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

SIGMA 21 (2025), 068, 25 pages      arXiv:2311.06506      https://doi.org/10.3842/SIGMA.2025.068

From Toda Hierarchy to KP Hierarchy

Di Yang a and Jian Zhou b
a) School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, P.R. China
b) Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P.R. China

Received October 08, 2024, in final form July 27, 2025; Published online August 09, 2025

Abstract
Using the matrix-resolvent method and a formula of the second-named author on the $n$-point function for a KP tau-function, we show that the tau-function of an arbitrary solution to the Toda lattice hierarchy is a KP tau-function. We then generalize this result to tau-functions for the extended Toda hierarchy (ETH) by developing the matrix-resolvent method for the ETH. As an example the partition function of Gromov-Witten invariants of the complex projective line is a KP tau-function, and an application on irreducible representations of the symmetric group is obtained.

Key words: Toda hierarchy; KP hierarchy; matrix-resolvent method; complex projective line; Gromov-Witten invariant.

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