nLab tau-function

This page is about the $\tau$-functions in the study of solitons and integrable models, as invented by Kyoto school: isomonodromic $\tau$-function and Sato-Segal-Wilson $\tau$-function for integrable hierarchies. There will be a separate entry for a completely different notion of Ramanujan tau function?.

Motivation

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.

Formal idea

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.

Other formal interpretations

$\tau$-functions appear as dependent variables in Hirota bilinear equations associated to integrable hierarchies.

Tau functions can also be associated to the isomonodromic problem?s.

References

The $\tau$-function has first been introduced in the formalism of Clifford group? of free fermions in study of holonomic quantum fields and in relation to isomonodromic problems

• M. Jimbo M, T. Miwa, K. Ueno, Monodromy preserving deformation of linear ordinary differential equations with rational coefficients, I.. Physica D 2, 306–352 (1981)

• 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)

Wikis: wikipedia

The $\tau$-function for KP hierarchy has originally being studied in

• Masaki Kashiwara, Tetsuji Miwa, The $\tau$-function of the Kadomtsev-Petviashvili equation transformation groups for soliton equations, I, Proc. Japan Acad. Ser. A Math. Sci. 57, N. 7 (1981), 337-386 euclid

The background on the context of Sato-Segal-Wilson Grassmanian?s is in

• Andrew Pressley, Graeme Segal, Loop groups, Oxford University Press (1988)
• G. Segal, G. Wilson, Loop groups and equations of KdV type, Inst. Hautes Etudes Sci. Publ. Math. 61 (1985) 5–65.

Reviews include

• A. Alexandrov, A. Zabrodin, Free fermions and tau-functions, J.Geom.Phys. 67 (2013) 37-80 arxiv/1212.6049 doi
• A. Zabrodin, Bethe ansatz and Hirota equation in integrable models, arxiv/1211.4428

The connection to 2D gravity is elucidated in

• Victor Kac, Albert Schwarz, Geometric interpretation of the partition function of 2D gravity Phys. Lett. B 257:3-4 (1991) 329–334 pdf

Some special cases of $\tau$-functions include

• J. Palmer, Determinants of Cauchy–Riemann operators as τ-functions, Acta Appl. Math. 18 (1990), 199-223.

Isomonodromic interpretation is stemming already from the works of holonomic fields, and isomonodromic $\tau$-functions are studied also in

• Bertola, Eynard, Harnad, Semiclassical orthogonal polynomials, matrix models and isomonodromic tau functions, nlin.SI/0410043

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

• Jan Ambjørn, Leonid Chekhov, The matrix model for dessins d’enfants, arxiv/1404.4240

A connection to elementary geometry is studied (with nice reference list) in

• Adam Doliwa, Desargues maps and the Hirota-Miwa equation, pdf

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

• Mattia Cafasso, P. Gavrylenko, O. Lisovyy, Tau functions as Widom constants, Commun. Math. Phys. 365, 741–772 (2019) doi

• 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

• John Harnad, Ferenc Balogh, Tau functions and their applications, Cambridge Studies in Math. Physics 2021

category: physics, representation theory, Lie theory