68 research outputs found

    A new realization of rational functions, with applications to linear combination interpolation

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    We introduce the following linear combination interpolation problem (LCI): Given NN distinct numbers w1,…wNw_1,\ldots w_N and N+1N+1 complex numbers a1,…,aNa_1,\ldots, a_N and cc, find all functions f(z)f(z) analytic in a simply connected set (depending on ff) containing the points w1,…,wNw_1,\ldots,w_N such that ∑u=1Nauf(wu)=c. \sum_{u=1}^Na_uf(w_u)=c. To this end we prove a representation theorem for such functions ff in terms of an associated polynomial p(z)p(z). We first introduce the following two operations, (i)(i) substitution of pp, and (ii)(ii) multiplication by monomials zj,0≤j<Nz^j, 0\le j < N. Then let MM be the module generated by these two operations, acting on functions analytic near 00. We prove that every function ff, analytic in a neighborhood of the roots of pp, is in MM. In fact, this representation of ff is unique. To solve the above interpolation problem, we employ an adapted systems theoretic realization, as well as an associated representation of the Cuntz relations (from multi-variable operator theory.) We study these operations in reproducing kernel Hilbert space): We give necessary and sufficient condition for existence of realizations of these representation of the Cuntz relations by operators in certain reproducing kernel Hilbert spaces, and offer infinite product factorizations of the corresponding kernels

    Unified Approach to Thermodynamic Bethe Ansatz and Finite Size Corrections for Lattice Models and Field Theories

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    We present a unified approach to the Thermodynamic Bethe Ansatz (TBA) for magnetic chains and field theories that includes the finite size (and zero temperature) calculations for lattice BA models. In all cases, the free energy follows by quadratures from the solution of a {\bf single} non-linear integral equation (NLIE). [A system of NLIE appears for nested BA]. We derive the NLIE for: a) the six-vertex model with twisted boundary conditions; b) the XXZ chain in an external magnetic field hzh_z and c) the sine-Gordon-massive Thirring model (sG-mT) in a periodic box of size \b \equiv 1/T using the light-cone approach. This NLIE is solved by iteration in one regime (high TT in the XXZ chain and low TT in the sG-mT model). In the opposite (conformal) regime, the leading behaviors are obtained in closed form. Higher corrections can be derived from the Riemann-Hilbert form of the NLIE that we present.Comment: Expanded Introduction. Version to appear in Nucl. Phys. B. 60 pages, TeX, Uses phyzz

    New critical matrix models and generalized universality

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    We study a class of one-matrix models with an action containing nonpolynomial terms. By tuning the coupling constants in the action to criticality we obtain that the eigenvalue density vanishes as an arbitrary real power at the origin, thus defining a new class of multicritical matrix models. The corresponding microscopic scaling law is given and possible applications to the chiral phase transition in QCD are discussed. For generic coupling constants off-criticality we prove that all microscopic correlation functions at the origin of the spectrum remain in the known Bessel universality class. An arbitrary number of Dirac mass terms can be included and the corresponding massive universality is maintained as well. We also investigate the critical behavior at the edge of the spectrum: there, in contrast to the behavior at the origin, we find the same critical exponents as derived from matrix models with a polynomial action

    Q-operators in the six-vertex model

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    In this paper we continue the study of QQ-operators in the six-vertex model and its higher spin generalizations. In [1] we derived a new expression for the higher spin RR-matrix associated with the affine quantum algebra Uq(sl(2)^)U_q(\widehat{sl(2)}). Taking a special limit in this RR-matrix we obtained new formulas for the QQ-operators acting in the tensor product of representation spaces with arbitrary complex spin. Here we use a different strategy and construct QQ-operators as integral operators with factorized kernels based on the original Baxter's method used in the solution of the eight-vertex model. We compare this approach with the method developed in [1] and find the explicit connection between two constructions. We also discuss a reduction to the case of finite-dimensional representations with (half-) integer spins.Comment: 18 pages, no figure

    Nonsemisimple Fusion Algebras and the Verlinde Formula

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    We find a nonsemisimple fusion algebra F_p associated with each (1,p) Virasoro model. We present a nonsemisimple generalization of the Verlinde formula which allows us to derive F_p from modular transformations of characters.Comment: LaTeX (amsart, xypic, times), 35p
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