The exact Schur index of $\mathcal{N}=4$ SYM

Abstract

The Witten index counts the difference in the number of bosonic and fermionic states of a quantum mechanical system. The Schur index, which can be defined for theories with at least $\mathcal{N}=2$ supersymmetry in four dimensions is a particular refinement of the index, dependent on one parameter $q$ serving as the fugacity for a particular set of charges which commute with the hamiltonian and some supersymmetry generators. This index has a known expression for all Lagrangian and some non-Lagrangian theories as a finite dimensional integral or a complicated infinite sum. In the case of $\mathcal{N}=4$ SYM with gauge group $U(N)$ we rewrite this as the partition function of a gas of $N$ non interacting and translationally invariant fermions on a circle. This allows us to perform the integrals and write down explicit expressions for fixed $N$ as well as the exact all orders large $N$ expansion.

11 pages; v2: typos corrected

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• Localization on Hopf surfaces
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Cited by in corpus (36)

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• Finite $N$ Corrections to the Superconformal Index of S-fold Theories
• The $\mathcal{N}=2$ Schur index from free fermions
• Finite-$N$ corrections to the M-brane indices
• The exact Schur index in closed form
• $\mathcal{N}=2^{\ast }$ Schur indices
• Twisted M2 brane holography and sphere correlation functions
• Exact Stringy Microstates from Gauge Theories
• Finite $N$ corrections to the superconformal index of toric quiver gauge theories
• The $\mathcal{N}=4$ Schur index with Polyakov loops
• Decomposition of BPS Moduli Spaces and Asymptotics of Supersymmetric Partition Functions
• Exact $\mathcal{N}=2^{\ast }$ Schur line defect correlators
• Finite $N$ corrections to the superconformal index of orbifold quiver gauge theories
• Schur Indices of Class $\mathcal{S}$ and Quasimodular Forms
• More on holographic correlators: Twisted and dimensionally reduced structures
• Modular Anomaly Equation for Schur Index of $\mathcal{N}=4$ Super-Yang-Mills
• Surface defects, flavored modular differential equations and modularity
• Dualities and flavored indices of M2-brane SCFTs
• Modularity and 4D-2D spectral equivalences for large-N gauge theories with adjoint matter
• BPS Partition Functions for S-folds
• The giant graviton expansion in $Ad{S}_5\times S^5$
• Closed form fermionic expressions for the Macdonald index
• High-temperature asymptotics of the 4d superconformal index
• M2-branes and $\mathfrak{q}$-Painlevé equations
• Root of unity asymptotics for Schur indices of 4d Lagrangian theories
• ADHM Wilson line defect indices
• The NS limit of the 5D Superconformal Index
• The Giant Graviton Expansion
• Finite-$N$ superconformal index via the AdS/CFT correspondence
• Localization and wall-crossing of giant graviton expansions in AdS${}_5$
• The Hyperbolic Asymptotics of Elliptic Hypergeometric Integrals Arising in Supersymmetric Gauge Theory
• Giant Gravitons and Volume Minimisation
• $SL(2,\mathbb{Z})$ dualities of boundary conditions in Abelian M2-brane SCFTs