configurations that satisfy (2.29)

9 Note that both the measure and the action Seff[a] are invariant under this change of variables.

61 (and thus removing by hand all the flat directions), provided that we supplement the effective action Seff[a] by a term SFp[a] exhibited in (2.32) that comes from the

Faddeev-Popov procedure. Just like Seff[a], SFP[a] can be expanded in powers of a:

1 SFp[a] = - Tr' log (-D (A)D(A)p) + O(a2 ). (2.33)

The term linear in a in this expansion vanishes by an argument based on the symmetries of the background (2.18) similar to the one that showed that the linear term in a in (2.26) vanished.

Before presenting the answer for the free energy, we note that the factor of

1/Vol(H(A)) should be ignored. This factor would be relevant if we computed the partition function on S 2 x S1, where the S' circle has circumference 3, because in this case every generator of HWA) would contribute a factor proportional to 3 to Vol(HWA)).

Thus, log Za[A] would receive a contribution proportional to log 3 from every generator of HA). However, these contributions disappear when we consider the limit in (2.9).

Evaluating (2.31) in the saddle point approximation, we have

F[A] = N1 Fo[A] + SF[A] + 0(1/N1 ), (2.34) JF[A] = FFP[A] + Fgauge[A] , where Fo[A] is the fermion determinant

Fo[A] = - T log(j$ ) . (2.35)

6F[A] denotes the subleading term in the free energy. It is a sum of two terms, namely FFP[A], which is the Faddeev-Popov determinant

1 FFP[A] = - Tr'log (-D()D , (2.36) 2 Tr' lo 7(ADA

62 and Fauge[A], which is the gauge fluctuation determinant

1 Fgauge[A] = -Tr'log K . (2.37) 2

The fermion, Faddeev-Popov, and gauge field determinants will be calculated in Sections 2.4.1, 2.4.2, and 2.4.3, respectively.

2.4 Functional determinants

2.4.1 The fermion determinant

We now start by calculating more explicitly the leading term in (2.34), the fermion determinant Fo[A] defined in (2.35). This term arises from evaluating the partition function Zo[A] = e-N Fo[A] of non-interacting fermions in the background (2.18). See (2.21). Examining the action So[A; 4, V)/] more closely, we see that because we have taken the monopole background (2.18) to be diagonal in the color indices, it is not only the fermions of different flavor that decouple from one another, but also those of different color. Each fermion 4', is only coupled to an Abelian monopole background of charge qa. Because the fermions are non-interacting, the ground state energy can be written as

N. FO[A = Fo(qa) , (2.38) a=1 where F(q) denotes the ground state energy of a single fermion in an Abelian monopole background of charge q. The quantity F(q) = - log Z(q) can be computed from the partition function corresponding to this free fermion,

Z(q) = DOD4fes(V ) , S(q) = Jds Vg(x)Oty(iV, + qAN'))V. (2.39)

This computation was performed in [13,21] as part of studying the scaling dimensions

63 of monopole operators in U(1) gauge theory with N fermion flavors. We now review this computation briefly, partly because such a review will keep our presentation self- contained, and partly because in doing so we will also introduce some notation that will become useful in the following sections.

In order to evaluate the integral in (2.39), we should first decompose the fermion field 0 into a suitable basis of spinor fields. Since translations in the Euclidean time direction are a symmetry of the action, it is convenient to consider modes with harmonic time dependence, 4 oc e-w. Finding a basis for the angular dependence of 0 requires more thought. If 0 were instead a complex scalar experiencing the same monopole flux qAU(l), one could use the basis of monopole harmonics Yq,em" (i), which were defined in [16,83] (see also Appendix A.1) as eigenfunctions of the gauge- covariant Laplacian on S 2 . Here, f > IqI is the angular momentum, and mt ranges from -f through f. To find a complete basis for a field of a different spin s, we can work in the frame (2.16) obtained by conformal transformation from R3 and expand every component of the spin s field in terms of the monopole harmonics Yq,tm(h).

More conveniently, we can use the usual angular momentum addition rules to work in a basis where the quantum numbers are {j,m, e, s}, j being the total angular momentum and m the eigenvalue of J 3 . For spinor fields where s = 1/2, we have j =f - 1/2 or j = f + 1/2, and we can define

= Yqj- -1 1 Tq,jm , 2|+ + , S2 2) (2.40) j+1-m -q3j 2(j+1) q,j+I,m-. . Sq,jm = , =2--2 jq9 +1+m y 2(j+1) q,j+!,m+ 1 where Yq,em are the scalar monopole harmonics. Note that for total angular momentum j = jqj - 1/2 we have only the Sq,jm harmonics, which in this case have orbital angular momentum f = Iqi, while for larger j we have both Sq,jm, with orbital angular

64 momentum f = j+1/2, and Tq,jm, with orbital angular momentum f = j - 1/2. For q = 0, the Sq,m start at j = 1/2.

Expanding the fermion O(x) in the basis (2.40),

O(x) = Z t (qj'52(w)Sjm() + T'-(w)Tqjm(f)) e-iwT , (2.41) j 2Jqj-1/2 M=-j with anti-commuting coefficients 'IV (w) and M (w), one finds that the action S(q) defined above can be written in almost diagonal form, because the gauge-covariant Dirac operator only mixes the modes Sqjm and T,jm with the same j and m [13,21]: t ( >1 C3m'' )JM (2.42)

Here, Gqj(w) is a matrix that can be identified with the inverse propagator. When j > Iql - 1/2, dj (w) is a 2 x 2 matrix given by [13,21]

qw_ - + -q2

Gq,j(w) = 2

-z i+ bj)Fj+ ly- 2 \ TI \ ; +2 (2.43) when j = |q| - 1/2, Tq,jm does not exist, and Gq,j(w) should be thought of as a 1 x 1 matrix equal to the bottom-right entry of (2.43), namely q5j(w) = qw/ qi.

The path integral (2.39) becomes a Gaussian integral over the Grassmannian coefficients V and ' T). Performing this integral yields

dw Fo(q)=- 2-7 (2j+1) log W+ + -q2 -2 2ql(log4, j=lq\+i (2.44)

65 where the first term corresponds to j > jqj - 1/2 while the second term represents the contribution from j = tqI - 1/2. As one can easily check, the arguments of the logarithms are nothing but det Gq,j (w) , with Gq,j(w) defined in (2.43). The pre- factors of the logarithms in (2.44) come from summing over the allowed values of m.

The integrals over w are divergent, but they can be regularized by analytic continuation,

2 8 log (w2 + b2) & (W2 + b ) S=o = IbI. (2.45) J27r J 27rds

Using this identity to define the regularized expression, the free energy reduces to

00 / \ Fo(q) = - E (2j + 1) (j + -q2. (2.46) i=lql-'

This sum is still divergent and can be regularized by various methods, such as by the

Abel-Plana summation formula as in [13] or by zeta-function regularization as in [79].

The result can be written in terms of an absolutely convergent sum and the Hurwitz zeta function ((s, a) = _'(n + a)- as

Fo(q)=- (2j + 1) (j 2 - q2 - (2j + 1)2 + q2 j=lql-' (2.47) q(2q - 1)(q + 2) 6

In Section 2.6 we will tabulate this sum for a few values of q; see Table 2.1. Knowing

F(q), one can easily calculate the leading term in the large Nf expansion of the ground state energy in the presence of our GNO background using (2.38).

Now that we have a handle on the leading order computation let us move on to the next-to-leading order contribution.

66 2.4.2 The Faddeev-Popov determinant

The next-to-leading order computation of the free energy has two contributions given by the second and third terms in (2.34). The second term represents the contribution from the Faddeev-Popov ghosts, while the third term comes from the determinant of the gauge field fluctuations. Of these two contributions, the Faddeev-Popov one is considerably simpler, because it involves the determinant of a local operator, and we will discuss it first.

The Faddeev-Popov contribution to the ground state energy,

1 FFP log Det' (-D (A)D(A),p) , (2.48)

can be written as -log ZFP [A], where ZFP [A] is the partition function for an anti- commuting scalar ghost field c valued in the Lie algebra of the gauge group:

1 3XC l1 -Sghost ~ 2 ZFP[A] = Dc IsghostSghost = 2 / d g(x)3 01A ab [ ab a,b=1 (2.49)

Evaluated in the GNO monopole background (2.18), the ghost action becomes

N, 3 Sghost d X g(X) [ -Z(qa - qb)Auj)] Cab 2 (2.50) a,b=1

The interpretation of this formula is that the diagonal components caa are free real scalar fields, while the off-diagonal components cab, a : b, whose complex conju- gates are Cba = Cab*, are free complex scalar fields experiencing an Abelian monopole background (qa - qb)AU(l).

To diagonalize the action (2.50) we should expand the ghost fields ca in terms of the monopole harmonics YQ,jm introduced in the previous section, with Q = qa - qb.

67 Explicitly, writing

cab(X) = C '()Yqa -q,JM(h)e (2.51) W 2J,M and using the fact that the monopole spherical harmonics YQ,JM have eigenvalue

2 2 J(J + 1) - Q under the gauge-covariant Laplacian [16,83] on S , we can put the ghost action in the form

Sghost1 d I C (Q)12 [Q2 J(J+ 1) -(qa -qb) 2 ] (2.52) a,b=1 J,M

Note that here the sum over J runs only from j9a - qbI to infinity, and the sum over

M runs from -J to J, as appropriate for the spin-J representation of SU(2).

The contribution to the free energy can now be computed by integrating over the

Grassmannian coefficients CjM in (2.49). Because the ghost fields Cab do not mix, the result will be a sum

N. FFP [A] = E FFp (qa,qb) a,b=1

FFp(q, q (2J + 1) log (J + 1) - Q2 +2 Q=-q-q' J=IQI (2.53)

We will postpone evaluating this expression until after combining it with the contribution coming from the gauge field fluctuations in Subsection 2.4.4.

68 2.4.3 The gauge fluctuations determinant

We now turn our attention to the third term in (2.34), which is also the hardest to compute. 10 Combining (2.37) and (2.24), we can write this term as

Nc Fgauge[A] = E Fgauge(qa, qb) a,b=1 (2.54) 1 Fgauge(q, q') = Tr' log ]Cqqf 2qq where we recall that the quantity Kqq can be written in terms of the Green's function

Gq(x, x') of a single fermion in an Abelian monopole background with charge q as

K",(x, x') = -r(y"Gq(x, ')y'Gt,(x, x')). (2.55)

Therefore, in order to evaluate (2.54), we should first write down an explicit expression for Gq(x, x'), and then describe how to use it to construct Cqq, and find its eigenvalues.

In evaluating (2.54), we will not be able to find simple analytical formulae such as (2.47) or (2.53), and instead we will have to resort to numerics. In the rest of this section we aim to provide enough details on the steps one has to take in implementing these numerics, and the cross-checks that can be performed. We will postpone the numerical results until Section 2.6.

Green's Functions

The expression for the Green's function Gq(x, y) of a single fermion in a charge q Abelian monopole background can be read off from Fourier transforming back to position space the inverse Gq,j(w) = Gq,j(w)-' of the expression in (2.43). It is not

1OWe develop a method slightly different from the calculation for N, = 1, IqI = 1/2 done in [211. That approach, although less straightforward, has the advantage of producing simpler formulae than the approach of this chapter. However, it seems hard to generalize the method of [21] to the present case.

69 hard to check that this inverse is

Gqj (w) 2 Gq') w2 + (j + 1/2)2- q ' with Gq,j(w) as in (2.43). The quantity Gq,j(w) is a 2 x 2 matrix if j > jqj - 1/2.

When j = q- 1/2, it is a 1 x 1 matrix equal to q/(Iq Iw). The Green's function is then

Gq(x, X') =(X) 0 (X')

Tt (W') (2.57) = - (T,jm(h) Sqjm(fl)) Gqj(w)e-i-'T')

where it should be understood that when j = qj - 1/2 and Gq,j(w) is a 1 x 1 matrix, we should only consider the Sq,jm modes.

The Green's function (2.57) is a 2 x 2 matrix whose indices are the spinor indices that we have been consistently suppressing. By combining (2.57) with the explicit expressions for the monopole harmonics (2.40), one can see that each entry in Gq(x, X') can be written as a sum over products of two monopole spherical harmonics:

'~ (x, x') = ~ Z gM;,11(T - T1 (2.58)~0Yt/ j,M~efEj-j'

with coefficients g him-,e(T - ') that can be easily worked out by integrating with respect to w in (2.57). (In (2.58), mi = 1/2 or -1/2 if i = 1 or i = 2, and similarly for Mk.)

The details of expression (2.58) are crucial for calculating the numerical value of the monopole scaling dimensions, but not so essential if one is only concerned with understanding the general structure of the calculation. For understanding the

70 structure of the calculation, we can write down (2.58) schematically as

00 G,(x, x') = (2 x 2 matrix oc Y... (h)Y(*...(f')) , (2.59) j=JqJ-1/2

where we emphasized that Gq is a 2 x 2 matrix (because it has spin indices). Each

entry can be written as a single infinite sum of products between a monopole harmonic

with charge q at A and the conjugate of a monopole harmonic with charge q at h'.

There are additional finite sums in (2.59) that have not been indicated.

The careful reader may notice that the position space Green's function Gq (x, X') is not unique, because one needs to specify a pole-passing prescription in performing the w integral in (2.57). While usually in Euclidean signature the poles of the propagator

are off the real axis and the Fourier transform provides a well-defined position-space

Green's function, in our case we have zero-energy modes that generate a pole for the

propagator on the real axis. We choose the prescription for passing around this pole given by principal value integration, whereby

d - J - -- sgn(r - r') . (2.60) 27r W 2

This prescription respects CP invariance of the monopole vacuum, in which the Green's function is an expectation value.

The kernel IC

Now that we have an expression for the Green's function, it is a straightforward matter to write down the kernel K (x, y) = -Tr(-yG(x, y)'"G ,(x, y)) and compute its eigenvalues. Since the Green's function is a sum over products of two monopole harmonics, the kernel /Cqq, is a sum over products of four spherical harmonics

00 00 Cqq, (x, x') = (3 x 3 matrix oc Y,... (h)Yq,,... (ft')Y *...(f')Y* (i)), j=JqJ--! j'=\q'J-6 (2.61)

71 where we emphasize that this kernel can be written as a 3 x 3 matrix (since the tangent indices p and v run over three values), and that each entry of this matrix contains two infinite sums of products involving four spherical harmonics. It is straight-forward, but tedious, to work out the precise form of ACqq, (x, x') given the precise form of the Gq(x, X').

The object K",(x, x') should be thought of as an integration kernel that acts on

a space of vector fields aO(x) on S2 x R by

[Kqqa]"(x) = d x'v -g(x,)t(x, x')a,(x'). (2.62)

Actually, the expansion (2.61) reveals that the a"(x) must not be regular vector

fields on S2 x R, but rather sections of a more complicated vector bundle. Indeed, if

we pass from the North chart where AU(1) = (1 - cos 9)d# to the South chart where

A(s = (-1-cos O)d#, a scalar monopole harmonic Yqfm picks up a phase, Y() =

Y,(Nh)e-21qO, as appropriate for how a field with charge q should transform under a

gauge transformation A = A + dA, with A = -24. Consequently, we have

KYv(S)qqtql (X, X,) = KYV(N) Xe(.3 i (2.63)

Imposing the condition that both aP(x) and [Cqqa]1'(x) transform in the same way

when passing from the North to South chart (because otherwise Kqq, would not be a

well-defined operator), we see that (2.62) implies

2 ap(s)(x) = a (N) - iQ, Q /q- (2.64)

In other words, a"(x) carries charge Q under AU).

That Kqq, (x, x') acts on vector fields carrying charge Q could have been anticipated

from the form of the effective action (2.26). Indeed, in (2.26) we see that Kqq, (with

q = q. and q' = q.) is multiplied on the right by the a' b components of the gauge field fluctuations. In the background AUM), these components carry precisely charge

72 Q = q.- q= q - q, as can be seen by an argument similar to the one in Section 2.4.2 that showed that the ghost fields cab carry charge Q as well. We will call the off- diagonal components a ab W bosons in the following.

We are interested in finding the eigenvalues of the kernel Aqq,. To do so, we should make use of translational and rotational symmetry. Translational symmetry in the Euclidean time direction means that if we expand aP(x) in Fourier modes, the kernel

Kqq, will not mix modes with different frequencies. Similarly, rotational symmetry

2 along the S directions implies that ICqq, only mixes modes that transform in the same representation of SU(2),ot. As per (2.64) above, a good basis for the angular dependence of a"(x) is given by the vector monopole harmonics with charge Q. We saw in the spinor case that we can construct spinor monopole harmonics from scalar harmonics with any given charge Q. A similar construction can be performed for the vector monopole harmonics [84] (see also Appendix A.1). While in the spinor case we had two independent sets of spinor harmonics, Tq,jm and Sq,jm with orbital angular momentum t = j + 1/2 and j - 1/2, respectively, we now have three sets of vector harmonics with total angular momentum quantum numbers (J,M) and orbital angular momentum L:

UQJM(f), L = J+l, J >ifQ JJQJ , if JQJ = 0 or 1/2

VQ',M (,) , L = J, J > IQ, if Q >0, (2.65) 1, if QI=0,

WQJM(ft), L=J-1, J IQI+-1.

That the orbital angular momentum is L means that in a frame basis one can write down the components of the harmonics (2.65) in terms of scalar monopole harmonics

YQ,LML, where ML E {M - 1, M, M + 1}. Since we must always have L > IQ1, we obtain the allowed ranges in (2.65). Note that the harmonics with J = IQI - 1 are defined only when IQ| > 1; these harmonics will play an important role shortly.

73 We can thus expand a in terms of vector monopole harmonics (2.65) and Fourier modes in r:

a(x) = > aM(Q)UQJM(h) + ajm(Q)VQ,jm(h) + ajm(Q)WQ,JM(h)] e-QT Jm (2.66)

with coefficients aum(Q), ajm(Q)) and ajm(Q). Then the operator ACqq, is almost

diagonal and mixes together only modes with the same J, M, and Q:

dx d x' g(x)Vg(x')a,(x)*C"(x, x')av(x') t aim(Q) aim(Q) dQ Kj(2.67) E ajm (Q) Kq A,(i) M(Q) . J,M

aw(tiwGm(p) JMy(Q)j

When J ;> IQ + 1 there are three such modes for each M, ICqq, acts in this 3- dimensional space as the 3 x 3 matrix K ,(Q). When J = IQ, the modes corresponding to WQ,JM are absent, and K ,(Q) is a 2 x 2 matrix. Lastly, when J = IQI -1, both VQ,JM and WQ,JM are absent, and K ,(Q) is a 1 x 1 matrix. Because of rotational invariance, K ,(Q) is independent of M.

From (2.67) it is not hard to extract an inversion formula for the components of the matrix K ,(Q):

[K q(Q)]xy2IicrJ(Q - Q')

= d x 3d x''g(x)) (x')XQ",JM(f)*K ,4v(xx')Ym (')eirnr',

(2.68) where X, Y E {U, V, W} denote the indices of K 1 ,(Q). This expression is rather

unwieldy, especially after plugging in the explicit formula (2.61) for the kernel KIqq,

74 which yields, schematically,

Kq ,(Q) = J Cdd...(Q)Y ,....()Y* ...(O)Y4,....(")Y*...(')Yq,,....(f)YQ,...( .i 3, (2.69)

This formula for K ,(Q) involves two angular integrals over a product of six monopole spherical harmonics, two infinite sums exhibited explicitly in (2.69), as well as several finite sums that were omitted.

We are not discouraged and still determined to evaluate (2.69) as efficiently as we can. We can simplify (2.69) by using rotational invariance, which, as mentioned above, implies that K ,(Q) is independent of M. So we might as well compute

K q,(Q) after averaging over M. Writing (2.68) as

[(K ,(Q)]xy27r6(l - Q')

= J 1 ( 3 3 d x d x' /g7Y g(x'Xj(f)*K,,,,x, IY,y(ftei, -2J +1 1I \I4,i 1IIIX'IQ4JV\ M=-JfJV..4 (2.70) and plugging in the explicit form of Cqq, we again obtain an expression of the schematic form in (2.69). This time, however, because we averaged over M, this expression is rotationally-invariant and the integrand depends only on the relative angle between ft and ft'. The integral with respect to ft is therefore independent of ft', so we can choose f' to point in the direction and replace the integral with respect to ft' by a factor of 47r. Using

Yq,im(s) = 3 q,-m + (2.71) 4ir we get rid of three of those pesky monopole harmonics in (2.69). The remaining angular integral over the product of three harmonics can be evaluated using some

75 properties of monopole harmonics [83].

Yq,,m(h)* = (-1)q+mY-,,,-m(h) (2.72) and

fdn Y,tm(h)Y,,,/M1(h)Yq/f'PM//(h)

21 + 1)(2' + 1)(21" + 1) 4ir V q' q ") m (2.73)

where is the Wigner 3j symbol. After using these identities, K,, (Q) can be put in the form:

+ 3'+i 1(Q) 2' L (2 E' L K ,() =ZZ 2 S 2,...(Q) -q' -Q -q + 5q q' - Sq Q (2.74) where L = J - 1, J, or J + 1 (depending on which component of K ,(Q) we are computing), and where the coefficients C ... (Q) have fairly complicated expressions 33 that will not be reproduced here. The lesson to be learned is that one can write

K1 ,(Q) explicitly in terms of two potentially infinite sums (over j and j') and several finite sums over products of 3j symbols. In fact, for fixed j and J the sum over j' is finite, because the 3j symbol vanishes if its arguments are not triangular. There is therefore only a single infinite sum over j in the expression for Kq ,(Q), which can be

evaluated using zeta-function regularization. The terms in the infinite sum are shown

in an example in Figure 2-1 for one matrix element.

76 0.00 % 0 0.01 * q 1, q' 1 2 S1, J 352 0.02

0.03

10 20 30 40 Figure 2-1: We plot the terms in the infinite sum over j (2.74) that give the matrix element [K4J,(Q)]G for q = -1, q' = 1/2, Q = 1, and J = 35/2. We show the stage of the calculation where all the finite sums (over Jq, 1', 1, and j') in (2.74) have been done and only the infinite sum over j remains. The dots represent the actual terms in the sum, while the solid line is the asymptotic expansion of the summand to Q(1/j 8 ) that we determined analytically. We perform the infinite sum by zeta- function regularization of the asymptotic form for j > jc, where jc is the value below which we use the numerical values of the terms in the sum. We check the numerical precision by changing j, and we reach our goal of 10-1 precision by choosing jc ~ 40. This precision is needed to get the free energy with 10-3 precision.

Properties of Kj, (Q)

On general grounds, the matrix K ,(Q) should satisfy certain properties that can be used as checks on the explicit formulae (2.74). For instance, K 1, (0) is Hermitian, K't,(Q) = K,,(Q), and due to invariance of the monopole background under CP, one can show that

[K ,q(Q)]xy = [K'q,,q_,(Q)]xy = (-1)--[K!,(Q)]xy. (2.75)

Here, the range of X, Y E {U, V, W} depends on whether K q,(Q) is a 3 x 3, 2 x 2, or 1 x 1 matrix.

It follows from gauge invariance that K ,(Q) has a zero eigenvalue." To leading

"When J = IQI - 1 KJ,(Q) is a 1 x 1 matrix that does not vanish. For J = IQ it is a 2 x 2 and for J > IQI +1 it is a 3 x 3 matrix with one zero eigenvalue. In the following we assume J > IQI +1.

77 order in the large Nf expansion the current conservation equation takes the form:

0 = D A)ji = alX - i [A, = [- i(qb - qa) AU()] ja, (2.76) where we dropped terms proportional to the gauge fluctuation a and used that the monopole background is diagonal in the gauge indices. The gauge kernel, K"', defined in (2.22) is a current two point function. Applying (2.76) to the second current in K," we get

o = + i(q - q')A 71)] ",(x,x') (2.77) where we used (2.24). Note that in this Ward identity the delta function is absent, as follows from Lorentz covariance and dimensional analysis. From (2.77) we determine the eigenvector with zero eigenvalue of K , (Q). Indeed, inverting (2.68) we obtain:

K (x,x') = d2Q ( XiJM(h) [K ,, ()]xy Y 5JMfih) JM X,YE{U,V,W} (2.78)

and then acting with the derivative in (2.77), we obtain:

o = ( [K ,(Q)]xy + iQA Y"JM(h')* ei' . (2.79) YE{U,V,W} -

We compute the divergence of vector spherical harmonics in Appendix A.1.4. Thus, the eigenvector with zero eigenvalue of K,, (Q) is:

(+ iQ) ( (J+1)(2J+1)

K ,(Q) Q(1 - iQ) = 0 . (2.80) J(J+ 1)

2 2 J - Q J(2J+1)

78 This property provides an essential check of our numerical results.

The same result can be understood in a different way. The gauge field effective action should be gauge invariant, hence pure gauge modes should be zero eigenvectors of the real space kernel. We set the gauge fluctuation to be pure gauge by taking a= D(A)YQ,je-in in (2.62) to get

0 = [AKqq, (D(A)YQJe-iT')' (x) = dox' g(x')AC,(xx') (A)y (2)e- (2.81)

Plugging in for C" the formula (2.78) we obtain for the zero eigenvector of Kj , (Q) the following expression:

UQjM(h) ei'r

(DbA)YQ,JM (ft)e I dx WQyM (i) ein'r

n'tr W",m (h) * e

2 (L+1)2 -Q -(L+ (L + 1)(2L+1)1)2L)

Q(1 -if) = 27r(Q - Q') V/L(L +1)

(2 _Q2 (L +1M) (2L +1) (2.82)

This expression agrees with (2.80). What this formula says is that the zero eigenvector is a pure gauge mode D (A)YQ,jme-6Q in the vector monopole harmonic basis. The calculation above is by no means an independent derivation of (2.80), as current conservation follows from gauge invariance.

