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

COMBINATORIAL STACKS AND THE FOUR-COLOUR THEOREM

Abstract. We interpret the number of good four-colourings of the faces of a trivalent, spherical polyhedron as the 2-holonomy of the 2-connection of a fibered category, ϕ, modeled on Rep f (sl2) and defined over the dual triangulation, T. We also build an sl2-bundle with connection over T, that is a global, equivariant section of ϕ, and we prove that the four-colour theorem is equivalent to the fact that the connection of this sl2-bundle vanishes nowhere. This interpretation is proposed as a first step toward a cohomological proof of the four-colour theorem. Content

Extracting vibrational anharmonicities from short driven molecular dynamics trajectories

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