Toda lattice field theories, discrete W algebras, Toda lattice hierarchies and quantum groups

In analogy with the Liouville case we study the $sl_3$ Toda theory on the lattice and define the relevant quadratic algebra and out of it we recover the discrete $W_3$ algebra. We define an integrable system with respect to the latter and establish the relation with the Toda lattice hierarchy. We compute the the relevant continuum limits. Finally we find the quantum version of the quadratic algebra.Comment: 12 pages, LaTe

Noncommutative gauge theory for Poisson manifolds

A noncommutative gauge theory is associated to every Abelian gauge theory on a Poisson manifold. The semi-classical and full quantum version of the map from the ordinary gauge theory to the noncommutative gauge theory (Seiberg-Witten map) is given explicitly to all orders for any Poisson manifold in the Abelian case. In the quantum case the construction is based on Kontsevich's formality theorem.Comment: 12 page

Noncommutative Geometry Framework and The Feynman's Proof of Maxwell Equations

The main focus of the present work is to study the Feynman's proof of the Maxwell equations using the NC geometry framework. To accomplish this task, we consider two kinds of noncommutativity formulations going along the same lines as Feynman's approach. This allows us to go beyond the standard case and discover non-trivial results. In fact, while the first formulation gives rise to the static Maxwell equations, the second formulation is based on the following assumption $m[x_{j},\dot{x_{k}}]=i\hbar \delta_{jk}+im\theta_{jk}f.$ The results extracted from the second formulation are more significant since they are associated to a non trivial $\theta$-extension of the Bianchi-set of Maxwell equations. We find $div_{\theta}B=\eta_{\theta}$ and $\frac{\partial B_{s}}{\partial t}+\epsilon_{kjs}\frac{\partial E_{j}}{\partial x_{k}}=A_{1}\frac{d^{2}f}{dt^{2}}+A_{2}\frac{df}{dt}+A_{3},$ where $\eta_{\theta}$, $A_{1}$, $A_{2}$ and $A_{3}$ are local functions depending on the NC $\theta$-parameter. The novelty of this proof in the NC space is revealed notably at the level of the corrections brought to the previous Maxwell equations. These corrections correspond essentially to the possibility of existence of magnetic charges sources that we can associate to the magnetic monopole since $div_{\theta}B=\eta_{\theta}$ is not vanishing in general.Comment: LaTeX file, 16 page

The Fuzzy Kaehler Coset Space by the Fedosov Formalism

We discuss deformation quantization of the Kaehler coset space by using the Fedosov formalism. We show that the Killing potentials of the Kaehler coset space satisfy the fuzzy algebrae, when the coset space is irreducible.Comment: 12 pages, no figure, Latex; reference added and typos correcte

Entropy of random coverings and 4D quantum gravity

We discuss the counting of minimal geodesic ball coverings of $n$-dimensional riemannian manifolds of bounded geometry, fixed Euler characteristic and Reidemeister torsion in a given representation of the fundamental group. This counting bears relevance to the analysis of the continuum limit of discrete models of quantum gravity. We establish the conditions under which the number of coverings grows exponentially with the volume, thus allowing for the search of a continuum limit of the corresponding discretized models. The resulting entropy estimates depend on representations of the fundamental group of the manifold through the corresponding Reidemeister torsion. We discuss the sum over inequivalent representations both in the two-dimensional and in the four-dimensional case. Explicit entropy functions as well as significant bounds on the associated critical exponents are obtained in both cases.Comment: 54 pages, latex, no figure

BPS Black Hole Degeneracies and Minimal Automorphic Representations

We discuss the degeneracies of 4D and 5D BPS black holes in toroidal compactifications of M-theory or type II string theory, using U-duality as a tool. We generalize the 4D/5D lift to include all charges in N=8 supergravity, and compute the exact indexed degeneracies of certain 4D 1/8-BPS black holes. Using the attractor formalism, we obtain the leading micro-canonical entropy for arbitrary Legendre invariant prepotentials and non-vanishing D6-brane charge. In particular, we find that the N=8 prepotential is given to leading order by the cubic invariant of $E_6$. This suggests that the minimal unitary representation of $E_8$, based on the same cubic prepotential, underlies the microscopic degeneracies of N=8 black holes. We propose that the exact degeneracies are given by the Wigner function of the $E_8(Z)$ invariant vector in this automorphic representation. A similar conjecture relates the degeneracies of N=4 black holes to the minimal unipotent representation of $SO(8,24,Z)$.Comment: 36 pages, uses JHEP3.cls; v5: the "extra charge", which was, mistakenly identified as angular momentum, is now correctly identified as NUT charge (see historical note on p29), a few other cosmetic changes to

Triangulated surfaces and polyhedral structures

In this chapter we introduce the foundational material that will be used in our analysis of triangulated surfaces and of their quantum geometry. We start by recalling the relevant definitions from Piecewise–Linear (PL) geometry, (for which we refer freely to [20, 21]). After these introductory remarks we specialize to the case of Euclidean polyhedral surfaces whose geometrical and physical properties will be the subject of the first part of the book. The focus here is on results which are either new or not readily accessible in the standard repertoire. In particular we discuss from an original perspective the structure of the space of all polyhedral surfaces of a given genus and their stable degenerations. This is a rather delicate point which appears in many guises in quantum gravity [6], and string theory, and which is related to the role that Riemann moduli space plays in these theories. Not surprisingly, the Witten–Kontsevich model [10] lurks in the background of our analysis, and some of the notions we introduce may well serve for illustrating, from a more elementary point of view, the often deceptive and very technical definitions that characterize this subject. In such a framework, and in the whole landscaping of the space of polyhedral surfaces an important role is played by the conical singularities associated with the Euclidean triangulation of a surface.We provide, in the final part of the chapter, a detailed analysis of the geometry of these singularities