79 Eigenvalues and determinant of gauge field fluctuations

Having computed K ,,(Q), it is now easy to find the eigenvalues of the kernel K",(x, x'): they are simply the eigenvalues of Kq, (9) for every J and Q, and they come with multiplicity 2J + 1. To have the terms in (2.54) we need to compute:

Tr'log Kqq, = J> 1 (2J+ 1) log det'K,() , (2.83) J=11-1 where det' indicates that we should only take the product of the non-zero eigenvalues.

As shown in (2.80) the presence of pure gauge modes result in one zero eigenvalue for

Kq, (Q), and we have to drop the zero eigenvalue as our prescription is to integrate only over gauge inequivalent configurations.

The matrix Kq ,(Q) is not necessarily positive definite. If it has a negative eigenvalue that signals an instability, the corresponding gauge fluctuation gives a wrong sign Gaussian integral in the partition function and makes the free energy complex.

In Section 2.5 we discuss the instances when this happens. For illustration, we plot some of the low J eigenvalues in two examples. Figure 2-2a shows a stable monopole background, while Figure 2-2b an unstable one.

The expression (2.83) is not yet ready to be put on a computer due to various divergences. We find it convenient to combine it with the Faddeev-Popov determinant and introduce a UV cutoff first.

Note that (2.75) implies that Tr'log Kqq, = Tr' log ?Cqq, which further implies

Fgauge(q, q') = Fgauge(q', q) . (2.84)

This property is the consequence of CP invariance. It is also not hard to show that

Fgauge(q, q') = Fgauge(-q, -q').

80 ALgauge 0.25- J = Q + 1, Larger Eigenvalue

J=Q -I

0.05

.0 0.5 1.0 1.5 2.0

(a) A stable example with q = -1, q' = 1/2.

0.20ue- J= Q + 1, Larger Eigenvalue J = Q + 1. Smaller Eigenvalue 0.15- J=Q J=Q -I 0.10

0.05-

0 .0 0 - - - - ' - - - ' - - - - ' ----' 1.0 1.5 2. ' -0.05- (b) An unstable example with q = 1/2, q' = 3/2. The instability is indicated in orange.

Figure 2-2: The eigenvalues of K ,(Q) for some example q, q' and low J values as a function of Q. Zero eigenvalues corresponding to pure gauge modes are omitted. Note that the eigenvalues are monotonic in J and Q, hence it suffices to examine the Q = 0 behavior of the lowest J mode for stability. Also note that in both examples IQI > 1 and the two lowest lying J modes have one non-zero eigenvalue, while higher J modes come with two eigenvalues. (The smaller number of eigenvalues corresponds to the reduced size of the matrix K I(Q).)

2.4.4 Combining the subleading terms in the free energy

In the 1/N expansion of the free energy (2.34) there are two terms at O(N?) order, the ghost and the gauge fluctuation contribution. Both contributions involve a sum

81 of N terms; see (2.53) and (2.54). Each term takes the form in (2.53) and (2.83):

FFp(q, q') 2 2,7:(J+ 1) lgfI+1 24g J=IQI (2.85)

Fgauge(q, q') Tr'log Kqqi = d- (2J+ 1)log det'K ,(Q) 2 2 27r J~~j-1qq where we used the notation Q = q-q'. These expressions only determine a meaningful free energy if KJ(Q) only has positive eigenvalues apart from the zero eigenvalue corresponding to pure gauge modes for all J and Q. If there is a negative eigenvalue, there is an instability that will be discussed in Section 2.5.

Firstly, let us consider the large J, Q behavior of the eigenvalues A auge(Q) of K ,(Q), the product of which gives det'K, (Q). For J and Q large we get

J(J+ 1) - Q2 + Q 2 (2.86) Agauge(Q) asymp( - 16 which gives a divergence when integrated over Q and summed over J. We notice however, the appearance of the Faddeev-Popov determinant (2.91). Note that the ghost determinant comes with a negative sign and we end up with the ratio inside the logarithm

SF(q, q') FFp(q, q') + Fgauge(q, q') 1 d (2±)0g det'K ,(Q) (2.87) (2 J +1)log q+... 2 27 J=IQI+ J(J + 1) - Q2 + Q2 where we introduced the notation 6F(q, q') for the sum of the gauge and ghost contributions and the dots stand for the low J modes that do not pair up nicely with the 1 ghosts.12 The two beautifully combine to give a well-behaved result for large J, Q. In Figure 2-3 we show that the eigenvalues of the gauge kernel, Ajgauge(Q) asymptote

12The J = -1 and J = IQI cases has to be treated separately with zeta function regularization. 13Note that according to (2.86) the integrand goes to the constant - ZJ-IQI+1 (2J + 1) log 256. In zeta-function regularization, the Q integral of a constant vanishes, hence constants do not give any contribution.

82 Agauge / Aasymp

1.04' %%J=1/2 - --- J=3/2 1.02 - J=5/2

1.00 J=7/2 - J=9/2 0.98-- J=11/2

0.96 J=13/2

0 5 10 15

Figure 2-3: We plot the ratio of the non-zero eigenvalues Agauge (Q) of the gauge kernel divided by their asymptotic behavior A'ymp(2). We chose q = -1, q' = 1/2 for this example. Because IQI = 3/2 the J = 1/2, 3/2 modes contribute one eigenvalue, while for higher J eigenvalues come in pairs. We used the same colors to plot the pair of eigenvalues for these higher J modes. Because the ghosts give a contribution proportional to Aymp(Q) this plot shows that the low energy modes are the most important in determining the free energy.

to (2.86). Because [AJymp(q)] 2 asymptotes to that of the ghost contribution we see that the main contribution to the free energy is from the low energy modes. At high energies the ghosts cancel the contribution coming from the gauge fluctuations.

To complete the evaluation of the subleading terms we have to introduce a cutoff that makes the integral definite and the sum finite. Because for large J and Q we are probing the UV of the field theory where it should not matter what manifold we are working on, we use a relativistic cutoff

J(J+ 1) - Q2 + 02 < A(A + 1) . (2.88)

With this cutoff the sum and the integral in (2.87) are convergent. Evaluating (2.87) for different A and extrapolating to A -+ oo we obtain our final result for the subleading term in the free energy. An example is given in Figure 2-4. We give a systematic collection of results in Section 2.6.

83 2.4.5 Summary and an example

In this subsection we summarize the key formulae in the evaluation of the S 2 x R free energy. We repeat the 1/Nf expansion of the free energy (2.34)

F[A] = Nf Fo[A] + SF[A] + (1/Nf). (2.89)

Fo[A] is the fermion determinant in the monopole background given by (2.38) and (2.47):

Nr Fo [A] = (Fo(qa) a=1

1) - q2 - (2j + 1) 2 + (2.90) FO(q) = - E (2j + ( 2 q2 )

q(2q - 1)(q+ 2) 6

6F[A] is the sum of the gauge and ghost contributions. The Faddeev-Popov determinant is given by (2.53), while the determinant of gauge fluctuation is obtained by (2.54) and (2.83):

Ne 6F[A] = 6F(qa,qb)6 a,b=1 6F(q, q') FFp(q, q') + Fgauge(q, q') dQ (2.91) FFp(q,q') -1 log [J(J+i1)-2 2 Q2J+1) +Q ] J=IQI

Fgauge(q, q') STr' log Kqq = 1 (2J+ 1)logdet'Kjq,(Q) 2 2J2r J=IQI-1

We combine the subleading terms before evaluating (2.87) numerically.

In this subsection we examine an example in more detail to illustrate some of the steps sketched in the previous subsections. We make the simple choice G = U(2) and

84 qi = 1/2, q2 = -1. The leading contribution is (2.90):

Fo[A] = Fo(qi) + F(q2) = 0.265 + 0.673 = 0.938 , (2.92) where we numerically evaluated (2.90). The list of Fo(q)'s will be given in Table 2.1.

The subleading term is a sum of four terms

5F[A] = 6F(q1 , q1) + &F(q1 , q2 ) + F(q2 , q1) + 6F(q2 ,q2) . (2.93)

We pick 6F(q2, q1) from this sum to illustrate the calculation. The ghost contribution is known analytically (2.91). To calculate the gauge contribution we need to determine the gauge kernel Kj,qq() numerically. For every J and Q we need to construct this matrix. This construction involves an infinite sum, and the procedure is explained around Figure 2-1. In that figure we display the matrix element [Ki,1 (Q)] u for a representative choice of J and Q. We need to know this kernel to 10-12 precision.

Gauge invariance determines the eigenvector with zero eigenvalue of the matrix

Kjq2 '(Q) analytically (2.80). This eigenvector provides a powerful check of the result and whether we indeed achieved the precision advertised. It turns out that matrices Kq,(Q) can be reused in the computation for general gauge groups discussed in Section 2.8.1.

We calculate the eigenvalues of the matrices KjqI (Q) numerically. Figure 2-3 shows a few eigenvalues for our choice of qi, q2 . We drop the zero eigenvalue, and combine the ghost and gauge contributions as explained in Subsection 2.4.4. Finally, we calculate the sum over J and the integral over Q in (2.91) for different UV cutoffs A defined in (2.88). All the divergences have been regularized in previous steps in zeta- function regularization and the free energy is finite as we take 1/A -+ 0. 6F(q2, q1) as a function of 1/A is plotted in Figure 2-4.

The terms that can appear in (2.91) for jqj, jq'j < 2 will be presented in Table 2.2.

85 SF(-1,1/2)

-0575

-0580

-0585

-05901

-0595

-0.6"0 - 1/A 0A6.0 0.1 02 0.3 0.4 0.5

Figure 2-4: We plot the subleading term in the free energy, SF(q, q') for q = -1, q' = 1/2 as a function of the cutoff A. We extrapolate to 1/A -+ 0 by fitting the data points by a second order polynomial. Our results are reliable to 10- precision.

We can find the terms needed in (2.93) from that table:

JF[A] = -0.0383 - 0.574 - 0.574 - 0.194 = -1.38 . (2.94)

We conclude that in U(2) gauge theory the dimension of the GNO monopole operator with charges qi = 1/2, q2 = -1 is:

A = 0.938 Nf - 1.38 + 0 (1/Nf) . (2.95)

We discuss the results for monopole operator dimensions more systematically in Sec- tion 2.6.

2.5 Stability

In the previous section we studied the effective action for the gauge field fluctuations in the presence of a GNO monopole background (2.18). We noticed that the effective action for the W bosons (off-diagonal components of the gauge field) is not always positive-definite (see Figure 2-2), which is to say that certain classical monopole

86 backgrounds are unstable. In this section, we discuss this instability in more detail and characterize which sets of GNO charges yield an unstable background.

The instability of certain GNO backgrounds should come as no surprise, as similar instabilities have been studied in related examples. Indeed, it is well-known that GNO monopoles in Yang-Mills theory in flat space are generically unstable [20,85]. To characterize the unstable configurations, recall that the GNO monopoles organize themselves into classes of topologically-equivalent backgrounds, where each class cor-

responds to an element of the first fundamental group of the gauge group, 7ri(G). In the case G = U(Nc), we have wr,(G) = Z, and there is a discrete topological charge that can be identified with the sum of the GNO charges,

Nc qtop = qa. (2.96) a=1

The monopoles that were shown in [20,85] to be unstable in Yang-Mills theory in flat space were those with Iqa - gbJ > 1 for at least one pair of GNO charges (qa, qb). It is not hard to convince oneself that each topological class with charge qtop contains precisely one stable monopole background.14 All the rest are unstable.

It is important to note that the flat-space instability of monopoles in Yang-Mills theory discussed in [20,85], as well as the instability we noticed in the previous section, occurs only at low frequencies and for W bosons with total angular momentum J =

Iqa - qbl - 1. (This is the lowest value of the total angular momentum provided that Iqa - qab > 1-see (2.65).) That the instability is at low frequency and low angular momentum means that it is a property of the infrared dynamics. Different non-Abelian gauge theories with different IR dynamics can therefore have different sets of stable/unstable GNO configurations. It just so happened that in the case of

Yang-Mills theory in flat space it was all the W bosons with J = Iqa - qbJ - 1 that were unstable. In a different theory, on the other hand, it could be that not all these lowest J modes are unstable. To assess stability, one has to examine the effective

14The stable background has qa = = [qt 0p/Nc] + 1/2 for a ; (q mod Nc) and a = - 1/2 for the other a > (q mod N,).

87 action for the gauge field fluctuations, as well as the fluctuations of other fields, and see whether there are any negative modes.

In the case studied in this chapter, namely the IR fixed point of QCD3 at large Nf, the question of stability is much richer than in pure Yang-Mills theory in flat space. 15

We find that in contrast to the pure Yang-Mills case, in QCD 3 there are multiple stable monopoles per topological class. As we will discuss later, even topologically trivial gauge groups such as SU(2) admit stable monopole backgrounds.

2.5.1 A systematic study of monopole stability in QCD 3

In performing a more systematic study of the instability of monopole backgrounds in large Nf QCD with gauge group U(Nc), let us first note that at leading order in Nf, where one can treat the gauge field as a background and ignore its fluctuations, there are no instabilities, as this is just a theory of non-interacting fermions. To decide whether or not a given GNO background is stable, it is important to consider the subleading 1/Nf effects described by the effective action for the gauge field fluctuations.

In the previous section we have developed a whole machinery needed to study the eigenvalues and eigenfunctions of the quadratic action for the fluctuations of the gauge field around a GNO monopole background (2.18). In brief, each component a,,(x) of the gauge field fluctuation can be expanded in terms of Fourier modes in the Euclidean time direction as well as the monopole vector harmonics UTT' Vg'J,

' 5 One could wonder how many stable monopoles there are in pure Yang-Mills theory on S2 x R. Since Yang-Mills theory in three dimensions is not conformal, one cannot simply borrow the flat space result, so a separate analysis is needed. We find that if the gauge group G = U(Nc), the quadratic action for the a b component of the gauge field fluctuations around the GNO monopole (2.18) has eigenvalues oc + J(J +1) - (q, - qb) 2 (for physical modes) or 0 (for pure gauge modes), where Q is the frequency and J is the total angular momentum. There is an instability at low Q for J = 1q. - qb - 1, so the situation is identical to that of Yang-Mills theory in flat space.

88 and W',JM defined in (2.65):

aI'ab) = J[,JM(Q)USab,JM() + ay'JM(Q)Voab,JM()

+ ab,JM()W'IJM e-T.

(2.97)

For the fluctuation a b(X), we should take Qab =qa - qb. The quadratic action for the coefficients ab,JM takes the form

abJ ab,Jm

. (2.98) aab,JM(Q) Kj b'y a,b=1 JM

aab J(Q),aW ()

The matrix Kj, (Q) can be computed tediously by following all the steps presented in the previous section. As demonstrated in the example presented in Figure 2-2, the

eigenvalues of Kjq (Q) increase with both Q and J. To check whether the action for a"b is positive-definite, it is therefore sufficient to calculate Kj.'q (Q) for Q = 0 and the lowest attainable value of J = Jab. If IQabI < 1, this lowest value is Jab = Pad;

if IQabI 1, it is Jab = IQabI - 1.

We have computed numerically KJab (0) for all possible values of qa and qg in the range -10 < qa, qb < 10. From our numerics, we find that it is only the modes with Jab = IQabI - 1 and IQadi > 1 that are sometimes unstable. We have plotted

Kaqb (0) in Figure 2-5 as a function of qa and qb. In Figure 2-6 we have indicated the stable region in black and the unstable region in orange. A GNO monopole

labeled by charges {qi, q2, - -- , qNc} is stable if every pair of charges lies in the stable region displayed on the plot.

As can be seen from Figure 2-6, we find two stable regions in the qa-qb plane. The

89 SAgauge

-10-5 qa

Figure 2-5: The lowest eigenvalue A = Kf' 1 (O) of the a component of the gauge field fluctuations around the GNO monopole background (2.18). We have marked explicitly the plane z = 0. The region where this eigenvalue dips below zero corresponds to an instability of aab. If this eigenvalue is positive, then the action for aab is positive-definite.

first such region is where

Iqa - qbI <1, (2.99) which, as mentioned above, is the same stability condition as in Yang-Mills theory in flat space. For these values of the charges there is no instability because the problematic mode with angular momentum J = Iqa - qa| - 1 is simply absent. The second stable region is new and unexpected. It occurs where qa and q6 are comparable in magnitude and of opposite sign. Asymptotically, at large values of qa and qb we can estimate that the second stable region is where

-tan73 0 < $ < - tan170 . (2.100) qa

The existence of the second stable region implies that there are several stable GNO monopoles per topological class. Indeed, the first stable region alone implies that each topological class contains at least one stable GNO monopole, and hence

90 qb 0....e. :0* * * **t**o***** **.* ** -0 * 0 Fur26:Amar tftestabity f GN m 0e00oooe. rfedrs,wh*hgs{ ,..***.t..... tsste there.t.at a0l We s.eoeoe 0o.0.e = esq - s e.heefeetonfomai*txtTh n e *t repndas ngt e f wit(, J )o =0e 0r -. K T p() filedeeakercesrpd aes5 (0*f q) forwh ee monopetheset aseOdsteregon ooogi cassmn*F t e tee 0 U(2)e gag e t the e tGoopoesaredexeby par f GNo 0g (, q ae harge.= + , te .. 0 .nubesabeGN npeshtr topologically..tss . .t0oo .0. 0let rg ( )

o**e...... ,...... e...... 00 * 00****0*0S * 00 0 @ *, ************** 0 - *|Q*|** 1,* tae* .*** a** . 1, Unstable .. o Oa O .1. ta

*....e oo0e .. e0 *e..e...... ee....o.e*

fat ero -ga ass n*er ta an 0i0nite mer stabeGN pobe th.eopeswth : 0e0 0 x ie wh sp n oo0 e.~ eV.. *e- 0a O COSS *@eeeee 0 0000 0

th*oetvleO f J is gOO@Sg|. ... oo~~eo~

filled~~~~~~~corspnblc to vaue 0ice of (0 q0 fo whc thr is no suc negative

th loes au of J, iStab 0 I.l 1 ntbe ~b ,Sal

ie. 2.2effectve actoong,o * h a neg m Jevauti-. = The oeuand

thFistne of6 Asmar econd th stableit reio mean thtmhrecnpbes mor thn onostale

wth chNge Imonopo.es are indexedby arvddal pairofGOcags (q, q) andrtespn toogca chr isd black cicls and +tr is usa inintehuerwe stablne GNO monopoles hathae

Figur2-6:aitooeogicytivifo tAt summar because peel oftea onopoiveoes siity rgo withGN.O moops.tiAllJ = 0b li in the regio GNt(210)and monopole factlerylak topologc clssn thi thlesoy conain an infinhichthe ube nof stablnegNOiv monoes,orafid)blc bhecausferecee irces, and sbitythe onoole unalse wihnixd otwisea. file We onlacdote cles Qab wistha discus~, lo sint45e intedte- ones phlnes anwila necessaril lie e in ~ -1 hl o the regio (210osmttialntlres

Havingle finis the setabiloityaalyclss. woeaowmplet U2)ue thesoof Shec-

tion2.2 Iln thet follong, essaalyzeiheMinkosk time00 evoluto.c Int quantumq

91 theory all field modes fluctuate about the monopole background. If the background is stable, at large Nf the wave functional is supported on gauge configurations close to the background, as the typical size of the fluctuations is 0(1/ Nf). Conversely, if the background is unstable, the unstable mode grows exponentially. The wave functional that started out as having delta function support on the background spreads, and ends up with broad support. In this case, it is more useful to decompose the wave functional in terms of energy eigenstates. The spread in energy will be wide, comparable to the potential energy difference to a nearby local minimum. 16 The Euclidean path integral will be dominated by the lowest energy eigenstate at the bottom of a nearby local minimum, as a result we get the free energy of another monopole. Note that there is no topological obstruction to this scenario, as every topological sector has at least one stable monopole background in it.

One could still consider the disorder operators corresponding to unstable monopole backgrounds. The decomposition of the corresponding state into energy eigenstates translates into this disorder operator being a sum of operators that have a big range of scaling dimensions. In correlation functions, at long distances such a disorder operator would behave as the operator with the lowest scaling dimension from this sum, i.e. another monopole operator in the same topological sector (or the identity).

2.6 Monopole operator dimensions

In this section we collect the results for monopole operator dimensions in U(Nc) gauge theories. We first exhibit the QED case N, = 1 explicitly in Subsection 2.6.1, and then we present the results for N, > 2 in Subsection 2.6.2. For details on how to obtain the results collected in this section we refer the reader to the example in section 2.4.5.

16For very late times tunneling has to be taken into account. We neglect tunneling effects in this discussion.

92 2.6.1 Monopole operator dimensions in QED

In U(1) gauge theory all monopole backgrounds are stable because the monopole charge is a topological quantum number. To obtain the scaling dimensions of the corresponding monopole operators, one should simply set N, = 1 in the formulae summarized in Section 2.4.5. See Table 2.1 for the scaling dimensions Aq of the monopole operators with |qj < 5/2.

|qj Aq 0 0 1/2 0.265 Nf - 0.0383 + O(1/Nf)

1 0.673 N1 - 0.194 + O(1/Nf) 3/2 1.186 Nf - 0.422 + O(1/Nf) 2 1.786 Nf - 0.706 + O(1/N) 5/2 2.462 Nf - 1.04 + 0(1/Nf)

Table 2.1: Monopole operator dimension Aq for monopole charge q in U(1) gauge theory.

Part of these results are not new: the O(Nf) contributions to the scaling dimensions given in Table 2.1 were first obtained in [13], while the subleading correction to the dimension of the monopole operators with jqj = 1/2 was also obtained in [21].

2.6.2 Monopole operator dimensions in U(Nc) QCD

As mentioned before, in U(Nc) gauge theory not all GNO backgrounds A specified by the charges {qi, q2 , ... , qNcj are stable. Stability is a dynamical question, and we presented the criterion for stability in Section 2.5-see Figure 2-6. For the stable backgrounds, we can compute the scaling dimension A = F[A] of the corresponding operators using the formulae (2.89), (2.90), and (2.91), which we repeat here for the reader's convenience:

F[A] = Nf Fo[A] + 6F[A] + 0(1/N), (2.101)

93 where

Ne Nr Fo [A] = S Fo(qa) , 6F[A] = 5 6F(qa, qb)- (2.102) a=1 a,b=1

The numerical values of Fo(q) are the same as the coefficients of Nf in the expressions for Aq given in Table 2.1. For 6F(q, q'), see Table 2.2. Note that not all the entries in Table 2.2 are numerical; some of them are instead orange dots, which indicate an instability. According to the recipe of Section 2.5, if such a term features in the second sum in (2.102), the corresponding monopole background is unstable and does not correspond to a monopole operator with well-defined scaling dimension. Using the values listed in Tables 2.1 and 2.2 one can determine the dimension of any monopole operator with GNO charges obeying Iqal 5 2. For higher GNO charges one has to construct larger tables. Note that the subleading terms in Table 2.1 are equal to the diagonal entries in Table 2.2. Note also that Table 2.2 has a reflection symmetry about the diagonal

F(q, q') = F(q', q) , (2.103) as a consequence of CP symmetry (2.84), as well as a reflection symmetry about origin, 6F(q, q') = 6F(-q, -q'). Let us consider a few examples:

" If we take {qi, q2, , qN {- /2, 0,... , 0, we have

Fo[A] = FO(1/2) = 0.265 ,

6F[A] = 6F(1/2, 1/2) + 2(Nc - 1) 6F(1/2, 0) , (2.104)

A = 0.265 Nf - 0.0383 - (N, - 1) 0.516 + O(1/Nf).

This monopole operator has the smallest dimension among all.

" If we instead took {qi, q2 , ... , qNc , 0,... , 0}, we would find that there is no monopole operator with this GNO charge and well-defined scaling dimension,

94 -2 -3/2 -1 -1/2 0 1/2 I 3/2 2 q

2 -1.90 -1.63 -1.52 -2.16 * -0.857 -0.706 3/2 -1.63 -1.26 -1.04 -1.05 * 0 -0.592 -0.422 -0.857 1 -1.52 -1.04 -0.730 -0.574 0 -0.386 -0.194 -0.592 0 1/2 -2.16 -1.05 -0.574 -0.338 -0.258 -0.0383 -0.386 * 0 0 0 -0.258 0 -0.258 * * -1/2 * -0.386 -0.0383 -0.258 -0.338 -0.574 -1.05 -2.16 -1 9 -0.592 -0.194 -0.386 * -0.574 -0.730 -1.04 -1.52 -3/2 -0.857 -0.422 -0.592 0* -1.05 -1.04 -1.26 -1.63 -2 -0.706 -0.857 -2.16 -1.52 -1.63 -1.90

Table 2.2: 6F(q, q') for various values of q and q'. The orange dots mean that the corresponding W boson is unstable.

because F(1, 0) = ..

Finally, if we consider {qi, q2 ,... , qN} = {1/2, 0, ... ,0, -1/2}, the corresponding dimension is

Fo[A] = 2Fo(1/2) = 0.530 , JF[A] = 23F(1/2, 1/2) + 26F(1/2, -1/2) + 4(Nc - 2) 6F(1/2, 0) , (2.105)

A = 0.530Nf - 0.753 - (Nc - 2) 1.06 + O(1/Nf).

2.7 Other quantum numbers of monopole opera-

tors

In the previous sections we calculated the energy of the ground state 17 on S 2 x R localized around the GNO saddle (2.2). This computation used the equivalence

17As explained before, the use of the term "ground state" is not necessarily appropriate. We are talking about the lowest energy states whose wavefunction at large N is highly peaked around the saddle (2.2).

95 between the ground state energy and the thermal free energy at zero temperature, and as such does not tell us much about the properties of the ground state, or equivalently, about the quantum numbers of the operator corresponding to it. In this section we fill this gap. Of course, the results presented here will only be valid for the GNO saddles that do not have any unstable directions. The states on S 2 must transform in representations of the conformal group and of the flavor symmetry group. The flavor symmetry group of a theory of Nf fermions and gauge group U(Nc) is SU(Nf). The conformal group on S2 x R is SO(4, 1), regardless of whether the R coordinate is Lorentzian or Euclidean time. We choose to work in Euclidean signature, even though time evolution is a unitary transformation on the Hilbert space of states only in Lorentzian signature. We expect the bare monopole operators that we studied in the previous sections to be conformal primaries. We will now determine their spin and SU(Nf) quantum numbers.

