The Schur index is a powerful tool to probe the spectrum and dualities of four-dimensional <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">N</mi><mo>=</mo><mn>2</mn></math> superconformal field theories, deeply related to two-dimensional vertex operator algebras (VOAs). In this paper, we compute the Schur index in closed form for two series of non-Lagrangian theories. We explore and classify the Argyres-Douglas (AD) theories <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>D</mi><mi>p</mi><mi>b</mi></msubsup><mo stretchy="false">(</mo><msub><mi mathvariant="fraktur">sl</mi><mi>N</mi></msub><mo>,</mo><mo stretchy="false">[</mo><mi>Y</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></math> realized as the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></math> gauging of two AD matter theories, where we identify several infinite families with interesting central charge relations analogous to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>a</mi><mrow><mn>4</mn><mi mathvariant="normal">d</mi></mrow></msub><mo>=</mo><msub><mi>c</mi><mrow><mn>4</mn><mi mathvariant="normal">d</mi></mrow></msub></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">N</mi><mo>=</mo><mn>4</mn></math> theories. We focus on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>D</mi><mrow><mi>N</mi><mo>−</mo><mn>4</mn></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="fraktur">sl</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">[</mo><mi>N</mi><mo>−</mo><mn>4</mn><mo>,</mo><mn>4</mn><mo stretchy="false">]</mo><mo stretchy="false">)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>D</mi><mrow><mi>N</mi><mo>−</mo><mn>2</mn></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="fraktur">sl</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo><mo>,</mo><mspace linebreak="goodbreak"></mspace><mo stretchy="false">[</mo><mi>N</mi><mo>−</mo><mn>3</mn><mo>,</mo><mn>3</mn><mo stretchy="false">]</mo><mo stretchy="false">)</mo></math> and compute their flavored and unflavored Schur and Wilson line indices in compact form. We also explore their large-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi></math> behavior and show that they arise as special limits of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></math> SQCD flavored index, also analogous to the relation among the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>a</mi><mrow><mn>4</mn><mi mathvariant="normal">d</mi></mrow></msub><mo>=</mo><msub><mi>c</mi><mrow><mn>4</mn><mi mathvariant="normal">d</mi></mrow></msub></math> theories. We also generalize the elliptic function integration formula in the presence of higher-order poles to compute in closed form the partially flavored indices of the Minahan-Nemeschansky <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>E</mi><mn>6</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>E</mi><mn>7</mn></msub></math> theories. Our results point to a universal structure underlying the residues of elliptic integrands, Wilson loop indices, and nonvacuum modules of the corresponding VOAs.

China

Yiwen Pan

Peihe Yang

Exact Schur index in closed form for non-Lagrangian theories

The infinite series of 4D <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">N</mi><mo>=</mo><mn>2</mn></math> superconformal field theories with central charge relation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>a</mi><mrow><mn>4</mn><mi mathvariant="normal">D</mi></mrow></msub><mo>=</mo><msub><mi>c</mi><mrow><mn>4</mn><mi mathvariant="normal">D</mi></mrow></msub></math> are closely related to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">N</mi><mo>=</mo><mn>4</mn></math> super Yang-Mills theory. In this paper, we study the modular properties of their associated vertex operator algebras <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="double-struck">V</mi><mo stretchy="false">[</mo><msub><mi mathvariant="script">T</mi><mrow><mi>p</mi><mo>,</mo><mi>N</mi></mrow></msub><mo stretchy="false">]</mo></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="script">T</mi><mrow><mi>p</mi><mo>,</mo><mi>N</mi></mrow></msub></math> are those <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>a</mi><mo>=</mo><mi>c</mi></math> theories with an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></math> gauge group. We exploit the closed-form formula for the Schur index of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">N</mi><mo>=</mo><mn>4</mn></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></math> theories <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="script">T</mi><mrow><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></mrow></msub></math> to derive the space of characters of the vertex operator algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="double-struck">V</mi><mo stretchy="false">[</mo><msub><mi mathvariant="script">T</mi><mrow><mi>p</mi><mo>,</mo><mi>N</mi></mrow></msub><mo stretchy="false">]</mo></math> and the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>S</mi></mrow></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>T</mi></mrow></math> matrices, and find the (nonmonic) modular linear differential equations that constrain the module characters when possible. We investigate the geometric interpretation of some of these modular data from the viewpoint of 4D mirror symmetry. Using insights from the flavored modular differential equation and defect index, we investigate a map between module characters of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="double-struck">V</mi><mo stretchy="false">[</mo><msub><mi mathvariant="script">T</mi><mrow><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">]</mo></math> and those of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="double-struck">V</mi><mo stretchy="false">[</mo><msub><mi mathvariant="script">T</mi><mrow><mi>p</mi><mo>,</mo><mi>N</mi></mrow></msub><mo stretchy="false">]</mo></math>.

