The resolution of field identification fixed points in diagonal coset theories

TL;DR

This work resolves the fixed-point problem for diagonal coset conformal field theories by implementing field identification directly on branching spaces via diagram automorphisms that satisfy a factorization property. Twining characters and orbit Lie algebras are used to compute both the coset characters and the modular $S$-matrix, showing that fixed-point splitting corresponds to eigenspaces under the identification group, with the coset chiral algebra effectively fixed by simple-current actions. The authors derive explicit expressions for the fixed-point characters and the $S$-matrix in terms of orbit Lie-algebra data and identification-group characters, and validate the framework on diagonal $B_n$ embeddings by connecting results to ${f Z}_2$ orbifolds of a compact boson. The approach generalizes to generalized diagonal cosets and clarifies the structure of the coset chiral algebra as the fixed-point subalgebra, providing a robust algebraic handle on a previously subtle issue with fixed points and modular invariants.

Abstract

The fixed point resolution problem is solved for diagonal coset theories. The primary fields into which the fixed points are resolved are described by submodules of the branching spaces, obtained as eigenspaces of the automorphisms that implement field identification. To compute the characters and the modular S-matrix we use `orbit Lie algebras' and `twining characters', which were introduced in a previous paper (hep-th/9506135). The characters of the primary fields are expressed in terms of branching functions of twining characters. This allows us to express the modular S-matrix through the S-matrices of the orbit Lie algebras associated to the identification group. Our results can be extended to the larger class of `generalized diagonal cosets'.