A non-rational CFT with c = 1 as a limit of minimal models

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.

A common limit of super Liouville theory and minimal models

“…Further checks on the theory should be performed, in particular one should analyse whether the theory is crossing symmetric. For the bosonic counterpart of the theory, this has been analysed in [5], on the one hand analytically for specific four-point correlators, and on the other hand numerically for the generic case. Similar checks could also be performed in the supersymmetric case, numerical checks could use the recently found recursion relations for superconformal blocks [28,29].…”

“…At some points of the moduli space, the theory is rational, and it was argued in [4] that these are already all rational theories at c = 1. That there are further non-rational models that cannot be obtained from the free boson was shown by Runkel and Watts [5] by explicitly constructing a new not even quasi-rational theory at c = 1.…”

“…Later in [6], Schomerus considered a continuation of Liouville theory to central charge c = 1, and found that it agrees with the Runkel-Watts theory. In [5], the authors also constructed boundary states for this theory as a limit of minimal model boundary states (see also [7]). It was then understood in [8] that these boundary states can be obtained from the ZZ boundary states [9] in Liouville theory.…”

“…Because of field identifications, we can restrict attention to labels (r, s) satisfying r ≥ st (where r = st can only be satisfied for the Ramond ground state, which exists for even p). Let us rewrite 5) where the first term is an even (odd) integer in the Neveu-Schwarz (Ramond) sector, and the second term ranges between 0 and 2. For a given d we look for d rs that approximate d within a small interval of size ǫ, so we introduce the set…”

“…For bosonic minimal models, both options have been investigated leading to one discrete family of boundary conditions [7,5], and to one continuous family [8]. For the supersymmetric minimal models we shall only consider the case when the boundary labels are taken to be fixed.…”