2.7.1 Quantum numbers of monopole operators in QED

Before studying the quantum numbers of the GNO monopoles in QCD, it is instructive to study the same question in the QED case, N, = 1, where the monopole operators are labeled by the charge q E Z/2 and heuristically create the background (2.1). The quantum numbers of the monopole operators in QED were calculated in [13] for IqI = 1/2. In this section we present the quantum numbers for arbitrary q. 18 The result we will find is that the monopole operator of charge q transforms as a Lorentz scalar, and as an irreducible representation of SU(Nf) given by the rectangular Young diagram with Nf/2 rows and 2jqI columns:

Nf/2 { . (2.106)

21qj

We now explain the derivation of this result. Because at large Nf the fluctuations of the gauge field around the background

18Our work corrects a slight error in the analysis of [13] for general q.

96 (2.1) are suppressed, we should start by canonically quantizing the theory of free fermions in this background, and worry later about the effects of having a dynamical gauge field. The fermionic modes can be found by solving the Dirac equation

(i + A)0 = 0 . (2.107)

To solve this equation, one can begin by expanding V) in terms of the spinor harmonics Sq,jm(ft) and T,jm(ft), and Fourier modes in time, as in Section 2.4.1:

( t)= S S "CjSq,jm() +c "TTq,m(i) e~W (2.108) j=ql-. m=-j with arbitrary w to be determined by solving the Dirac equation. When j ; I + 1/2, the Dirac equation has two solutions (for every j, m and flavor a) with energy iwj = ±Eq,j, where

Eq, = (j + 1/2)2 - q2 . (2.109)

We can denote by c±)athe linear combinations of (c and cjm corresponding to energy ±Eq,j. In the quantum theory, the c a become anti-commuting annihilation operators for the corresponding modes. Generically, there are 2j + 1 such operators for both choices of sign and every a and j.

The case j = jq| - 1/2 is special because the spinor monopole harmonics Tq,jm are absent, and the Dirac equation implies that the Sq,jm modes have energy iwj = 0.

In the quantum theory, the coefficients c(s)a which in this case we denote by cjm for brevity, become annihilation operators for these zero-energy modes. For each flavor, there are 2j + 1 = 2 IqI zero-energy modes transforming in the spin IqI - 1/2 representation of the SU(2)rot rotation group. There are a total of 2 IqI Nf zero-energy modes when we consider all of the flavors.

If there had been no zero-energy modes, the situation would have been quite simple. The theory would have had a unique rotationally-invariant vacuum1Q),

97 corresponding to a Dirac sea filled with particles with negative energy and containing no particles with positive energy. In other words, this vacuum should be annihilated by all annihilation operators for positive energy modes and by all creation operators for all negative energy modes:

cMa|) = cj;~QtIG) = 0, i> |ql - . (2.110)

One could build up the states in the Hilbert space by adding particles with positive energy or removing particles with negative energy.

The existence of 2 IqI Nf zero-energy modes, however, means that the theory of

free fermions in the monopole background (2.1) has, in fact, not just one, but 2 2|qlNf degenerate ground states that satisfy the condition (2.110). Let us call the Hilbert space spanned by these ground states g. One of the states in g is the Fock vacuum I), defined by (2.110) together with the requirement that it should be annihilated by the annihilation operators cjm with j = |q. - :

1 CimIQ) = 0 , j = q - -. (2.111) 2

The other linearly independent states in g can be obtained by acting with any number of creation operators ct (with j = I- ), on the rotationally-invariant Fock vacuum IQ). The full Hilbert space of the theory R is obtained by acting on the states of g with any number of creation operators cm for positive-energy modes and annihilation operators c m)c for negative energy modes (j > IqI - 1).

This description of the Hilbert space W is correct assuming the gauge field is a background field. For us, however, the gauge field is dynamical and its effect is to remove some of the states in ?- (and consequently some of the ground states in !). In the path integral language the gauge field appears in the action only as a Lagrange multiplier that imposes the constraint j(x) = 0. What we mean by this constraint

98 is that all correlation functions of the current should vanish in the full theory

0 = (j1(X1) j12(X2) ... j1n(Xn)) full theory - (2.112)

This equation looks perplexing at first sight, as we spent most of the chapter determining the gauge kernel K""(x, y) = -(j(X)j"(y))conn. The resolution of this puzzle is that K"" is determined by the current-current correlator in the free fermion theory, where the gauge field is treated as a background. In canonical quantization language we have a constrained system; the canonical momenta conjugate to a"(x) vanish identically, and we should not define any oscillator modes in the gauge sector. Instead, in analogy with the Gupta-Bleuler prescription, we should require that the positive and zero energy part, j, of the current operator j,(x) =: V51t(x)-y,,04(x) : annihilates all physical states Ix):

i(+) (X)Ix) = 0. (2.113)

This requirement reduces the Hilbert space 71 introduced above to a smaller one

'Wphys, and the 22lqlNf-dimensional space of ground states g to 9 phys- Understaning the quantum numbers of the monopole operators means understanding what gphys is, and how the SU(Nf) flavor symmetry and the rotation group SU(2)rot acts on it. The (2.113) condition guarantees that (2.112) is satisfied. Of course, the expectation value has to be taken between states in 'Wphys-1 9 Using (2.108) and the definition of j"(x), one can obtain an explicit expression for the current operator in terms of oscillators:

+() cc, - CJmm) S,(,jm(i)'ySq,jm(n) + (non-zero modes), (2.114) where we wrote down explicitly only the contributions from the oscillators corresponding to the zero-energy modes with j = IqI - 1. (From here on, it should be

19 1t is easy to check that imposing only the strictly positive energy part of j1 to annihilate physical states is not sufficient to ensure (2.112). We have to require the stronger condition (2.113).

99 understood that j = IqI - 1/2 and that m runs over -j through j unless otherwise specified.) The quantity C appearing in (2.114) is a c-number corresponding to a possible normal-ordering ambiguity when taking the product of cg9 with c,'m,. Such a normal ordering ambiguity is present only when j = j', m = m', and a = a', because any given cjm anti-commutes with all the other fermionic creation and annihilation operators except for cjm.

The normal ordering constant is determined by CP-invariance to be C = Nf/2. Indeed, creating a zero-energy mode is related by CP to destroying a zero-energy mode, and if we want to quantize the theory in a CP-invariant way, we better treat the creation and annihilation operators for zero-energy modes on equal footing. Doing so means that instead of the expression in the parenthesis in (2.114) we should have written

c-ic.c ,,-c Cmc (2.115) -J 3m jm' im)

Anti-commuting the two factors in the second term, summing over a, and comparing with (2.114) yields C = Nf/2.

Using the explicit expressions for the spinor monopole harmonics, it is straightforward to find the explicit position dependence of the zero-mode contribution to the current operator (2.114). For instance, when q = 1/2, the zero modes have spin j = 0. In the North chart, the expression for the only spinor harmonic with j = 0 is

1Cos 0 S1 =o-2 (() (2.116) e=1 sin

After plugging this expression into (2.114), a little algebra shows that the zero modes do not contribute to jo(x) and jO(x), while the charge density p(x) = jT (x) is

q = : p(x) = 7( - Nf/2) + (non-zero modes), (2.117)

100 where K c'ce is the operator that counts the total number of excited zero-energy modes. More generally, the operator that counts the total number of fermions in the zero-mode sector is

S cm m (2.118) m

That the charge density p(x) annihilates the states means that !phys consists only of the states in 9 for which k = Nf/2.

Similarly, when q = 1, the zero modes have spin j = 1/2. In the North chart, the spinor harmonics are

1 ( e sin 0 1 + cos 9219 S1 11(i) = , S1 1 1 (L) = I. (2.119) '22 l/8e2 (1 - cos0) '2 2 e' sin )

Again, using these expressions one can show that the zero modes do not give any contributions to jo(x) and jO(x), while the charge density is

q=1: p(x)= 1 Y()*( - N) - 1 Y-m (h)* Sm + (non-zero modes).

(2.120)

Here, K is the fermion number operator in the zero-mode sector defined in (2.118) and Sm is the total spin of the zero-energy modes organized as states in the spin-1 angular momentum basis:

S, = -ct C1 1 = SX - iSy ,

S= c ilc' - c = .1 -v 2Sz, (2.121) 72 2 2 22 2 2' 2

S_ 1 = cot icia = X - is, . 2' 2 2 2

The requirement that the charge density should vanish for any physical states implies

that 9 phys consists only of the states in 9 that satisfy K = Nf and S= 0.

101 The expressions (2.117)-(2.120), as well as the characterization of !phys as a sub- space of g, generalize to arbitrary q in the following way. From the creation and annihilation operators in the zero-energy mode sector one can construct SU(Nf)- singlet operators of the form

o = "mM't3 ' (2.122) m,m'

where mm' is a 21 q x 2 1qI Hermitian matrix. There are (2 |qI) 2 linearly independent such matrices, and hence (2 |qj) 2 linearly independent operators 0, which organize themselves according to irreducible representations of the rotation group SU(2 )rt. The representations that appear are precisely those in the product of two spin-j irreps of SU(2), namely all the ones whose spin is between f = 0 and f = 2j. If we denote the spin-f operators by Ojm,, then OoO is proportional to the total fermion number K, 61m is proportional to the total spin Sm, and so on. The expression for the charge density operator in (2.117) and (2.120) then gener- alizes to

21q|-1 p(x) = ( - Iqj Nf) + S 5 Yjm- ()(O'm, + (non-zero modes), (2.123)

where Ym (h) are the usual spherical harmonics, and the operators O9 m, come with specific normalizations. The precise normalization of Oem, is not essential for the argument we are about to make. That the charge density annihilates all the states

means that out of the 2 2JqlNf degenerate ground states in g we should only consider the ones where

91X) = IqI Nf IX) and Ofm, IX) = 0, for t > 1 and all me. (2.124)

The first requirement in (2.124) means that gphys contains only states of the form

IqjNf

3c4 iC|). (2.125) i=1

102 where we act with precisely Iqi Nf fermion creation operators (out of the total of 2 1qI Nyf) on the Fock vacuum I2) defined in (2.110)-(2.111). The second requirement in (2.124) requires more thought. It can be understood most simply by enlarging the

SU(2)rot symmetry to an SU(2 qi) symmetry, where the (21ql) 2 - 1 operators 61m, generate SU(2 jql) and the states ct IQ) transform in the fundamental representation. The second condition in (2.124) means that all operators (2.122) where Omm', is a traceless Hermitian matrix should annihilate the physical states IX)-20 This is just the requirement that, infinitesimally, Ix) should be invariant under SU(2jql) transformations. In other words, 9phys consists of the states of g that are of the form (2.125) and, in addition, are also SU(2 IqI) singlets.

Each fermionic creation operator transforms as a fundamental of SU(N), so we are looking for singlets under SU(2 IqJ) in the product of IqI Nf fundamentals of SU(Nf). There is a further wrinkle, however. As we are considering anti-commuting creation operators, the states must be totally antisymmetric.

To count how many such states there are and see how they transform under SU(Nf), it is convenient to introduce a bigger group that contains both SU(Nf) and SU(2 qJ): if we make a list of all the zero-energy mode creation operators ciM, we can consider SU(2 IqI Nf) transformations under which cgi form a fundamental vector. Similarly, the annihilation operators cym transform in the anti-fundamental representation of the same SU(2 Iq Nf) group.

The benefit of considering this larger group is that constructing totally antisymmetric states is simple. The states of Gphys are constructed by decomposing the antisymmetric products of jqI N1 fundamentals of SU(2 IqI Nf) under SU(2 IqI) x SU(Nf) and selecting those which are singlets under the SU(2 IqI) factor. We therefore need to identify all the SU(2 IqI) singlets in the decomposition of the rank-JqI N1 totally

20Note that the non-traceless part of j is included in the operator 9.

103 antisymmetric representation of SU(2 IqI Nf),

IIq| Nf (2.126)

under

SU(2 |q| Nf) D SU(2 Iq|) x SU(Nf). (2.127)

Such a group theory exercise is common in atomic physics where one needs to construct a totally anti-symmetric wavefunction for several identical particles with given angular momentum and spin. In general, the rank r anti-symmetric representation of SU(NM) decomposes under SU(N) x SU(M) as the sum (see, for example, [86])

ED (,/) (2.128) V over all possible irreps with Young diagrams v with a total of r boxes (whose con- jugates are denoted by I), such that v has at most N rows and F has at most M rows. Each ordered pair (v, F) appears precisely once in this decomposition. For our problem, we have

Jq+ Nf 2 -- -2, (2.129)

Nf /2 21ql where on the RHS the first Young diagram of any given pair corresponds to SU(2 jqJ) and the second to SU(Nf). Of this infinite sum, we want to pick out the terms for which the first factor is an SU(2 JqJ) singlet. Only the diagram explicitly exhibited in (2.129) has this property. Consequently, the states of gphys transform as the SU(Nf)

104 Nf = 2 N 1 = 4 Nf = 6

q = 1/2 D-(2) (6) (20) -..

q = 1 ZD| |(3) (20) (175) ...

q = 3/2 zp p(4) B -(50) P7(980) ...

Table 2.3: The transformation properties of the first few (bare) monopole operators under the flavor SU(Nf) global symmetry of QED3 with Nf flavors. The dimensions of the irreps were calculated using (2.131). All these monopole operators are singlets under spatial rotations. irrep whose Young diagram is a rectangle with Nf/2 rows and 2 IqI columns:

Nf/2{IH (2.130)

(Recall that we should only consider an even number of flavors in order to avoid a parity anomaly.) The dimension of this irrep is

2 2 ((ii + N)!) (2.131) j= ((i + Nf /2)!)

See Table 2.3 for a few examples.

This discussion also shows that the physical ground states gphys are singlets under the rotation group SU(2)rot. Indeed, SU(2)rot can be embedded as a subgroup of SU(2 Iqi), and the states that are SU(2 IqI) singlets must also be SU(2)rot singlets.

We have thus found the quantum numbers of the (bare) monopole operators of charge q in QED with Nf flavors. Their topological charge is q, and their conformal dimensions were computed in the previous section at large N1 (see Table 2.1). In this section we determined that these operators transform as the irrep (2.130) (see

105 also Table 2.3) under the flavor SU(Nf) symmetry and as singlets under the SU(2)rot group of spatial rotations.

2.7.2 Quantum numbers of monopole operators in U(Nc) QCD

The careful analysis of the previous section can be generalized to the more complicated GNO monopole operators in U(Nc) QCD with Nf flavors in the fundamental representation. As in the QED case, when Nf is large we can start by quantizing the theory of free fermions in the GNO background (2.2), and then we can take into account the effects of having a dynamical gauge field. The result is that the monopole operators now transform in an irrep of the SU(Nf) flavor symmetry corresponding to a Young diagram with Nf/2 rows and 2 EI q,Iqa boxes in each row,

Nf /2{7§j1(2.132)

2 EZaqaI where {qa} is the set of GNO charges. In addition, the monopole operators are singlets under the SU(2)rot rotation group. We obtained the same result in the Abelian case, but now Iqj is replaced with Ea Iqa. The rest of this section provides the derivation of these quantum numbers in the non-Abelian case.

In the non-Abelian case, the fermions 4#a,a carry a color index a = 1,..., N. in addition to the flavor index a. In the GNO background (2.2), the action for ba'" is the same as that of a QED fermion in an Abelian monopole background (2.1) with charge q = qa. We therefore have 2 1qal Nf zero energy modes for each value of a, with some corresponding creation operators c and annihilation operators c a (here, ja = Iqal - 1/2 and ma ranges from -ja through ja). In addition to the zero energy modes, we also have positive and negative energy modes. As in the Abelian case, we can define the vacuum in the non-zero mode sector by requiring that all positive-energy annihilation operators and all negative-energy creation operators annihilate this vacuum. These conditions leave 2 2Nf ZalqaI degenerate ground states

(as appropriate for having 2Nf E qaI fermionic oscillators with zero energy) that

106 span a Hilbert space g. This Hilbert space has a Fock vacuum IQ), which by definition is annihilated by all C'. The other linearly independent states in g can be constructed by acting with any number of fermionic creation operators c',, on The analysis so far did not take into account the dynamical gauge field, which, as in the QED case, acts as a Lagrange multiplier that imposes the constraint that the positive and zero energy part of the current, jba+)(x), should annihilate all physical states. This constraint reduces g to a smaller Hilbert space 9 phys, whose transformation properties under the flavor group SU(Nf) and the rotation group SU(2)rt we need to understand, as each state in gphys corresponds to a monopole operator.

The creation operators transform as fundamentals under SU(Nf). In the Abelian case, we saw that it was useful to consider SU(2 IqI) acting on the creation operators. The condition of vanishing current in that case translated into two constraints that select the states of 9 phys from g. The first constraint required the number of creation operators be equal to JqI Nf, and the second required that the states in 9phys be singlets under this SU(2 Iql). Similarly, in the case of U(Nc) we can consider the action of SU(2 Ea IqaI) on the set of all creation operators of a fixed flavor. Note that this group mixes fermions of different color and spin. We will argue that the condition of vanishing current in this case translates into the following two constraints: each state

in 9 phys is created by acting with Ea qaI Nf creation operators on the Fock vacuum, and it should transform as a singlet under SU(2 Ea lqa ). The problem of finding physical states is then just the same group theory problem we solved in the Abelian case with IqI replaced by Ea Iq,I. As in the Abelian case, each diagonal component ja1t imposes the constraint that the number of creation operators of color a equal JqaI Nf and that the physical

states IX) are invariant under SU(2 lqal). It will be more useful, however, to consider the overall constraint coming from Eajjli, which says that the total number of generators of all colors is Ea IqaI Nf. The other constraints coming from jia( imply that the physical states are invariant under U(1)Nc- 1 X Ha SU(2 Iqa 1).21 We will

2 1The factors of U(1) come from the separate particle number constraints for each color, with one removed corresponding to the total particle number.

107 now argue that invariance of the physical states under this latter group enhances to invariance under a full SU(2 aqaI) when one also examines the off-diagonal generators j(+) with b = a.

For simplicity we start by considering N, = 2, where the discussion above implies that the conditions coming from j(+) and j require invariance under U(1) x

SU(2 Iqil) x SU(2 jq2j). Recall that, in general, U(1) x SU(M) x SU(N) is a maximal subgroup of SU(M + N). Therefore, if a state is a singlet under U(1) x SU(M) x SU(N) and is annihilated by any other generator of SU(M + N), it is automatically a singlet under the whole SU(M + N). For the N, = 2 case, the off diagonal current jl, provides at least one additional condition independent from the ones coming from j and j , which required that the physical states be annihilated by the generators of U(1) x SU(2 1q 1) x SU(2 1q2 1). As such, the physical states must be singlets under the full SU(2 Iqil + 2 1q2 I). For the general case, U(1)Nc-1 X Ha SU(2 IqaI) is not quite a maximal subgroup of SU(2 Ea Iqa|). Instead, for each pair of indices (a, b) with a =A b, the subgroup

U(1) x SU(2 IqaI) x SU(2 IqbI) C SU(2 qaI + 2 Iqb) is maximal.2 2 Repeating the argument above from the N, = 2 case, the off-diagonal current j>, b / a, is non- vanishing, so it provides an additional constraint on the physical states beyond the invariance under U(1) x SU(2 qa|) x SU(2 IqbI) required by (,/ and '(. The physical states Ix) are therefore singlets under SU(2 qaI + 2 1qb1) for every pair (a, b). Iterating this procedure for all pairs of color indices leads to the singlet condition under the full SU(2 Ea IqaI).

Putting everything together, the states in 9 phys are the SU(2 E.Iqa I) singlets in the decomposition of the totally anti-symmetric tensor of SU(2N Za IaI) with

Nf Ea IqaI indices under SU(Nf) x SU(2 Za IqaI) It follows that the states of 9phys transform under SU(N) as the irrep (2.132). The corresponding monopole operators are singlets of SU(2)rot because SU(2)rot is embedded in SU(2 Za IqaI) and we selected only the SU(2 Ea IqaI) singlets.

22 We only need to consider pairs of indices, (a, b), where qa # 0 and qb = 0. For vanishing charge the SU(2 jqi) factor is not present.

108 A generalization of these results to more complicated groups and/or representations of the fermion flavors is left for future work.

2.8 Monopoles in general gauge theories

In this section we generalize the computation of the dimension of GNO monopole operators, as well as the stability analysis included in Section 2.5, to arbitrary gauge groups. We will see that the computation proceeds analogously to the U(Nc) case, and, moreover, no new ingredients are needed. In particular, to complete the study of gauge field fluctuations around a monopole background in QCD 3 with gauge group G, all that is needed are the properties of the kernel Cqq, (x, x') analyzed in the U(Nc) case. In gauge theory with gauge group G, the most general monopole background centered at the origin is

A = H(±1 - cos O)dp, (2.133) where H is an element of the Lie algebra of G, and the two signs correspond to the North and South charts. As explained in [19], each such configuration is gauge equivalent to one where H points along the Cartan, namely

i=1 where r is the rank of the gauge group and hi are the Cartan generators. There is still some remaining gauge redundancy in (2.134), as the Weyl group acts non-trivially on the qj, and we should hence consider configurations of the form (2.134) only as equivalence classes under the action of the Weyl group. The Dirac quantization condition is [19]

IW)R = e47ri H R e R (2.135)

109 for any state Iw)R in any representation R of G. Here, w is the weight vector corre-

sponding to Iw)R (such that hjlw)R = wilw)R), and so it belongs to the weight lattice of G. The quantization condition (2.135) implies that q - w E Z/2 for any w. The set of all q with this property form themselves a lattice that can be identified with a rescaled version of the weight lattice of a dual group d. The group d is referred to as the GNO dual (or Langlands dual) of G.

2.8.1 Anomalous dimensions for general groups

In general, the fermions transform in some representation R of G. Let us denote the states of this representation by 1w), suppressing from now on the index R that we introduced above. In terms of these states, the fermions can be decomposed as

(x) = ZbW)(x) jW, (2.136) wER

with 4w(x) being anti-commuting spinor coefficients. To avoid clutter, the flavor and spinor indices are suppressed. Like in the U(Nc) case, having Nf flavors of fermions has the only effect of multiplying the gauge field effective action by a factor of Nf.

Similarly, the gauge field background and fluctuations can be decomposed in terms of the states in the adjoint representation of G. Some of the components point along the Cartan generators hi, and some along the root directions E.:

T A = q - hAU(l, a = Z a hi + E a. E.. (2.137) i=1 aEroots

The hi and EQ are defined such that they satisfy the standard commutation relations

[hj, hy] = 0 [hi, Ea] = ai Ea. (2.138)

As in the U(Nc) case, the large Nf expansion is equivalent to an expansion in the gauge field fluctuations a,. To leading order in Nf we can thus treat the gauge field

110 as a background and write the action for the fermions in the background A as

So [A; V), d3X~fg- E 7/4 iV + q w qU(1)>)bw. (2.139) I] = W

Since this action does not mix fermions with different weights w, the Green's function takes the form

(Vipw(x) ),(x')) = 6wwGq.w(x, x') , (2.140) where Gq.w(x, x') is as defined in (2.23).

The corrections to (2.139) come from the coupling between the fermions and the gauge fluctuations, which is

Sint [a, Of, 0] = jd3x V9 E (w1w',Iw)jww,,, , (2.141)

In complete analogy with the discussion of Section 2.3.2, we can obtain the effective action for the gauge field fluctuations by integrating out the fermions. As in (2.22), it will be useful to define the kernel

V Ku"vv,., (X, Y) UVIU Wjsw(Y))conn I (2.142) and rewrite this kernel in terms of the single fermion Green's function using (2.140). We have

Ku",w,(x, y) = Nf 6v.6u, C",L,q.u(x, y) , (2.143) with K .qb (x, y) defined in (2.25). Finally, using (2.141), we can write the effective

111 action for the gauge fluctuations as

Seff[a] = Nf tr log(i ) +S([a] + - --

S([2] = d 3xd 3 y Vfg(x) /g (y) 1:(w'l a,(x) Iw) KM",.(x, y) (w Iav (y) Iw'). w,1j! (2.144)

This expression is the analog of (2.26) from the U(Nc) case. We can be more explicit and decompose the gauge fluctuations in terms of internal directions as in (2.137). Using the fact that the states 1w) are orthonormal, we can write

(w'|a"(x)Iw) = all -w Jw + E a' (w'|Ec w),) (2.145) aEroots where the short-hand notation a w = ai wi involves only a sum over the Cartan components. Combining (2.145) with (2.144), we see that the cross terms between the fluctuations along the Cartan and root directions are proportional to (wIEIw), which vanishes, so these two sets of fluctuations decouple from each other. Furthermore, the gauge field fluctuations in different root directions do not mix either, because

if # a or &' w - a, (w'|EaIw)(wIEa3 Iw') = I I(w - aE.Iw)12 if 3 = -a and w' = w - a. (2.146)

We finally obtain:

Sf [a] = +d dy yx g(x) g(y) aj,g(x) (wwim "wq.w(x, y) aj,,(y) 2 1 -~~i,j=1 ( -

+ 1 aa,p(x) |(w-a|Ea lw)1 2 ( X y) a.,g(y). Y aEroots \ qw~.wc)(7Y (2.147)

Note that this expression can be computed using only the kernel KI"A" (x, y) analyzed

112 in Section 2.4.

