This page is about the $\tau$-function in the study of solitons and integrable models, as invented by Kyoto school. There will be a separate entry for a different notion of Ramanujan tau function?.
Quoting Zabrodin, arxiv/1211.4428,
“In nature”, $\tau$-functions or their logarithms appear as partition functions, different kinds of correlators and their generating functions, and effective actions as functions of coupling constants.
Sato and Graeme Segal have formulated the $\tau$-function as the Fredholm determinant of certain 1-parameter family of operators on separable Hilbert space. See also closely related page Sato Grassmannian.
$\tau$-functions appear as dependent variables in Hirota bilinear equations associated to integrable hierarchies.
Tau functions can also be associated to the isomonodromic problem?s.
The $\tau$-function has first been introduced in the formalism of Clifford group? of free fermions in study of holonomic quantum fields
Mikio Sato, Tetsuji Miwa, Michio Jimbo
E. Date, Michio Jimbo?, Masaki Kashiwara, T. Miwa, Transformation groups for soliton equations,IV A new hierarchy of soliton equations of KP-type,Physica D4, 343-365 (1982)
M. Jimbo, T. Miwa, Solitons and infinite dimensional Lie algebras,Publ. RIMS Kyoto Univ. 19, 943-1001 (1983)
The $\tau$-function for KP hierarchy? has originally being studied in
The background on the context of Sato-Segal-Wilson Grassmanian?s is in
Reviews include
The connection to 2D gravity is elucidated in
Some special cases of $\tau$-functions include
Isomonodromic interpretation is stemming already from the works of holonomic fields, and isomonodromic $\tau$-functions are studied also in
Witten conjecture is about the equivalence of two approaches to gravity, and boils down to a connection of the $\tau$-function for the KP hierarchy and theory of Riemann surfaces. It has been proved by Kontsevich who also introduced related family of matrix models.
Maxim Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys. 147 (1992), no. 1, 1–23, euclid, author’s pdf
A.Alexandrov, Enumerative geometry, tau-functions and Heisenberg-Virasoro algebra, arxiv/1404.3402
In this paper we establish relations between three enumerative geometry tau-functions, namely the Kontsevich-Witten, Hurwitz and Hodge tau-functions. The relations allow us to describe the tau-functions in terms of matrix integrals, Virasoro constraints and Kac-Schwarz operators. All constructed operators belong to the algebra (or group) of symmetries of the KP hierarchy.
Connection of $\tau$-functions to dessins d'enfants is discussed in
A connection to elementary geometry is studied (with nice reference list) in
Recent works on $\tau$-function include
J. W. van de Leur, A. Yu. Orlov, Pfaffian and determinantal tau functions I, arxiv/1404.6076
Mattia Cafasso, Chao-Zhong Wu, Tau functions and the limit of block Toeplitz determinants, arxiv/1404.5149
J. Harnad, Eunghyun Lee, Symmetric polynomials, generalized Jacobi-Trudi identities and τ-functions, arxiv/1304.0020
S. Natanzon, A. Zabrodin, Formal solution to the KP hierarchy, arxiv/1509.04472
There is now a book
Last revised on February 3, 2021 at 04:00:45. See the history of this page for a list of all contributions to it.