Related papers: On spectral theory of quantum vertex operators
We provide the mathematical foundation for the $X_m$-Jacobi spectral theory. Namely, we present a self-adjoint operator associated to the differential expression with the exceptional $X_m$-Jacobi orthogonal polynomials as eigenfunctions.…
The Nekrasov partition function in supersymmetric quantum gauge theory is mathematically formulated as an equivariant integral over certain moduli spaces of sheaves on a complex surface. In ``Seiberg-Witten Theory and Random Partitions'',…
We construct bosonized vertex operators (VOs) and conjugate vertex operators (CVOs) of $U_q(su(2)_k)$ for arbitrary level $k$ and representation $j\leq k/2$. Both are obtained directly as two solutions of the defining condition of vertex…
A class of non-linear eigenvalue problems defined in the form of operator polynomials is investigated. The problems are related to wave equations which appear in a relativistic quantum field theory. Spectral asymptotics for this class are…
Let Uq(g) be the quantum affine superalgebra associated with an affine Kac-Moody superalgebra g which belongs to the three series osp(1|2n)^(1),sl(1|2n)^(2) and osp(2|2n)^(2). We develop vertex operator constructions for the level 1…
The spectral problem for the high order differential operator with singular weight is considered. If the weight is a generalized derivative of self-similar function with zero spectral degree the asymptotics of eigenvalues is obtained. They…
The aim of this article is to provide a rigorous-but-simple steps to prove the hermiticity of the volume operator of Rovelli-Smolin and Ashtekar-Lewandowski using the angular momentum approach, as well as pointing out some subleties which…
We give a detailed description of the resolution of the identity of a second order $q$-difference operator considered as an unbounded self-adjoint operator on two different Hilbert spaces. The $q$-difference operator and the two choices of…
We consider scalar two-dimensional quantum field theories with the factorizing S-matrix which has poles in the physical strip. In our previous work, we introduced the bound state operators and constructed candidate operators for observables…
The two-parameter quantum vertex operator representation of level-one is explicitly constructed for $U_{r,s}(C^{(1)}_n)$ based on its two-parameter Drinfeld realization we give. This construction will degenerate to the one-parameter case…
We prove a conjecture of Dekimpe, De Rock and Penninckx concerning the existence of eigenvalues one in certain elements of finite groups acting irreducibly on a real vector space of odd dimension. This yields a sufficient condition for a…
Let $\omega\subset\mathbb{R}^n$ be a bounded domain with Lipschitz boundary. For $\varepsilon>0$ and $n\in\mathbb{N}$ consider the infinite cone $\Omega_{\varepsilon}:=\big\{(x_1,x')\in (0,\infty)\times\mathbb{R}^n: x'\in\varepsilon…
We establish the equivalence between the refined topological vertex of Iqbal-Kozcaz-Vafa and a certain representation theory of the quantum algebra of type W_{1+infty} introduced by Miki. Our construction involves trivalent intertwining…
The $J$-matrix method is extended to difference and $q$-difference operators and is applied to several explicit differential, difference, $q$-difference and second order Askey-Wilson type operators. The spectrum and the spectral measures…
We study tensor products of infinite dimensional representations (not corepresentations) of the $\mathrm{SU}(2)$ quantum group. Eigenvectors of certain self-adjoint elements are obtained, and coupling coefficients between different…
We determine the exchange relations of the level-one q-vertex operators of the quantum affine superalgebra $U_q[\hat{gl(N|N)}]$. We study in details the level-one irreducible highest weight representations of $U_q[\hat{gl(2|2)}]$, and…
We construct a realization of the elliptic quantum algebra $U_{q,p}(\hat{sl_N})$ for any given level $k$ in terms of free boson fields and their twisted partners. It can be considered as the elliptic deformation of the Wakimoto realization…
We propose a conjectural correspondence between the spectra of the Bethe algebra for the quantum toroidal $\mathfrak{gl}_2$ algebra on relaxed Verma modules, and $q$-hypergeometric opers with apparent singularities. We introduce alongside…
We construct explicitly the $q$-vertex operators (intertwining operators) for the level one modules $V(\Lambda_i)$ of the classical quantum affine algebras of twisted types using interacting bosons, where $i=0, 1$ for $A_{2n-1}^{(2)}$,…
We consider intertwining relations of the augmented $q$-Onsager algebra introduced by Ito and Terwilliger, and obtain generic (diagonal) boundary $K$-operators in terms of the Cartan element of $U_{q}(sl_2)$. These $K$-operators solve…