Exact solutions for some sigma models in quantum field theory

TL;DR

This paper explores exact solutions in WZNW sigma models, constructs the KZB connection, and demonstrates the self-duality of Hitchin systems at genus two, advancing understanding of quantum integrable systems and their geometric quantization.

Contribution

It provides explicit constructions of the KZB connection and Hamiltonians for WZNW models, and proves the self-duality of Hitchin systems at genus two, with quantization insights.

Findings

Constructed the KZB connection in genus zero and one.

Explicit Hamiltonians in involution for genus zero, one, and two.

Proved the self-duality of Hitchin systems at genus two.

Abstract

We invistigate exact solutions for the two-dimensional quantum field theories called Wess-Zumino-Novikov-Witten (WZNW) models. A WZNW model is a sigma model whose classical fields are applications from a bidimensional space-time (a Riemann surface in the euclidian case) to a Lie group, the target space. We construct (and we compute in genus zero and one) the metric connection, called the Knizhnik-Zamolodchikov-Bernard (KZB) connection, on the bundle of conformal blocks of the WZNW model. The KZB connection may be viewed as a quantization of Hitchin integrable systems whose configuration space is the moduli space of principal holomorphic bundles over a Riemann surface and whose phase space is the (holomorphic) cotangent bundle to the configuration space. For these systems, we construct explicitly a complete familly of Hamiltonians in involution in genus zero, one and two, with (complex) group SL(2) for the last case. The main result is the self-duality property of the Hitchin system at genus two, that is the invariance of the Hamiltonians with respect to the interchange of positions and momenta in the phase space. We finally realize the (geometric) quantization of the Hitchin systems.