# nLab tau-function

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?.

## 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 Segal have formulated the $\tau$-function as the Fredholm determinant of certain 1-parameter family of operators on separable Hilbert space.

#### 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 problems.

## References

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

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 infinite Grassmanians 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

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

• 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

Last revised on September 16, 2015 at 11:19:31. See the history of this page for a list of all contributions to it.