Vertex operator algebras, Higgs branches, and modular differential equations

Journal of High Energy Physics, Aug 2018

Abstract Every four-dimensional \( \mathcal{N}=2 \) 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 \( \mathcal{N}=2 \) 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-Cvitanović exceptional series” of simple Lie algebras, several families of Argyres-Douglas theories, an assortment of class \( \mathcal{S} \) theories, and \( \mathcal{N}=2 \) super Yang-Mills with \( \mathfrak{s}\mathfrak{u}(n) \) gauge group for small-to-moderate values of n.

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Vertex operator algebras, Higgs branches, and modular differential equations

Journal of High Energy Physics August 2018, 2018:114 | Cite as Vertex operator algebras, Higgs branches, and modular differential equations AuthorsAuthors and affiliations Christopher BeemLeonardo Rastelli Open Access Regular Article - Theoretical Physics First Online: 21 August 2018 Received: 15 June 2018 Accepted: 31 July 2018 29 Downloads 1 Citations Abstract Every four-dimensional \( \mathcal{N}=2 \) 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 \( \mathcal{N}=2 \) 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-Cvitanović exceptional series” of simple Lie algebras, several families of Argyres-Douglas theories, an assortment of class \( \mathcal{S} \) theories, and \( \mathcal{N}=2 \) super Yang-Mills with \( \mathfrak{s}\mathfrak{u}(n) \) gauge group for small-to-moderate values of n. Keywords Conformal and W Symmetry Conformal Field Theory Extended Supersymmetry Supersymmetric Gauge Theory  ArXiv ePrint: 1707.07679 Download to read the full article text Notes Open Access This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. References [1] C. Beem, M. Lemos, P. Liendo, W. Peelaers, L. Rastelli and B.C. van Rees, Infinite chiral symmetry in four dimensions, Commun. Math. Phys. 336 (2015) 1359 [arXiv:1312.5344] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar [2] P. Liendo, I. Ramirez and J. Seo, Stress-tensor OPE in N = 2 superconformal theories, JHEP 02 (2016) 019 [arXiv:1509.00033] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar [3] M. Lemos and P. 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Christopher Beem, Leonardo Rastelli. Vertex operator algebras, Higgs branches, and modular differential equations, Journal of High Energy Physics, 2018, 114, DOI: 10.1007/JHEP08(2018)114