Coset construction of logarithmic minimal models: branching rules and branching functions

TL;DR

This paper extends logarithmic minimal models to higher fusion levels using coset constructions, providing explicit branching rules and functions that describe their structure and representations.

Contribution

It introduces an infinite hierarchy of logarithmic conformal field theories at higher fusion levels and derives explicit branching rules and functions for their characters.

Findings

Logarithmic minimal models extend to higher fusion levels n=1,2,3,...

All models are realized as diagonal cosets involving fractional levels

Explicit branching rules and functions are derived for these models

Abstract

Working in the Virasoro picture, it is argued that the logarithmic minimal models LM(p,p')=LM(p,p';1) can be extended to an infinite hierarchy of logarithmic conformal field theories LM(p,p';n) at higher fusion levels n=1,2,3,.... From the lattice, these theories are constructed by fusing together n x n elementary faces of the appropriate LM(p,p') models. It is further argued that all of these logarithmic theories are realized as diagonal cosets (A_1^{(1)})_k \oplus (A_1^{(1)})_n / (A_1^{(1)})_{k+n} where n is the integer fusion level and k=np/(p'-p)-2 is a fractional level. These cosets mirror the cosets of the higher fusion level minimal models of the form M(m,m';n), but are associated with certain reducible representations. We present explicit branching rules for characters in the form of multiplication formulas arising in the logarithmic limit of the usual Goddard-Kent-Olive coset construction of the non-unitary minimal models M(m,m';n). The limiting branching functions play the role of Kac characters for the LM(p,p';n) theories.