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

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arxiv 1507.08659 v2 pith:6WG5EMFD submitted 2015-07-30 hep-th

Jun Bourdier , Nadav Drukker , Jan Felix This is my paper · ORCID
classification hep-th
keywords indexmathcalexactparticularschursomesupersymmetrytheories
verification ladder T0 review T1 audit T2 compute T3 formal
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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.

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Cited by 5 Pith papers

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  1. Interior zeros of supersymmetric indices

    hep-th 2026-07 accept novelty 7.0 of 10

    Interior zeros of a supersymmetric index exist iff its arithmetic coefficients δ(ν) grow exponentially at rate set by the nearest zero, detectable from finitely many q-series terms.

  2. Line operator indices of S-fold theories

    hep-th 2026-07 conditional novelty 6.0 of 10

    Line-operator Schur indices for S-fold theories are matched to Wilson-'t Hooft indices in rank-2 N=4 SYM once giant graviton corrections are included, with new fivebrane-junction indices derived for k=3,4,6.

  3. The giant graviton expansion in AdS$_5\times$SE$_5$

    hep-th 2026-07 conditional novelty 6.0 of 10

    Giant-graviton fluctuations on AdS5 imes SE5 are governed by a conical Fock-Darwin system whose lowest Landau level yields the protected finite-N index of the dual N=1 SCFT for T1,1.

  4. Character Identities Between Affine and Virasoro Vertex Operator Algebra Modules

    math.QA 2025-11 conditional novelty 6.0 of 10

    Admissible-level affine sl2 modules and rational Virasoro minimal-model modules have matching characters under the substitution (w,q) -> (q^{+/-1/2}, q^3).

  5. The Superconformal Index and Black Hole Instabilities

    hep-th 2025-02 reject novelty 5.0 of 10

    The two-angular-momentum superconformal index of N=4 SYM has degeneracy tails beyond the entropy of the single AdS5 black hole, interpreted as grey galaxy signatures.

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