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 a 1 , 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.

Generalized multiple q-zeta values and characters of vertex algebras

“…We would like to express our gratitude to the organizers for the invitation. After this paper was finished, we were informed by Palash Singh about two very recent papers [12,47] dealing with indices of S-class theories.…”

Snowmass White Paper on SCFTs

“…This conjecture, which has been shown [278] to follow from a previous conjecture by Arakawa [330], carries deep implication; the VOAs which arise from the cohomological reduction of N = 2 SCFTs would then be of a special type known as "quasi-lisse" [330], a property which ensures that their vacuum characters satisfy a linear modular differential equation [331][332][333]. The representation theory of χ[T ], which is seldomly rational, can be complicated and it remains an open problem to understand which characters participate in the modular property of the vacuum character; these and related issues have been recently investigated in [334][335][336][337]. In recent years a varieties of techniques have been employed to compute χ[T ] in a large set of examples [279,331,332,[338][339][340][341][342][343][344][345][346][347][348][349] and it is worthwhile noticing that surprisingly often, in the Argyres-Douglas case, χ[T ] is an affine Kac-Moody at boundary admissible level [342,343,350,351].…”

Modular Anomaly Equation for Schur Index of $\mathcal{N}=4$ Super-Yang-Mills

“…According to the localization method, the unflavored Schur index of the N = 4 SU (N ) super-Yang-Mills theory can be written in terms of a unitary matrix integral. As in the literature [14,15], it is convenient to treat the even and odd ranks of the gauge groups separately. We consider first the simpler SU (2N + 1) case.…”

“…The results were organized into nice formulas in terms of quasi-modular forms in [14,15]. We can list the formulas in term of Eisenstein series for the first few orders (2.4)…”

“…In this paper we will simply consider the unflavored index without the extra fugacity. Some remarkable (quasi)-modular properties of the index are studied recently in [14,15] in the context of a larger class of theories, based on some earlier works in e.g. [16,17].…”