Limits and Degenerations of Unitary Conformal Field Theories

TL;DR

The paper develops an intrinsic framework to study degeneration and limits of two-dimensional conformal field theories (CFTs) by defining convergent sequences of CFTs and extracting geometric data from their limits via Connes' non-commutative geometry. It shows that, in degenerating limits, the vacuum sector can yield a commutative zero-mode algebra whose spectrum defines a target manifold $M$ with a dilaton-corrected metric, and it connects this to a limiting spectral triple when possible. The approach is illustrated with concrete examples: torus models and their orbifolds, where Narain lattices control convergent behavior and boundary degenerations; and most prominently with the A-series of unitary Virasoro minimal models ${\mathcal M}(m,m{+}1)$, whose large-$m$ limit yields a geometric interpretation on the interval $[0,\pi]$ with a specific dilaton, encoded via Chebyshev polynomials. The work thus links CFT degenerations to classical geometric limits, clarifying how boundary points of CFT moduli spaces can encode geometric data and D-brane structures, and it lays groundwork for extending these ideas to WZW models and broader coset theories.

Abstract

In the present paper, degeneration phenomena in conformal field theories are studied. For this purpose, a notion of convergent sequences of CFTs is introduced. Properties of the resulting limit structure are used to associate geometric degenerations to degenerating sequences of CFTs, which, as familiar from large volume limits of non-linear sigma models, can be regarded as commutative degenerations of the corresponding ``quantum geometries''. As an application, the large level limit of the A-series of unitary Virasoro minimal models is investigated in detail. In particular, its geometric interpretation is determined.