Seminarium KMMF "Teoria Dwoistości"

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 first part of the talk, conceived as an elementary introduction tothe applications of the theory of gerbes and associated structures in thecontext of two-dimensional field theory (and to be further expatiated uponat the seminar ``Methods of Geometry in Physics'' of 24 XI 2010), thesource of gerbe-theoretic structures in the lagrangean approach shall beindicated, their naturality shall be discussed and certain significantconsequences of their existence shall be enumerated. Thus, in particular,the action functional of the sigma model with a generic configurationbundle shall be re-derived, the concept of transgression, whichcanonicallyassociates to the gerbe structure a pre-quantum bundle of the model, shallbe presented, the notion of a conformal defect, intimately related to thatof a string-theory duality, shall be introduced, and, finally, the gaugeprinciple for groupoidal symmetries of the string, illustrating the ideaof categorial descent, shall be formulated. Time permitting, we shallembellish the general discourse with a handful of explicit workingconstructions.