The singular support of the Ising model

TL;DR

This paper provides a new Fermionic sum expression for the Ising model's vertex algebra character, describes its singular support, and establishes new Rogers-Ramanujan type identities with combinatorial interpretations.

Contribution

It introduces a novel Fermionic quasiparticle sum for the Ising model, characterizes its singular support, and proves new q-series identities related to Virasoro modules.

Findings

Explicit monomial basis for the Ising model

Description of the singular support as a subscheme

Three new Rogers-Ramanujan-Slater type q-series identities

Abstract

We prove a new Fermionic quasiparticle sum expression for the character of the Ising model vertex algebra, related to the Jackson-Slater $q$-series identity of Rogers-Ramanujan type and to Nahm sums for the matrix $\left( \begin{smallmatrix} 8 & 3 \\ 3 & 2 \end{smallmatrix} \right)$. We find, as consequences, an explicit monomial basis for the Ising model, and a description of its singular support. We find that the ideal sheaf of the latter, defining it as a subscheme of the arc space of its associated scheme, is finitely generated as a differential ideal. We prove three new $q$-series identities of the Rogers-Ramanujan-Slater type associated with the three irreducible modules of the Virasoro Lie algebra of central charge $1/2$. We give a combinatorial interpretation to the identity associated with the vacuum module.