Related papers: $W_{1+\infty}$ flows and multi-component hierarchy…
It was known through the efforts of many works that the generating functions in the closed Gromov-Witten theory of $K_{\mathbb P^2}$ are meromorphic quasi-modular forms basing on the B-model predictions. In this article, we extend the…
A new representation of the 2N fold integrals appearing in various two-matrix models that admit reductions to integrals over their eigenvalues is given in terms of vacuum state expectation values of operator products formed from…
Elements of a global operator approach to the WZWN theory for compact Riemann surfaces of arbitrary genus $g$ are given. Sheaves of representations of affine Krichever-Novikov algebras over a dense open subset of the moduli space of Riemann…
We generalize in a combinatorial way the notion of the affine energy function of type $A$ to the case of a more general class of modules over a general linear Lie superalgebra $\mathfrak{g}$ based on a Howe duality of type…
We construct a family of irreducible representations of the quantum continuous $gl_\infty$ whose characters coincide with the characters of representations in the minimal models of the $W_n$ algebras of $gl_n$ type. In particular, we obtain…
We construct a quadratic basis of generators of matrix-extended $\mathcal{W}_{1+\infty}$ using a generalization of the Miura transformation. This makes it possible to conjecture a closed-form formula for the operator product expansions…
We give a combinatorial expansion of a Schubert homology class in the affine Grassmannian Gr_{SL_k} into Schubert homology classes in Gr_{SL_{k+1}}. This is achieved by studying the combinatorics of a new class of partitions called…
We show that the system of the standard one-component KP hierarchy endowed with a special infinite set of abelian additional symmetries, generated by squared eigenfunction potentials, is equivalent to the two-component KP hierarchy.
We construct the multi-variable realizations of the $W_{1+\infty}$ algebra such that they lead to the $W_{1+\infty}$ $n$-algebra. Based on our realizations of the $W_{1+\infty}$ algebra, we derive the $W_{1+\infty}$ constraints for the…
Recently, Jack polynomials have been proposed as natural generalizations of Z_k Read-Rezayi states describing non-Abelian fractional quantum Hall systems. These polynomials are conjectured to be related to correlation functions of a class…
This paper is devoted to constructing a quantum version of the famous KP hierarchy, by deforming its second Hamiltonian structure, namely the nonlinear $\hat{W}_{\infty}$ algebra. This is achieved by quantizing the conformal noncompact…
The property of some finite W algebras to be the commutant of a particular subalgebra of a simple Lie algebra G is used to construct realizations of G. When G=so(4,2), unitary representations of the conformal and Poincare algebras are…
A set of functions is introduced which generalizes the famous Schur polynomials and their connection to Grasmannian manifolds. These functions are shown to provide a new method of constructing solutions to the KP hierarchy of nonlinear…
Given a closed symplectic manifold $(M,\omega)$ we introduce a certain quantity associated to a tuple of conjugacy classes in the universal cover of the group ${\hbox{\it Ham}} (M,\omega)$ by means of the Hofer metric on ${\hbox{\it Ham}}…
We establish an operator--theoretic correspondence between periodic Bernoulli kernels and Hermite polynomials, framed through the umbral calculus and a quantum analogy. Starting from the analytic master function $F^\ast$, the periodic…
We study a unitary matrix model of the Gross-Witten-Wadia type, extended with the addition of characteristic polynomial insertions. The model interpolates between solvable unitary matrix models and is the unitary counterpart of a deformed…
We construct the $W_{1+\infty}$ 3-algebra and investigate the relation between this infinite-dimensional 3-algebra and the integrable systems. Since the $W_{1+\infty}$ 3-algebra with a fixed generator $W^0_0$ in the operator Nambu 3-bracket…
Given a semisimple compact Lie group $G$ and a nonzero dominant integral weight $\lambda$, the highest weight $G_q$-modules $V_{n\lambda}$ form a subproduct system of finite dimensional Hilbert spaces. Using a conjectural asymptotic…
A topological description of various generalized function algebras over corresponding basic locally convex algebras is given. The framework consists of algebras of sequences with appropriate ultra(pseudo)metrics defined by sequences of…
The multi-indexed Jacobi polynomials are the main part of the eigenfunctions of exactly solvable quantum mechanical systems obtained by certain deformations of the P\"oschl-Teller potential (Odake-Sasaki). By fine-tuning the parameter(s) of…