For any finite simple graphG = (V,E), the discrete Dirac operator D = d+d∗ and the Laplace-Beltrami operator L = dd∗+d∗d = (d+d∗)2 = D2 on the exterior algebra bundle Ω = ⊕Ωk are finite v × v matrices, where dim(Ω) = v = ∑ k vk, with vk = dim(Ωk) denoting the cardinality of the set Gk of complete subgraphs Kk of G. We prove the McKean-Singer formula χ(G) = str(e−tL) which holds for any complex ...

We present a signature formula for compact 4k-manifolds with corners of codimension two which generalizes the formula of Atiyah, Patodi and Singer for manifolds with boundary. The formula expresses the signature as a sum of three terms, the usual Hirzebruch term given as the integral of an L-class, a second term consisting of the sum of the eta invariants of the induced signature operators on t...

In this expository article, we survey index theory of Dirac operators using the Gauss-Bonnet formula as the catalyst to discuss index formulas on manifolds with and without boundary. Considered in detail are the Atiyah-Singer and Atiyah-Patodi-Singer index theorems, their heat kernel proofs, and their generalizations to manifolds with corners of codimension two via the method of ‘attaching cyli...

The Atiyah-Singer index theorem gives a topological formula for the index of an elliptic differential operator. The topological index depends on a cohomology class that is constructed from the principal symbol of the operator. On contact manifolds, the important Fredholm operators are not elliptic, but hypoelliptic. Their symbolic calculus is noncommutative, and is closely related to analysis o...

We define generalized Atiyah-Patodi-Singer boundary conditions of product type for Dirac operators associated to C∗-vector bundles on the product of a compact manifold with boundary and a closed manifold. We prove a product formula for the K-theoretic index classes, which we use to generalize the product formula for the topological signature to higher signatures.

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...