Structure constants for the D-series Virasoro minimal models

TL;DR

The paper determines all bulk and boundary structure constants for D-series Virasoro minimal models on the upper half-plane by first solving the boundary CFT data and then extending to the bulk in a manner consistent with modular invariance. It derives genus-zero sewing constraints tailored to Virasoro minimal models, uses cylinder partition functions to fix boundary content, and expresses all constants through F-matrices, revealing a manifest Z2 symmetry tied to the D-diagram. The results include explicit formulas for boundary constants, bulk-boundary couplings, and bulk constants, with real-valued forms achievable in suitable bases; the maximal bulk content matches modular-invariant spectra, and numerical checks demonstrate consistency with the sewing constraints. The work highlights novel features of D-series boundary theories, such as i-type vs n-type boundaries and the Pasquier-algebra structure on disc partition functions, and connects to A-series structures through even subsectors and symmetry considerations.

Abstract

In this paper expressions are given for the bulk and boundary structure constants of D-series Virasoro minimal models on the upper half plane. It is the continuation of an earlier work on the A-series. The solution for the boundary theory is found first and then extended to the bulk. The modular invariant bulk field content is recovered as the maximal set of bulk fields consistent with the boundary theory. It is found that the structure constants are unique up to redefinition of the fields and in the chosen normalisation exhibit a manifest Z_2-symmetry associated to the D-diagram. The solution has been subjected to random numerical tests against the constraints it has to fulfill.