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The exact Schur index in closed form

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arxiv 2112.09705 v4 pith:ZZ3AOH4Z submitted 2021-12-17 hep-th math-phmath.MP

Yiwen Pan , Wolfger Peelaers This is my paper · ORCID
classification hep-thmath-phmath.MP
keywords indexschurtheoriesformclassclosedexactflavored
verification ladder T0 review T1 audit T2 compute T3 formal
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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.

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