A note on the identity module in $c=0$ CFTs

TL;DR

This paper elucidates the structure of the identity module in $c=0$ logarithmic CFTs, specifically for polymers and percolation, revealing a rank-3 Jordan cell and a common logarithmic coupling.

Contribution

It determines the detailed structure of the identity module in $c=0$ CFTs, including the OPEs and Jordan cell structure, using bootstrap and conformal invariance methods.

Findings

Identified a rank-3 Jordan cell involving $Tar{T}$

Derived the OPE structure up to level 2 for polymers and percolation

Found a universal non-chiral logarithmic coupling $a_0=-25/48

Abstract

It has long been understood that non-trivial Conformal Field Theories (CFTs) with vanishing central charge ($c=0$) are logarithmic. So far however, the structure of the identity module -- the (left and right) Virasoro descendants of the identity field -- had not been elucidated beyond the stress-energy tensor $T$ and its logarithmic partner $t$ (the solution of the "$c\to 0$ catastrophe"). In this paper, we determine this structure together with the associated OPE of primary fields up to level $h=\bar{h}=2$ for polymers and percolation CFTs. This is done by taking the $c\to 0$ limit of $O(n)$ and Potts models and combining recent results from the bootstrap with arguments based on conformal invariance and self-duality. We find that the structure contains a rank-3 Jordan cell involving the field $T\bar{T}$, and is identical for polymers and percolation. It is characterized in part by the common value of a non-chiral logarithmic coupling $a_0=-{25\over 48}$.