Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 060, 29 pages      arXiv:1703.00232
Contribution to the Special Issue on Recent Advances in Quantum Integrable Systems

Integrability, Quantization and Moduli Spaces of Curves

Paolo Rossi
IMB, UMR5584 CNRS, Université de Bourgogne Franche-Comté, F-21000 Dijon, France

Received February 28, 2017, in final form July 25, 2017; Published online July 29, 2017

This paper has the purpose of presenting in an organic way a new approach to integrable $(1+1)$-dimensional field systems and their systematic quantization emerging from intersection theory of the moduli space of stable algebraic curves and, in particular, cohomological field theories, Hodge classes and double ramification cycles. This methods are alternative to the traditional Witten-Kontsevich framework and its generalizations by Dubrovin and Zhang and, among other advantages, have the merit of encompassing quantum integrable systems. Most of this material originates from an ongoing collaboration with A. Buryak, B. Dubrovin and J. Guéré.

Key words: moduli space of stable curves; integrable systems; cohomological field theories; double ramification cycle; double ramification hierarchy.

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