Bundle gerbes are a higher version of line bundles, we present nonabelian bundle gerbes as a higher version of principal bundles. Connection, curving, curvature and gauge transformations are studied both in a global coordinate independent formalism and in local coordinates. These are the gauge fields needed for the construction of Yang-Mills theories with 2-form gauge potential. a

The Razumov-Stroganov conjecture relates the ground-state coefficients in the periodic even-length dense O (1) loop model to the enumeration of fully-packed loop configurations on the square, with alternating boundary conditions, refined according to the link pattern for the boundary points. Here we prove this conjecture, by means of purely combinatorial methods. The main ingredient is a generalization of the Wieland proof technique for the dihedral symmetry of these classes, based on the 'gyration' operation, whose full strength we will investigate in a companion paper.

Abstract. We derive a matrix product formula for symmetric Macdonald polynomials.Our results are obtained by constructing polynomial solutions of deformed Knizhnik-Zamolodchikov equations, which arise by considering representations of the Zamolodchikov-Faddeev and Yang-Baxter algebras in terms of t-deformed bosonic operators. These solutions are generalised probabilities for particle configurations of the multi-species asymmetric exclusion process, and form a basis of the ring of polynomials in n variables whose elements are indexed by compositions. For weakly increasing compositions (anti-dominant weights), these basis elements coincide with non-symmetric Macdonald polynomials. Our formulas imply a natural combinatorial interpretation in terms of solvable lattice models. They also imply that normalisations of stationary states of multi-species exclusion processes are obtained as Macdonald polynomials at q = 1.

It is shown that the reduced particle dynamics of 2+1 dimensional gravity in the maximally slicing gauge has hamiltonian form. This is proved directly for the two body problem and for the three body problem by using the Garnier equations for isomonodromic transformations. For a number of particles greater than three the existence of the hamiltonian is shown to be a consequence of a conjecture by Polyakov which connects the auxiliary parameters of the fuchsian differential equation which solves the SU (1, 1) Riemann-Hilbert problem, to the Liouville action of the conformal factor which describes the space-metric.We give the exact diffeomorphism which transforms the expression of the spinning cone geometry in the Deser, Jackiw, 't Hooft gauge to the maximally slicing gauge. It is explicitly shown that the boundary term in the action, written in hamiltonian form gives the hamiltonian for the reduced particle dynamics.The quantum mechanical translation of the two particle hamiltonian gives rise to the logarithm of the Laplace-Beltrami operator on a cone whose angular deficit is given by the total energy of the system irrespective of the masses of the particles thus proving at the quantum level a conjecture by 't Hooft on the two particle dynamics.The quantum mechanical Green's function for the two body problem is given.