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For an arbitrary solution to the KdV hierarchy, the generating series of logarithmic derivatives of the tau-function of the solution can be expressed by the basic matrix resolvent via algebraic manipulations. Based on this we develop in…

The Brezin-Gross-Witten tau function is a tau function of the KdV hierarchy which arises in the weak coupling phase of the Brezin-Gross-Witten model. It falls within the family of generalized Kontsevich matrix integrals, and its…

This is a summary of a recursive construction of solutions of the hbar-dependent KP hierarchy. We give recursion relations for the coefficients X_n of an hbar-expansion of the operator X = X_0 + \hbar X_1 + \hbar^2 X_2 + ... for which the…

The loop equations in the $U(N)$ lattice gauge theory are represented in the form of constraints imposed on a generating functional for the Wilson loop correlators. These constraints form a closed algebra with respect to commutation. This…

A $(q,t)$-deformation of the 2d Toda integrable hierarchy is introduced by enhancing the underlying symmetry algebra $\mathfrak{gl}(\infty)\simeq \text{q-W}_{1+\infty}$ to the quantum toroidal $\mathfrak{gl}(1)$ algebra. The…

In a previous paper we constructed all polynomial tau-functions of the 1-component KP hierarchy, namely, we showed that any such tau-function is obtained from a Schur polynomial $s_\lambda(t)$ by certain shifts of arguments. In the present…

Using the matrix-resolvent method and a formula of the second-named author on the $n$-point function for a KP tau-function, we show that the tau-function of an arbitrary solution to the Toda lattice hierarchy is a KP tau-function. We then…

For two solutions of the WDVV equations that are related by the inversion symmetry, we show that the associated principal hierarchies of integrable systems are related by a reciprocal transformation, and the tau functions of the hierarchies…

The CKP hierarchy is one important sub-hierarchy of the KP hierarchy, which is quite special due to its tau function. Here we construct the tau functions for the constrained CKP hierarchy…

Matrix elements in different representations are connected by quadratic relations. If matrix elements are those of a $\textit{group element}$, i.e. satisfying the property $\Delta(X) = X\otimes X$, then their generating functions obey…

Virasoro constraints are applied to degree zero Gromov-Witten theory of weighted projective stacks $\mathbb{P}(1,N)$ and $\mathbb{P}(1,1,N)$ to obtain formulas of descendant cyclic Hurwitz-Hodge integrals in higher genera.

Let v be a real polynomial of even degree, and let \rho be the equilibrium probability measure for v with support S; so that v(x)\geq 2\int \log |x-y| \rho (dy)+C_v for some constant C_v with support S. Then S is the union of finitely many…

We will describe algebro-geometric solutions of the KdV hierarchy whose $\tau$-functions in addition satisfy a generalization of the Virasoro constraints (and, in particular, a generalization of the string equation). We show that these…

Using the free fermions technique and non-abelian bosonization rules we introduce the multi-component Pfaff-Toda hierarchy. The tau-function is defined as vacuum expectation value of a Clifford group element of the algebra of…

We introduce the tau-function of a rational d-connection and its isomonodromy transformations. We show that in a continuous limit our tau-function agrees with the Jimbo-Miwa-Ueno tau-function, compute the tau-function for the isomonodromy…

For a large class of hierarchies of integrable equations admitting a classical $r-$matrix, we propose a construction for the Virasoro algebra actionon the Lax operators which commutes with the hierarchy flows. The construction relies on the…

The algebraic structures of integrable hierarchies play an important role in the study of soliton equations. In this paper, we use splitting theory to give a matrix representation of a constrained CKP hierarchy, which can be considered as a…

The Witten-Kontsevich theorem states that a certain generating function for intersection numbers on the moduli space of stable curves is a tau-function for the KdV integrable hierarchy. This generating function can be recovered via the…

We prove the conjectural relationship recently proposed in [9] between certain special cubic Hodge integrals of the Gopakumar--Mari\~no--Vafa type [17, 28] and GUE correlators, and the conjecture proposed in [7] that the partition function…

For each partition p of an integer N \geq 2, consisting of r parts, an integrable hierarchy of Lax type Hamiltonian PDE has been constructed recently by some of us. In the present paper we show that any tau-function of the p-reduced…