Solving two-dimensional conformal field theories using the bootstrap approach
Mercredi 07/09/2022, 14:00
Amphi Claude Bloch, Bât. 774, Orme des Merisiers
This thesis consists of three main results. Firstly, we build logarithmic representations of the Virasoro algebra at generic central charge by using derivatives of primary fields (null or not). In the case of null fields, the resulting representations are parametrized by the logarithmic couplings, which can be completely determined by using the existence of the degenerate fields. We also write down closed expressions for four-point conformal blocks of these logarithmic representations. As an application, logarithmic representations, generated by the first-order derivative of null fields, complete the determination of the action of the Virasoro algebra on the spectra of CFTs describing the critical points of the Potts model and the $O(n)$ model in two dimensions, also known as the Potts and $O(n)$ CFTs.
Secondly, we initiate a systematic study of generic four-point functions of the Potts and $O(n)$ CFTs at generic central charge. Four-point functions of these two CFTs are subject to two constraints: the crossing-symmetry equation and constraints from their global symmetries. We then solve the crossing-symmetry equation for several of their four-point functions. For the $O(n)$ CFT, we find that solutions to the crossing-symmetry equation are always consistent with $O(n)$ symmetry of the $O(n)$ CFT. In the case of the Potts CFT, there however can be extra solutions, which are inconsistent with $S_Q$ symmetry of the Potts CFT and do not yet have clear interpretations. In particular, for both CFTs, we have determined their numbers of crossing-symmetry solutions, several exact spectra, several analytic formulae of their four-point structure constants, and a few corresponding fusion rules. We also discuss our preliminary results on bootstrapping the $O(n)$ CFT at $n=0$, which corresponds to the critical self-avoiding random walk in two dimensions. Thirdly, we consider rational limits of four-point functions of the so-called generalized minimal models. We find that the resulting four-point functions can have our logarithmic representations propagating in their channels. These four-point functions also lead to non-chiral fusion products, whose chiral projection coincides with some fusion rules of chiral logarithmic minimal models proposed by P. Mathieu and D. Ridout. This suggests that there may exist logarithmic minimal models in the bulk.