It is customary to couple a quantum system to external classical fields. One application is to couple the global symmetries of the system (including the Poincare symmetry) to background gauge fields (and a metric for the Poincare symmetry). Failure of gauge invariance of the partition function under gauge transformations of these fields reflects 't Hooft anomalies. It is also common to view the ordinary (scalar) coupling constants as background fields, i.e. to study the theory when they are spacetime dependent. We will show that the notion of 't Hooft anomalies can be extended naturally to include these scalar background fields. Just as ordinary 't Hooft anomalies allow us to deduce dynamical consequences about the phases of the theory and its defects, the same is true for these generalized 't Hooft anomalies. Specifically, since the coupling constants vary, we can learn that certain phase transitions must be present. We will demonstrate these anomalies and their applications in simple pedagogical examples in one dimension (quantum mechanics) and in some two, three, and four-dimensional quantum field theories. An anomaly is an example of an invertible field theory, which can be described as an object in (generalized) differential cohomology. We give an introduction to this perspective. Also, we use Quillen's superconnections to derive the anomaly for a free spinor field with variable mass. In a companion paper we will study four-dimensional gauge theories showing how our view unifies and extends many recently obtained results.

/pdf/anomalies-in-the-space-of-coupling-constants-and-their-4p05o74f4n.pdf

Anomalies in the Space of Coupling Constants and Their Dynamical Applications I

It is customary to couple a quantum system to external classical fields. One application is to couple the global symmetries of the system (including the Poincar\'{e} symmetry) to background gauge fields (and a metric for the Poincar\'{e} symmetry). Failure of gauge invariance of the partition function under gauge transformations of these fields reflects 't Hooft anomalies. It is also common to view the ordinary (scalar) coupling constants as background fields, i.e. to study the theory when they are spacetime dependent. We will show that the notion of 't Hooft anomalies can be extended naturally to include these scalar background fields. Just as ordinary 't Hooft anomalies allow us to deduce dynamical consequences about the phases of the theory and its defects, the same is true for these generalized 't Hooft anomalies. Specifically, since the coupling constants vary, we can learn that certain phase transitions must be present. We will demonstrate these anomalies and their applications in simple pedagogical examples in one dimension (quantum mechanics) and in some two, three, and four-dimensional quantum field theories. An anomaly is an example of an invertible field theory, which can be described as an object in (generalized) differential cohomology. We give an introduction to this perspective. Also, we use Quillen's superconnections to derive the anomaly for a free spinor field with variable mass. In a companion paper we will study four-dimensional gauge theories showing how our view unifies and extends many recently obtained results.

/pdf/anomalies-in-the-space-of-coupling-constants-and-their-41f74bja8k.pdf

Anomalies in the Space of Coupling Constants and Their Dynamical Applications II

We use supersymmetric localization to calculate correlation functions of half-BPS local operators in 3d N=4 superconformal field theories whose Lagrangian descriptions consist of vectormultiplets coupled to hypermultiplets. The operators we primarily study are certain twisted linear combinations of Higgs branch operators that can be inserted anywhere along a given line. These operators are constructed from the hypermultiplet scalars. They form a one-dimensional non-commutative operator algebra with topological correlation functions. The 2- and 3-point functions of Higgs branch operators in the full 3d N=4 theory can be simply inferred from the 1d topological algebra. After conformally mapping the 3d superconformal field theory from flat space to a round three-sphere, we preform supersymmetric localization using a supercharge that does not belong to any 3d N=2 subalgebra of the N=4 algebra. The result is a simple model that can be used to calculate correlation functions in the 1d topological algebra mentioned above. This model is a 1d Gaussian theory coupled to a matrix model, and it can be viewed as a gauge-fixed version of a topological gauged quantum mechanics. Our results generalize to non-conformal theories on S^3 that contain real mass and Fayet-Iliopolous parameters. We also provide partial results in the 1d topological algebra associated with the Coulomb branch, where we calculate correlation functions of local operators built from the vectormultiplet scalars.

