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 $G$

Weighted composition operators as Daugavet centers

We investigate the norm identity $\|uC_\phi + T\| = \|u\|_\infty + \|T\|$ for classes of operators on $C(S)$, where $S$ is a compact Hausdorff space without isolated point, and characterize those weighted composition operators which satisfy this equation for every weakly compact operator $T : C(S)\to C(S)$. We also give a characterization of such weighted composition operator acting on the disk algebra $A(D).$Comment: 18 page

The supports of higher bifurcation currents

Let (f_\lambda) be a holomorphic family of rational mappings of degree d on the Riemann sphere, with k marked critical points c_1,..., c_k, parameterized by a complex manifold \Lambda. To this data is associated a closed positive current T_1\wedge ... \wedge T_k of bidegree (k,k) on \Lambda, aiming to describe the simultaneous bifurcations of the marked critical points. In this note we show that the support of this current is accumulated by parameters at which c_1,..., c_k eventually fall on repelling cycles. Together with results of Buff, Epstein and Gauthier, this leads to a complete characterization of Supp(T_1\wedge ... \wedge T_k).Comment: 13 page