Modularity, 4d mirror symmetry, and vertex operator algebra modules of 4D <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="script">N</mi><mo>=</mo><mn>2</mn></mrow></math> SCFTs with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>a</mi><mo>=</mo><mi>c</mi></math>

In this paper, we propose a generalization to the holomorphic modular bootstrap approach featuring flavored modular differential equations arising from null states of the algebra. The approach can be applied to rational and the more general quasilisse chiral algebras that arise in the study of four-dimensional superconformal field theories. The transformation property of the flavored equations reveals hidden and universal effective action of the modular group and spectral flow on the null states. As shown in examples, the equations compute the full ordinary and nonordinary spectra both in the untwisted and the twisted sector.

Chenxi Zeng

Holomorphic quasimodular bootstrap

In this paper we analytically explore the modularity of the flavored Schur index of 4d <math><mi>N</mi><mo>=</mo><mn>2</mn></math> superconformal field theories. We focus on the <math><msub><mi>A</mi><mn>1</mn></msub></math> theories of class-<math><mi>S</mi></math> and <math><mi>N</mi><mo>=</mo><mn>4</mn></math> theories with <math><mi>S</mi><mi>U</mi><mo>(</mo><mi>N</mi><mo>)</mo></math> gauge group. We work out the modular orbit of the flavored index and defect index, compute the dimension of the space spanned by the orbit, and provide complete basis for computing modular transformation matrices. The dimension obtained from the flavored analysis predicts the minimal order of the unflavored modular differential equation satisfied by the unflavored Schur index. With the help of modularity, we also study analytically the high temperature asymptotics of the Schur index. In the high temperature limit <math><mi>τ</mi><mo>→</mo><mo>+</mo><mi>i</mi><mn>0</mn></math>, we identified the (defect) Schur index of the genus-zero <math><msub><mi>A</mi><mn>1</mn></msub></math> theories of class <math><mi>S</mi></math> with the <math><msup><mi>S</mi><mn>3</mn></msup></math> partition function of the <math><mi>S</mi><mi>U</mi><mo>(</mo><mn>2</mn><mo>)</mo><mo>×</mo><mi>U</mi><mo>(</mo><mn>1</mn><msup><mo>)</mo><mi>n</mi></msup></math> star-shaped quiver (with Wilson line insertion). In the identification, we observe an interesting relation between the linear-independence of defect indices and the convergence of the Wilson line partition functions.

Modularity of the Schur index, modular differential equations, and high-temperature asymptotics

We study two-dimensional <math><mi>N</mi><mo>=</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>)</mo></math> and <math><mi>N</mi><mo>=</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>4</mn><mo>)</mo></math> theories derived from compactifying class <math><mi>S</mi></math> theories on <math><msup><mi>S</mi><mn>2</mn></msup></math> with a topological twist. We present concise expressions for the elliptic genera of both classes of theories, revealing the topological quantum field theory structure on Riemann surfaces <math><msub><mi>C</mi><mrow><mi>g</mi><mo>,</mo><mi>n</mi></mrow></msub></math>. Furthermore, our study highlights the relationship between the left-moving sector of the (0,2) theory and the chiral algebra of the four-dimensional <math><mi>N</mi><mo>=</mo><mn>2</mn></math> theory. Notably, we propose that the (0,2) elliptic genus of a theory of this class can be expressed as a linear combination of characters of the corresponding chiral algebra.

