Tau Functions and their Applications

  title={Tau Functions and their Applications},
  author={J. Harnad and Ferenc Balogh},
Tau functions are a central tool in the modern theory of integrable systems. This volume provides a thorough introduction, starting from the basics and extending to recent research results. It covers a wide range of applications, including generating functions for solutions of integrable hierarchies, correlation functions in the spectral theory of random matrices and combinatorial generating functions for enumerative geometrical and topological invariants. A self-contained summary of more… 
Fredholm Pfaffian ${\tau}$-functions for orthogonal isospectral and isomonodromic systems
We extend the approach to τ -functions as Widom constants developed by Cafasso, Gavrylenko and Lisovyy to orthogonal loop group Drinfeld-Sokolov hierarchies and isomonodromic deformations systems.
Sparse deformations of determinant expansions and integrability of set functions
We introduce an algebraic criterion for the reduction of Z-valued maps of subsets to corresponding Z-valued maps of elements of a given set. The algebraic structure is given by the expansion of the
Sparse deformations of determinant expansions: hyperdeterminants and complexity of set functions
We introduce an algebraic structure to check the complexity of Z-valued set functions, which is interpreted as a combinatorial form of integrability. The algebraic structure is based on the expansion
Invertible minor assignment: sparse deformations of determinant expansions and their hyperdeterminants
We introduce an algebraic model based on the expansion of the determinant of two matrices, one of which is generic, to check the additivity of Z-valued set functions. Each individual term of the
Rationally weighted Hurwitz numbers, Meijer G-functions and matrix integrals
The quantum spectral curve equation associated to KP $\tau$-functions of hypergeometric type serving as generating functions for rationally weighted Hurwitz numbers is solved by generalized
Fermionic approach to bilinear expansions of Schur functions in Schur $Q$-functions
An identity is derived expressing Schur functions as sums over products of pairs of Schur $Q$-functions, generalizing previously known special cases. This is shown to follow from their
New solvable matrix models III
We present a family of matrix models whose perturbation series in the coupling constants are written as the series in projective Schur functions over strict partitions and are examples of the tau
Notes about the KP/BKP correspondence
  • A. Orlov
  • Physics, Mathematics
    Theoretical and Mathematical Physics
  • 2021
I present a set of remarks related to joint works [13],[14],[15],[34]. These are remarks of polynomials solutions, application of the Wick theorem, examples of creation of polynomial solutions with
Isotropic Grassmannians, Plücker and Cartan maps
This work is motivated by the relation between the KP and BKP integrable hierarchies, whose $\tau$-functions may be viewed as sections of dual determinantal and Pfaffian line bundles over infinite
BKP Hierarchy, Affine Coordinates, and a Formula for Connected Bosonic $N$-Point Functions
  • Zhiyuan Wang, Chenglang Yang
  • Physics, Mathematics
  • 2022
We derive a formula for the connected n-point functions of a taufunction of the BKP hierarchy in terms of its affine coordinates. This is a BKPanalogue of a formula for KP tau-functions proved by