/pdf/a-one-dimensional-theory-for-higgs-branch-operators-4jgezkpods.pdf

A one-dimensional theory for Higgs branch operators

We compute 1/λ corrections to the four-point functions of half-BPS operators in SU(N) $$ \mathcal{N} $$
 = 4 super-Yang-Mills theory at large N and large ’t Hooft coupling λ = $$ {g}_{\mathrm{YM}}^2N $$
 using two methods. Firstly, we relate integrals of these correlators to derivatives of the mass deformed S4 free energy, which was computed at leading order in large N and to all orders in 1/λ using supersymmetric localization. Secondly, we use AdS/CFT to relate these 1/λ corrections to higher derivative corrections to supergravity for scattering amplitudes of Kaluza-Klein scalars in IIB string theory on AdS5× S5, which in the flat space limit are known from worldsheet calculations. These two methods match at the order corresponding to the tree level R4 interaction in string theory, which provides a precise check of AdS/CFT beyond supergravity, and allow us to derive the holographic correlators to tree level D4R4 order. Combined with constraints from [1], our results can be used to derive CFT data to one-loop D4R4 order. Finally, we use AdS/CFT to fix these correlators in the limit where N is taken to be large while gYM is kept fixed. In this limit, we present a conjecture for the small mass limit of the S4 partition function that includes all instanton corrections and is written in terms of the same Eisenstein series that appear in the study of string theory scattering amplitudes.

/pdf/mathcal-n-4-super-yang-mills-correlators-at-strong-coupling-3cfawn34f4.pdf

$\mathcal{N}=4$ Super-Yang-Mills Correlators at Strong Coupling from String Theory and Localization

We study the O(N) and Gross-Neveu models at large N on AdSd+1 background. Thanks to the isometries of AdS, the observables in these theories are constrained by the SO(d, 2) conformal group even in the presence of mass deformations, as was discussed by Callan and Wilczek [1], and provide an interesting two-parameter family of quantities which interpolate between the S-matrices in flat space and the correlators in CFT with a boundary. For the actual computation, we judiciously use the spectral representation to resum loop diagrams in the bulk. After the resummation, the AdS 4-particle scattering amplitude is given in terms of a single unknown function of the spectral parameter. We then “bootstrap” the unknown function by requiring the absence of double-trace operators in the boundary OPE. Our results are at leading nontrivial order in $$ \frac{1}{N} $$
 , and include the full dependence on the quartic coupling, the mass parameters, and the AdS radius. In the bosonic O(N) model we study both the massive phase and the symmetry-breaking phase, which exists even in AdS2 evading Coleman’s theorem, and identify the AdS analogue of a resonance in flat space. We then propose that symmetry breaking in AdS implies the existence of a conformal manifold in the boundary conformal theory. We also provide evidence for the existence of a critical point with bulk conformal symmetry, matching existing results and finding new ones for the conformal boundary conditions of the critical theories. For the Gross-Neveu model we find a bound state, which interpolates between the familiar bound state in flat space and the displacement operator at the critical point.

/pdf/a-study-of-quantum-field-theories-in-ads-at-finite-coupling-257ogbrwyc.pdf

A study of quantum field theories in AdS at finite coupling

We consider the correlation functions of Coulomb branch operators in four-dimensional $$ \mathcal{N} $$
 = 2 Superconformal Field Theories (SCFTs) involving exactly one antichiral operator. These extremal correlators are the “minimal” non-holomorphic local observables in the theory. We show that they can be expressed in terms of certain determinants of derivatives of the four-sphere partition function of an appropriate deformation of the SCFT. This relation between the extremal correlators and the deformed four-sphere partition function is non-trivial due to the presence of conformal anomalies, which lead to operator mixing on the sphere. Evaluating the deformed four-sphere partition function using supersymmetric localization, we compute the extremal correlators explicitly in many interesting examples. Additionally, the representation of the extremal correlators mentioned above leads to a system of integrable differential equations. We compare our exact results with previous perturbative computations and with the four-dimensional tt
 ∗ equations. We also use our results to study some of the asymptotic properties of the perturbative series expansions we obtain in $$ \mathcal{N} $$
 = 2 SQCD.

