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Giovanni Felder

Publications and source records attributed to Giovanni Felder.

At least 73 records · Page 4Linked to original sources
arXiv2001

q-deformed KZB heat equation: completeness, modular properties and SL(3,Z)

We study the properties of one-dimensional hypergeometric integral solutions of the q-difference ("quantum") analogue of the Knizhnik-Zamolodchikov-Bernard equations on tori. We show that they also obey a difference KZB heat equation in the modular parameter, give formulae for modular transformations, and prove a completeness result, by showing that the associated Fourier transform is invertible. These results are based on SL(3,Z) transformation properties parallel to those of elliptic gamma functions.

math.QA↗
arXiv2001

Action of Coxeter groups on m-harmonic polynomials and KZ equations

The Matsuo-Cherednik correspondence is an isomorphism from solutions of Knizhnik-Zamolodchikov equations to eigenfunctions of generalized Calogero-Moser systems associated to Coxeter groups G and a multiplicity function m on their root systems. We apply this correspondence to the most degenerate case of zero spectral parameters. The space of eigenfunctions is then the space H_m of m-harmonic polynomials, recently introduced in math-ph/0105014. We compute the Poincare' polynomials for the space H_m and of its isotypical components corresponding to each irreducible representation of the group G. We also give an explicit formula for m-harmonic polynomials of lowest positive degree in the S_n case.

math.QA↗
arXiv2001

From local to global deformation quantization of Poisson manifolds

We give an explicit construction of a deformation quantization of the algebra of functions on a Poisson manifolds, based on Kontsevich's local formula. The deformed algebra of functions is realized as the algebra of horizontal sections of a vector bundle with flat connection.

math.QA↗
arXiv2001

Poisson sigma models and deformation quantization

This is a review aimed at a physics audience on the relation between Poisson sigma models on surfaces with boundary and deformation quantization. These models are topological open string theories. In the classical Hamiltonian approach, we describe the reduced phase space and its structures (symplectic groupoid), explaining in particular the classical origin of the non-commutativity of the string end-point coordinates. We also review the perturbative Lagrangian approach and its connection with Kontsevich's star product. Finally we comment on the relation between the two approaches.

hep-th↗
arXiv2001

On the AKSZ formulation of the Poisson sigma model

We review and extend the Alexandrov-Kontsevich-Schwarz-Zaboronsky construction of solutions of the Batalin-Vilkovisky classical master equation. In particular, we study the case of sigma models on manifolds with boundary. We show that a special case of this construction yields the Batalin-Vilkovisky action functional of the Poisson sigma model on a disk. As we have shown in a previous paper, the perturbative quantization of this model is related to Kontsevich's deformation quantization of Poisson manifolds and to his formality theorem. We also discuss the action of diffeomorphisms of the target manifolds.

math.QA↗
arXiv2000

The q-deformed Knizhnik-Zamolodchikov-Bernard heat equation

We introduce a q-deformation of the genus one sl_2 Knizhnik-Zamolodchikov-Bernard heat equation. We show that this equation for the dependence on the moduli of elliptic curves is compatible with the qKZB equations, which give the dependence on the marked points.

math.QA↗
arXiv2000

Correlation functions and boundary conditions in RCFT and three-dimensional topology

We give a general construction of correlation functions in rational conformal field theory on a possibly non-orientable surface with boundary in terms of 3-dimensional topological quantum field theory. The construction applies to any modular category. It is proved that these correlation functions obey modular and factorization rules. Structure constants are calculated and expressed in terms of the data of the modular category.

hep-th↗
arXiv2000

Deformation quantization with traces

In the present paper we prove a statement closely related to the cyclic formality conjecture. In particular, we prove that for a divergence-free Poisson bivector field on R^d, the Kontsevich star-product with the harmonic angle function is cyclic. We also prove a globalization of this theorem in the case of arbitrary Poisson manifolds and prove a generalization of the Connes-Flato-Sternheimer conjecture on closed star-products in the Poisson case.

math.QA↗
arXiv2000

Poisson sigma models and symplectic groupoids

We consider the Poisson sigma model associated to a Poisson manifold. The perturbative quantization of this model yields the Kontsevich star product formula. We study here the classical model in the Hamiltonian formalism. The phase space is the space of leaves of a Hamiltonian foliation and has a natural groupoid structure. If it is a manifold then it is a symplectic groupoid for the given Poisson manifold. We study various families of examples. In particular, a global symplectic groupoid for a general class of two-dimensional Poisson domains is constructed.

