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

Abstract

It has long been understood that non-trivial Conformal Field Theories (CFTs) with vanishing central charge (c=0c=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 TT and its logarithmic partner tt (the solution of the ``c\to 0c→0 catastrophe’’). In this paper, we determine this structure together with the associated OPE of primary fields up to level h=\bar{h}=2h=h‾=2 for polymers and percolation CFTs. This is done by taking the c\to 0c→0 limit of O(n)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}TT‾, and is identical for polymers and percolation. It is characterized in part by the common value of a non-chiral logarithmic coupling a_0=-\frac{25}{48}a0=−2548.