Bundle gerbes and moduli spaces

In this paper, we construct the index bundle gerbe of a family of self-adjoint Dirac-type operators, refining a construction of Segal. In a special case, we construct a geometric bundle gerbe called the caloron bundle gerbe, which comes with a natural connection and curving, and show that it is isomorphic to the analytically constructed index bundle gerbe. We apply these constructions to certain moduli spaces associated to compact Riemann surfaces, constructing on these moduli spaces, natural bundle gerbes with connection and curving, whose 3-curvature represent Dixmier–Douady classes that are generators of the third de Rham cohomology groups of these moduli spaces.

► Bundle gerbes are degree 3 analogs of complex line bundles. ► The paper constructs an analytically defined index bundle gerbe. ► A geometrically defined caloron bundle gerbe is also constructed. ► It is proved in the paper that these bundle gerbes are isomorphic. ► The constructions apply to moduli spaces associated to compact Riemann surfaces.