The exact Schur index in closed form

Abstract

The Schur limit of the superconformal index of a four-dimensional N = 2 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 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 S of type a1, we put forward a conjecture for the unflavored Schur indices of all N=4 super Yang-Mills theories with gauge group SU(N), 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.

65 pages; v2: minor clarifications, refs added; v3: typos corrected; v4: typos corrected

References in corpus (61)

• N=2 dualities
• An Index for 4 dimensional Super Conformal Theories
• Infinite Chiral Symmetry in Four Dimensions
• Gauge Theories and Macdonald Polynomials
• $\mathcal{N}=1$ supersymmetric indices and the four-dimensional A-model
• An N=2 Superconformal Fixed Point with E_6 Global Symmetry
• S-duality in N=2 supersymmetric gauge theories
• The 4d Superconformal Index from q-deformed 2d Yang-Mills
• S-duality and 2d Topological QFT
• 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
• On the Superconformal Index of Argyres-Douglas Theories
• The Superconformal Index of the E_6 SCFT
• Chiral algebras of class S
• Modular Data and Verlinde Formulae for Fractional Level WZW Models I
• 5d and 4d SCFTs: Canonical Singularities, Trinions and S-Dualities
• More on the N=2 superconformal systems of type $D_p(G)$
• Deformation quantization and superconformal symmetry in three dimensions
• Surface Defects and Chiral Algebras
• Superconformal indices of generalized Argyres-Douglas theories from 2d TQFT
• Infrared Computations of Defect Schur Indices
• Infinitely many N=2 SCFT with ADE flavor symmetry
• Vertex operator algebras of Argyres-Douglas theories from M5-branes
• Chiral algebra of Argyres-Douglas theory from M5 brane
• Chiral Algebras for Trinion Theories
• VOAs labelled by complex reflection groups and 4d SCFTs
• Factorisation and holomorphic blocks in 4d
• The exact Schur index of $\mathcal{N}=4$ SYM
• Joseph ideals and lisse minimal W-algebras
• Line Operator Index on S1 $\times$ S3
• A gravity interpretation for the Bethe Ansatz expansion of the $\mathcal{N}=4$ SYM index
• Modular differential equations and null vectors
• Superconformal surfaces in four dimensions
• A Bethe Ansatz type formula for the superconformal index
• Higgs branch localization of $\mathcal{N}=1$ theories on $S^3\times S^1$
• The Superconformal Index of Class S Theories of Type D
• 2d TQFT structure of the superconformal indices with outer-automorphism twists
• Schur index of the $\mathcal{N}=4$ $U(N)$ SYM via the AdS/CFT correspondence
• On a modular property of N=2 superconformal theories in four dimensions
• Torus n-Point Functions for $\mathbb{R}$-graded Vertex Operator Superalgebras and Continuous Fermion Orbifolds
• Schur correlation functions on $S^3\times S^1$
• SCFT/VOA correspondence via $\Omega$-deformation
• A "Lagrangian" for the E7 Superconformal Theory
• The $\mathcal{N}=2$ Schur index from free fermions
• Surface defects, the superconformal index and q-deformed Yang-Mills
• Surface Defect Indices and 2d-4d BPS States
• Chiral Algebras, Localization and Surface Defects
• Infinitely many 4d $\mathcal{N}=2$ SCFTs with $a=c$ and beyond
• Superconformal index of low-rank gauge theories via the Bethe Ansatz
• An N=1 Lagrangian for an N=3 SCFT
• Schur Indices of Class $\mathcal{S}$ and Quasimodular Forms
• 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
• Hidden exceptional symmetry in the pure spinor superstring
• Peculiar Index Relations, 2D TQFT, and Universality of SUSY Enhancement
• Schur correlation functions from q-deformed Yang-Mills

Cited by in corpus (8)

• $\mathcal{N}=2^{\ast }$ Schur indices
• Schur Indices of Class $\mathcal{S}$ and Quasimodular Forms
• Large $N$ and large representations of Schur line defect correlators
• Surface defects, flavored modular differential equations and modularity
• Root of unity asymptotics for Schur indices of 4d Lagrangian theories
• Realizing giant valley polarization effect based on monolayer altermagnets
• A landscape of 4d N=1 SCFTs with a=c
• A New Determination of the (Z,A) Dependence of Coherent Muon-to-Electron Conversion