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

Abstract

We propose a novel modular anomaly equation for the unflavored Schur index in the $\mathcal{N}=4$ $SU(N)$ super-Yang-Mills theory. The vanishing conditions overdetermine the modular ambiguity ansatz from the equation, thus together they are sufficient to recursively compute the exact Schur indices for all $SU(N)$ gauge groups. Using the representations as MacMahon's generalized sum-of-divisors functions and Jacobi forms, we then prove our proposal as well as elucidate a general formula conjectured by Pan and Peelaers.

14 pages. v2: journal version, some improvements, references added

References in corpus (4)

• The 4d Superconformal Index from q-deformed 2d Yang-Mills
• 5-dim Superconformal Index with Enhanced En Global Symmetry
• Infinitely many 4d $\mathcal{N}=2$ SCFTs with $a=c$ and beyond
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Cited by in corpus (6)

• $\mathcal{N}=2^{\ast }$ Schur indices
• Surface defects, flavored modular differential equations and modularity
• Root of unity asymptotics for Schur indices of 4d Lagrangian theories
• A landscape of 4d N=1 SCFTs with a=c
• Hawking-Page transition on a spin chain
• $SL(2,\mathbb{Z})$ dualities of boundary conditions in Abelian M2-brane SCFTs