Having found the effective action for the gauge field, we can now compute the free energy on S 2 x R by evaluating the path integral on this space in the saddle point approximation. Let us first examine the fermion determinant term in (2.144), as this term gives the leading contribution to the free energy. As can be seen from (2.139), So[A; V), 44] decomposes into a sum where each fermion 0,/- is only coupled to an Abelian monopole background of charge q -w. In analogy with (2.38), we obtain

Fo[A] = E Fo(q -w) , (2.148) where Fo(q) is the same quantity as defined in (2.47) that is equal to the ground state energy of a single fermion in an Abelian monopole background of charge q. To leading order in Nf, the ground state energy on S2 x R, or equivalently the scaling dimension of the corresponding GNO monopole operator, equals

A = NfFo[A] + O(Ny). (2.149)

As in the U(Nc) case, the O(NO) contribution to the scaling dimension (2.149) receives contributions both from the Faddeev-Popov ghosts and from the gauge field fluctuations. Let us start by examining the Faddeev-Popov contribution, which can be computed from a generalization of the path integral in (2.49):

3 ZFP[A] = Dc e-Sghost Sghost = ! X lg K&c - i[Ac, C] IAC - i[A., c],

(2.150) where c is an anti-commuting scalar ghost valued in the Lie algebra. In (2.150), (-1.) is the standard inner product on the Lie algebra, defined such that (see for example [87])

(hiIhj) = J6j , (hilEa) = 0 , (EaIEa) = Jp6 . (2.151)

Transforming in the adjoint representation of the gauge group, the ghosts c can be

113 decomposed just like the gauge field fluctuations in (2.137) into components along the Cartan and components along the root directions:

r c= c hi + 1 c Ec. (2.152) i=1 aEroots

Using the commutators (2.138) and the normalization conditions (2.151), we obtain the analog of (2.50):

r 3 Sghost = 1 I X g(x) l 2 U(1) C1 2 (2.153) [i=1 aEroots I Following the same steps as in Section 2.4.2, we find that the ghost contribution to F is:

FFP[A] = - r (2J + 1) log [j(J + 1) + Q 2] [J=O + E (2J + 1) log [J(J+ 1) - (q . a)2 + Q2] aEroots J=q.a (2.154)

As can be seen from the first line of this expression, the r ghosts in the Cartan directions give equal contributions.

To obtain the full O(N2) correction to (2.149), we should also include the contribution coming from the gauge field fluctuations. We can split this contribution as

Fgauge [Al = FCartan + Froot(a) , (2.155) aEroots where Feartan is obtained by integrating out the Cartan components of the gauge field, and each Froot(a) comes from the component along the root a.

In general, all the Cartan contributions mix with one another. Performing the

114 same decomposition in vector spherical harmonics as in Section 2.4.3, we obtain

Fcatan 1T log WiW ,Cq.w,q.w / W(2.156)

= J(2J+1)ogdet' (Z wiwK .w,q.w(Q) J=1

where K is the same object as the one appearing in the U(Nc) result (2.85). For each J

and Q we now have to calculate the determinant of the matrix E WjWy Kq.,q.w (). The dimension of this matrix is 3r x 3r because there are 3 spatial directions (or equivalently, there are 3 vector harmonics UQ,JM, VQ,JM, and WQ,JM), and r Cartan elements. As a sanity check, we note that all fluctuations in the Cartan directions are regular vector fields on S2 x R, and therefore they can be decomposed in terms of the Q = 0 vector spherical harmonics. The total angular momentum J hence takes integer values. We also note that E wiw, K' .,q.w(Q) has r vanishing eigenvalues, as required by gauge invariance, and that the contribution of the r uncharged ghosts exhibited explicitly in the first line of (2.154) cancels the integrand in (2.156) asymptotically at large J and Q, just as in the U(Nc) case.

As mentioned before, the fluctuations in the root directions decouple and can be examined individually. Each such fluctuation gives a contribution equal to

Froot(a) I Tr' log (z |(w - aIEaIw)| 2

E (2J+1)logdet' ( I(w- aEaW)2 K.w,.(w_)(Q) )

(2.157)

Unlike in the Cartan case, here we need to add the matrices K instead of taking their tensor product, so in evaluating (2.157) we only need to calculate the eigenvalues of a 3 x 3 matrix. A sanity check in this case is that the vector spherical harmonics that appear in the decomposition of Kq.w,q.(w-a) have the same Q = q- w - q- (w - a) = q- a for all the terms of the sum over w in (2.157). For every root there is a ghost that

115 cancels the integrand in (2.157) asymptotically at large J and Q. We should emphasize that in preparing the results presented in Section 2.6 in the U(Nc) case, we calculated the matrices Ki,(Q) for all jqj 2 and Iq'I 2. It is not hard to use the same matrices combined with the needed group theory data in (2.156)-(2.157) in order to calculate the scaling dimension of any monopole operator that has Iq -wl < 2 for all weights w of the matter representation.

We can also see easily which GNO backgrounds are unstable. The instability only arises for the gauge fluctuations in the root directions for which Iq - al > 1. For the lowest J = Iq - al - 1 mode, K j7(q)a ( is simply a number, as the VQ,JM and WQ,jm modes in (2.65) are absent. Hence the condition of stability is

0 < > (w - aIEaIw)12 K"*( -'w)(0) - (2.158)

Here, K 4 _a) is evaluated at Q = 0 because, as one can check, it is a monotonically increasing function of Q; it is thus sufficient to check its sign for Q = 0. The expression on the right-hand side of this equation can be easily evaluated in particular cases. We now provide a few examples.

2.8.2 Examples

In this subsection we use the formulae derived in the previous subsection to obtain some of the monopole dimensions for various gauge groups. We start with G = U(Nc) and demonstrate the equivalence of the results obtained in the previous subsection with those in the previous parts of the chapter. We then move on to discuss several gauge groups with rank r = 1 and 2.

Another perspective on U(Nc) QCD with Nf fundamental fermions

As a first example, let us see how the G = U(Nc) results presented in Sections 2.5 and 2.6 fit within the general group framework of this section. The Nf fermions transform

116 in the fundamental representation of U(Nc), whose weight vectors w are

Wa = ea, (2.159) where a = 1, ... , Nc, and the ea form the standard unit frame in RNc. In components, eq = 6j. The adjoint representation has N, Cartan elements hi, as well as N- roots

a a= =e - e , a:34 b. (2.160)

(We could identify the Cartan elements with aa = 0.) Since the set of all possible weights is the lattice ZNc, the Dirac quantization condition implies that q E (Z/2)Nc In other words, the GNO monopoles are indexed by N, charges q, that are half- integers. Using the weights (2.159) and roots (2.160) in (2.148), (2.154), (2.156), and (2.156), one can straightforwardly reproduce the U(Nc) formulae in (2.90) and (2.91).

In doing so, it is helpful to note that in (2.157) we have I(wC - aabl EabI Wc) = 6bc

SU(2) QCD with fundamental fermions

Our second example is where G = SU(2) with Nf fermions transforming in the fundamental representation of SU(2). The group SU(2) has rank r = 1, so its roots and weights are simply numbers. The weights of the fundamental representation are

2 1 w1 =-, =-- . (2.161) 2 2

The adjoint consists of a Cartan element and two generators with roots

a1 = w1 - w 2 = 1, a2l 2 _ w = -1. (2.162)

The weight lattice is generated by the fundamental weights (2.161) and is therefore Z/2. The monopole charges that satisfy the condition (2.135) are all q E Z, modulo

117 the Weyl group-see Figure 2-7. The Weyl group consists of Z2 reflections about the origin, so the monopoles with charge q and -q should be identified. Concretely, we can think of a monopole with charge q as the background where

q/2 0 A = (±1 - cos 9)do. (2.163) 0 -q/2

Figure 2-7: The weight lattice of SU(2) (blue dots) as well as the lattice of all possible monopole charges (dotted circles). The monopole charges are defined modulo the action of the Weyl group, which in this case is Z2 and acts as reflections about the origin.

All these monopole backgrounds are topologically trivial because 7r 1(SU(2)) is also trivial. However, they are all stable because the stability condition (2.158) reduces to Kq/2,-q/2(O) > 0, which can be seen to be true from Figure 2-6. The scaling dimensions are

A = 2Fo(q/2)Nf + [6F(q/2, q/2) + 26F(q/2, -q/2)] + 0(1/N1 ). (2.164)

The numerical values of Fo(q/2) as well as 6F(q/2, tq/2) can be read out from Ta- bles 2.1 and 2.2. See Figure 2-8 for specific examples.

SO(3) QCD with fundamental fermions

Our third example is where the gauge group is G = SO(3). The Lie algebra of SO(3) is identical to that of its double covering, which is SU(2), but SO(3) has fewer allowed representations than SU(2). In particular, the spinor representation considered in the previous subsection is not a representation of SO(3). The smallest irrep of SO(3) is

118 Symbol A * 0.530Nf - 0.713 + O(1/Nf) A- & &% IV~. MW 60- * L 1.35Nf - 1.65 + O(1/Nf) * 2.37Nf - 2.95 + O(1/Nf) A 3.57Nf - 4.51 + 9(1/Nf)

Figure 2-8: The SU(2) monopoles appearing as black dotted circles in Figure 2-7. In the presence of Nf fundamental fermions these backgrounds are all stable, and we list the scaling dimensions A of the corresponding monopole operators.

the fundamental, which in this case is the same as the adjoint. The weights are

w 1 = 1, w 2 = 0, w 3 = -1, (2.165) where w 2 corresponds to the Cartan element, and wi and w3 to the roots. The weight lattice of SO(3) is generated by the fundamental weights (2.165), so it can be identified with Z. It is a subset of the weight lattice of SU(2). The quantization condition (2.135) implies that in SO(3) the allowed values of q are q E Z/2, and not just q E Z as was the case for SU(2). See Figure 2-9. As in the case of SU(2), the

qiw

Figure 2-9: The SO(3) weight lattice (blue dots) and its dual lattice (dashed circles). The weight lattice is a sublattice of the SU(2) weight lattice in Figure 2-7. The dual lattice contains more monopole charges q than the dual lattice of SU(2). As in the SU(2) case, the Weyl acts by reflections about the origin, so it provides the identification q - -q on the set of monopole charges.

Weyl group acts by reflections about the origin, so we should identify the monopoles with +q. Unlike the case of SU(2), however, we now have a non-trivial fundamental group, as 7r 1(SO(3)) = Z2 . The topological charge is (2q) mod 2, so the extra values of q that are allowed in SO(3) but not allowed in SU(2) correspond to topologically non-trivial monopole backgrounds.

119 Symbol A 10.530Nf - 0.554 + 0(1/Nf) 0 Unstable

Figure 2-10: The SO(3) monopoles appearing as black dotted circles in Figure 2-9. Here, we consider these backgrounds in the presence of N1 fermions transforming in the three-dimensional fundamental representation of SO(3). The orange circles correspond to unstable backgrounds. For the stable backgrounds (represented in black by various shapes), we list the scaling dimensions A of the corresponding monopole operators.

The stability condition (2.158) reduces to KO,0 (0) > 0 in this case. As can be seen from Figure 2-6, the only stable monopole background is that with Iql = 1/2. Note that this stable monopole background is also topologically non-trivial. The scaling dimension of the corresponding monopole operator is

A = 2Fo(1/2)Nf + /2, 0)] + 0(1/Nf). (2.166) [SF(1/2, 1/2) + 26F(I

See Figure 2-10.

SU(3) QCD with fundamental fermions

Our next example is QCD with gauge group SU(3) and N1 fermions in the fundamental representation. The rank of SU(3) is r = 2, so the roots and weights are points in R2 The weights of the fundamental representation are

= 1 2 3 = (2.167) , w = (0, (2 -2 6 3)

The adjoint consists of two Cartan generators, as well as six roots given by

1 -4 1 ± (2.168) (1,7 0) , E ( 2' 2 J

120 The weight lattice is generated by the fundamental weights (2.167). Dirac quantization implies that the monopole charges belong to a lattice generated by

1 f3 (2.169) (1,0) , 2 2

This lattice is the weight lattice of the GNO dual group SU(3)/Z 3. See Figure 2-11.

q 2 /W 2 Wi * * *1 0 0 0 0 0 3 0 0 We 0 S 0 * 0 S 0 0 S 0 0 2 0 0 We *1 6 WI WI S WI . 0 0 S 0 S 0 0 WI * 0 S et S 0 0 0 0 .0 ~av~ S S 0 0 0 S 0 * 0 WI S 0 0 0 * 0 0 0 WI * 0 Wi S .- 2 * S 0 0 0 0 0 0 0 * * 0 0 0 -34 0 0 0 0 .4 0 0 WI WI We * WI Figure 2-11: The weight lattice of SU(3) (blue dots) as well as the lattice of all possible monopole charges (dotted circles). The monopole charges are defined modulo the action of the Weyl group, which in this case is S3 and is generated by 120 degree rotations as well as reflections about the q2 axis.

In this case, the Weyl group is S3 and is generated by rotations of 27r/3 and reflections about the q2 axis. The independent monopoles are thus given by the points (qi, q2) in the monopole charge lattice modded out by the action of S3. See Figure 2-12 for which of these monopole configurations are stable and for the scaling dimensions of the monopole operators corresponding to the stable backgrounds pictured. Since 7r,(SU(3)) is trivial, none of these monopoles carry any non-trivial topological quantum numbers.

121 q2 o 0 0 o 0 0 3 0 * 0 0 o * Symbol A *A * 0 0 0 *0 A * 0.530Nf - 1.75 + O(l/Nf)

l 1.20Nf - 2.55 + O(1/Nf) ...... * 2.12Nf - 5.38 + 0 (1l/Nf) + 0 *-1 0 0 + A 3.13Nf - 7.20 + O(1/Nf)

A * O _f 0 + A 0 Unstable - o 0 * 0 0 -3 0 0 0 0 0 0

Figure 2-12: The SU(3) monopoles appearing as black dotted circles in Figure 2-11. Here, we consider these backgrounds in the presence of Nf fermions transforming in the three-dimensional fundamental representation of SU(3). The orange circles correspond to unstable backgrounds. For the stable backgrounds (represented in black by various shapes), we list the scaling dimensions A of the corresponding monopole operators.

Sp(4) QCD with fundamental fermions

We can also consider the gauge group G = Sp(4) and Nf fermions transforming in the four-dimensional fundamental representation of this gauge group. The rank of 2 Sp(4) is r = 2, so again the roots and weights are points in R . The weights of the fundamental representation are

1 0 1 ) (2.170) t V2- .

The adjoint consists of two Cartan elements as well as eight roots:

Q{ 1 i (V2, ± 0, ,2) (2.171) t v2 V2 0) ,1

122 The charge lattice in this case is the same as the weight lattice, and is generated by:

1 0 ,2 ) 1, oj1v, ). . (2.172)

After scaling, the charge basis vectors also generate the weight lattice of SO(5), which indeed is the GNO dual of Sp(4). See Figure 2-13.

NJ 1 'I 0t WI W' W

F 2 V. 147 TV & qllwl

-1 WIWI*~NJ p I WI WI W

Figure 2-13: The weight lattice of Sp(4) (blue dots) as well as the lattice of all possible monopole charges (dotted circles). The monopole charges are defined modulo the 3 action of the Weyl group, which in this case is (Z 2 ) and is generated by reflections about the qi axis, q2 axis, and the line that makes a 45 degree angle with the qi axis.

Sp(4) is simply connected, so all GNO monopoles have trivial topological charge. The stability of various monopoles along with their scaling dimensions are included in Figure 2-14.

123 q2 Symbol A 0 0 E * 0 0.530N, - 1.75 + O(l/Nf) a V o 0 N 0 V 0 l 1.06Nf - 2.18 + O(1/Nf) 0 V A +1 * 1.88Nf - 4.29 + O(1/Nf) 0o 0 0 +* 0 A 2.69Nf - 5.16 + O(1/Nf) ...... W - I'W -~~' '~2 M .o 3 V 3.72Nf - 7.87 + O(1/Nf) 0 0+ 13 + 0 0 o 4.75Nf - 9.27 + O(1/Nf) O VA+ * A V 0 * 5.95Nf - 12.4 + O(1/Nf) * o V 0 o V 0 N _ 7.15Nf - 14.2 + O(1/Nf) * O O _ 0o 0 N o Unstable

Figure 2-14: The Sp( 4 ) monopoles appearing as black dotted circles in Figure 2-13. Here, we consider these backgrounds in the presence of Nf fermions transforming in 4 the four-dimensional fundamental representation of Sp( ). The orange circles correspond to unstable backgrounds. For the stable backgrounds (represented in black by various shapes), we list the scaling dimensions A of the corresponding monopole operators.

SO(5) QCD with fundamental fermions

Moving onto G = SO(5) with Nf fundamental fermions, the weights of the fundamental representation are:

t (1,0) , t(0,1), (0,0). (2.173)

The adjoint consists of two generators in the Cartan, as well as generators with roots

(2.174) t (1, 1) , i (1, -1)), + (1, 0) , + (0, 1) .

The charge lattice is generated by:

,0 , 0, ) . (2.175)

124 See Figure 2-15.

q2/W2 0 0) 0. 0 0 0 0 *

~.. *...' C'

~~1 ~ Ci

~~1 ~*i Ci Ci

... q J. - ". 1;

o c~ ~.i

o C' '.a

I 0 . 0 Ce -It Figure 2-15: The weight lattice of SO(5) (blue dots) as well as the lattice of all possible monopole charges (dotted circles). The monopole charges are defined modulo the action of the Weyl group, which, as in the Sp(4 ) case, can be identified with the 3 (Z 2 ) generated by reflections about the qi axis, q2 axis, and the line that makes a 45 degree angle with the qi axis.

In this case the fundament group is non-trivial, 7r1 (SO(5)) = Z 2 . The topological charge of a monopole with GNO charges qi, q2 is (2q, + 2q2) mod 2. For SO(5), there are only two stable monopoles. Monopoles of various charges are plotted in Figure 2-16.

G2 QCD with fundamental fermions

Lastly, we consider G = G 2 and Nf fermions transforming in the seven-dimensional fundamental representation of G2. The weights of the fundamental representation are

12 2v 12 + (0, 1) (0,0) . (2.176) t 2 ' 2) 2 ' 2,'

125 q2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x S0 0 0 0 0 0 0 Symbol A 0 0 0 04 0 0 0 0 0 * 0.530Nf - 1.59 + O(l/Nf) Af. M . .1!!I. a,. 1Wa. - 1 - M - . - M7 '11 1 MW Y L 1.06Nf - 1.86 + O(1/Nf)

00 0 0 4 30 0 0 0 Unstable 0 0 0 0-s 0 0 0 0 0 0 0 0 0 0 0 00-1 0 0 0 0 0 0 0

Figure 2-16: The SO(5) monopoles appearing as black dotted circles in Figure 2-15. Here, we consider these backgrounds in the presence of Nf fermions transforming in the five-dimensional fundamental representation of SO(5). The orange circles correspond to unstable backgrounds. For the stable backgrounds (represented in black by various shapes), we list the scaling dimensions A of the corresponding monopole operators.

The adjoint representation is fourteen-dimensional and consists of two Cartan elements as well as the roots

1 12 3 'I t(0, 1) ' 2 3 ' 2) 2 2) 2 2) (2.177) 3 t ,3 ± (v, 0) 2) ,

The set of all possible monopole charges is generated by the vectors

vf3 1 ( 1\) (2.178) ,30) '2 ' 2J.

After scaling and rotating, the charge lattice is identical to the weight lattice, reflect- ing the fact that G2 is its own GNO dual. See Figure 2-17. Here, the Weyl group is D6 (the dihedral group with 12 elements), which is generated by rotations by 60

126 degrees as well as reflections about the line that makes 45 degrees with the qi axis.

q2/w2 0*000 0 0 0 WI o c * o o o o

WI ~-' 0 * 00 o

-2' '~1 *t-2 q 1/w

oS o1 0. O 1 0 WI ~.I ?.~I

0 0 -* ~,-I *.~l *

S O 1 [ 0 Wi 0 0

0 0 0 -* 0 0 *

Figure 2-17: The weight lattice of G2 (blue dots) as well as the lattice of all possible monopole charges (dotted circles). The monopole charges are defined modulo the action of the Weyl group, which in this case is D6 (the dihedral group of order 12) and is generated by 60 degree rotations as well as reflections about the line that makes a 45 degree angle with the qi axis.

G2 has a trivial fundamental group, and so there is no topological charge. The stability of monopoles with different GNO charges as well as the dimensions of the operators corresponding to the stable backgrounds are given in Figure 2-18.

127 q2 0 00 X)00 0

0 00 0 00 00

Symbol A 0 0 0 0 0 0 0 0 I' qW 1W I 1.06N -280+0(1/N) o Unstable 0 0 0 * * 0 0 0 0 0 0 -0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - 0 0 0

Figure 2-18: The G2 monopoles appearing as black dotted circles in Figure 2-17. Here, we consider these backgrounds in the presence of Nf fermions transforming in the seven-dimensional fundamental representation of G2. The orange circles correspond to unstable backgrounds. For the stable backgrounds (represented in black by various shapes), we list the scaling dimensions A of the corresponding monopole operators.

SU(3) QCD with adjoint and fundamental fermions

While in all of our previous examples, the matter fields were in the fundamental representation of the gauge group, we can also consider matter in other representations. In this example we consider fermions that transform in the adjoint representation of SU(3). The weights of the adjoint are just the root vectors (2.168). The set of possible monopoles is independent of the matter representations, and so the charge lattice is still generated by (2.169) and is shown in Figure 2-11. 2 3 Unfortunately, for Nf copies of the adjoint, there are no stable monopoles. The absence of stable monopoles does not mean that the adjoint representation is unin- teresting, however. There is no reason to restrict to matter in an irreducible representation, and we can consider theories with nadjNf adjoint fermions, and nfundNf fundamentals. For nadj < nfund this theory should have many stable monopoles, as

23 The absence of stable monopoles is a common feature of larger representations of any gauge group.

128 is the case for SU(3) with only fundamental matter. 2 4 For nadj > nfund the theory should behave more like the theory with only adjoint matter, and have no stable monopoles. Below we plot the number of stable monopoles as a function of the ratio nfund/fnadj. For small values of this ratio, the specific monopoles which become stable are shown.

24SU(3) with only fundamental matter has infinitely many stable monopole backgrounds.

129 q2 Number of stable monopoles 1o1

0 2_0_00_ 0 0 01'0 0 0 ...... q1 8 -!3 Z 2 - ' o 0 01 0 0 0 o 0 0 2( 0 0 0 o 0 0 0o 0 0 -3 o o 0 10 0 0

q2 IV 0 0 0 4 3 0 0 0 IV 0 0 0 0 0 0 0 0 61 .0 0 0 -- V q, - 3 -2 -71 T 2 3 II 0 0 Al 0S 0 0 0 0 0- 0 0 0 I -- /~. 0 0 0 *0 0 0 5 10 15 20 25 30 35 -l -3 0 0 0 0 0 0

q2 q2 12 0 0 0 4 0 *| _I 0 0 * 0 0 0 0 O 0 0 0V 0 0 0 * 0 A 0 0 0 0 A A + 0 * S a * A

0 0 0 0 0 0I- q q, 0 0 0 22 ~'1K~r~r0 0 0 -' - 1 T -2 -1 0 oa 4 a 0 0 0 01- 1 0 + 0 i A 0 02_ 0 0 A A + 0 1 0 * A 0 0 0 0 0 0 0 0 * * 0 0 -3 -3 -3 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 , 0 0 0 Figure 2-19: In the top right corner we show the number of (inequivalent) stable monopoles for SU(3) gauge theory with nfndNf fundamental fermions and najNf adjoint fermions as a function of the ratio nfund/nadj. The solid line is divided into five regions that correspond to the diagrams on the left and bottom, where we show explicitly which monopoles are stable in each region. The dashed line is a continuation of the solid line for larger values of nfund/naj, but in this region we do not show explicitly which monopoles are stable.

130 2.9 Discussion

2.9.1 Summary

In this chapter we studied properties of monopole operators in non-supersymmetric

QCD3 and QED3 with Nf fermion flavors. We worked in the limit of large Nf, where gauge field fluctuations are suppressed and where the theory flows in the infrared to an interacting CFT. At this infrared fixed point, we used the state-operator correspondence to first define the monopole operators in terms of energy eigenstates on S 2 x R and then to study their transformation properties under the conformal and flavor symmetry groups. As we emphasized in Section 2.2, associating energy eigenstates with certain GNO monopole backgrounds can. be done cleanly in the limit of large Nf, provided that these GNO backgrounds are stable saddle points of the effective action for the gauge field fluctuations. We obtain three main results. Our first result is that only certain GNO monopole backgrounds are stable saddle points in the CFT. In general, stability is a dynamical issue that can only be decided by studying the effective potential for the gauge field fluctuations. We provided the criterion for stability in Section 2.5 in the case where the gauge group is U(Nc) (see Figure 2-6 for a summary plot). We later generalized this criterion to theories with other gauge groups in Section 2.8. In all these theories, we were thus able to identify precisely for which sets of GNO charges one can define independent monopole operators, at least at large N1 . We found that many, but not all, GNO backgrounds in each topological sector are stable. For every stable background there is a Fock space of energy eigenstates on S2 x R whose wavefunctions are localized around that background. Each such energy eigenstate corresponds to an operator on R3 . We further focused on the lowest energy eigenstate within every Fock space and studied its quantum numbers. We referred to the operator that corresponds to this state as a bare monopole operator. Our second result is that we computed the scaling dimensions of the bare GNO monopole operators in the 1/Nf expansion. The scaling dimension is a quantity determined by the dynamics. It equals the ground state energy on S2 x R in the

131 GNO monopole background. We obtained the ground state energy by evaluating the path integral on S2 x R to subleading order in the 1/N expansion. For large N the monopole operators have O(Nf) dimension

A = Nf Fo + 5F + O(1/Nf) (2.179)

Explicit results for A for various GNO charges can be found in Section 2.6 for U(Nc) theories, and in Section 2.8 for general gauge groups. We expect the results we obtained from the large N1 expansion to be reliable down to fairly small values N1 > JF/Fo = 0(1). This expectation is supported by the high accuracy of large Nf computations for supersymmetric theories [88], where the answer can be compared to exact results.

Our third result is that for the case where the gauge group is G = U(Nc), we calculated all the other quantum numbers of the bare monopole operators. We found that these operators are all spin singlets and that they transform in the irreducible representations of the SU(Nf) flavor symmetry group given by the Young diagrams

N/2{IE (2.180) 2 E.|qal

where the GNO charges are {qi, q2, ... , qN}. We can therefore completely characterize the quantum numbers of the bare monopole operators in U(Nc) QCD with Nf fundamental flavors: We know their topological charge, scaling dimension, spin, and representation of the flavor symmetry group. It would be very interesting to generalize this analysis to other gauge groups and matter representations.

