Rafael Diaz

Quantum Symmetric Functions

ABSTRACT We study quantum deformations of Poisson orbivarieties. Given a Poisson manifold (ℝ m , α) we consider the Poisson orbivariety ( . The Kontsevich star product on functions on (ℝ m ) n induces a star product on functions on ( . We provide explicit formulae for the case 𝔥×𝔥/𝒲, where 𝔥 is the Cartan subalgebra of a classical Lie algebra 𝔤 and 𝒲 is the Weyl group of 𝔥. We approach our problem from a fairly general point of view, introducing Polya functors for categories over nonsymmetric Hopf operads.

Rational combinatorics

We propose a categorical setting for the study of the combinatorics of rational numbers. We find combinatorial interpretation for Bernoulli and Euler numbers and polynomials.

Feynman-Jackson integrals

Abstract We introduce perturbative Feynman integrals in the context of q-calculus generalizing the Gaussian q-integrals introduced by Díaz and Teruel. We provide analytic as well as combinatorial interpretations for the Feynman-Jackson integrals.

An example of Feynman–Jackson integrals

We review the construction of a q-analogue of the Gaussian measure. We apply that construction to obtain a q-analogue of Feynman integrals and to compute explicitly an example of such integrals.

Surface-invariants in 2D classical Yang-Mills theory

We study a method to obtain invariants under area-preserving diffeomorphisms associated to closed curves in the plane from classical Yang-Mills theory in two dimensions. Taking as starting point the Yang-Mills field coupled to nondynamical particles carrying chromo-electric charge, and by means of a perturbative scheme, we obtain the first two contributions to the on-shell action, which are area-invariants. A geometrical interpretation of these invariants is given.

On N-differential graded algebras

Abstract We introduce the concept of N -differential graded algebras ( N -dga), and study the moduli space of deformations of the differential of an N -dga. We prove that it is controlled by what we call the ( M , N ) -Maurer–Cartan equation.

Quantum Product of Symmetric Functions

We provide an explicit description of the quantum product of multisymmetric functions using the elementary multisymmetric functions introduced by Vaccarino.