A non-rational CFT with c=1 as a limit of minimal models
I. Runkel, G. M. T. Watts
TL;DR
This work proposes that the $c\to1$ limit of unitary minimal models $M_p$ yields a new non-rational CFT $M_\infty$ with a continuous bulk spectrum. The theory hosts primaries $\phi_x$ with weights $h_x = x^2/4$ for $x>0$, excluding positive integers, and its bulk OPE is an integral over $z$ with structure constants $c(x,y,z)=P(x,y,z)\exp(Q(x,y,z))$, where $P$ is a symmetric step function and $Q$ is specified by a convergent integral (requiring analytic continuation for larger $x,y,z$). The authors provide explicit formulas for $c(x,y,z)$, show positivity, and test crossing symmetry analytically (in special cases) and numerically, while also identifying a discrete set of boundary states corresponding to $c=1$ boundary conditions. The limiting theory is not a free boson and is neither rational nor quasi-rational, bearing Liouville-like features and suggesting connections to Liouville theory and minimal-model boundary RG flows. Overall, the work furnishes concrete data for a novel $c=1$ CFT and outlines a framework for exploring its bulk-boundary structure and consistency checks.
Abstract
We investigate the limit of minimal model conformal field theories where the central charge approaches one. We conjecture that this limit is described by a non-rational CFT of central charge one. The limiting theory is different from the free boson but bears some resemblance to Liouville theory. Explicit expressions for the three point functions of bulk fields are presented, as well as a set of conformal boundary states. We provide analytic and numerical arguments in support of the claim that this data forms a consistent CFT.