These results are interesting in their own right as they teach us about the operator content of QCD3. Using the knowledge that we gained, it is desirable to understand the role that the monopole operators play in the dynamics of the theory.

132 2.9.2 Confinement and chiral symmetry breaking

From our results, we can learn about the following three theories, one of which describes confinement:

I. We can consider Yang-Mills theory coupled to Nf flavors of massless fermions. This theory is super-renormalizable and asymptotically free, so it is well-defined up to arbitrarily high energies. If we wish, we could think of it as an effective field theory at large distances that arises from a lattice Hamiltonian that does not allow mass terms for the fermions.

II. We can also consider a non-trivial interacting CFT. At large Nf, we can define this CFT by erasing the Yang-Mills term from the action of (I), as we did throughout this chapter. This description of the CFT should make sense as long as this CFT can be achieved from (I) without any fine tuning, which should

happen for all Nf greater than or equal to some number N d*"f that we will estimate shortly. Below Nj~onf, a non-trivial CFT may still exist, but a good description for it may not be readily available.

III. A confined or partially confined theory, potentially with some number of Gold- stone bosons coming from spontaneous flavor symmetry breaking. The description of this theory is intentionally vague, as it should be viewed just as an alternative to (II) for describing the IR physics of (I).

Recall that we restrict our discussion to the case where the number of fermions is even, because otherwise we would necessarily be breaking parity [81,82]. The first question we can ask is: When is the infrared physics of generic RG flows starting from (I) described by the deconfined CFT (II), and when is it not? In other words, we should estimate Nd*nf from the fact that for Nf ;> N deconf, all monopole operators should be irrelevant, i.e. their scaling dimensions should be greater than three. From (2.179) we find:

Nf -O F . (2.181)

133 Here, the values of FO and 6F correspond to the monopole operator with the lowest scaling dimension for a given gauge group and matter content. See Table 2.4 for a

Gauge group N "o

U(1) 12 U(2) 14 SU(2) 8 SO(3) 8 SU(3) 10 Sp(4) 10 SO(5) 10

G 2 6

Table 2.4: Estimates of the smallest number of fermions, Nfeconf for which the IR of QCD 3 with gauge group G is in a deconfined quantum critical point. Results are listed for various rank one and two gauge groups.

few particular cases. As can be seen from this table, N econf is smaller for groups with fewer monopoles. We stress that the estimate (2.181) as well as the numbers given in Table 2.4 are

not relying on any assumptions about the physics at Nf < N econf. All we can tell for sure is that in this case, Yang-Mills theory with Nf fermions does not generically flow to the deconfined CFT (II).

It is possible to obtain an independent estimate of N d,,"f if we make some as-

sumptions about what happens for Nf < N econf. We will do so only in the cases where the gauge group is G = U(1) and U(2), and leave a more extensive analysis for future work. As reviewed in [89], one expects that below Nof"", the SU(N) global flavor symmetry should be broken to SU(Nf/2) x SU(Nf/2) x U(1). 2 5 A simple

25 1n [90,91] it is shown that if the number of fermions is N1 > 6 there must be massless particles in the infrared. It is likely that these particles are Goldstone bosons corresponding to the symmetry breaking pattern mentioned in the main text. This symmetry breaking pattern is usually referred to as chiral symmetry breaking, even though there is no chiral symmetry for fermions in three dimensions. The name "chiral symmetry" comes from the fact that if the same theory were realized

134 computation shows that the number of Goldstone bosons is

NG 2 (2.182) 2

As pointed out in [89], such a symmetry breaking pattern is constrained by the F- theorem [92-95] ,26 which states that any three-dimensional Lorentz-invariant RG flow from a UV CFT to an IR CFT should satisfy

FUV ;> FIR, (2.183)

where Fuv (FIR) is the S3 free energy of the UV (IR) CFT.

To use the F-theorem, we can consider starting with Yang-Mills theory with Nf N econf - 2 fermions, which by assumption is the largest value of N for which the IR theory consists of N2/2 Goldstone bosons. It is likely that the same IR theory of Goldstone bosons can be obtained by starting with the CFT in (II) with N'eonf fermions and giving masses to two of them. These masses should be of opposite sign in order to preserve parity. The latter flow is the one for which we will use the F- theorem. The F-theorem should of course hold for the flow from Yang-Mills theory with Nf fermions to the theory of N'/2 Goldstone bosons as well, but in the UV Yang-Mills theory is not conformal and should be assigned Fuv = oo.

For the flow between the deconfined CFT with N econf fermions and the IR theory of N2/2 Goldstone bosons, Fuv can be read off from the results of [96]:

N2 grN+deconf Vol(U(Nc)) Fuv ( + 87r2 Nc Neconf Vol (U(1))NC + O(1/Nd"onf),

(2.184)

in 4d by pairing up the N1 Weyl spinors into N1 /2 Dirac spinors, then the broken symmetry would be chiral. 2 6 3 Note that this F stands for the S free energy, and should not be confused with the S2 x R partition function that was discussed in this chapter.

135 where for N, = 1, 2 we should use Vol (U(1)) = 27r and Vol (U(2)) = 87r3. Because the IR theory is a CFT of Nf/2 free scalar fields, we have

N2 FIR = N Fscalar , (2.185) 2 with Fscaiar ~~0.0638 being the S3 free energy of a single real scalar field [93]. Using (2.184) and (2.185), we see that the F-theorem inequality (2.183) holds

2 7 for N donf < 12 in the U(1) case and NeConf < 20 in the U(2) case. This result is consistent with, but less precise than, the values N eonf - 12 and N econf - 14 for G = U(1) and U(2), respectively, that we obtained from studying the scaling dimensions of the monopole operators. It should be stressed that the SU(Nf) global symmetry cannot be broken by the same monopoles which cause confinement. If an operator spontaneously breaks a symmetry, it necessarily is not a singlet under this symmetry, and so will not be generated under RG flow. The connection, therefore, between the prediction from chiral symmetry-breaking and that from monopole relevance is not clear.

2.9.3 QED and and algebraic spin liquids

The analysis of the previous subsection on the minimal value of Nf for which Yang- Mills theory with Nf fermions flows to a deconfined phase considered only "generic" such RG trajectories. This analysis can be refined in the case where ri(G) is non- trivial-and so certain monopole operators carry topological charges-by restricting our attention to RG trajectories that are invariant under a subgroup of the corresponding topological symmetry. Under this extra assumption, Yang-Mills theory with Nf fermions flows generically to a deconfined CFT provided that the monopole operators that are invariant under the above subgroup are irrelevant; it does not matter whether the other ones are relevant or not. Consequently, the values of Nyeconf in

271n the U(1) case, [89] obtained Nyeconf < 14 from considering a flow between supersymmetric QED3 and the symmetry broken phase of N'/2 Goldstone bosons. The bound Nfeco"f < 12 that we obtain is more constraining than that of [89f, because we start from a UV theory with fewer degrees of freedom.

136 this case would be smaller than the values obtained in the previous section. The QED case N, = 1 provides a nice example relevant to algebraic spin liquids

[47,97]. In this case the topological symmetry is U(1)t0 p, and the topological charge is qtop = q E Z/2. It was suggested in [98] that if U(1) QED with Nf = 4 fermions can be obtained as an effective theory of a spin system on the Kagome lattice, the lattice symmetries are embedded into a Z3 subgroup of U(1)t0 p. So let us restrict our attention to RG trajectories that preserve this Z3 subgroup as a symmetry. Under the generator of this Z3 symmetry, a monopole of charge q is multiplied by a phase equal to e4'7riq/3, so only monopole operators with q E 3Z/2 are invariant. If all fermion mass terms are also forbidden by the lattice regularization, it then follows that the IR theory is a deconfined CFT provided that all the monopole operators with q E 3Z/2 are irrelevant. According to Table 2.1, these monopole operators have scaling dimension greater than 3 for Nf > 4. This bound is less restrictive than Nf > 12, which is what we obtained in the previous section by requiring that all monopole operators should be irrelevant.

137 138 Chapter 3

Chiral Gauge Theory for Stringy Backgrounds

In this chapter, we move down to two dimensions, using strongly coupled, chiral, two-dimensional gauge theories, we construct candidate low-energy conformal field theories that describe string backgrounds outside of the traditional Calabi-Yau framework.

3.1 Introduction

String theory has taught us a great deal about the quantum geometry of Calabi-Yau (CY) manifolds. Central to this progress is the gauged linear sigma model (GLSM), a formalism which translates quantum computations in Kdhler geometries into free- field-theory calculations in an auxiliary gauge theory [23]. Of course, CYs form a set of measure zero in the full space of string compactifications, so it is natural to wonder what we can say about the quantum geometry of more general non-CY manifolds. This question is particularly natural in the heterotic string, where a worldsheet analysis should suffice. This motivates us to search for GLSMs for non-Ksihler' manifolds, with the goal of using them to study the quantum geometry of more general

'The term "balanced" is probably more appropriate [99], since all such 4d K = 1 compactifications come from balanced manifolds which may or may not be Kahler. The term "non-Kifhler" has become standard, however, emphasizing that these manifolds need not be Kahler.

139 geometries. At first glance, this seems quite challenging. Mathematically, the basic structure of a GLSM is a Kdhler quotient of flat space, which naively should not be much help in getting a non-Kihler manifold. Meanwhile, if the geometry is not Kdhler, target space SUSY requires non-trivial 3-form torsion, H, which must satisfy the Green-Schwarz (GS) Bianchi identity,

dH = a' (trR A R -TrF A F). (3.1)

Correspondingly, the B-field itself must transform non-trivially under the full set of spacetime gauge transformations. Furthermore, dH -/ 0 implies a non-trivial dilaton profile [32], so that the worldsheet conformal symmetry is only realized non linearly. Finally, the fact that tree- and one-loop effects compete in (3.1) means that some cycles may be frozen near string scale, making a large-radius limit problematic. It is difficult to see how all these effects could be incorporated into a GLSM. Indeed, considerable effort has been devoted to adding bells and whistles to the GLSM to mock these effects up [100-105]. In this chapter, we demonstrate a general mechanism for generating the modified Bianchi identity and related quantum effects in a garden-variety (0, 2) GLSM.2 Surprisingly, this mechanism does not require any new ingredients. Rather, simple quantum effects in every (0, 2) GLSM generate all the necessary features dynamically. The basic mechanism first appeared in the study of 4d chiral gauge theories [41,42]. In a chiral gauge theory, integrating out heavy fermions in chiral representations of the gauge group will generically generate anomalous Green-Schwarz terms in the action. These terms are essential for canceling the anomaly of the surviving light fermions. In our (0, 2) GLSMs, such GS terms precisely generate the corrections to the B-field transformation law which are required to satisfy the Bianchi identity, and which lead to non-trivial H-flux and non-Kifhler hermitian metric. The central claim of this chapter is that all of the features of a generic non-

2For useful introductions and reviews of (0,2) sigma models see [106,107].

140 Kdhler model with H-flux can in fact be found within a standard (0, 2) GLSM. As we shall see, non-trivial H-flux and a modified transformation law for the B-field are automatically generated as needed by the mechanism sketched above. Indeed, we will find that all of the various previously-preposed quasi-linear mechanisms for generating H-flux in special cases emerge naturally when studying the low energy effective physics of standard (0,2) GLSMs.

The rest of this chapter is organized as follows. In Section 2, we demonstrate how the GS mechanism in a GLSM generates precisely the anomalous transformation law for the B-field needed to ensure cancellation of the sigma model anomalies of the IR NLSM. Along the way, we demonstrate that the GS models previously constructed can thus be UV completed into completely pedestrian (0, 2) GLSMs; conversely, at low energies and at generic points in the (0, 2) moduli space, a general (0, 2) GLSM reduces to such an anomalous GS effective GLSM. In Section 3, we will study a particularly simple class of such effective GLSMs in which the axial couplings are entirely linear, and use them to explore the quantum consistency and semi-classical geometry of such GLSMs. Furthermore, we find unanticipated topological constraints on the existence of such models. The central ingredient in these models is a set of Green-Schwarz axions, Y, playing the role of Stiickelberg fields for the anomalous gauge symmetries. This allows us to avoid the subtleties associated with logarithms and address both classical and quantum properties of the models. As we shall see, these theories show every sign of running to good IR CFTs - more precisely, these theories enjoy a (0, 2) superconformal algebra which closes in Q+-cohomology, as was previously shown for the T 2 models in [101]. While the models on which we focus are not generic, the lessons we learn can be readily applied to more general GLSMs with non-trivial H-flux. We close in Section 4 with a summary and list of future directions.

141 3.2 Generating dH in a (0, 2) GLSM

The goal of the present section is to demonstrate that all the ingredients necessary to study models with non-trivial dH are already present in a garden-variety (0, 2) GLSM. We begin by recalling how dH =4 0 and the modified Bianchi identity arise in a general (0, 2) NLSM. We then review how these effect can be incorporated into quasi-linear (0, 2) GLSMs by hand via Green-Schwarz anomaly cancellation, as first demonstrated in [100,103]. We then argue that such quasi-linear models arise as effective descriptions of totally standard (and non-anomalous) (0, 2) GLSMs at generic points in their moduli space. Concretely, moving along the (0,2) moduli space mod- ifies masses for chirally-gauged fermions in the GLSM; integrating out the heaviest fermions then generates non-linear Green-Schwarz terms in the action which realize the non-trivial dH. This explains for example how dH $ 0 is generated in (0, 2) deformations of (2,2) GLSMs, such as deformations of the tangent bundle. We also comment briefly on the effective geometry of such models.

3.2.1 Torsion in (0, 2) NLSMs

Before we address the GLSM, let's recall how the modified Bianchi identity and the c2 constraints arise in a non-linear sigma model (NLSM) with (0, 2) supersymmetry. The action for a (0, 2) NLSM is given by [108]:

11 L = - (G + B 5) o9+'&0 q$0'+ - (G - B ) a_#1a35 2 11 1 2 1 11

2~ (+Gi(i + Fc±{IL&'9 /4 + +i41 1 Op ++

j (flAP(&+7y + AI A CaO/I7) + _Y nAB(P+ Y- + A? 0 +s~

+ WA- + . .. (3.2) where ( JK + GIL HLJK = GIL (KGJL+ OJBKL ~ OKBJL) and the ... refer to four-fermi terms which we will not need for the moment. By construction, this action is invariant under target space local Lorentz transfor-

142 mations, 3 target space gauge transformations, and target space Kalb-Ramond transformations:

-+ ()

y_ -+U()B-

B - B+ dwl. (3.3)

Note that plugging these transformations into the Lagrangian does not leave the action invariant - rather, we must supplement these transformations on the worldsheet fields with a corresponding change in the background fields, G, B, n, A, and W. Only under this combined transformation is the Lagrangian, classically, invariant.

Quantum mechanically, however, this is not in general a symmetry of the theory due to an anomaly in the fermionic measure [109, 110]. Nonetheless, these theories can be improved, order by order in a' [25-27], so as to respect a modified symmetry. More precisely, while the theory is not invariant under (3.3), it is invariant under a slightly different symmetry whose transformation rule for B is modified at order a',

JBj ~ a' (a[I GMAM - 0 w0) (3.4)

where 6 ab, M parameterize infinitesimal local Lorentz transformations and gauge transformations respectively. With this transformation law, Ho = dB is no longer gauge invariant. Correspondingly, we must also modify the definition of H at order

H = Ho + a' (Q3 (W) -Q 3 (A)) , (3.5)

where Q3 are the Chern-Simons three forms for the spin and gauge connections. The

3 For the local Lorentz transformations it is convenient to introduce a vielbein, ey and define V a)

143 redefined H gives the modified Bianchi identity,

dH = a'(trR A R - TrF A F). (3.6)

This is the worldsheet manifestation of the Green-Schwarz mechanism [40]. Note that if we wish to construct a theory with a non-anomalous symmetry that reduces to the classical transformations as a' -+ 0, these modifications are unavoidable. As emphasized in [27] care must be used to see that supersymmetry is preserved. In principle, then, there is no obstruction to studying non-Kdhler geometries with intrinsic torsion via a worldsheet NLSM - the NLSM itself is perfectly well-posed, supersymmetric, and non-anomalous under all the symmetries of interest. In practice, however, this construction is not very useful for many computational purposes. First, as usual with a NLSM, most things we would like to compute end up depending on the physical metric; however, finding a solution to the Einstein equation and modified Bianchi is even more difficult than finding a compact Calabi-Yau metric (and indeed we still do not have a general proof of the existence of solutions to these equations except in a few very special cases [111]). Second, since the Bianchi identity mixes orders in a', it is not clear when, or if, perturbation theory around a classical solution even makes sense. Computationally, then, the NLSM is just not enough.

3.2.2 Adding dH to a (0, 2) GLSM by hand: the Green Schwarz mechanism

The obvious question, then, is how to implement H-flux and the modified transformation law for the B-field in a computationally effective GLSM. Ideally, this would produce a GLSM which manifestly reduces to an NLSM of the above form at low energies. To begin, consider a (0, 2) GLSM whose classical action reduces, at lower energies, to an effective NLSM for a complex manifold, X, with left-moving fermions valued in sections of a holomorphic vector bundle, Vx. Importantly, since all the geometry and topology of the IR CFT is generated by the gauge action, all of the potential

144 anomalies are similarly embedded within the gauge group [112]. This allows us to lift the problem of tracking anomalies in the NLSM to the easier problem of identifying gauge anomalies in the UV-free GLSM. This suggests that the anomalous Lorentz transformation law for the B-field in the NLSM should lift to an anomalous gauge transformation law for the B-field in the GLSM.

Our first job, then, is to locate the B-field in the GLSM. Happily, this is well- understood physics: the B-field in such a GLSM is controlled [23,113] by the axial part of the FI coupling, Lo = 0a F+_. The resulting spacetime B-field is then B = 0"Wa, where the w' are the (1, 1)-forms on the target space corresponding to the gauge field strengths, F+a. More generally, the hermitian (1, 1) form (which becomes the complexified Kdhler class on the (2,2) locus) is J = J+iB = taWa, where ta = ra+i6a are the FI parameters in the superpotential. Since Wa is closed and 0 is constant, H=dB=0. To introduce non-trivial H-flux, then, we can simply promote the Oa to dynamical fields such that H = dB = d6a A Wa does not vanish identically [100,114,115]. Each such dynamical Oa then represents a coordinate on an S1 on the target space of the GLSM. 4 Notably, this also generates the dilaton gradients we expect from supergravity, as shown e.g. in [114]. Unfortunately, gauge invariance and single-valuedness of the action require that d (d6a) = 0, and thus dH = 0.

This suggests a simple way [100-103] to build a GLSM with dH # 0: let Oa be shift-charged under the worldsheet gauge symmetry, Oa - +a + Qab ab. The resulting B field now transforms non-trivially under the gauge symmetry,

B -c-+ B + Qaba bWb (3.7)

The worldsheet gauge-invariant H-flux is thus of the form

H = (d~a + QabAb) A a. (3.8)

4 1n the UV 0 provides a new S' in the field space, in the IR this becomes an S' on the target space of the NLSM to which the theory flows, with the gluing in of the S' specified by the gauge action.

145 Taking a further exterior derivative then gives

dH = Qabwa A b, (3.9)

where wb -= dAb is the 2-form representing the gauge field strength under which 0 is charged.

By making 9 charged, however, we have rendered the classical action (in particular, the axial term LO) non-gauge-invariant. Fortunately, the variation of the axial term is precisely of the form of a 2d anomaly: under a gauge transformation with parameter a, J64 ocl Qab aa F . (3.10)

It is thus possible to cancel the gauge-variation of the classical action against a quantum anomaly in the measure a la Green and Schwarz [100]', with L providing the classical Green-Schwarz term.

This raises an obvious question: what is the gauge anomaly in a classical (0, 2) GLSM measuring? It has long been understood that the gauge anomaly in a standard (0, 2) GLSM is in fact a probe of the sigma model anomaly in the target space, as follows. The anomaly in 2d comes from a di-angle diagram and thus defines a quadratic form on gauge fields: A = Aab FT, F_. Since the gauge fields represent

the pullback to the worldsheet of 2-forms Wa in the target space, FTL_ = q*wa, the anomaly thus defines a 4-form on the target, A = AabWaWb. A short computation (see e.g. [100,103]) then verifies that the corresponding 4-form is the RHS of the heterotic Bianchi identity, A = [trR A R - TrF A F].6 Choosing the gauge-transformation of our dynamical theta angle to cancel the worldsheet anomaly then ensures that

5Note that adding a GS term is not possible in a (2,2) model. Corespondingly, the fermionic spectrum is necessarily non-chiral, which forbids any gauge anomaly. Such models necessarily have dH =0. 'Technically, this result uses the natural Kdhler structure on the toric variety. In the presence of torsion, the physical metric will in general not be this Kdhler metric. They will differ, however, only by terms proportional to a, the loop counting parameter in the worldsheet; for the Bianchi identity above, we need only the leading order result. Note that this argument is not reliable away from large radius - however, away from large radius, the geometric picture is itself not reliable so we should focus instead on the quantum consistency of the gauge theory.

146 we satisfy the target space Bianchi identity, with the vanishing of the net anomaly four-form in cohomology corresponding to an integrability condition for a smooth H. Rather poetically, then, the Green-Schwarz mechanism on the target space pulls back to the GLSM as a Green-Schwarz mechanism on the worldsheet. This mechanism was first used in [100] to build GLSMs for non-Ksihler T2 -fibrations over K3 (which geometries were first studied via supergravity in [111,116-118]) and was subsequently exploited to study LG-orbifold points in the moduli space [103], to compute the stringy spectrum [101], and generalized to non-abelian GLSMs [102].

3.2.3 On the geometry of GS GLSMs

We would like to argue that the Green-Schwarz mechanism for the worldsheet gauge theory, together with the anomalous transformation law for the B-field under worldsheet gauge variations, reduces precisely to the anomalous transformation law for the B-field and the corresponding Green-Schwarz mechanism of the NLSM. To see that this indeed works out, we need to think more carefully about the effective geometry of the GS GLSM. Consider an arbitrary, potentially anomalous, (0, 2) GLSM. At the classical level, this defines a classical NLSM by solving the equations of motion for all massive fields and evaluating the action on-shell. By construction, the target space X of the resulting NLSM is simply the quotient of the flat target space of the UV free fields

(call it Z) by the (complexified) gauge group, G, of the GLSM, i.e. X = Z'/G, where Z' is Z with the fixed locus of G removed. If there is a superpotential turned on, we further restrict to the vanishing locus of the superpotential in X, X w=o. The result is a completely garden-variety classical (0, 2) NLSM. Quantum mechanically, of course, this NLSM may be anomalous via the standard sigma model anomaly. The key observation [112] is that, due to linearity of the UV theory, the sigma model anomaly of the NLSM is embedded in the gauge anomaly of the GLSM. Moreover, from general properties of the classical geometry associated to a GLSM, we have a direct map between forms in the classical geometry, X, and gauge field strengths in the GLSM. This allows us to translate the anomalous gauge-

147 transformation law of the axial GS term in the GLSM directly to an anomalous transformation law of the B-field in the resulting classical NLSM. It also allows us to relate the gauge-anomaly of the fermion measure in the GLSM to the sigma- model anomaly of the fermion measure in the NLSM. The conditions for cancellation of the GLSM anomaly in the UV then directly map to the conditions required for cancellation of the sigma model anomaly in the IR NLSM as discussed above. Of course, the above analysis only works at the classical level - matching the two theories precisely would require showing that the full one-loop effective actions, including un-protected kinetic terms, precisely match up to finite renormalizations. The corresponding calculation is straightforward in principle, but quite involved in practice, so we will not attempt it here. Instead, we make the following observation. Consider a classical NLSM constructed from the gauge-invariant on-shell action of a classical GLSM. In general, both the NLSM and GLSM will have quantum anomalies. As described above, we now have two ways of building non-anomalous models: we can cancel the gauge anomaly in the GLSM via a GS term for the worldsheet gauge symmetry; or we can cancel the sigma model anomaly of the NLSM directly using standard NLSM techniques as described above. The question is whether these two quantum modifications are equivalent or not - more precisely, whether they lie in the same universality class in the deep IR. This can be represented in the following diagram:

Classical GLSM aImprove GLSM

On-Shell (3.11)

Classical NLSM 0""" (0,2) NLSM

While it remains technically possible for these models to be inequivalent, being different would mean there is a new way to deal with sigma model anomalies which is distinct from anything done before. We consider this highly unlikely, and thus happily conjecture that the non-anomalous GLSM and NLSM thus defined in fact flow to the same CFT in the IR. Finally, it is important to note that these GLSMs with non-trivial GS anomaly

148 terms are not in fact linear. When the mass of the heavy fermions is taken to infinity, however, corresponding to running to a boundary of the vector bundle moduli space, the only remaining non-linearity appears in the GS term itself, which will generically include globally ill-defined logs. As we shall see in the next section, this need not cause us to panic. However, various of the standard moves used in studying GLSMs must be considered with some care in this quasi-linear setting. Happily, as we shall now explore in detail, we can in fact embed these quasi-linear models within completely standard (0, 2) GLSMs, with the quasi-linear models arising as low-energy effective descriptions of part of the moduli space. The question thus reduces to asking whether the corresponding limits of the vector bundle moduli are well-behaved.

