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# Gen KUROKI【黒木 玄】, Asst. Professor

## Research Field

Representation theory of infinite dimensional Lie algebras and quantum groups and its application to mathematical physics.

## Research Interests

Representation theoretic study of conformal field theories and quantum integrable systems Quantum field theories with infinite dimensional symmetries generated by the Virasoro algebra are called conformal field theories. Recently various conformal field theories are formulated by representation theory of vertex algebras.

A conformal field theory with an affine Lie algebra symmetry is called a Wess-Zumino-Wittem model (WZW model). Its correlation functions satisfy the Knizhnik-Zamolodchikov equation (KZ equation), which is the fundamental equation of motion of the model and has multi-variable hypergeometric integral solutions deduced by bosonization of affine Lie algebras.

Each WZW model is associated to a pair of a finite-dimensional complex simple Lie algebra and a complex parameter k (called a level). If the parameter k is at the special value called the critical level, then the associated WZW model describes certain quantum integrable systems.
It is conjectured by Beilinson-Drinfeld that such quantum integrable systems represent the ``automorphic side'' of the geometric Langrands program.

The classical limit of the KZ equation coincides with the nonlinear differential equation characterizing the isomonodromic deformation of meromorphic connections with regular singularities (Reshetikhin, Harnad). In other words, the WZW model quantize the isomonodromic deformation of meromorphic connection with regular singularities. It is an important and fundamental problem to extend the theory of WZW models to irregularly singular cases. Discrete symmetries (Backlund transformations) play important roles in the theory of isomonodromic deformations and hence expected to do so in the theory of WZW models. Painleve systems is closely related to isomonodromic systems and hence the quantization of Painleve systems might be closely related to conformal field theory.

## Remarks

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