The use of bundle gerbes and bundle gerbe modules is considered as a replacement for the usual theory of Clifford modules on manifolds that fail to be spin. It is shown that both sides of the Atiyah-Singer index formula for coupled Dirac operators can be given natural interpretations using this language and that the resulting formula is still an identity.

In this paper, we rigorously construct 2d Liouville Quantum Field Theory on the Riemann sphere introduced in the 1981 seminal work by Polyakov Quantum Geometry of bosonic strings. We also establish some of its fundamental properties like conformal covariance under PSL2(C)-action, Seiberg bounds, KPZ scaling laws, KPZ formula and the Weyl anomaly (Polyakov-Ray-Singer) formula for Liouville Quant...

Several proofs have been published of the modZ gluing formula for the h-invariant of a Dirac operator. However, so far the integer contribution to the gluing formula for the h-invariant is left obscure in the literature. In this article we present a gluing formula for the h-invariant which expresses the integer contribution as a triple index involving the boundary conditions and the Calderón pr...

Several proofs have been published of the modZ gluing formula for the η–invariant of a Dirac operator. However, so far the integer contribution to the gluing formula for the η–invariant is left obscure in the literature. In this article we present a gluing formula for the η–invariant which expresses the integer contribution as a triple index involving the boundary conditions and the Calderón pr...

We give a new proof of the Jantzen sum formula for integral representations of Chevalley schemes over Spec Z. This is done by applying the fixed point formula of Lefschetz type in Arakelov geometry to generalized flag varieties. Our proof involves the computation of the equivariant Ray-Singer torsion for all equivariant bundles over complex homogeneous spaces. Furthermore, we find several expli...

The Legendre-Fenchel transform is a classical piece of mathematics with many applications. In this paper we show how it arises in the context of category theory using categories enriched over the extended real numbers R := [−∞,+∞]. A key ingredient is Pavlovic’s “nucleus of a profunctor” construction. The pairing between a vector space and its dual can be viewed as an R-profunctor; the construc...