Surface defects, flavored modular differential equations and modularity

Abstract

Every 4d $\mathcal{N}=2$ SCFT $\mathcal{T}$ corresponds to an associated VOA $\mathbb{V}(\mathcal{T})$, which is in general non-rational with a more involved representation theory. Null states in $\mathbb{V}(\mathcal{T})$ can give rise to non-trivial flavored modular differential equations, which must be satisfied by the refined/flavored character of all the $\mathbb{V}(\mathcal{T})$-modules. Taking some $A_1$ theories ${\mathcal{T}}_{g,n}$ of class-$\mathcal{S}$ 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 $\mathbb{V}(\mathcal{T})$-module characters. These equations transform almost covariantly under modular transformations, ensuring the presence of logarithmic solutions which may correspond to characters of logarithmic modules.

arXiv comment76 pages, 3 figures

References in corpus (64)

• Liouville Correlation Functions from Four-dimensional Gauge Theories
• N=2 dualities
• Localization of gauge theory on a four-sphere and supersymmetric Wilson loops
• An Index for 4 dimensional Super Conformal Theories
• Infinite Chiral Symmetry in Four Dimensions
• Loop and surface operators in N=2 gauge theory and Liouville modular geometry
• The Moduli Space of N=2 SUSY QCD and Duality in N=1 SUSY QCD
• Gauge Theories and Macdonald Polynomials
• The 4d Superconformal Index from q-deformed 2d Yang-Mills
• Bootstrapping the superconformal index with surface defects
• Schur Indices, BPS Particles, and Argyres-Douglas Theories
• W Symmetry in six dimensions
• 2d Index and Surface operators
• Vertex operator algebras, Higgs branches, and modular differential equations
• 4d Index to 3d Index and 2d TQFT
• On the Superconformal Index of Argyres-Douglas Theories
• Affine SL(2) conformal blocks from 4d gauge theories
• Chiral algebras of class S
• Exceptional Indices
• Surface Defects and Chiral Algebras
• Superconformal indices of generalized Argyres-Douglas theories from 2d TQFT
• Infrared Computations of Defect Schur Indices
• Vertex operator algebras of Argyres-Douglas theories from M5-branes
• M2-brane surface operators and gauge theory dualities in Toda
• Chiral algebra of Argyres-Douglas theory from M5 brane
• Chiral Algebras for Trinion Theories
• VOAs labelled by complex reflection groups and 4d SCFTs
• AGT on the S-duality Wall
• The exact Schur index of $\mathcal{N}=4$ SYM
• Joseph ideals and lisse minimal W-algebras
• Modular differential equations and null vectors
• sl^(2)_{-1/2}: A Case Study
• Superconformal surfaces in four dimensions
• On Irregular Singularity Wave Functions and Superconformal Indices
• Modular anomaly equation, heat kernel and S-duality in N=2 theories
• 2d TQFT structure of the superconformal indices with outer-automorphism twists
• Instanton Expansions for Mass Deformed N=4 Super Yang-Mills Theories
• On a modular property of N=2 superconformal theories in four dimensions
• The Liouville side of the Vortex
• Torus n-Point Functions for $\mathbb{R}$-graded Vertex Operator Superalgebras and Continuous Fermion Orbifolds
• Schur correlation functions on $S^3\times S^1$
• Towards a Classification of Two-Character Rational Conformal Field Theories
• Intersecting Surface Defects and Two-Dimensional CFT
• Surface defects, the superconformal index and q-deformed Yang-Mills
• The exact Schur index in closed form
• Chiral Algebras, Localization and Surface Defects
• Ghost Systems: A Vertex Algebra Point of View
• Generalized matrix models and AGT correspondence at all genera
• Schur sector of Argyres-Douglas theory and $W$-algebra
• Rational CFT With Three Characters: The Quasi-Character Approach
• On the Correspondence between Surface Operators in Argyres-Douglas Theories and Modules of Chiral Algebra
• Bootstrapping Fermionic Rational CFTs with Three Characters
• Classifying three-character RCFTs with Wronskian Index equalling $0$ or $2$
• Needles in a haystack: An algorithmic approach to the classification of 4d $\mathcal{N}=2$ SCFTs
• Schur Indices of Class $\mathcal{S}$ and Quasimodular Forms
• Wronskian Indices and Rational Conformal Field Theories
• Modular Anomaly Equation for Schur Index of $\mathcal{N}=4$ Super-Yang-Mills
• AGT/Z${}_2$
• Modular and duality properties of surface operators in N=2* gauge theories
• Hecke Relations, Cosets and the Classification of 2d RCFTs
• A geometric free field realisation for the genus-two class $\mathcal{S}$ theory of type ${\mathfrak{a}}_1$
• Defects, modular differential equations, and free field realization of N = 4 VOAs
• The Chiral Algebra of Genus Two Class $\mathcal{S}$ Theory
• Deformation quantizations from vertex operator algebras

Cited by in corpus (4)

• On Classification of Fermionic Rational Conformal Field Theories
• Bridging 4D QFTs and 2D VOAs via 3D high-temperature EFTs
• Rényi Entropy with Surface Defects in Six Dimensions
• Quasi-lisse vertex (super)algebras