3.2.4 Generating dH in a garden-variety (0,2) GLSM

Surprisingly, the GS couplings studied above, and the non-trivial dH they represent, have been hiding in plain sight in almost all (0, 2) GLSMs studied to date. For example, consider a (0, 2) GLSM for a Calabi-Yau, X, at standard embedding, Vx = Tx. Thanks to (3.1), standard embedding ensures that dH = 0. Crucially, such models generically contain vector bundle moduli which deform the vector bundle away from standard embedding, Vx = Tx. Away from standard embedding, however, we no longer have trR A R = TrF A F. Consistency of the IR CFT, as reviewed above, then requires that dH # 0. But the CY GLSM we started with has no anomaly, and no GS terms, from which to derive a non-trivial dH. Where is the H-flux hiding in this simple GLSM? The key observation is that deforming the vector bundle boils down, in the GLSM, to tuning a set of Yukawa couplings which control the masses of a host of charged chiral fermions. When we study the system at energies beneath the mass of some particular pair of fermions, it is appropriate to integrate them out. However, if these heavy fermions transform in chiral representations of the gauge group (as for example does the top quark in the standard model), then integrating them out leaves us with an apparently anomalous spectrum of light fermions. However as observed in [42] in the context of the standard model below the top mass, this would-be anomaly

149 is precisely cancelled by a GS term generated when integrating out the heavy chiral fermions. We are thus left with an effective GLSM in a Higgs phase with an anomalous spectrum of light fermions plus a GS term which ensures cancellation of the anomaly. This is precisely the form of the models constructed by hand above. It is useful to see this happen in detail in a simple example. Consider a garden- variety (0, 2) GLSM for a Calabi-Yau, X. Since the specific model will not be important for what follows, we will be relatively schematic for the moment, focusing only on the details we will need. For clarity of presentation, we restrict attention to a model with a single U(1). The field content includes a set of chiral superfields, 4D, a set of fermi fields, FA, and a U(1) vector with fermi field strength T. Let the charges of the chiral and fermi fields be Q, and qA respectively (some of which may vanish), and let the fermi multiplets satisfy the chiral constraint, D+L-A = EA (DI), as usual. The corresponding Lagrangian has kinetic terms

2f (ieQIV~ qAV1- ' 1 2 Lkin = - d 0 D- + e^ IF-AJ-A + 2 -T , (3.12) 2 j y e I4e ,

together with a chiral superpotential of the form,

Lw =- d+ A ]LAJA(4) + 1tT) + h.c. (3.13)

Here, the JA(D) are holomorphic functions of the oD, with net gauge charge -qA such that the superpotential is gauge-invariant. Chirality of the superpotential then requires that EA JA = 0. The conventions and component expansions are listed in Appendix (B.1).7 Since the fermions in such a (0, 2) gauge theory live in chiral representations of the gauge group, there will in general be a gauge anomaly. Under a general gauge transformation with gauge parameter A, the effect of the anomaly is to shift the action by, Oc f dO+j AT + h.c. , A= QiQI - EqAqA, (3.14) I A 7A useful tool for performing (0, 2) component expansions in mathematica with examples may be found at http: //www. mit . edu/-edyer/code. html.

150 as can be verified by a standard one-loop calculation. As discussed above, we are interested in UV GLSMs which are completely free of anomalies, so we hereby demand that A = 0, as is the case, for example, at standard embedding.

Among the terms that appear in the action are a host of Yukawa couplings,

I d9+ (DHgqFQ-q + h.c. , (3-15) where 4Dq is a chiral field of charge q, FQq is a fermi field of charge Q - q, '1)H is a chiral field of charge -Q which gets a non-zero vev (01qH 0) on the chosen patch of the target space, and p is a tunable modulus. Such terms can arise, for example, from

IPAJ^A(4) terms in the action.' Below the scale set by p(Oq$Hj0), we can integrate out '1 q and FQ-q to get an effective action for the remaining light modes.

For example, consider a (0, 2) model for a Calabi-Yau at standard embedding, corresponding to a (2,2) point in the moduli space. At such points, the mass terms for the fermions are non-chiral,

Lyuk = v J d+ (a1G FLP + Hi-G) + h.c. oc 91G (7r-04 + ?PUr+) + .(3.16)

where the 7r± have charge -Q, the V)+ have charge 1, and G = 0 defines a hypersurface in X. Since the scalar coefficient is the same for both pairs of fermions, the masses are the same, so while integrating out either pair of unequally charged fermions would generate an anomaly, the two pairs are degenerate so there is no regime in which it makes sense to integrate out one but leave the other.

On the other hand, if we turn on a deformation which breaks this accidental (2,2) supersymmetry, these two chiral mass terms will no longer be degenerate. A simple

such (0, 2) deformation, for example, replaces &1G in the FI term with a general

function, (01 G + p JI). The coupling p represents a modulus of the vector bundle, with the precise geometry of this deformation being encoded in the functional form

8 Similar (non-superpotential) terms arise from or EAr^ terms deriving from the kinetic terms for the fermi multiplets; the basic effect is directly analogous (though details can be phenomenologically different), so we focus for simplicity on the superpotential case for the moment.

151 of J1 . Adding this deformation into the action then gives,

Lyuk oc &1G (7_0i) + (O1G + p JI) (0,+) + ... (3.17)

When IL = 0, we recover (2,2) supersymmetry and a non-chiral mass spectrum. When p is non-zero, on the other hand, the masses of the two pairs of chiral fermions will be different. When [ is large we will generally find that one pair is heavy and should be integrated out while the other remains light and fluctuates. At energies well below the mass of the heavy fermions, the spectrum of surviving light fermions is thus explicitly anomalous.

As the heavy fermions are gauge-charged, integrating them out requires some care. One method is to use the non-vanishing Higgs field to change variables to uncharged fields,

4DIQq FD =((DH)'QF (3.18) which we can then integrate out without concern. However, in doing this change of variables we pick up a Jacobian factor of the form,

DqEqDrQ-q = Dpff> e 6f 1f('H)T (3.19) where C is such that the gauge variation of the resulting GS term precisely cancels the effective action of the remaining light fermions. (See [41,42] for a lovely example of this effect from integrating out the heavy top quark in the standard model.) These anomaly-canceling GS terms then descend, in the deep IR, to non-trivial dH satisfying the Bianchi identity, as discussed in Sec. 2 and in [100,103].

Crucially, for the special case of standard embedding (or indeed at any (2, 2) point in the (0, 2) moduli space), the Yukawa couplings are tightly constrained such that the physical fermion masses and interactions are non-chiral. There is thus never any regime of parameter space or energy in which the set of fermions above, or below, a given mass is chiral. When integrating out the massive fermions, then, all such

152 H-flux generating terms must cancel.

Note that the resulting effective model has an interesting limit in which we send our bundle parameter to infinity, p -+ oo. In this limit, the massive fermions entirely decouple and we are left with precisely the theory we started with, minus the fields

Note, too, that the resulting models should be treated with some caution. In particular, the (0, 2) GLSM with which we began is as well-behaved as one could hope. Fore example, it enjoys a non-singular topological chiral ring which varies smoothly with the moduli, and the unbroken (0,2) supersymmetry together with the linear model structure ensre that world sheet instantons do not generate a spacetime superpotential [120,121]. However, once we take the limit p -+ oo, the assumptions going into those arguments no longer trivially hold, so these results must be re-evaluated. Doing so, however, is hard: since the GS terms are non-linear, and indeed logarith- mic, computing the quantum OPEs does not trivialize in the UV. This motivates us to search for more tractable variants of these models which have more gentle, and computable, physics in the UV.

153 3.3 Verifying Quantum Consistency in a Special Class of Models

As we have seen, moving to a generic point in the (0, 2) moduli space leaves us with a non-Kihler manifold supporting non-trivial H-flux. Around such points, it is sometimes convenient to integrate out the heavy fermions, generating a quasi-linear Green-Schwarz model in which the structure of the H-flux is more manifest. The price of this simplicity is losing the manifest good-behavior of the original GLSM. It is thus illuminating to verify that these effective GS models are in fact good quantum- consistent GLSMs which flow to candidate CFTs in the IR.

In this section we marshall evidence that these GS effective theories do, in fact, flow to good quantum-consistent CFTs in the IR. For simplicity, we focus on a special class of such GS models which are fully linear (avoiding the logarithms which generically appear in such effective models). Borrowing a technique from Silverstein and Witten [122], and following a similar analysis to that in [103,114], we identify a chiral left-moving conformal algebra in the UV that has all the properties needed to flow to a left-moving conformal algebra in the IR. The existence of this algebra is equivalent to the vanishing of anomalies for specific left moving and right moving symmetries. We briefly review the justification of this technique. We then present the details of the calculation in the case of a single U(1) gauge group, then generalize the results to the case of multiple U(1)s. We also compute and record the left and right central charges, and vector bundle rank, of the IR fixed point theory as a function of the charges in the original quasi-linear model. We begin by identifying the linear models of interest.

3.3.1 The Models

Consider the following models, which will be the focus of the rest of this section.

As above, we begin with chiral superfields, 4D, fermi fields, I'_A, and a set of U(1) vectors with fermi field strengths, Ta, under which the matter fields have charges Qa

154 and qA. The fermi multiplets again satisfy the chiral constraint, D+F-A = EA ('1 ). The Lagrangian includes kinetic terms,

2 Lkin =- 1 fd 0 ( " Aj-A + 1- (3.20)

where the ea are the gauge couplings, together with a chiral superpotential of the form,

Lw = -J d+ LA jA (D) + 1taT a + h.c. (3.21)

Here, the JA(JD) are holomorphic functions of the 4b, with net gauge charge -qA such that the superpotential is gauge-invariant. Chirality of the superpotential then requires that EAJA = 0. The conventions and component expansions are listed in Appendix (B.1).

Since the fermions live in chiral representations, the gauge symmetry will again be

anomalous with anomaly matrix Aal = QaQb - q"aql. Instead of setting the anomaly to zero, however, we now add to the model a set of Green-Schwarz axions, Y, with shift-charges Qa,

6AY = -iQaAa (3.22)

D-Y = oYi+ (o- Va +iAa), (3.23) 2 standard gauge-invariant kinetic terms,

2 2 Ly = J d 9+k (Y +Y + Q aVa)D-(Yi - Y1) (3.24) and a set of non-gauge-invariant Green-Schwarz terms,

LGS ±aC2 b]lVaAb - al a + (3.25)

The d, specify the axial couplings of the axions to the gauge fields. Note that the anomaly itself is strictly symmetric, while the axial term in LGS has a priori no symmetry. The purpose of the VA term in LGS is to cancel the antisymmetric part

155 of the axial term, leaving the symmetric part to cancel the quantum anomaly. The Y fields may be thought of as Stiickelberg fields for the anomalous gauge multiplets. In particular, for any anomalous U(1) we can introduce a shift charged field, Y, together with suitable GS couplings, so that the anomaly cancels the variation of the classical action. Note that setting the gauge Y = 0 we return to the original anomalous theory, though now with an explicit mass term arising as the legacy of Y's kinetic term. For more details about quantizing anomalous theories and applying the Stiickelberg method to an anomalous action in the non-supersymmetric case see [123].

Example: A Single U(1)

To gain a little familiarity with these shift fields, let's take a brief look at the classical geometry in a simple gauge invariant case, a single U(1) and shift field of charge, Q = 1, but no GS coupling, and then highlight the subtlety when including the anomaly. The action and component expansions are given in Appendix (B.2). For this simple model, the bosonic potential is given by:

U (Z QI,0I12 + k2(y + 2)-r + E JAI2 + EA12. (3.26) A A

The classical moduli space is obtained by restricting the field configuration to mini- mize the bosonic potential: D = 0, JA = 0, and EA = 0. In focusing on the D-term constraint, we see that there are no compact models with a single U(1) and a shift field. There always exists a runaway direction. The vanishing of JA and EA cannot help with compactness, as the holomorphic hyper-surface of a non-compact complex manifold is either a discrete set of points or non-compact. Now consider the same model as above, but with an anomalous fermion content cancelled by a GS term for Y,

LGS _ dO+TY + h.c. (3.27)

One might naively be tempted to plug in the Wess-Zumino (WZ) component expan-

156 sion for T and Y.

LGS -- (D(y + g) - iF+-(y - y) +iA-X++iiA\+) 2 (3.28)

Yielding,

e2 D = 2 (1]Q 012 + (k 2 - d)(y + # ) I

This doesn't make sense classically, however. The Lagrangian is not gauge invariant, and so fixing WZ gauge in the classical action is not possible.

It is of course possible to do away with gauge symmetry completely and write down the full component expansion. After appropriate field redefinitions, the component action is:

IGS = + - iF+-(y - 9) iA-X+ 2 (D(y ) + i -+)

+ - (D(s + 9) - iF+-(s - 9) + iA-(+ + i-(+) . (3.29) 2

Where s and (+ encode the unfixed parts of the gauge field. This is going too far, however. This action, and the corresponding modified D term,

e2 D = - (ZQI,12+(k2 - d)(y + 9) +C(s + g) - r), (3.30)

should not be thought of as a good starting place for analyzing the topology and geometry. In writing these classical expressions down we have neglected crucial one loop effects, not least of which is the fact that the full theory is gauge invariant.

157 Y multiplets vs T 2 multiplets

In the models introduced in this section, the Green-Schwarz axion field is a (0, 2) chiral boson with a shift gauge symmetry:

Y = y + V20+X+ - ij+O+y, e" E C*. (3.31)

This is a particular supersymmetric completion of the GS axion. Instead, we could have chosen a different completion. For example, the torsion multiplet, E in [100, 101,103] of the T 2 models is another completion,

2 E = d + VfO++ - i9+ +a+O, 9 = 01 +i9 2 E T . (3.32)

Choosing a different supersymmetric completion of the GS axion has a natural interpretation in terms of quotient actions of the target space. fi = (0, 2) ensures that the target space is a complex manifold; precisely what the complex structure is follows from the action of SUSY on the real scalars; fixing this SUSY action then fixes the action of the complexified gauge group, and thus determines the topology of the quotient. Inequivalent SUSY completions of the GS-axion thus correspond to different quotient actions. The effective geometry of the T 2 multiplet is analyzed carefully in [100].

3.3.2 Methodology

The algebra of a (0, 2) superconformal field theory consists of a left moving stress tensor, 'TL, a right moving stress tensor, T, right moving supercurrents, g:, and a right moving R current JR. We will rely on the existence of an additional left moving current, JL, to have some control in the IR. These operators satisfy the following OPE algebra:

158 JL(X-)JL(y-) rl-$ JR(X')JR(Y') - 2

JR(X')gR'(Y') -±+Y

gR+(X+)g;i(y+) 2 (x - +)'+(+y

+ X+)± 2JRY+ ) (3.33)

TL~~x)TI,(y)~~~2 TL(!r)T(X)Ty) + _R +2T(+

TL7L(x-) (X I (X--y- )2 RX)T( ) - 4~W(x- T +)' (+y

++ 9TL(Y-) -Y

,. JL()L + '9-JL(y-) 3R(Y+) O+JR(Y+) 2 + TL@§-)JL(y-) (X--y-) 2 X--y- TR (X+)JR (Y+) (X+-y+) + +Y

In this section we identify theories which are believed to flow to such superconformal field theories with nontrivial central charge and vector bundle rank at their IR fixed point. We find these theories by constructing models that posses such an algebra in cohomology even in the UV. Though this is neither necessary nor sufficient to guarantee the existence of the IR algebra, it is usually taken as strong motivation [101,102, 114, 122].

The (0, 2) supersymmetry algebra contains the anti-commutation relation:

{Q+,Q+} = 2P+. (3.34)

9 1t is conceivable that the UV (0, 2) algebra does not describe the IR fixed point; further operators may appear near the fixed point, the theory may become trivial, etc....

159 As such, elements of Q+ cohomology are in one to one correspondence with left moving 10 ground states. Furthermore, correlators of cohomology elements are protected. This motivates searching for the left moving components of the superconformal algebra in cohomology. It turns out that it is more convenient to identify states in D+ cohomology rather than Q+ cohomology. This is not a problem, as the two operators are conjugate. Even after finding candidate chiral currents, one might imagine that calculating the OPEs would be difficult in the presence of a superpotential and gauge interactions. As it turns out this is not an issue. Due to the magic of supersymmetry, there is a dramatic simplification when considering left moving ground states. As explained in [122], the superpotential comes with a dimension-full parameter, M. By power counting, any term in the operator algebra that contains a factor of A must also contain a factor of x 2 and so vanishes in the x+ -+ 0 limit, while the gauge interactions flow away in the UV. This means that we can use the free field OPEs in calculating the algebra. So far this has just been a discussion of the left moving part of the algebra. As we will see in the following sections, however, the existence of this left moving algebra relies heavily on having a non anomalous R-symmetry. Once this R-symmetry is discovered, the rest of the right moving algebra is guaranteed, as long as supersymmetry is preserved. In the next few sections we walk through the construction of the currents and the calculation of their algebra for the models of interest.

3.3.3 Gauge Invariant Model

To begin with, let's examine the case of a single U(1) and shift field coupled in a gauge invariant fashion. We consider a model with fairly generic field content:

4, F-A, Y (3.35)

10Imagine that the action depends on some parameter, t, multiplying a Q+ commutator, L = £o+t[Q+,0], then correlators of Q+ cohomology elements are independent of t. A specific realization of this will allow us to compute the OPEs.

160 with the usual chirality constraints

+I D+Y =0, D+L-A = V 2 EA. (3.36)

For now, we will focus on a single U(1) gauge field. The action is

L J d20+ i eQIVD-5 + i k(Y + Y + V)D-(Y - Y) + E e ^V)

A J d9+ AF-Aj + h.c. (3.37 2 A

Here EA EA JA = 0 in order to preserve supersymmetry. This action is classically gauge invariant. In what follows we will write down the conditions for the gauge symmetry to be anomaly free.

Equations of Motion

The equations of motion for these models are:

U+ (AVL-) = l jA (3.38)

5+ (eQIVPA) = -41 E L.AIjA + iv/ e^Vr-AaIEA (3.39) A A k2 D+ (-Y) J +ei v EA ^ -A YEA (3.40) A

+ - Q D +EeAt-A -A + kQ2+i I A

Global Symmetries

We are interested in models which posses global symmetries, JDI -+ e-ia'E4D, in addition to the gauge symmetry. In particular we want candidate U(1)L and U(1)R symmetries. We need further constraints on EA and jA for our theory to be invariant under these global symmetries. First of all, IF-A and jA must have opposite charges; jA (Y - irE, e-i'f() - eW/AEJA; while EA and rA must have the same charge; EA (Y - iKe, e-iaEPI) = e-iAEEA. These relations imply the following quasi-

161 homogeneity conditions.

aS&IaI JA + KOYJ A + IAJA = 0 (3.42)

5 O11&9EA + KOyEA - /AEA = 0 (3.43)

Where K is the charge of the shift field, Y, a1 are charges for the chiral scalars, LDI, and 3A are charges for the fermi multiplets, F-A. One needs a little more care for the R-symmetry due to the fact that both 0+ and

D+ have R charge +1. Preserving R-symmetry requires F-A JA to have R charge +1.

Thus the jA have R charges -OAR + 1, while the EA have R charges OAR + 1. The quasi-homogeneity conditions corresponding to the R-symmetry are:

a 1 + KRYJ A + AR _ )jA

5&pIIo9IEA + I Ra9YEA - (OAR~ + 1)EA 0

Now that we have our equations of motion and quasi-homogeneity conditions we can begin to search for the chiral superconformal algebra. In particular we will identify chiral superfield currents J+ and T++, whose lowest components are gauge invariant conserved currents which satisfy the conformal OPE relations.

We begin by discussing the various U(1) currents in our theory,

U(1) Currents

In our class of models, there are three U(1) symmetries of particular interest: U(1)G gauge, U(1)R and global U(1)L symmetry. Each of these plays a critical role in constructing the IR theory and the (0, 2) superconformal algebra. The gauge symmetry effects a quotient of the target space, and ensuring gauge invariance at the quantum level is crucial. The U(1)L symmetry descends to the IR and can be used for the GSO projection in string backgrounds. As we will see, having a non-anomalous U(1)R current is essential for constructing the chiral, left moving stress tensor.

162 Let us start by exploring the gauge current.

2 + ==Q 1 1 D- 1 -- qA'-AI-A - ik D_9 (3.44) A

jG = i Qj1+O1 - QQi +j + ik 2 P+y (3.45)

Note, the current is neutral, so it is conserved both partially and covariantly. The superfield completion is

2 JG= (-i eQIVQiD i + i e Ar _ A + k EDY) (3.46) \ A

J( B+ QiND+ (eQIv&i) + k2 D+ (Y + . (3.47)

It is nice to see that J- is D+ trivial. Thus the interesting contribution of the gauge current in cohomology comes purely from JG.

In order to check the chirality of JG, recall the EOM:

1 2 + (0-y) = 5 eQIVQi4DjD_4 + i S ^AP-AIP-A + k TY = i4q.,48) 22 A

JG is exact up to equations of motion. DU+JG = 0 follows from the fact that B2 = 0. Another way to see this is to apply T+ directly to JG. Using the classical EOM and the quasi-homogeneity conditions, one can explicitly check that J is chiral.

For any other global symmetry, in particular for U(1)L, the procedure to find a gauge invariant chiral current is similar and we get:

J =+ eQIVa i D .. + i e ^ + k iLD _Y . (3.49) A

Classical chirality, U+JA|EOMs = 0, can be checked using the EOM and the quasi- homogeneity conditions.

The R-current is a little trickier, as the component fields come with different R

163 charges. The lowest component of the R-current is:

3 ij+ = # --i E5 AY-Y-Ak-A+k2 RD- + (3.50) A 2 iij = -( (aI)+01 - i (a - 1) +±i+±)- k2RD+y - ik +X+ (3.51) which is gauge invariant and conserved.

Further Modification of Chiral Currents

A classically chiral current for a given symmetry, Js+, can fail to be chiral in a quantum theory. The supersymmetric extension of the chiral anomaly, known as the Konishi anomaly [124], in 1+1 dimensions 1, tells us that

- As T (D+J+) = ( 7 )(3.52)

where As is the anomaly coefficient for the given symmetry. Generically it is not possible to remove this anomaly by redefining the current in a manner that preserves gauge invariance. As noted in [114], however, when there is a shift charged field in the game we can remove the anomaly. This freedom to redefine the current is easy to understand by looking at the bosonic components. In two dimensions the anomaly is given by o9j" oc c""o,A,. It is possible to define a new conserved current, j'" = j#- Ed"'A, but this is not gauge invariant. If we have a shift field, 0, at our disposal we can do better and define, j" = j" - "'(o9,0 + A,). This is gauge invariant and conserved. This new current corresponds to adding a term of

the form O&"F,, to the quantum effective action and then improving the resulting current. Note, that this addition to the action is just the form of a Green-Schwarz term. The superfield completion of this effect is:

(3.53) J + s, i+ + i 47rApnDY.

1See, for example Appendix C of [114] and section 3 of [120].

164 The modified currents is no longer (classically) chiral:

T. (3.54) B+ is++ i 47r DY = 8,7r

If we choose ANs = As, however, the classical non-chirality of this modified current exactly cancels the one-loop contribution12.

Stress Tensor

Now that we have understood the U(1) currents, let's move on to constructing the chiral stress tensor. The unimproved stress tensor can be obtained from Ndether's procedure, however this is not gauge invariant. Using integration by parts, the gauge invariant superfield completion of the stress tensor is:

+ i e* ^V,-AE- _A + k2EYDY . T + - Ta _ + eQIVD_4DD_di S 2 I4e A

This is equivalent to promoting the gravitational stress tensor to superspace. Acting with T+ on To+ and using the EOM and quasi-homogeneity conditions we get:

1 +T+ 2 (-A). (3.55)

Thus To+ is not quite chiral. We would like to improve it to a chiral stress tensor without ruining the conservation. To achieve this, recall that a conserved current JP can be improved to J'" by adding the divergence of an antisymmetric operator.

J't = J1 + &vKl"t1 , - j't = a, (Jt + 0,K EAv]) = 0 (3.56)

12For some theories, further modification of currents is possible. Adding iID_Y, for instance, works for theories in which JA and EA are independent of Y as the EOMs give T5+ (D-Y) = 0. One example of this kind of theory is the HK [114] model.

165 We want to find a K"] that renders T++ chiral. In our case this will be of the form,

CAW&LY.

As demonstrated by Silverstein and Witten [122], given an R-symmetry, finding K"I" is a simple matter. Consider the following, non-conserved, current:

R ~-Z e I0Z - E qAV (OA 1)-AL-A k2 R_Y . (3.57) ( I A

Where ai are the R-charges of the chiral scalars, #3 are the R-charges of the left- moving fermions in the fermi multiplets, and ,R denotes the R-charge of the shift field.

We emphasize that this is NOT the R-current. This would be a regular chiral current for a global symmetry under which all components of each superfield transform with the same charge. However, as the quasi-homogeneity conditions corresponding to the R-symmetry are different from those for a standard global symmetry, the given current is not quite chiral.

+R+ EOMs = -2/ J-AJA, (3.58)

In fact, J1+ is not chiral in exactly the correct way to compensate for the non-chirality of TO+. Now, we are able to construct a chiral stress tensor:

T++ = + 0T +

1 ( - rv-qAV ~AD-7 F-A + k2D2Y)2 -2 (Te TOa-T + E eQ ' DADA + j ~ 2 I A

+ (4& e a.e - -|+i WA + 1)-A-A + k2 R hY I A (3.59)

166 such that

15+T++ EOMs = 0- (3-60)

The lowest component is:

-++_ , D-IID-I + i IY-AD_--_A + k-y-Y + 2ED-\I SA

+8 a((a( o'b#D-4I+ i + 1)A-A7-- + k2 r YD~) (3.61) A which we identify as flowing to the IR stress tensor TL.

To summarize, exploiting the existence of an R-symmetry and its quasi-homogeneity conditions, we are able to identify a chiral, gauge invariant, and left moving stress tensor.

Operator Product Expansion for Chiral Operators

Now that we have constructed candidate chiral operators, let's check the operator algebra. We consider a larger algebra, containing the gauge currents as well as the stress tensor and U(1)L current. The OPEs of the gauge current give the gauge and chiral anomalies. When calculating the Operator Product Expansion of the currents and stress tensor, we may use the free field OPEs for our component fields.

In the UV, where e -+ 0, we can rescale our gauge field strengthT -+ eT to go to a free theory. The chiral operators in the free theory and the R-current are given below. We have not included the potential modifications of section (3.3.3) because, as mentioned, they correspond to using the Green-Schwarz mechanism, which we address in the next section.

