The Trans-Carpathian Seminar on Geometry & Physics

It has, by now, been rather well understood that the proper language inwhich to give a lagrangean formulation of the two-dimensional non-linearsigma model, regarded as a classical description of the critical bosonicstring (and of relevance in the study of certain condensed-matter systems,as well as models of statistical physics), is that of the theory ofgerbes. These latter are higher-cohomological structures with adifferential-geometric realisation whose rôle in string theory isanalogous to that played by fibre bundles in the modelling of the dynamicsof a charged pointlike particle moving in an external electromagneticfield. The talk, to be regarded as a fairly general overview of the stateof art in this field of mathematical physics in two dimensions, aspires tooutline some basic aspects of gerbe theory relevant to the classical andquantum description of poly-phase string world-sheets and string-theorydualities, laying due emphasis on the higher-categorial structure thatunderlies the theory of gerbes.In the second part of the talk, in which we take up (and brieflyrecapitulate beforehand) the subject introduced at the KMMF seminar``Theory of Duality'' of 18 XI 2010 with view to a detailed account of thegeneralised geometry underlying the groupoidal symmetries of the sigmamodel, the emergence, in the study of (infinitesimal) rigid symmetries ofthe two-dimensional field theory, of algebro-differential structures akinto the gerbe-twisted Courant algebroid shall be demonstrated and thenaturality of these structures in the context of the canonical descriptionof the two-dimensional field theory, and - in particular - that of theattendant gauge principle, shall be discussed. More specifically, we shallexamine the geometry of a family of generalised tangent bundles over theconfiguration bundle of the sigma model in the presence of the full2-category of bundle gerbes over it, establish its direct relation to theNoether (canonical) description of the rigid symmetries of the sigmamodel, and formulate a universal gauge principle for the two-dimensionalfield theories in hand based on the concept of categorial descent and thenotion of a principal bundle with a structural groupoid over thetwo-dimensional spacetime.