Homotopy and duality in non-Abelian lattice gauge theory

We propose an approach of lattice gauge theory based on a homotopic interpretation of its degrees of freedom. The basic idea is to dress the plaquettes of the lattice to view them as elementary homotopies between nearby paths. Instead of using a unique $G$-valued field to discretize the connection 1-form, $A$, we use an \AG-valued field $U$ on the edges, which plays the role of the 1-form \ad_A, and a $G$-valued field $V$ on the plaquettes, which corresponds to the Faraday tensor, $F$. The 1-connection, $U$, and the 2-connection, $V$, are then supposed to have a 2-curvature which vanishes. This constraint determines $V$ as a function of $U$ up to a phase in $Z(G)$, the center of $G$. The 3-curvature around a cube is then Abelian and is interpreted as the magnetic charge contained inside this cube. Promoting the plaquettes to elementary homotopies induces a chiral splitting of their usual Boltzmann weight, $w=v\bar{v}$, defined with the Wilson action. We compute the Fourier transform, $\hat{v}$, of this chiral Boltzmann weight on $G=SU_3$ and we obtain a finite sum of generalized hypergeometric functions. The dual model describes the dynamics of three spin fields : $\lambda_P\in{\hat{G}}$ and $m_P\in{\hat{Z(G)}}\simeq\Z_3$, on each oriented plaquette $P$, and \epsilon_{ab}\in{\hat{\OG}}\simeq\Z_2, on each oriented edge $(ab)$. Finally, we sketch a geometric interpretation of this spin system in a fibered category modeled on the category of representations of $G$

Two-dimensional parallel transport : combinatorics and functoriality

We extend the usual notion of parallel transport along a path to triangulated surfaces. A homotopy of paths is lifted into a fibered category with connection and this defines a functor between the fibers above the boundary paths. These "sweeping functors" transport fiber bundles with connection along a surface whereas usual connections transport a group element along a path. We show that to get rid of the parametrization, we must use Abelian degrees of freedom. In the general, non-Abelian case, we conjecture that the smooth limit of this construction provides us with representations of the group of diffeomorphisms of the swept surface. Applications to gauge theories are proposed.Comment: 18 pages ; v2 : interpretation of abelianisation changed and typos correcte

Bilayers in Four Dimensions and Supersymmetry

I build $N=1$ superstrings in $\Bbb R^4$ out of purely geometric bosonic data. The world-sheet is a bilayer of uniform thickness and the $2D$ supercharge vanishes in a natural way.Comment: 4 pages,Latex, no figur

Markov Chains and Dynamical Systems: The Open System Point of View

This article presents several results establishing connections be- tween Markov chains and dynamical systems, from the point of view of open systems in physics. We show how all Markov chains can be understood as the information on one component that we get from a dynamical system on a product system, when losing information on the other component. We show that passing from the deterministic dynamics to the random one is character- ized by the loss of algebra morphism property; it is also characterized by the loss of reversibility. In the continuous time framework, we show that the solu- tions of stochastic dierential equations are actually deterministic dynamical systems on a particular product space. When losing the information on one component, we recover the usual associated Markov semigroup

The Langevin Equation for a Quantum Heat Bath

We compute the quantum Langevin equation (or quantum stochastic differential equation) representing the action of a quantum heat bath at thermal equilibrium on a simple quantum system. These equations are obtained by taking the continuous limit of the Hamiltonian description for repeated quantum interactions with a sequence of photons at a given density matrix state. In particular we specialise these equations to the case of thermal equilibrium states. In the process, new quantum noises are appearing: thermal quantum noises. We discuss the mathematical properties of these thermal quantum noises. We compute the Lindblad generator associated with the action of the heat bath on the small system. We exhibit the typical Lindblad generator that provides thermalization of a given quantum system.Comment: To appear in J.F.

Stochastic Master Equations in Thermal Environment

We derive the stochastic master equations which describe the evolution of open quantum systems in contact with a heat bath and undergoing indirect measurements. These equations are obtained as a limit of a quantum repeated measurement model where we consider a small system in contact with an infinite chain at positive temperature. At zero temperature it is well-known that one obtains stochastic differential equations of jump-diffusion type. At strictly positive temperature, we show that only pure diffusion type equations are relevant