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\section*{tau-function}

This page is about the $\tau$-function in the study of [[soliton]]s and [[integrable model]]s, as invented by Kyoto school. There will be a separate entry for a different notion of [[Ramanujan tau function]].

\hypertarget{motivation}{}\subsection*{{Motivation}}\label{motivation}

Quoting Zabrodin, \href{http://arxiv.org/abs/1211.4428}{arxiv/1211.4428},

\begin{quote}%
``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.


\end{quote}
\hypertarget{formal_idea}{}\subsection*{{Formal idea}}\label{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.

\hypertarget{other_formal_interpretations}{}\paragraph*{{Other formal interpretations}}\label{other_formal_interpretations}

$\tau$-functions appear as dependent variables in [[Hirota bilinear equation]]s associated to integrable hierarchies.

Tau functions can also be associated to the isomonodromic problems.

\hypertarget{references}{}\subsection*{{References}}\label{references}

The $\tau$-function has first been introduced in the formalism of [[Clifford group]] of free fermions in study of [[holonomic quantum field]]s

\begin{itemize}%
\item Mikio Sato, Tetsuji Miwa, Michio Jimbo

\end{itemize}
The $\tau$-function for [[KP hierarchy]] has originally being studied in

\begin{itemize}%
\item [[Masaki Kashiwara]], [[Tetsuji Miwa]], \emph{The $\tau$-function of the Kadomtsev-Petviashvili equation transformation groups for soliton equations, I}, Proc. Japan Acad. Ser. A Math. Sci. \textbf{57}, N. 7 (1981), 337-386 \href{http://projecteuclid.org/euclid.pja/1195516327}{euclid}

\end{itemize}
The background on the context of [[Sato-Segal-Wilson Grassmanian]]s is in

\begin{itemize}%
\item Andrew Pressley, Graeme Segal, \emph{Loop groups}, Oxford University Press (1988)
\item G. Segal, G. Wilson, \emph{Loop groups and equations of KdV type}, Inst. Hautes Etudes Sci. Publ. Math. 61 (1985) 5--65.

\end{itemize}
Reviews include

\begin{itemize}%
\item A. Alexandrov, A. Zabrodin, \emph{Free fermions and tau-functions}, J.Geom.Phys. \textbf{67} (2013) 37-80 \href{http://arxiv.org/abs/1212.6049}{arxiv/1212.6049} \href{http://dx.doi.org/10.1016/j.geomphys.2013.01.007}{doi}
\item A. Zabrodin, \emph{Bethe ansatz and Hirota equation in integrable models}, \href{http://arxiv.org/abs/1211.4428}{arxiv/1211.4428}

\end{itemize}
The connection to 2D [[gravity]] is elucidated in

\begin{itemize}%
\item [[Victor Kac]], [[Albert Schwarz]], \emph{Geometric interpretation of the partition function of 2D gravity} Phys. Lett. B 257 (1991), no. 3-4, 329–334 \href{https://web.phys.ntu.edu.tw/string/matrix/PhysLettB257(1991}{pdf}329\_Kac\_Schwarz\_GeometricInterpretationOfThePartitionFunctionOf2DGravity.pdf)

\end{itemize}
Some special cases of $\tau$-functions include

\begin{itemize}%
\item J. Palmer, \emph{Determinants of Cauchy--Riemann operators as $\tau$-functions}, Acta Appl. Math. 18 (1990), 199-223.

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

\begin{itemize}%
\item Bertola, Eynard, Harnad, \emph{Semiclassical orthogonal polynomials, matrix models and isomonodromic tau functions}, \href{http://arxiv.org/abs/nlin.SI/0410043}{nlin.SI/0410043}

\end{itemize}
[[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.

\begin{itemize}%
\item [[Maxim Kontsevich]], \emph{Intersection theory on the moduli space of curves and the matrix Airy function}, Comm. Math. Phys. \textbf{147} (1992), no. 1, 1--23, \href{http://projecteuclid.org/euclid.cmp/1104250524}{euclid}, \href{http://193.51.104.7/~maxim/TEXTS/intersection_theory_6.pdf}{author's pdf}


\item A.Alexandrov, \emph{Enumerative geometry, tau-functions and Heisenberg-Virasoro algebra}, \href{http://arxiv.org/abs/1404.3402}{arxiv/1404.3402}



\end{itemize}
\begin{quote}%
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.


\end{quote}
Connection of $\tau$-functions to [[dessins d'enfants]] is discussed in

\begin{itemize}%
\item Jan Ambj\o{}rn, Leonid Chekhov, \emph{The matrix model for dessins d'enfants}, \href{http://arxiv.org/abs/1404.4240}{arxiv/1404.4240}

\end{itemize}
A connection to elementary geometry is studied (with nice reference list) in

\begin{itemize}%
\item Adam Doliwa, \emph{Desargues maps and the Hirota-Miwa equation}, \href{https://www.newton.ac.uk/preprints/NI09049.pdf}{pdf}

\end{itemize}
Recent works on $\tau$-function include

\begin{itemize}%
\item J. W. van de Leur, A. Yu. Orlov, \emph{Pfaffian and determinantal tau functions I}, \href{http://arxiv.org/abs/1404.6076}{arxiv/1404.6076}


\item [[Mattia Cafasso]], Chao-Zhong Wu, \emph{Tau functions and the limit of block Toeplitz determinants}, \href{http://arxiv.org/abs/1404.5149}{arxiv/1404.5149}


\item J. Harnad, Eunghyun Lee, \emph{Symmetric polynomials, generalized Jacobi-Trudi identities and $\tau$-functions}, \href{http://arxiv.org/abs/1304.0020}{arxiv/1304.0020}


\item S. Natanzon, A. Zabrodin, \emph{Formal solution to the KP hierarchy}, \href{http://arxiv.org/abs/1509.04472}{arxiv/1509.04472}



\end{itemize}
category: physics, representation theory, Lie theory

[[!redirects tau-function]]



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