/pdf/correlation-functions-of-coulomb-branch-operators-4hzgmarycs.pdf

Correlation functions of Coulomb branch operators

We consider the correlation functions of Coulomb branch operators in four-dimensional N=2 Superconformal Field Theories (SCFTs) involving exactly one anti-chiral operator. These extremal correlators are the "minimal" non-holomorphic local observables in the theory. We show that they can be expressed in terms of certain determinants of derivatives of the four-sphere partition function of an appropriate deformation of the SCFT. This relation between the extremal correlators and the deformed four-sphere partition function is non-trivial due to the presence of conformal anomalies, which lead to operator mixing on the sphere. Evaluating the deformed four-sphere partition function using supersymmetric localization, we compute the extremal correlators explicitly in many interesting examples. Additionally, the representation of the extremal correlators mentioned above leads to a system of integrable differential equations. We compare our exact results with previous perturbative computations and with the four-dimensional tt^* equations. We also use our results to study some of the asymptotic properties of the perturbative series expansions we obtain in N=2 SQCD.

/pdf/correlation-functions-of-coulomb-branch-operators-555dxf3rw8.pdf

Correlation Functions of Coulomb Branch Operators

We study the phases of the SU(N1) × SU(N2) gauge theory with a bifundamental fermion in 3+1 dimensions. We show that the discrete anomalies and Berry phases associated to the one-form symmetry of the theory allow for several topologically distinct phase diagrams. We identify several limits of the theory where the phase diagram can be determined using various controlled approximations. When the two ranks are equal N1 = N2, these limits all lead to the same topology for the phase diagram and provide a consistent global understanding of the phases of the theory. When N1 ≠ N2, different limits lead to distinct topologies of the phase diagram. This necessarily implies non-trivial physics at some intermediate regimes of parameter space. In the large N1,2 limit, we argue that the topological transitions are accounted for by a (non-supersymmetric) duality cascade as one varies the parameters of the theory.

/pdf/the-bi-fundamental-gauge-theory-in-3-1-dimensions-the-vacuum-2s2b4tmwar.pdf

The Bi-Fundamental Gauge Theory in 3+1 Dimensions: The Vacuum Structure and a Cascade

We study the phases of the SU(N1)X SU(N2) gauge theory with a bi-fundamental fermion in 3+1 dimensions. We show that the discrete anomalies and Berry phases associated to the one-form symmetry of the theory allow for several topologically distinct phase diagrams. We identify several limits of the theory where the phase diagram can be determined using various controlled approximations. When the two ranks are equal N1=N2, these limits all lead to the same topology for the phase diagram and provide a consistent global understanding of the phases of the theory. When the ranks are different, different limits lead to distinct topologies of the phase diagram. This necessarily implies non-trivial physics at some intermediate regimes of parameter space. In the large N1,N2 limit, we argue that the topological transitions are accounted for by a (non-supersymmetric) duality cascade as one varies the parameters of the theory.

/pdf/the-bi-fundamental-gauge-theory-in-3-1-dimensions-the-vacuum-3ieqrd8r0y.pdf

In this paper we study supersymmetric field theories on an AdS
 p
 × S
 q
 spacetime that preserves their full supersymmetry. This is an interesting example of supersymmetry on a non-compact curved space. The supersymmetry algebra on such a space is a (p − 1)-dimensional superconformal algebra, and we classify all possible algebras that can arise for p ≥ 3. In some AdS
 3 cases more than one superconformal algebra can arise from the same field theory. We discuss in detail the special case of four dimensional field theories with $$ \mathcal{N}=1 $$
 and $$ \mathcal{N}=2 $$
 supersymmetry on AdS
 3 × S
 1.

/pdf/supersymmetric-field-theories-on-ads-p-s-q-6ceevbed87.pdf

Avner Karasik

Papers

Correlation functions of Coulomb branch operators

Correlation Functions of Coulomb Branch Operators

The Bi-Fundamental Gauge Theory in 3+1 Dimensions: The Vacuum Structure and a Cascade

The Bi-Fundamental Gauge Theory in 3+1 Dimensions: The Vacuum Structure and a Cascade

Supersymmetric field theories on AdS p × S q