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Vertex operator algebras, Higgs branches, and modular differential equations

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arxiv 1707.07679 v4 pith:LSU2QFDG submitted 2017-07-24 hep-th

Vertex operator algebras, Higgs branches, and modular differential equations

classification hep-th
keywords operatorvertexalgebraassociatedtheoriesalgebraicalgebrasdifferential
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved
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Every four-dimensional ${\cal N}=2$ superconformal field theory comes equipped with an intricate algebraic invariant, the associated vertex operator algebra. The relationships between this invariant and more conventional protected quantities in the same theories have yet to be completely understood. In this work, we aim to characterize the connection between the Higgs branch of the moduli space of vacua (as an algebraic geometric entity) and the associated vertex operator algebra. Ultimately our proposal is simple, but its correctness requires the existence of a number of nontrivial null vectors in the vacuum Verma module of the vertex operator algebra. Of particular interest is one such null vector whose presence suggests that the Schur index of any ${\cal N}=2$ SCFT should obey a finite order modular differential equation. By way of the "high temperature" limit of the superconformal index, this allows the Weyl anomaly coefficient $a$ to be reinterpreted in terms of the representation theory of the associated vertex operator algebra. We illustrate these ideas in a number of examples including a series of rank-one theories associated with the "Deligne-Cvitanovi\'c exceptional series" of simple Lie algebras, several families of Argyres-Douglas theories, an assortment of class ${\cal S}$ theories, and ${\cal N}=4$ super Yang-Mills with $\mathfrak{su}(n)$ gauge group for small-to-moderate values of $n$.

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Cited by 6 Pith papers

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    hep-th 2026-07 conditional novelty 7.0

    Higgsless Lagrangian N=2 SCFTs are sparse (free vectors, one SO/USp quiver family, three sporadics); the VOAs of two sporadics are strongly finite and logarithmic.

  2. Vertex Superalgebras for Hypertoric Varieties and 3d Abelian Gauge Theories

    math.QA 2026-06 unverdicted novelty 7.0

    Constructs ħ-adic sheaves of vertex superalgebras on hypertoric varieties, proves the associated affine variety recovers the singular hypertoric one, establishes the 3d Higgs branch conjecture for abelian cases, and s...

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  4. On non-relativistic integrable models and 4d SCFTs

    hep-th 2026-04 unverdicted novelty 6.0

    Generalized Schur indices of N=2 class S theories are expressed using eigenfunctions of non-relativistic elliptic Calogero-Moser models, with extensions claimed for N=1 SCFTs via limits of models like Inozemtsev.

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