Satoshi Nawata

Jiahao Zheng

Class <math><mi>S</mi></math> theories on <math><msup><mi>S</mi><mn>2</mn></msup></math>

4D <math><mi>N</mi><mo>=</mo><mn>2</mn></math> superconformal field theories and their invariants can be often enriched by nonlocal Bogomol’nyi-Prasad-Sommerfield operators. In this paper we study the flavored Schur index of several types of <math><mi>N</mi><mo>=</mo><mn>2</mn></math> superconformal field theories with and without line operators, using a series of new integration formula of elliptic functions and Eisenstein series. We demonstrate how to evaluate analytically the Schur index for a series of <math><msub><mi>A</mi><mn>2</mn></msub></math> class-<math><mi>S</mi></math> theories and the <math><mi>N</mi><mo>=</mo><mn>4</mn></math> <math><mi>S</mi><mi>O</mi><mo>(</mo><mn>7</mn><mo>)</mo></math> theory. For all <math><msub><mi>A</mi><mn>1</mn></msub></math> class-<math><mi>S</mi></math> theories we obtain closed-form expressions for <math><mi>S</mi><mi>U</mi><mo>(</mo><mn>2</mn><mo>)</mo></math> Wilson line index, and ’t Hooft line index in some simple cases. We also observe the relation between the line operator index with the characters of the associated chiral algebras. The Wilson line index for some other low rank gauge theories is also studied.

Zhaoting Guo

Yutong Li

Yufan Wang

<math><mi>N</mi><mo>=</mo><mn>2</mn></math> Schur index and line operators

Flavored modular differential equations sometimes arise from null states or their descendants in a chiral algebra with continuous flavor symmetry. In this paper we focus on Kac-Moody algebras <math><msub><mover><mi>g</mi><mo>^</mo></mover><mi>k</mi></msub></math> that contain a level-four null state <math><mo>|</mo><msub><mi>N</mi><mi>T</mi></msub><mo>⟩</mo></math> which implements the nilpotency of the Sugawara stress tensor. We study the properties of the corresponding flavored modular differential equations, and show that the equations exhibit almost covariance under modular <math><mi>S</mi></math>-transformation, connecting null states and their descendants at different levels. The modular property of the equations fixes the structure of <math><mi>g</mi></math> and the level <math><mi>k</mi></math>, as well as the flavored characters of all the highest weight representations. Shift property of the equations can generate nonvacuum characters starting from the vacuum character.

Flavored modular differential equations

Every 4D <math><mi>N</mi><mo>=</mo><mn>2</mn></math> superconformal field theory <math><mi>T</mi></math> corresponds to an associated vertex operator algebra <math><mi>V</mi><mo>(</mo><mi>T</mi><mo>)</mo></math>, which is in general nonrational with a more involved representation theory. Null states in <math><mi>V</mi><mo>(</mo><mi>T</mi><mo>)</mo></math> can give rise to nontrivial flavored modular differential equations, which must be satisfied by the refined/flavored character of all the <math><mi>V</mi><mo>(</mo><mi>T</mi><mo>)</mo></math> modules. Taking some <math><msub><mi>A</mi><mn>1</mn></msub></math> theories <math><msub><mi>T</mi><mrow><mi>g</mi><mo>,</mo><mi>n</mi></mrow></msub></math> of class <math><mi>S</mi></math> as examples, we construct the flavored modular differential equations satisfied by the Schur index. We show that three types of surface defect indices give rise to common solutions to these differential equations and therefore are sources of <math><mi>V</mi><mo>(</mo><mi>T</mi><mo>)</mo></math>-module characters. These equations transform almost covariantly under modular transformations, ensuring the presence of logarithmic solutions which may correspond to characters of logarithmic modules.

Haocong Zheng

Surface defects, flavored modular differential equations, and modularity

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.

United Kingdom

Wolfger Peelaers

Exact Schur index in closed form

For all 4D <math><mi>N</mi><mo>=</mo><mn>4</mn></math> Super-Yang-Mills theories with simple gauge groups <math><mi>G</mi></math>, we show that a set of residues of the integrands in the <math><mi>N</mi><mo>=</mo><mn>4</mn></math> Schur indices, which are related to Gukov-Witten-type surface defects in the theories, equal the vacuum characters of <math><mrow><mi>rank</mi></mrow><mi>G</mi></math> copies of <math><mi>b</mi><mi>c</mi><mi>β</mi><mi>γ</mi></math> systems that provide the free field realization of associated <math><mi>N</mi><mo>=</mo><mn>4</mn></math> vertex operator algebras in [F. Bonetti, C. Meneghelli, and L. Rastelli, J. High Energy Phys. 05 (2019) 155]. This result predicts that these residues, as module characters, are additional solutions to the flavored modular differential equations satisfied by the original Schur index. The prediction is verified in the <math><mi>G</mi><mo>=</mo><mi>S</mi><mi>U</mi><mo>(</mo><mn>2</mn><mo>)</mo></math> case, where an additional logarithmic solution is constructed.