Modular Invariance of Characters for Affine Lie Algebras at Subprincipal Admissible Levels

TL;DR

The paper addresses modular invariance of normalized characters for subprincipal admissible levels of affine Lie algebras, establishing that the span of these characters is invariant under $\mathrm{SL}_2(\mathbf{Z})$ with an explicit transformation formula. It develops a framework expressing numerators $A_{\Lambda+\rho}$ through Jacobi theta forms and derives their modular behavior, enabling a complete SL2(Z) transformation analysis for subprincipal weights. The authors prove Theorem 5.1, which shows the $\mathrm{SL}_2(\mathbf{Z})$-invariance of the span of subprincipal normalized characters and extend these modularity results to quantum Hamiltonian reduction $H_f(\Lambda)$ and the modular invariance of certain vertex algebras, including subprincipal analogues of $V_k(\overline{\mathfrak g})$ and $\widetilde W_k(\overline{\mathfrak g},f)$. The work broadens the understanding of modularity beyond principal levels and connects representation theory of affine and twisted affine algebras with W-algebra constructions and vertex algebra modular invariance.

Abstract

We prove that the span of normalized characters of subprincipal admissible modules over an affine Lie algebra of subprincipal admissible level $k$ is $SL_2(\mathbf{Z})$-invariant and find the explicit modular transformation formula.

Theorems & Definitions (6)

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• proof : Proof of Theorem 5.1.
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