Lax matrix solution of c = 1 conformal field theory

Abstract

To a correlation function in a two-dimensional conformal field theory with the central charge c = 1, we associate a matrix differential equation Ψ ′ = LΨ, where the Lax matrix L is a matrix square root of the energy-momentum tensor. Then local conformal symmetry implies that the differential equation is isomonodromic. This provides a justification for the recently observed relation between four-point conformal blocks and solutions of the Painlevé VI equation. This also provides a direct way to compute the threepoint function of Runkel-Watts theory -the common c → 1 limit of Minimal Models and Liouville theory.

Liouville theory with a central charge less than one

“…And we numerically find that crossing symmetry is not obeyed. Another idea is to replace the non-analytic factor σ with any function of σ, as was suggested in [17] in the case c = 1. Since σ takes only two values, this amounts to considering three-point functions of the type…”

Integrable Differential Systems of Topological Type and Reconstruction by the Topological Recursion

“…. (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19) i.e.Ŵ n 's are linear combinations ofŴ k,n 's with coefficients independent of K. This is particularly convenient since it means that every statement about the analytic structure of theŴ n 's is immediately transmitted toŴ k,n and vice-versa.…”

The geometry of Casimir W-algebras

“…Next, let us test our conjecture in the case g = sl 2 and N = 3. In this case, the correlation function is the explicitly known Liouville three-point function at c = 1, and we have found a suitable basis [2]. However, this basis is ad hoc and cannot easily be generalized.…”

“…Of course, the interesting applications would mostly be confined to the case g = sl 2 of the Virasoro algebra, since most interesting CFTs have no larger W-algebra symmetry. But the generalization to Walgebras is essential for understanding geometrical structures, which were not discovered in previous work on the Virasoro case [2].…”

“…In recent years, it has been found that two-dimensional conformal field theory with the central charge c = 1 can be formulated in terms of Fuchsian differential systems. In particular, this has led to the expression of Virasoro conformal blocks in terms of solutions of Painlevé equations [1], and to a new derivation of the three-point structure constant of Liouville theory [2].…”