167 - 2a~ j+ = O 1941--I+E q A-A,-A ik I A - ik2 L JL+ a 1,1 i01 + E IAY -A-Y-A I A 2 ++ k ay&- + ! O -\ A

+- +- I A ~ + 1)!-AY-A + k2RO ) E 11 - JR+ a 1 - + 1A-AI-A - ik2 % O4 + -+ - A 2 2 + ik IKRO±g - k ,j+X± (3.62) =R (a Sa - (a -IV)+,r~~)

168 The singular part of the operator product expansion for relevant operators are:

(QIQI - qAqA) j+(x) j+(Y) _ - )2 (- A /

qA3A jG(x) j-(y) - )2 (Q L~ (- _ A I (E aic~ j 111(X) jj+j(Y) - ( EA 1X )

j(x) j++(y) -) 2 1_ (x I A

R a - (a -_1)2_ I I X+

j+ (x) ++(Y) 2( y-) 3 ( Qa R Qi - I A - _ _R2 - - _y-)

2(x- y-)3 ( I 2- (x - )2 (x - y-) t++(W t++ (3(af - W) - 1) + 3(# )2) (Y 8(x--y- E(1 3

I t++ a ++ (3.63) 2 (X- - y-)2 (X- - y-)

Conditions for Conformality

As mentioned previously, classically chiral operators can be anomalous. In order to check the chirality at the quantum level one should investigate whether chirality holds within correlation functions. It can be shown that checking this is equivalent to the vanishing of the most singular terms in the gauge current OPE relations, see appendix B.4. Requiring the existence of a chiral, (0, 2) superconformal algebra yields

169 the following anomaly cancellation conditions:

U(1)G U(1)G: QiQi - qAqA = 0 I A U(1)G U(1)R: I A

U(1)G U(1)L: Qiac - 5qA = 0 I A U(1)L U(1)R: Lf (aR - 1 # A- = o. (3.64) I A

The charge and rank of the vector bundle in the IR theory may also be gleaned from the leading singularities of the OPEs.

rL JL(x) JL (Y) 2 (3.65) (x- - y-) 1 CL 2 T () TL(y) 4 + - 2T(y) + 1a T(y) (3.66) 2 (x- - y-) (X- - y-) (x-y) 1 ( R JR(X) J(y) (3.67)

The last equation is equivalent to calculating the leading coefficient of the j+j+ OPE and subtracting the leading coefficient of the j-j- OPE. This gives one third of the right moving central charge UR = ICR. It is conventional to normalize L = -2t++. Then the UV OPEs tell us that:

CL = 1)2-1 ( 13c 2) (3.68) I A

rL = (3.69) I A

CR = a AOR (3.70) I A

Note, CL - CR = EA 1 - EZ 1, which is an RG invariant quantity as expected.

170 3.3.4 Anomalous Model with Green-Schwarz Mechanism

Now that we have warmed up with the classically gauge invariant case, lets consider adding to the action the following non-gauge invariant piece, to incorporate the Green- Schwarz mechanism.

LGS = dO+Y-T + h.c. (3.71) 4

Here we take d to be real which represents a convention for the periodicity of the imaginary part of Y. The motivation behind adding this term is that this classical, gauge variant piece will cancel against the quantum anomaly. This is reasonable as the classical variation of the added term has exactly the same form as an anomaly.

We follow the same procedure as before, but now with the GS term. The equations of motion are slightly modified.

-jA U5+ (e^vqVrA (3.72)

U+ (eQIVD_4 j JA + iVE e^Avr (3.73) A A

2 k T+ (D-Y) i -A ACj - iV E eZ^V _A Ee - - T (3.74) A A

+1- 2 -S I + i E eAVqp -A -A + k 2XY I A

It is interesting to note that acting with D+ on the last equation and using the quasi-homogeneity conditions reveals:

dT = 0. (3.76)

When the shift field is charged and the GS mechanism is in play, the field strength multiplet vanishes on-shell.

171 Modification of Currents: Stress tensor and U(1) currents

The action is no longer gauge invariant. As a consequence, the canonical currents transform under the gauge variation. All is not lost, however. Thanks to the shift field we may improve the currents to be gauge invariant.

Let us consider the contribution from the GS term to the currents. An interesting feature of the GS term is that, though it destroys classical gauge invariance, it preserves the global part of the symmetry. The action shifts by a total derivative.

Recall that when the action changes by a total derivative under a symmetry transformation,

6L(x) = ajK",

the conservation of currents get shifted.

,,jm(x) = 6L(x) - 6S (x) = 4pK"(x)- 0a(x ) 6#a(x) J#a(x)

9,y (j"(x) - K"(x)) = 0. (3.77)

The shift field transforms as y -+ y - ir, and the GS term leads to the variation:

1 J =G rav - 0-v+), (3.78) 2

implying K+ = -!dv_. Thus the effect of the GS term on the currents is:

2 j+ _ j+ + Idrv_, (3.79)

which is not gauge invariant. This is plausible as the GS term is not gauge invariant. Luckily, as mentioned, the shift field allows us to improve the current to a gauge

172 invariant form.

j+ -+ j+ - idna-y (3.80)

The supersymmetric completion of this is:

J+ 3 J+ - idtD_ Y. (3.81)

Notice that this is exactly the form of the current modification in 3.3.3. Thus with the GS contribution our supersymmetric currents can be written as,

J+ Q1VQvQj4j)Aj eqAV qAr-A E-A + k2E)Y )y (.2

2 J + ( QIVo i'I-0hi i e ^ o -A-A + k 'CY + Y (3.83) zI A

These are gauge invariant and chiral, as before.

A little more care is required to construct the stress tensor. Recall that the canonical stress tensor is not always gauge invariant. In fact, it can in general only be improved to be gauge invariant on shell. Varying with respect to the metric, however, produces a gauge invariant, symmetric, conserved stress tensor. Indeed this is one way to derive the expression for TO++, (3.55). This method of constructing the stress tensor is particularly nice. The Green Schwarz term is metric-independent and so does not contribute to this definition of the stress tensor. The improvement term does change. Just as for the regular currents we have:

+ -+ J+-i DY. (3.84)

173 Putting this together yields an expression for T++

Tit ia- QI - -~q - _ 2 -r& -i + eQIVDibhi + i e*^Vp-AD -A + k2 _yE) \ I A

+ (- eQIc1Di + i Z ev RA + 1)-A-A + 2 R A (3.85)

The chirality of this expression is ensured by the vanishing of the field strength multiplet.

Conditions for Conformality

With the improved gauge invariant currents, we can obtain constraints on the existence of an IR conformal algebra. As in the gauge invariant case (B.4), we check that the chirality condition is not anomalous. The details of the calculation are relegated to Appendix (B.3), but the results are presented below. For our model with GS term to flow into a (0, 2) superconformal field theory, we require the following anomaly cancellation conditions:

U(1)G U(1)G : SIQI - qAqA - 2C = 0 I A

U(1)G U(1)R : Q I ~-~ ~ qAOA ~ 2 R I A

U(1)G U(1)L : E Qa' - 5 qA/A - 2CrL = 0 I A

U(1)L U(1)R : a L (aRf- 1) - # - 2Ld =0. (3.86) I A

When there is a non trivial IR theory, the central charges and the rank of the vector bundle can also be identified, including the modifications from the GS term, this

174 gives:

CL = (3(a - 1)2 - 1) + E (1 - 3()2) - 6d (3.87) I A rL E0a - 2 . (3.88) I A CR =2 A _2 (3.89) I A

3.3.5 Multiple U(1)s

Now that we have warmed up with a single gauge group and shift field, we are ready to take on the more general case. The action is written in equations (B.17-B.22), but we repeat it below.

1C = k +1 -+ 1W +,CGS + LFI (3-90)

Where:

Cgk = - d20+ a Ta (3.c a I 4a

20+ eQia 2 Y7 + QaV E)_ (y - £21 2 (y+ 1 a) 1 Y) + 5e A-A A

Lw f-= + I ALJ + h.c. (3.c 2 A

2 'GS = ~/ 0+ [a b]I _-ab al f dTaYl + h.c. (3. ab,l a,l

LFI ta do+Ta + h.c. (3.E 4aI

175 Equations of Motion

The equations of motion differ only by the addition of extra indices.

= -, JP D+ (e A aF-A) (3U

D+ (eQV-aD ) iIJ -- A + i- _ A ":-AalEA (3. A A

k2D+ (D-I) -A~lJ + iv r _QE A - dalx (3. A A a 1 + 2(kQ - _da)(y +y + QbV) + a blv 2e2 (D+Ta - D+Ta) bVbQI12 S I 1 b,1 (3.

dal ) (Y ~Th~) 5 (aQbIl 2e (D+-a + D+a-Ta) S I ?-Q I> + (kQ + I I b,1

- 2i5e q _Ar _A (3.1( A

Combining the last two of these yields:

1 +a-Ta = eQ0bQ(ID- + - daly) S a b A - iaVb) 2a S (kQiI 2 b,I ±i5:eA+i eil~gi-AIF-A (3.101) A

Notice that acting with D+ on the above equation, we find:

0 = Qf )bT (3.102)

Modification of U(1) currents

Now let us look at the various currents. Constructing the gauge invariant conserved currents is a little more involved. The results are:

j~a±-5 QVbQ D~I +I i eq~bqp +S +Ual&a + b1 a

T A

176 iJ= QaegVaLo (b1D 1 + iJ~ q -A -A + E (i2 ( KLl Yi 1 + b (Ab ~ 0-1 A

Here the Us are chosen to make the various currents gauge invariant:

Qb = EdblQa (3.103) ualUQt 1 S~ 1 = E dbl K L (3.104) 1 1

It is not true that every model admits Us satisfying these conditions. Consider, for example, a model consisting of two U(1)s, with a single shift charged field Y, charged under a single factor, Qa = (0,2), with a particular GS term specified by Oa = (1, 0). This gives the symmetric anomaly matrix and anti-symmetric matrix:

0 1 d(a Qb)=

0 1 d[aQb = (3.105) C-1 0 Bosonic component fields of the GS Lagrangian can be written as:

CGS = OE""Fu - 2cI"Aj,,A 2,. (3.106)

The variation of this term is symmetric and can be cancelled off of otherwise anomalous fermion content, rendering the total anomaly zero. Despite this apparent good behavior, the term contributes to the U(1) currents in a gauge variant fashion.

oc 9e"Ai +... (3.107)

Since Y doesn't shift under the first U(1), this cannot be improved to a gauge invariant expression. There exists no U for this theory.

177 The existence of gauge invariant currents imposes non-trivial constraints on the

matrix Cal. Of course, if Q is invertible, i.e. if there is a shift charged field for every

U(1) factor, then Ua and UL exist. These constrains should imply non-trivial restrictions on the types of geometries that can be realized via (0, 2) Green-Schwarz GLSMs. It would be interesting to understand the larger implications of these equations and whether they have a natural mathematical interpretation.

The stress tensor is defined in a similar fashion as before, however now the existence of a gauge invariant R current requires a UR. The stress tensor is given by:

1 V a -I a- T++ ( Z T'D - Ta +L eQ - + i S e -ADV-r-A + k DYD-Y a A +( -Ya +RAir. \ A A (9- IEeai 4DID -1AXYDI + i E e (WA + 1)L-A1LA +5 k A

+ L4 t-Is + 2 C rl (Ab - iDa/b)). (3.108

With:

EURlQb =5bl. (3.109) 1 1

T++ is manifestly gauge invariant. Chirality follows from the equations of motion and the quasi-homogeneity conditions.

Conditions on Quantum Chirality: Multiple U(1)s and Shift Fields

When there exist improved gauge invariant currents, we can use them to obtain constraints on the existence of an IR conformal algebra. The details of the calculation are in Appendix (B.3) and only the results are presented below.

178 The anomaly cancellation conditions for multiple U(1)s and shift fields are:

U(1)a U(1)b : QaQb - q'4qi - 2 u(a Qb) -0 I A

U() U(1)R : - Qi- q IR 2 d~xf= 0 I Qia I A U() U(1)L : I A

U(1)L U(1)R : aI a - I SL - A L3~~- ~,~ (1O I A 1

One can also calculate parameters of the CFT and central charges get contribution from multiple gauge generators and shift fields

CL V2 11) a - 1) #OR21) A - -6 ll Rr 2 a rL = (3.112) I A cR5 A a- (3.113) A a

Again CL - CR is manifestly invariant.

3.4 Conclusions

In this chapter, we have argued that non-Kiihler geometries with H-flux satisfying the modified Bianchi identity can be described by standard (0, 2) GLSMs without any additional structure. At low energies, the non-trivial H-flux is realized through a set of effective Green-Schwarz terms canceling an anomaly in the fermion measure. These Green-Schwarz terms arise by integrating out charged fermions which are massive in the local patch of the moduli space; the Green-Schwarz terms ensure the cancellation of the total anomaly after truncating to the (would-be anomalous) spectrum of surviving light fermions. This clarifies, for example, how dH -, 0 arises in (0, 2) deformations of (2,2) models corresponding to deformations of the vector bundle away from standard embedding. This improved understanding of the familiar (0, 2) GLSM

179 then allows us to realize various previously-constructed quasi-linear models for non- Kdhler manifolds with torsion [100, 102, 104, 105] as effective descriptions of certain patches of the moduli space of elementary (0, 2) GLSMs. As an independent check of the consistency of these effective descriptions, we studied the quantum consistency of a simple class of such quasi-linear models involving an anomalous gauge group together with anomaly canceling Green-Schwarz axions. Effectively, this linearizes the GS sector. This allowed us to show, following [101,122], that GS anomaly cancellation in these models ensures the existence of an off-shell .N = 2 superconformal multiplet whose OPE algebra closes correctly within Q±- cohomology. We then used these OPEs to compute the central charge of the (0, 2) SCFT to which the theory is expected to flow in the IR. Many questions remain about the geometry and moduli space of such (0, 2) GLSMs and their quasi-linear effective descriptions. At a technical level, it would be reassur- ing to explicitly derive the equivalence between the NLSM and GLSM completions of models with trR A R -A TrF A F by computing the full one-loop effective action of the GLSM and verifying that it is in the same universality class as the standard NLSM construction of [25, 27]. It would be surprising if these construction do not agree, since that would give us a new way to complete naively anomalous string theories to good compactifications. Much more interesting are questions about the global moduli space of generic (0, 2) GLSMs. What, in the GLSM, distinguishes models which topologically admit a Kdhler structure (such as small deformations of (2,2) models) from models which do not (such as the non-Ksihler T2-fibration GLSMs of [100])? Relatedly, when can two such models (one admitting a Kdhler structure, one not) be realized as different phases of a single underlying GLSM connected by a smooth quantum transition? More generally, what is the global moduli space of a (0, 2) GLSM and does it contain multiple inequivalent (2,2) sub-loci which are embedded smoothly? To this end, it would also be of great interest to be able to compute topological invariants for specific non-Ksihler manifolds using the structure of the (0, 2) GLSM.

Another interesting question is whether worldsheet duality along the lines of [113,

180 125-127] might be applicable to the GS axion models. On the surface it would seem less than useful - under such an abelian duality, axial couplings of a scalar to a gauge field are exchanged with canonical couplings, but in the present models our GS scalars are both axially and canonically coupled. However, the role of the anomaly may alter the naive dualization. Even if the anomaly just goes along for the ride, however, such dualities may prove helpful in patching together a clear picture of the moduli space of these theories. There are clearly many interesting geometric questions waiting to be attacked through more careful analysis of the torsional moduli space of (0, 2) GLSMs. We hope to return to them soon.

181 182 Appendix A

Monopole Harmonics

A.1 Definition and Properties of Monopole Har- monics

A.1.1 Scalar Harmonics

Definition

In this Appendix we review some properties of the monopole harmonics. We start

with the scalar harmonics introduced in [16,83]. The monopole harmonics Yq,m (h) are eigenfunctions of the angular momentum operator in the presence of a monopole background of charge q, (2.1). In this background, the angular momentum operator takes the form:

LZ = - q - 2q (A.1) L= _V2 + - 2 (cos0 - 1)Lz.

The scalar monopole harmonics are defined to satisfy the relations:

= £( + l)Yq,em(?), (A.2) LzYq,em(n^) = mY,em().

183 We can write the solutions to these equations explicitly in position space:1

(1 + x)-m p(-q-m,q-m) (cos )e(m+q)o. Y m(ft) = 2m1 (2( + 1)(f - m)!(f + m)! 'V r(t - )(t + ) (1-x)q+m f+m (A.3)

It is sometimes convenient to write the monopole harmonics in bra-ket notation.

Yq,tm(^) = (0, #0, m) . (A.4)

Identities

The scalar monopole harmonics have a number of useful properties [83]. Under charge conjugation the monopole harmonics transform as:

(A.5) q~fm(ii) = ( )q+m) Y-q,f,-m(t) .

When evaluated at the north pole, the scalar harmonics satisfy.

5 (A.6) Yq,i,m() = q,-m ' 4wr+

Gauge invariant products of monopole harmonics also satisfy integral relations. The monopole harmonics are normalized such that

2 Yq,tm(ii)J = 1. (A.7)

'Recall that ft is a unit vector parameterizing the two-sphere and just shorthand for 0, <.

184 The integral of a product of three monopole harmonics is given by

dftY,iM(f)Yq/ 7fm/ (h)Yq//j,/1/ (h)

M' £/ q (A.8) 1) f (2f= + )(2f'/ 1)(2/1"+ --1-1 ) (+/'+ 47r M mJC;qq' q"/ A.1.2 Spin s Harmonics

Now that we have the scalar monopole harmonics for arbitrary angular momentum, Yqm (h). It is easy to construct monopole harmonics of arbitrary spin using the Clebsch-Gordon decomposition. Explicitly, we have:

Isf; j, m)q S ( ;mmi1s £;i,m)!S,m,)0 ,me)q. (A.9) m 8 =-S mI=-f

Here, (s f; msmels t; j, m) is the usual Clebsch-Gordan coefficient, which can also be written in terms of the Wigner 3j symbol.

Ji j2 i (ji j2; m1m2|ji j2; j, M) = (-j1)2-+m 2j+1 . (A.10) Mil M2 -M

The Clebsch-Gordon coefficient is zero unless ji, j2, and j satisfy the triangle inequality, ji - j21 5 i5 ji + j2l. In (A.9) the scalar monopole harmonic |l, me)q vanishes unless ;> Iqi. Together these relations imply that the state Is t; j, m)q only exists for,

j ;> qI - s. (A.11)

In this paper, we often decompose fields of fixed spin, s, and total angular momentum, j, in terms of sums over orbital angular momentum, t. The only terms that contribute

185 have

lij - s1i t j+s, and f > jqj. (A.12)

For large j, (A. 12) gives 2s + 1 states. For smaller j there are fewer allowed values for f.

A.1.3 Spin 1/2 Harmonics

It is useful to have explicit expressions for the spin 1/2 monopole harmonics. The number of independent states depends on the value of j.

j =Iqj - 1/2

From (A.9) we see that there are two states with s = 1/2 for each j when j ;> qi, 11/2 j ± 1/2; j, m). In position space these take the explicit form:

2+±Tq j-1/2,m-1/2(f) (6, #11/2 j - 1/2; j, M) Tqjm(?) =-

=-1/2,m+1/2(ii)) (A.13)

(0, #11/2 j + 1/2; j,m) Sq,jm(f) -

2+2j -qj1/2,m+1/2() j = jqj - 1/2

For j = IqI only the single mode Sqjm(n) exists.

186 A.1.4 Spin 1 Harmonics

The spin 1 case is similar to the spin 1/2 case, except that now there are two special values of j, j = q - 1 and j = q. The vector harmonics take the form:

j > qI

U+m-1)(j+m) Y 2j(2j-1) ,

- 1; j, m) Wq,jm(n) = (i-m)(i~m)y (9,4#11 j V j(2j-1) q,j-

\ /

_(i-m+1)U+m)y 2j(j+) q,j,M-1

(0,1 j; j,m) Vjm(n)1 = (-m)(+ml)y 1 iii (A.14) i V(-, +j) q, j m

/U-M)U+m+1)y. g V' _2j_(4+1)+ q, ,M+1(

/j-m+1)(j-m+2) Y j~lm (2j+2)(23) q,+,-

(0, #11 j + 1; j,m) = Uq, jm(h) = _j-m+1) +1)Y j ++ 3) q,j+l,m

U(+m+l)(i+m+ 2 )y (2j+2)(2j+3) q,j+1,m+1(n,,4 j = jql

If j = jqj we only have the last two modes, Uqjm(i) and Ve jm(ft). If j = q = 0, only the Uqjm(ft) mode is non vanishing. j = jqj - 1

In the case j = IqI - 1 only the mode Uq,jm(iz) is non-zero. This mode plays a key role for the stability analysis of monopoles.

In order to check gauge invariance of the effective action, (2.20), it is useful to have

187 an expression for the divergence of the harmonics. The gauge covariant divergence of the vector monopole harmonics with charge q is

(j + 1)2 _ 2 ,(e-wUjm(O,(ec~j(9' -WT 0))~))=(jiW) j(+j (j+ +1)-q 1)(2j+ 1)- Yq,jm(, ) ,

DA(e-iwrVtm(O, #)) = - iw - Yq,jm (0,)e , ((A.O15)

(j1 + j(2j+1)

188 Appendix B

(0, 2) Details

B.1 Conventions

B.1.1 Coordinates

We work in 1+1 space-time dimensions with coordinates x0, X1 and Lorentzian metric. For most of the calculations we use light-cone coordinates. The relevant formulae are:

(1c) = - 1 X*=1 (X0 ± X ), 09± = 90 + 1, g + = 6+- 1 (B.1)

We have superspace coordinates, 0+, and 0+. Integrals are normalized as:

dO++ = - d6+6+ = 1. (B.2) j d2 00+#+ I

We also define superspace operators:

a0 a Q+- a+ + i9+, Q+ + - i0+a+ (B.3)

D+ = 0+ - i +, D+ = + i+a+.

189 These satisfy:

D = 14 = 0 (B.4)

Q2= Q2 =0

{D+,D +} = 2ia+ {Q+, Q+} = -2ia+

{Q+, D+} = {Q+, D+} = 0.

When symmetrizing and anti-symmetrizing indices we take:

M[a,b] = I (Mab - Mba) , M(a,b) = (Mab + Mba) (B.5)

B.1.2 Superfields

A chiral superfield,

D =0 + /2+ + - iO+#+a+o (B.6)

F = --\20+G - f2+E - iO+#+a+-. (B.7)

We also use a shift charged chiral field, Y. We take it to have the component expansion:

Y = y + VO4+X+ - i9+6+a+y. (B.8)

In addition to the chiral fields there are gauge fields. As apposed to the (2, 2) case, which requires a single superfield, there are two (0, 2) vector fields, A, V. In Wess- Zumino gauge these have the component expansion.

V = o+U+v+ (B.9)

A = v_ + \0_++ 20+#+D. (B.10)

190 From this we can construct the fermionic, gauge invariant field strength, T.

T = D+(A - iaV) (B.11)

= -iV- (A + v0+(Fo1+ iD) - iO+6++A-)

Foi = ovi - 91vo is the gauge field strength.

We have both conventionally charged fields, 4, F, and shift charged fields, Y. The gauge transformations with chiral gauge parameter A are summarized bellow.

F -+ -iqAFp

Y -+ Y-iQA

V -+ V+i(A-A)

A A A+&_(A+A)

(B.13)

The different qs, Q, q, and Q represent our naming conventions for the charges of chiral, fermi, and shift fields respectively. To facilitate writing down a gauge invariant action in superspace, it is convenient to define the following covariant derivatives:

D-F = ( + q(oV + iA)) r (B.15) Q DY = aY + -(a-V +iA). 2

Lastly, for the FI parameter we use

t = r + i, 0 - 0 + 27r. (B.16)

191 B.2 Action

Here, for completeness, we record the full action we use for the shift charged field, and the complete component expansion for the gauge invariant case.

The action is given by:

L = Lgk+Lm+w+LGS+LFI (B.17)

12 where 9 k contains the gauge kinnetic terms, Lm contains the kinetic terms for the matter as well as the gauge interactions. Lw is the superpotential, LGS is the Green- Schwarz term , and the LFI is the conventional Fayet-Iliopoulos(FI) term.

Lgk =- d 20+ TaTa (B.18. a

Lm =-1 d 20+ e JE)1_(b +i Y Y + aa -(yl YI ) + E e A T

(B.19'

Lw d+ - IPA + h. c.- A

LGS - + 5[aQblVaE Ab dal J+T aYI + h.c. (B.21 abl a,l

L FI - E5ta d+Ta + h.c. (B.22. a

Restricting to the case of d = 0, i.e. the classically gauge invariant case, the component expansion in Wess-Zumino gauge can be organized as:

192 L = - L -- La + Li

_ aF01a + D+5,D_#+ k 2DD-y Lb = 2F1aF 21 a a

Lf = --aa+A-a + i V)+D-/+l + i S k +i-X+i+ i 5E -AD+7-A 2e 2a a A

+ ik 2 (AaX+i + -aX-t) al

L I 2 D1 2 La = 2e a + 1| (2|2 + Qa/k (Y1 + 9i) - ra ) + IGA|2+ " E(GAJA+ a a A N2-A

Li = 1jAV+1 , + SOJAX+l + h.c.) - '- 1 - x+jajEA+ E-A (1: OIEA

+ Qa (A1 -a)+ + h.c.) - |EA|2 I,a A

The auxiliary Lagrangian can be put in a more standard form by solving for the auxiliary fields.

