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 ${\Psi }^{\prime }=L\Psi$, 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 three-point function of Runkel-Watts theory -- the common $c\to 1$ limit of Minimal Models and Liouville theory.

20 pages, v3: Corrected sign mistakes in eqs. (4.35), (4.37), (4.42), (4.45) and (4.52). Conclusions unchanged

References in corpus (3)

• Search for the ${\Lambda }_b\to \Lambda {\eta }^{\prime }$ and ${\Lambda }_b\to \Lambda \eta$ decays with the LHCb detector
• Painlevé VI connection problem and monodromy of c=1 conformal blocks
• Seiberg-Witten equations and non-commutative spectral curves in Liouville theory

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