Combinatorial solutions to integrable hierarchies

Abstract

We give a review of modern approaches to constructing formal solutions to integrable hierarchies of mathematical physics, whose coefficients are answers to various enumerative problems. The relationship between these approaches and combinatorics of symmetric groups and their representations is explained. Applications of the results to constructing efficient computations in problems related to models of quantum field theories are given.

32 pages

References in corpus (2)

• On KP-integrable Hurwitz functions
• Hurwitz numbers and BKP hierarchy

Cited by in corpus (13)

• Ramifications of Hurwitz theory, KP integrability and quantum curves
• Quantization of Harer-Zagier formulas
• Combinatorics of Bousquet-Mélou-Schaeffer numbers in the light of topological recursion
• Hurwitz numbers from Feynman diagrams
• Genus expansion of matrix models and $\hslash$ expansion of KP hierarchy
• Hurwitz numbers and matrix integrals labeled with chord diagrams
• Matrix integrals and Hurwitz numbers
• Grothendieck's Dessins d'Enfants in a Web of Dualities. III
• Integrals of tau functions
• Notes about KP/BKP correspondence
• Combinatorics of KP hierarchy structural constants
• 2D Toda $\tau$ Functions, Weighted Hurwitz Numbers and the Cayley Graph: Determinant Representation and Recursion Formula
• Hyperelliptic sigma functions and Adler-Moser polynomials