A Laplace operator and harmonics on the quantum complex vector space

The aim of this paper is to study the q-Laplace operator and q-harmonic polynomials on the quantum complex vector space generated by z_i,w_i, i=1,2,...,n, on which the quantum group GL_q(n) (or U_q(n)) acts. The q-harmonic polynomials are defined as solutions of the equation Delta_qp=0, where p is a polynomial in z_i,w_i, i=1,2,...,n, and the q-Laplace operator Delta_q is determined in terms of q-derivatives. The q-Laplace operator Delta_q commutes with the action of GL_q(n). The projector H_{m,m'}: A_{m,m'} --> H_{m,m'} is constructed, where A_{m,m'} and H_{m,m'} are the spaces of homogeneous (of degree m in z_i and of degree m' in w_i) polynomials and homogeneous q-harmonic polynomials, respectively. By using these projectors, a q-analogue of the classical zonal spherical and associated spherical harmonics are constructed. They constitute an orthogonal basis of H_{m,m'}. A q-analogue of separation of variables is given. The quantum algebra U_q(gl_n), acting on H_{m,m'}, determines an irreducible representation of U_q(gl_n). This action is explicitly constructed. The results of the paper lead to the dual pair (U_q(sl_2), U_q(gl_n)) of quantum algebras.Comment: 26 pages, LaTe

Form factors of twist fields in the lattice Dirac theory

We study U(1) twist fields in a two-dimensional lattice theory of massive Dirac fermions. Factorized formulas for finite-lattice form factors of these fields are derived using elliptic parametrization of the spectral curve of the model, elliptic determinant identities and theta functional interpolation. We also investigate the thermodynamic and the infinite-volume scaling limit, where the corresponding expressions reduce to form factors of the exponential fields of the sine-Gordon model at the free-fermion point.Comment: 20 pages, 2 figure