math.SG↗
arXiv1999

The elliptic gamma function and SL(3,Z) x Z^3

The elliptic gamma function is a generalization of the Euler gamma function and is associated to an elliptic curve. Its trigonometric and rational degenerations are the Jackson q-gamma function and the Euler gamma function, respectively. The elliptic gamma function appears in Baxter's formula for the free energy of the eight-vertex model and in the hypergeometric solutions of the elliptic qKZB equations. In this paper, the properties of this function are studied. In particular we show that elliptic gamma functions are generalizations of automorphic forms of G=SL(3,Z) x Z^3 associated to a non-trivial class in H^3(G,Z).

math.QA↗
arXiv1999

Separation of variables for quantum integrable systems on elliptic curves

We extend Sklyanin's method of separation of variables to quantum integrable models associated to elliptic curves. After reviewing the differential case, the elliptic Gaudin model studied by Enriquez, Feigin and Rubtsov, we consider the difference case and find a class of transfer matrices whose eigenvalue problem can be solved by separation of variables. These transfer matrices are associated to representations of the elliptic quantum group $E_{τ,η}(sl_2)$ by difference operators. One model of statistical mechanics to which this method applies is the IRF model with antiperiodic boundary conditions. The eigenvalues of the transfer matrix are given as solutions of a system of quadratic equations in a space of higher order theta functions.

math.QA↗
arXiv1999

A path integral approach to the Kontsevich quantization formula

We give a quantum field theory interpretation of Kontsevich's deformation quantization formula for Poisson manifolds. We show that it is given by the perturbative expansion of the path integral of a simple topological bosonic open string theory. Its Batalin-Vilkovisky quantization yields a superconformal field theory. The associativity of the star product, and more generally the formality conjecture can then be understood by field theory methods. As an application, we compute the center of the deformed algebra in terms of the center of the Poisson algebra.

math.QA↗
arXiv1999

Resonance Relations for Solutions of the Elliptic QKZB Equations, Fusion Rules, and Eigenvectors of Transfer Matrices of Restricted Interaction-round-a-face Models

Conformal blocks for the WZW model on tori can be represented by vector valued Weyl anti-symmetric theta functions on the Cartan subalgebra satisfying vanishing conditions on root hyperplanes. We introduce a quantum version of these vanishing conditions in the sl(2) case. They are compatible with the qKZB equations and are obeyed by the hypergeometric solutions as well as by their critical level counterpart, which are Bethe eigenfunctions of IRF row-to-row transfer matrices. In the language of IRF models the vanishing conditions turn out to be equivalent to the sl(2) fusion rules defining restricted models.

math.QA↗
arXiv1997

Monodromy of solutions of the elliptic quantum Knizhnik-Zamolodchikov-Bernard difference equations

The elliptic quantum Knizhnik-Zamolodchikov-Bernard (qKZB) difference equations associated to the elliptic quantum group $E_{τ,η}(sl_2)$ is a system of difference equations with values in a tensor product of representations of the quantum group and defined in terms of the elliptic R-matrices associated with pairs of representations of the quantum group. In this paper we solve the qKZB equations in terms of elliptic hypergeometric functions and decribe the monodromy properties of solutions. It turns out that the monodromy transformations of solutions are described in terms of elliptic R-matrices associated with pairs of representations of the "dual" elliptic quantum group $E_{p,η}(sl_2)$, where $p$ is the step of the difference equations. Our description of the monodromy is analogous to the Kohno-Drinfeld description the monodromy group of solutions of the KZ differential equations associated to a simple Lie algebra in terms of the corresponding quantum group.

q-alg↗
arXiv1997

Elliptic quantum groups and Ruijsenaars models

We construct symmetric and exterior powers of the vector representation of the elliptic quantum groups $E_{τ,η}(gl_N)$. The corresponding transfer matrices give rise to various integrable difference equations which could be solved in principle by the nested Bethe ansatz method. In special cases we recover the Ruijsenaars systems of commuting difference operators.

q-alg↗
arXiv1996

Algebraic integrability of the two-body Ruijsenaars operator

We study the algebra of difference operators that commute with the two-body Ruijsenaars operator, a $q$-deformation of the Lamé differential operator, for generic values of the deformation parameter. The algebra is commutative. It is the algebra of polynomial functions on an affine hyperelliptic curve $Y^2=P(X^2)$. We also compute the difference Galois group of the eigenvalue problem.

q-alg↗
arXiv1996

The KZB equations on Riemann surfaces

In this paper, based on the author's lectures at the 1995 les Houches Summer school, explicit expressions for the Friedan--Shenker connection on the vector bundle of WZW conformal blocks on the moduli space of curves with tangent vectors at $n$ marked points are given. The covariant derivatives are expressed in terms of ``dynamical $r$-matrices'', a notion borrowed from integrable systems. The case of marked points moving on a fixed Riemann surface is studied more closely. We prove a universal form of the (projective) flatness of the connection: the covariant derivatives commute as differential operators with coefficients in the universal enveloping algebra -- not just when acting on conformal blocks.

hep-th↗