This article is cited in 30 papers

Combinatorial solutions to integrable hierarchies

M. È. Kazarian^ab, S. K. Lando<sup>b

a</sup> Steklov Mathematical Institute of Russian Academy of Sciences
^b National Research University Higher School of Economics

Abstract: This paper reviews modern approaches to the construction of formal solutions to integrable hierarchies of mathematical physics whose coefficients are answers to various enumerative problems. The relationship between these approaches and the combinatorics of symmetric groups and their representations is explained. Applications of the results to the construction of efficient computations in problems related to models of quantum field theories are described.
Bibliography: 34 titles.

Keywords: integrable systems, Kadomtsev–Petviashvili hierarchy, Toda lattice hierarchy, boson-fermion correspondence, Hurwitz numbers.

UDC: 512.542.74+515.162.2+517.958:530.145

MSC: Primary 14H70, 37K20, 81R12; Secondary 57R56

Received: 25.12.2014

DOI: 10.4213/rm9661

 English version: Russian Mathematical Surveys, 2015, 70:3, 453–482

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