The Schur limit of the superconformal index of a four-dimensional <math><mi>N</mi><mo>=</mo><mn>2</mn></math> superconformal field theory encodes rich physical information about the protected spectrum of the theory. For a Lagrangian model, this limit of the index can be computed by a contour integral of a multivariate elliptic function. However, surprisingly, so far it has eluded exact evaluation in a closed, analytical form. In this paper we propose an elementary approach to bring to heel a large class of these integrals by exploiting the ellipticity of their integrand. Our results take the form of a finite sum of (products of) the well-studied flavored Eisenstein series. In particular, we derive a compact formula for the fully flavored Schur index of all theories of class <math><mi>S</mi></math> of type <math><msub><mi>a</mi><mn>1</mn></msub></math>, we put forward a conjecture for the unflavored Schur indices of all <math><mi>N</mi><mo>=</mo><mn>4</mn></math> super Yang-Mills theories with gauge group <math><mi>S</mi><mi>U</mi><mo>(</mo><mi>N</mi><mo>)</mo></math>, and we present closed-form expressions for the index of various other gauge theories of low ranks. We also discuss applications to non-Lagrangian theories, modular properties, and defect indices.

China

Yiwen Pan

United Kingdom

Wolfger Peelaers

Exact Schur index in closed form

We revisit the leading irrelevant deformation of   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 Super Yang-Mills theory that preserves sixteen supercharges. We consider the deformed theory on S 3 × ℝ. We are able to write a closed form expression of the classical action thanks to a formalism that realizes eight supercharges off shell. We then investigate integrability of the spectral problem, by studying the spin-chain Hamiltonian in planar perturbation theory. While there are some structural indications that a suitably defined deformation might preserve integrability, we are unable to settle this question by our two-loop calculation; indeed up to this order we recover the integrable Hamiltonian of undeformed   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM due to accidental symmetry enhancement. We also comment on the holographic interpretation of the theory.

United States

João Caetano

Leonardo Rastelli

Maximally supersymmetric RG flows in 4D and integrability

We study the four-dimensional   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 superconformal field theories that describe D3-branes probing the recently constructed   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2   <math> <mi>S</mi> </math>  $$ \mathcal{S} $$ -folds in F-theory. We introduce a novel, infinite class of superconformal field theories related to   <math> <mi>S</mi> </math>  $$ \mathcal{S} $$ -fold theories via partial Higgsing. We determine several properties of both the   <math> <mi>S</mi> </math>  $$ \mathcal{S} $$ -fold models and this new class of theories, including their central charges, Coulomb branch spectrum, and moduli spaces of vacua, by bringing to bear an array of field-theoretical techniques, to wit, torus-compactifications of six-dimensional   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = (1, 0) theories, class   <math> <mi>S</mi> </math>  $$ \mathcal{S} $$  technology, and the SCFT/VOA correspondence.

Simone Giacomelli

Carlo Meneghelli

New   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 superconformal field theories from   <math> <mi>S</mi> </math>  $$ \mathcal{S} $$ -folds

In this note we address the question whether one can recover from the vertex operator algebra associated with a four-dimensional $$ \mathcal{N} $$ = 2 superconformal field theory the deformation quantization of the Higgs branch of vacua that appears as a protected subsector in the three-dimensional circle-reduced theory. We answer this question positively if the UV -symmetries do not mix with accidental (topological) symmetries along the renormalization group flow from the four-dimensional theory on a circle to the three-dimensional theory. If they do mix, we still find a deformation quantization but at different values of its period.

Deformation quantizations from vertex operator algebras

The Schur limit of the superconformal index of four-dimensional   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 superconformal field theories has been shown to equal the supercharacter of the vacuum module of their associated chiral algebra. Applying localization techniques to the theory suitably put on S 3 × S 1, we obtain a direct derivation of this fact. We also show that the localization computation can be extended to calculate correlation functions of a subset of local operators, namely of the so-called Schur operators. Such correlators correspond to insertions of chiral algebra fields in the trace-formula computing the supercharacter. As a by-product of our analysis, we show that the standard lore in the localization literature stating that only off-shell supersymmetrically closed observables are amenable to localization, is incomplete, and we demonstrate how insertions of fermionic operators can be incorporated in the computation.

Schur correlation functions on S 3 × S 1

Four-dimensional N $$ \mathcal{N} $$ = 2 superconformal quantum field theories contain a subsector carrying the structure of a chiral algebra. Using localization techniques, we show for the free hypermultiplet that this structure can be accessed directly from the path integral on the four-sphere. We extend the localization computation to include supersymmetric surface defects described by a generic 4d/2d coupled system. The presence of a defect corresponds to considering a module of the chiral algebra: our results provide a calculational window into its structure constants.

Sweden

Chiral algebras, localization and surface defects

We analyze intersecting surface defects inserted in interacting four-dimensional N = 2 $$ \mathcal{N}=2 $$ supersymmetric quantum field theories. We employ the realization of a class of such systems as the infrared fixed points of renormalization group flows from larger theories, triggered by perturbed Seiberg-Witten monopole-like configurations, to compute their partition functions. These results are cast into the form of a partition function of 4d/2d/0d coupled systems. Our computations provide concrete expressions for the instanton partition function in the presence of intersecting defects and we study the corresponding ADHM model.

Intersecting surface defects and instanton partition functions

We study Higgs branch localization of N = 2 $$ \mathcal{N}=2 $$ supersymmetric theories placed on compact Euclidean manifolds. We analyze the resulting localization equations in detail on the four-sphere and find that in this case the path integral is dominated by vortex-like configurations as well as singular Seiberg-Witten monopoles located at the north and south pole. The partition function is written accordingly.

Ellipsoid partition function from Seiberg-Witten monopoles

Four-dimensional N $$ \mathcal{N} $$ = 2 superconformal field theories have families of protected correlation functions that possess the structure of two-dimensional chiral algebras. In this paper, we explore the chiral algebras that arise in this manner in the context of theories of class S $$ \mathcal{S} $$ . The class S $$ \mathcal{S} $$ duality web implies nontrivial associativity properties for the corresponding chiral algebras, the structure of which is best summarized in the language of generalized topological quantum field theory. We make a number of conjectures regarding the chiral algebras associated to various strongly coupled fixed points.

Christopher Beem

CERN

Balt Rees

Chiral algebras of class S $$ \mathcal{S} $$

It was recently understood that one can identify a chiral algebra in any four-dimensional N = 2 $$ \mathcal{N}=2 $$ superconformal theory. In this note, we conjecture the full set of generators of the chiral algebras associated with the T n theories. The conjecture is motivated by making manifest the critical affine module structure in the graded partition function of the chiral algebras, which is computed by the Schur limit of the superconformal index for T n theories. We also explicitly construct the chiral algebra arising from the T 4 theory. Its null relations give rise to new T 4 Higgs branch chiral ring relations.