Crossing invariant correlation functions at $c=1$ from isomonodromic $\tau$ functions

Abstract

We present an approach that gives rigorous construction of a class of crossing invariant functions in $c=1$ CFTs from the weakly invariant distributions on the moduli space ${\mathcal{M}}_{0,4}^{SL(2,\mathbb{C})}$ of $SL(2,\mathbb{C})$ flat connections on the sphere with four punctures. By using this approach we show how to obtain correlation functions in the Ashkin-Teller and the Runkel-Watts theory. Among the possible crossing-invariant theories, we obtain also the analytic Liouville theory, whose consistence was assumed only on the basis of numerical tests.

39 pages, 4 figures, version in JHEP, fixed proof in sec. 6.4, updates in sec. 4.5, 4.6