Holomorphic quasi-modular bootstrap

TL;DR

This work extends holomorphic modular bootstrap to a flavored, quasi-modular setting by positing intrinsic constraints on a distinguished null state $|\mathcal{N}_T\rangle$ in a class of chiral algebras. It derives flavored modular differential equations with quasi-Jacobi coefficients and reveals a universal S-action that organizes the null-state orbit, enabling full determination of untwisted and twisted spectra in many cases, including the Deligne–Cvitanović series and $\widehat{\mathfrak{su}}(2)_k$ algebras. By coupling flavor refinements to modular transformations and spectral-flow translations, the authors construct flavor-augmented MLDEs, obtain twisted MLDEs, and demonstrate that quasi-modularity robustly constrains spectra of quasi-lisse algebras. The framework unifies modularity, null-state constraints, and spectral flow to predict entire spectra, including twisted sectors, with potential applications to 4d $\mathcal{N}=2$ SCFTs via associated chiral algebras and to the classification of non-rational RCFTs. Overall, the holomorphic quasi-modular bootstrap provides a powerful, predictive method for resolving spectra in non-rational or quasi-lisse chiral algebras through flavored MLDEs and universal null-state dynamics.

Abstract

Holomorphic modular bootstrap is an approach to classifying rational conformal field theories making use of the modular differential equations. In this paper we explore its flavored refinement. For a class of chiral algebras, we propose constraints on a special null state, which determine the structure of the algebra, and through flavored modular differential equations and quasi-modularity, completely fix the spectra in both the untwisted and twisted sector. Using the differential equations, we reveal hidden structures among null states of the chiral algebras under the modular group action and translation related to spectral flow.