2 2 La = - EIGA1 - S EA1 (B.27) a A A

Da I I-(?~ Q kQ2 (y (B.28) 2 -

GA - JA (B.29)

Thus the bosonic potential becomes

U =E EQai#i 2 + EQak2(y + 91) - ra + S lEA IB.30) A A A

193 B.3 OPEs

We consider a complex scalar with action Lb = -capoapo = f(+09-0+a-4 and a fermion with action, Lf = ic a/&- 3±. Also we consider the action of the form

S= ! d2 L(x) (B.31) 47r

Then the two point functions are:

1 = -log(x - y) 2 (B.32) C (B.33) cX y

B.3.1 Operator Product Expansion with single anomalous

U(1)

Let us list the currents and stress tensor that we want to calculate OPE's with

2 -G QIOI-b1 + qAk-A7-A ik + 47 9GY - i6y I A - Ld y 2 i 0 L -i IO +- A -A -A - 7L A

&q1qi+ i 5YAa-Y-A + k~ay~ + -Aa-A +Ia I A ) 9 R +E + 1)j'-A-Y-A + k 2 KR 4 a-'y + a 4- a I i5(O3RAR )

jR - qo- IA-A-A -_ 2rR& + -A- - RdY I A

2 R= i (a RoI I - i ( 1) +IV)b±) +i(k 2K~g+y + ik +X+) _ R

OPEs for the currents are:

194 2 QIQI - qAqA - 2) (x- -

j-(x)j-(y) Qi - qAA - 2C L (- _ - )2 (z A j+ (X)jL+(Y) a PLL - 2d ,sL (z i7 (x)i+ (Y) a R aRZ_,QR3R 2(KR)2 ) (- _ -)2 A

2 a Ra- (a R- 1)2 (X y 1+ I )

rl j 2(x- 3 (z I 2KR) A

1 j+G I -j+G 2 2 (x - y-) - (x -y-)

ell j - - 2 KLIR) 2(x- y-)3 ( a S -E IAL3 A 1 i 2 2(x--y-)2 2(x-- y-) t++ 1) +E(1-3(0)2) 6c(R )2 + - (x)t++(Y) 8(x- y-) 4 E (3( -1)2- A y A_ 1 ( t++ -t++ \ (B.31 2 (X- - Y-)2 (- _ _ .)

195 B.3.2 Operator Product Expansion with multiple U(1)s

After rescaling Ta -4 eaTa in the deep UV, the free operators we calculate OPEs are

~ I + I qA -AT-A - -1 i - al A I

-iI - + A -A7-A - -k 4 - -U & I A

+- ka&yll 2- I I A O-A + 1 'Aaa-ka I )

+ k i gRlay) 4- (I 1 A + 1)A A~->A A 1 -a A a O A -A1-A 2K~ a~- l + uRlay JRj+ =i R -E A!--Y- IyI-y Ld A- 2~ 2 I A 1 a

=R (a IO+ - i ( ~~) 79I+I) + (ki i +i ± + - Rigxi+ I 1 1 (B.36)

196 Then the free OPEs of the currents are:

1-1 aqb -2 Q) -j 2 (x A A 1 1+ ja+(X)jl+ Y 11_ M + uLlQa) j+(X)j+(y) Qia - qvAL/f~ - (ual (x -2 A I )

I I A A# - 21

j+ (X)i+ (Y Ra R 2 ( a I~a /RS \I I 11_ A a

j- (x)j- (y) a RaR 1)2 1) -+) (X+ I

3 -a aqo! - Pa(uaR + u1Qa)) j+(x)t++(y) - y-) \Ld 2(x- A

2 G 1 a2ja± 2 (x- _ - 2 2 (x--y-) L (u'L1M +UR j+( x)t+(y) -5OAOA 2(x- y-)3 a A111

1 j+ 1 aj+ 2 (x- - y-)2 2 (x- - y-)

t++ 1) + (1 - 3( )2) -6 '1 + 2-: (x)t++(Y) 8(x - ) (I A I I -1 t++ +-t++ + X 2 (X- -_y-)2 )

B.4 Quantum Chirality

As mentioned above, classically chiral operators can be anomalous. In this section we check that chirality holds within correlation functions. If we only focus on supersymmetric vacua then:

([I,J(x)] 0(y)) = -(J(x) [I-+, 0(y) ]) (B.38)

197 In order to verify chirality we want an operator 0(y), where the action of D+ is known, following [101], [122], we choose 0(y) = OT(y).

Using the equations of motion D+O-T = 2ie2 JG+(y). A necessary condition for quantum chirality is:

(D+J(x) a_.T(y)) -(J(x) D+O-f(y)|EOMs) = -2ie 2 = B.39)

Thus quantum chirality requires the vanishing of the leading singular part of the J JG+ OPE. Let us first investigate T++. The lowest component of T++ and JG give:

2G jG+(X ++ (Y ( IZeR Q A)A) 2 2(x - y-) - 2 (x - y 2 (x- y-) EA (B.40)

Therefore requiring U+T++ = 0, relies on having:

Q'a- QI -qA i3 = 0. (B.41) I I A

This condition is equivalent to a particular R-symmetry being non-anomalous. Thus quantum chirality of the left moving stress tensor requires a non-anomalous R-symmetry.

Next let's consider J+, which has an OPE with the gauge current:

jG+(X) jL+(Y) (X - y-) 2 A

Thus non-chirality of U(1)L boils down to the vanishing of an L-anomaly:

U+j+ oC) QICL _ 1 qAA) T. (B.42) I A

198 For the gauge current J+ to be a quantum chiral operator, the same analysis gives:

+ (ZQIQI - ZAA) T. (B.43) (I A

In short, the quantum chirality of a current requires the various anomaly coefficients vanish. The existence of a non-anomalous gauge current, L current and R current is necessary for quantum chirality of J+and T++, and tells us that the lowest components of these operators are elements of cohomology, which become the U(1)L current and left moving stress tensor for the IR CFT.

199 200 Bibliography

[1] H. David Politzer. Reliable perturbative results for strong interactions? Phys.Rev.Lett., 30:1346-1349, 1973.

[2] D.J. Gross and Frank Wilczek. Ultraviolet behavior of nonabelian gauge theories. Phys.Rev.Lett., 30:1343-1346, 1973.

[3] D.J. Gross and Frank Wilczek. Asymptotically Free Gauge Theories. 1. Phys.Rev., D8:3633-3652, 1973.

[4] Ethan Dyer, Mrk Mezei, and Silviu S. Pufu. Monopole Taxonomy in Three- Dimensional Conformal Field Theories. 2013.

[5] Allan Adams, Ethan Dyer, and Jaehoon Lee. GLSMs for non-Kahler Geome- tries. JHEP, 1301:044, 2013.

[6] Alexander M. Polyakov. Quark confinement and topology of gauge groups. Nucl.Phys., B120:429-458, 1977.

[7] Robert D. Pisarski. Chiral symmetry breaking in three-dimensional electrodynamics. Phys.Rev., D29:2423, 1984.

[8] Thomas W. Appelquist, Mark J. Bowick, Dimitra Karabali, and L.C.R. Wijew- ardhana. Spontaneous Chiral Symmetry Breaking in Three-Dimensional QED. Phys.Rev., D33:3704, 1986.

[9] Kenneth A. Intriligator and N. Seiberg. Mirror symmetry in three-dimensional gauge theories. Phys.Lett., B387:513-519, 1996.

[10] T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. A. Fisher. De- confined Quantum Critical Points. Science, 303:1490-1494, March 2004.

[11] T. Senthil, L. Balents, S. Sachdev, A. Vishwanath, and M. P. A. Fisher. Quantum criticality beyond the Landau-Ginzburg-Wilson paradigm. Phys.Rev., 70(14):144407, October 2004.

[12] Ganpathy Murthy and Subir Sachdev. Action of hedgehog instantons in the dis- ordered phase of the (2+ 1)-dimensional CPN-1 model. Nucl.Phys., B344:557- 595, 1990.

201 [13] Vadim Borokhov, Anton Kapustin, and Xin-kai Wu. Topological disorder operators in three-dimensional conformal field theory. JHEP, 0211:049, 2002.

[14] Vadim Borokhov, Anton Kapustin, and Xin-kai Wu. Monopole operators and mirror symmetry in three dimensions. JHEP, 0212:044, 2002.

[15] Max A. Metlitski, Michael Hermele, T. Senthil, and Matthew P.A. Fisher. Monopoles in CPN-1 model via the state-operator correspondence. Phys.Rev., B78:214418, 2008.

[16] Tai Tsun Wu and Chen Ning Yang. Dirac Monopole Without Strings: Monopole Harmonics. Nucl.Phys., B107:365, 1976.

[17] Paul A.M. Dirac. Quantized Singularities in the Electromagnetic Field. Proc.Roy.Soc.Lond., A133:60-72, 1931.

[18] Anton Kapustin. Wilson-'t Hooft operators in four-dimensional gauge theories and S-duality. Phys.Rev., D74:025005, 2006.

[19] P. Goddard, J. Nuyts, and David I. Olive. Gauge Theories and Magnetic Charge. Nucl.Phys., B125:1, 1977.

[20] Sidney R. Coleman. The magnetic monopole fifty years later. In Les Houches 1981, Proceedings, Gauge Theories In High Energy Physics, Part 1, pages 461- 552, 1982.

[21] Silviu S. Pufu. Anomalous dimensions of monopole operators in three- dimensional quantum electrodynamics. 2013.

[22] P. Candelas, Gary T. Horowitz, Andrew Strominger, and Edward Witten. Vac- uum Configurations for Superstrings. Nucl.Phys., B258:46-74, 1985.

[23] Edward Witten. Phases of N=2 Theories in Two-Dimensions. Nucl.Phys., B403:159-222, 1993.

[24] Abhijit Gadde, Sergei Gukov, and Pavel Putrov. Fivebranes and 4-manifolds. 2013.

[25] C.M. Hull and Edward Witten. Supersymmetric Sigma Models and the Het- erotic String. Phys.Lett., B160:398-402, 1985.

[26] C.M. Hull and P.K. Townsend. World Sheet Supersymmetry and Anomaly Cancellation In the Heterotic String. Phys.Lett., B178:187, 1986.

[27] Ashoke Sen. Local Gauge And Lorentz Invariance Of The Heterotic String Theory. Phys.Lett., B166:300, 1986.

[28] Daniel Harry Friedan. Nonlinear Models in Two + Epsilon Dimensions. Annals Phys., 163:318, 1985.

202 [29] D. Friedan. Nonlinear Models in Two Epsilon Dimensions. Phys.Rev.Lett., 45:1057, 1980.

[30] Luis Alvarez-Gaume, Daniel Z. Freedman, and Sunil Mukhi. The Background Field Method and the Ultraviolet Structure of the Supersymmetric Nonlinear Sigma Model. Annals Phys., 134:85, 1981.

[31] Jr. Callan, Curtis G., E.J. Martinec, M.J. Perry, and D. Friedan. Strings in Background Fields. Nucl.Phys., B262:593, 1985.

[32] Andrew Strominger. Superstrings with Torsion. Nucl.Phys., B274:253, 1986.

[33] Marcus T. Grisaru, A.E.M. van de Ven, and D. Zanon. Four Loop beta Func- tion for the N=1 and N=2 Supersymmetric Nonlinear Sigma Model in Two- Dimensions. Phys.Lett., B173:423, 1986.

[34] Shing-Tung Yau. Calabi's Conjecture and some new results in algebraic geometry. Proc.Nat.Acad.Sci., 74:1798-1799, 1977.

[35] Eugenio Calabi. On Khler manifolds with vanishing canonical class. Algebraic Geometry and Topology. A Symposium in Honor of S. Lefschetz, pages 78-89, 1957.

[36] K. Uhlenbeck and S. T. Yau. On the existence of hermitian-yang-mills connections in stable vector bundles. Communications on Pure and Applied Mathe- matics, 39(S1):S257-S293, 1986.

[37] Stephen L. Adler. Axial vector vertex in spinor electrodynamics. Phys.Rev., 177:2426-2438, 1969.

[38] J.S. Bell and R. Jackiw. A PCAC puzzle: pi0 -+ gamma gamma in the sigma model. Nuovo Cim., A60:47-61, 1969.

[39] Julian S. Schwinger. Gauge Invariance and Mass. 2. Phys.Rev., 128:2425-2429, 1962.

[40] Michael B. Green and John H. Schwarz. Anomaly Cancellation in Supersym- metric D=10 Gauge Theory and Superstring Theory. Phys.Lett., B149:117-122, 1984.

[41] Eric D'Hoker and Edward Farhi. Decoupling a Fermion Whose Mass Is Gener- ated by a Yukawa Coupling: The General Case. Nucl.Phys., B248:59, 1984.

[42] Eric D'Hoker and Edward Farhi. Decoupling a Fermion in the Standard Elec- troweak Theory. Nucl.Phys., B248:77, 1984.

[43] Gerard 't Hooft. Magnetic Monopoles in Unified Gauge Theories. Nucl.Phys., B79:276-284, 1974.

203 [44] Alexander M. Polyakov. Particle Spectrum in the Quantum Field Theory. JETP Lett., 20:194-195, 1974.

[45] Anton Kapustin and Edward Witten. Electric-Magnetic Duality And The Ge- ometric Langlands Program. Commun.Num. Theor.Phys., 1:1-236, 2007.

[46] Alexander M. Polyakov. Compact gauge fields and the infrared catastrophe. Phys.Lett., B59:82-84, 1975.

[47] M. Hermele, T. Senthil, M. P. A. Fisher, P. A. Lee, N. Nagaosa, and X.-G. Wen. Stability of U(1) spin liquids in two dimensions. Phys.Rev., B70:214437, 2004.

[48] J. Alicea. Monopole quantum numbers in the staggered flux spin liquid. Phys.Rev, B78(3):035126, 2008.

[49] N. Read and Subir Sachdev. Spin-Peierls, valence-bond solid, and Neel ground states of low-dimensional quantum antiferromagnets. Phys.Rev., B42:4568- 4589, 1990.

[50] N. Read and Subir Sachdev. Valence-bond and spin-Peierls ground states of low- dimensional quantum antiferromagnets. Phys.Rev.Lett., 62:1694-1697, 1989.

[51] R. K. Kaul and A. W. Sandvik. Lattice Model for the SU(N) N6el to Valence- Bond Solid Quantum Phase Transition at Large N. Physical Review Letters, 108(13):137201, March 2012.

[52] K. Damle, F. Alet, and S. Pujari. Neel to valence-bond solid transition on the honeycomb lattice: Evidence for deconfined criticality. ArXiv e-prints, February 2013.

[53] M. S. Block, R. G. Melko, and R. K. Kaul. Fate of CpN-1 fixed points with q-monopoles. ArXiv e-prints, July 2013.

[54] Ofer Aharony, Oren Bergman, Daniel Louis Jafferis, and Juan Maldacena. K= 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals. JHEP, 0810:091, 2008.

[55] Denis Bashkirov and Anton Kapustin. Supersymmetry enhancement by monopole operators. JHEP, 1105:015, 2011.

[56] Marcus K. Benna, Igor R. Klebanov, and Thomas Klose. Charges of Monopole Operators in Chern-Simons Yang-Mills Theory. JHEP, 1001:110, 2010.

[57] Andreas Gustavsson and Soo-Jong Rey. Enhanced K = 8 Supersymmetry of 8 ABJM Theory on R' and R /Z 2 . 2009.

[58] Davide Gaiotto and Daniel Louis Jafferis. Notes on adding D6 branes wrapping 3 RP in AdS4 x CP3. JHEP, 1211:015, 2012.

204 [59] Daniel Louis Jafferis and Alessandro Tomasiello. A Simple class of K = 3 gauge/gravity duals. JHEP, 0810:101, 2008.

[60] Daniel Louis Jafferis. Quantum corrections to K = 2 Chern-Simons theories with flavor and their AdS4 duals. 2009.

[61] Francesco Benini, Cyril Closset, and Stefano Cremonesi. Chiral flavors and M2-branes at toric CY4 singularities. JHEP, 1002:036, 2010.

[62] Sebastian Franco, Igor R. Klebanov, and Diego Rodriguez-Gomez. M2-branes on Orbifolds of the Cone over Q'"1'. JHEP, 0908:033, 2009.

[63] Dario Martelli and James Sparks. AdS4 /CFT 3 duals from M2-branes at hypersurface singularities and their deformations. JHEP, 0912:017, 2009.

[64] Yosuke Imamura. Monopole operators in K = 4 Chern-Simons theories and wrapped M2-branes. Prog.Theor. Phys., 121:1173-1187, 2009.

[65] Francesco Benini, Cyril Closset, and Stefano Cremonesi. Quantum moduli space of Chern-Simons quivers, wrapped D6-branes and AdS4/CFT3. JHEP, 1109:005, 2011.

[66] Daniel R. Gulotta, Christopher P. Herzog, and Silviu S. Pufu. From Necklace Quivers to the F-theorem, Operator Counting, and T(U(N)). JHEP, 1112:077, 2011.

[67] Daniel R. Gulotta, Christopher P. Herzog, and Silviu S. Pufu. Operator Counting and Eigenvalue Distributions for 3D Supersymmetric Gauge Theo- ries. JHEP, 1111:149, 2011.

[68] Marcus Benna, Igor Klebanov, Thomas Klose, and Mikael Smedback. Super- conformal Chern-Simons Theories and AdS4/CFT 3 Correspondence. JHEP, 0809:072, 2008.

[69] Ofer Aharony. IR duality in d = 3 K = 2 supersymmetric USp(2Nc) and U(Nc) gauge theories. Phys.Lett., B404:71-76, 1997.

[70] Ofer Aharony, Shlomo S. Razamat, Nathan Seiberg, and Brian Willett. 3d dualities from 4d dualities. JHEP, 1307:149, 2013.

[71] Ofer Aharony, Shlomo S. Razamat, Nathan Seiberg, and Brian Willett. 3d dualities from 4d dualities for orthogonal groups. 2013.

[72] Vadim Borokhov. Monopole operators in three-dimensional K = 4 SYM and mirror symmetry. JHEP, 0403:008, 2004.

[73] Eric D'Hoker. Monte Carlo study of three-dimensional QCD. Nucl.Phys., B180:341, 1981.

205 [74] Richard P. Feynman. The Qualitative Behavior of Yang-Mills Theory in (2+1)- Dimensions. Nucl.Phys., B188:479, 1981.

[75] Jan Ambjorn, Anthony J.G. Hey, and Steve Otto. String tensions for lattice gauge theories in (2 + 1) dimensions. Nucl.Phys., B210:347, 1982.

[76] Jan Ambjorn, P. Olesen, and C. Peterson. Three-dimensional lattice gauge theory and strings. Nucl.Phys., B244:262, 1984.

[77] Thomas Appelquist and Daniel Nash. Critical behavior in (2 + 1)-dimensional QCD. Phys.Rev.Lett., 64:721, 1990.

[78] Elihu Lubkin. Geometric definition of gauge invariance. Annals Phys., 23:233, 1963.

[79] Silviu S. Pufu and Subir Sachdev. Monopoles in 2 + 1-dimensional conformal field theories with global U(1) symmetry. 2013.

[80] A.J. Niemi and G.W. Semenoff. Axial Anomaly Induced Fermion Fraction- ization and Effective Gauge Theory Actions in Odd Dimensional Space-Times. Phys.Rev.Lett., 51:2077, 1983.

[81] A.N. Redlich. Parity Violation and Gauge Noninvariance of the Effective Gauge Field Action in Three-Dimensions. Phys.Rev., D29:2366-2374, 1984.

[82] A.N. Redlich. Gauge Noninvariance and Parity Violation of Three-Dimensional Fermions. Phys.Rev.Lett., 52:18, 1984.

[83] Tai Tsun Wu and Chen Ning Yang. Some properties of monopole harmonics. Phys.Rev., D16:1018-1021, 1977.

[84] Erick J. Weinberg. Monopole vector spherical harmonics. Phys.Rev., D49:1086- 1092, 1994.

[85] Richard A. Brandt and Filippo Neri. Stability analysis for singular non-Abelian magnetic monopoles. Nucl.Phys., B161:253-282, 1979.

[86] Jin-Quan Chen, Jialun Ping, and Fan Wang. Group Representation Theory for Physicists. World Scientific, 2nd edition, 2002.

[87] Howard Georgi. Lie algebras in particle physics. Westview Press, 1999.

[88] Benjamin R. Safdi, Igor R. Klebanov, and Jeongseog Lee. A Crack in the conformal window. 2012.

[89] Tarun Grover. Chiral symmetry breaking, deconfinement and entanglement monotonicity. 2012.

[90] Cumrun Vafa and Edward Witten. Restrictions on symmetry breaking in vector-like gauge theories. Nucl.Phys., B234:173, 1984.

206 [91] Cumrun Vafa and Edward Witten. Eigenvalue inequalities for fermions in gauge theories. Commun. Math. Phys., 95:257, 1984.

[92] Daniel L. Jafferis, Igor R. Klebanov, Silviu S. Pufu, and Benjamin R. Safdi. Towards the F-theorem: K = 2 field theories on the three-sphere. JHEP, 1106:102, 2011.

[93] Igor R. Klebanov, Silviu S. Pufu, and Benjamin R. Safdi. F-Theorem without supersymmetry. JHEP, 1110:038, 2011.

[94] Robert C. Myers and Aninda Sinha. Holographic c-theorems in arbitrary dimensions. JHEP, 1101:125, 2011.

[95] H. Casini and Marina Huerta. On the RG running of the entanglement entropy of a circle. Phys.Rev., D85:125016, 2012.

[96] Igor R. Klebanov, Silviu S. Pufu, Subir Sachdev, and Benjamin R. Safdi. En- tanglement entropy of 3-d conformal gauge theories with many flavors. JHEP, 1205:036, 2012.

[97] Walter Rantner and Xiao-Gang Wen. Spin correlations in the algebraic spin liquid: Implications for high-T superconductors. Phys.Rev., B66:144501, 2002.

[98] M. Hermele, Y. Ran, P. A. Lee, and X.-G. Wen. Properties of an algebraic spin liquid on the kagome lattice. Phys.Rev., B77:224413, 2008.

[99] ML Michelsohn. On the existence of special metrics in complex geometry. Acta Mathematica, 149(1):261-295, 1982.

[100] Allan Adams, Morten Ernebjerg, and Joshua M. Lapan. Linear Models for Flux Vacua. Adv. Theor.Math.Phys., 12:817-851, 2008.

[101] Allan Adams and Joshua M. Lapan. Computing the Spectrum of a Heterotic Flux Vacuum. JHEP, 1103:045, 2011.

[102] Allan Adams and David Guarrera. Heterotic Flux Vacua from Hybrid Linear Models. 2009.

[103] Allan Adams. Orbifold Phases of Heterotic Flux Vacua. 2009.

[104] Michael Blaszczyk, Stefan Nibbelink Groot, and Fabian Ruehle. Green-Schwarz Mechanism in Heterotic (2,0) Gauged Linear Sigma Models: Torsion and NS5 Branes. JHEP, 1108:083, 2011.

[105] Callum Quigley and Savdeep Sethi. Linear Sigma Models with Torsion. JHEP, 1111:034, 2011.

[106] Jacques Distler. Notes on (0,2) superconformal field theories. 1995.

207 [107] Jock McOrist. The Revival of (0,2) Linear Sigma Models. Int.J.Mod.Phys., A26:1-41, 2011.

[108] Stefan Groot Nibbelink and Leonhard Horstmeyer. Super Weyl invariance: BPS equations from heterotic worldsheets. JHEP, 1207:054, 2012.

[109] Gregory W. Moore and Philip C. Nelson. Anomalies In Nonlinear Sigma Models. Phys.Rev.Lett., 53:1519, 1984.

[110] Gregory W. Moore and Philip C. Nelson. The Etiology Of Sigma Model Anoma- lies. Commun.Math. Phys., 100:83, 1985.

[111] Ji-Xiang Fu and Shing-Tung Yau. The Theory of superstring with flux on non-Kahler manifolds and the complex Monge-Ampere equation. J.Diff.Geom., 78:369-428, 2009.

[112] Aneesh Manohar, Gregory W. Moore, and Philip C. Nelson. A Comment on Sigma Model Anomalies. Phys.Lett., B152:68, 1985.

[113] Kentaro Hori and Cumrun Vafa. Mirror symmetry. 2000.

[114] Kentaro Hori and Anton Kapustin. Duality of the Fermionic 2-D Black Hole and N=2 Liouville Theory as Mirror Symmetry. JHEP, 0108:045, 2001.

[115] Kentaro Hori and Anton Kapustin. World Sheet Descriptions of Wrapped NS Five-Branes. JHEP, 0211:038, 2002.

[116] Keshav Dasgupta, Govindan Rajesh, and Savdeep Sethi. M theory, orientifolds and G - flux. JHEP, 9908:023, 1999.

[117] Edward Goldstein and Sergey Prokushkin. Geometric model for complex nonKahler manifolds with SU(3) structure. Commun.Math.Phys., 251:65-78, 2004.

[118] Katrin Becker, Melanie Becker, Ji-Xiang Fu, Li-Sheng Tseng, and Shing-Tung Yau. Anomaly cancellation and smooth non-Kahler solutions in heterotic string theory. Nucl.Phys., B751:108-128, 2006.

[119] Eva Silverstein and Edward Witten. Criteria for conformal invariance of (0,2) models. Nucl. Phys., B444:161-190, 1995.

[120] Anirban Basu and Savdeep Sethi. World sheet stability of (0,2) linear sigma models. Phys.Rev., D68:025003, 2003.

[121] Chris Beasley and Edward Witten. Residues and world sheet instantons. JHEP, 0310:065, 2003.

[122] Eva Silverstein and Edward Witten. Global U(1) R symmetry and conformal invariance of (0,2) models. Phys.Lett., B328:307-311, 1994.

208 [123] John Preskill. Gauge anomalies in an effective field theory. Annals Phys., 210:323-379, 1991.

[124] K. Konishi. Anomalous Supersymmetry Transformation of Some Composite Operators in SQCD. Phys.Lett., B135:439, 1984.

[125] Martin Rocek and Erik P. Verlinde. Duality, quotients, and currents. Nucl.Phys., B373:630-646, 1992.

[126] David R. Morrison and M.Ronen Plesser. Towards mirror symmetry as duality for two-dimensional abelian gauge theories. Nucl.Phys.Proc.Suppl., 46:177-186, 1996.

[127] Allan Adams, Anirban Basu, and Savdeep Sethi. (0,2) duality. Adv. Theor.Math.Phys., 7:865-950, 2004.

209