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Register a new userA Short Proof of the Localization Formula for the Loop Space Chern Character of Spin Manifolds
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In this note, we give a short proof of the localization formula for the loop space Chern character of a compact Riemannian spin manifold M, using the rescaled spinor bundle on the tangent groupoid associated to M.
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We study super parallel transport around super loops in a quotient stack, and show that this geometry constructs a global version of the equivariant Chern character.
We introduce the notion of a {vartheta}-summable Fredholm module over a locally convex dg algebra {Omega} and construct its Chern character as a cocycle on the entire cyclic complex of {Omega}, extending the construction of Jaffe, Lesniewski and Osterwalder to a differential graded setting. Using this Chern character, we prove an index theorem involving an abstract version of a Bismut-Chern character constructed by Getzler, Jones and Petrack in the context of loop spaces. Our theory leads to a rigorous construction of the path integral for N=1/2 supersymmetry which satisfies a Duistermaat-Heckman type localization formula on loop space.
We give a motivic proof of a character formula for depth zero supercuspidal representations of $p$-adic SL(2). We begin by finding the virtual Chow motives for the character values of all depth zero supercuspidal representations of $p$-adic SL(2), at topologically unipotent elements. Then we find the virtual Chow motives for the values of the Fourier transform of all regular elliptic orbital integrals with depth zero in their Cartan subalgebra, at topologically nilpotent elements. Finally, we prove a character formula for depth zero supercuspidal representations by showing that the formula corresponds to three identities in the ring of virtual Chow motives over $mathbb{Q}$.
As the loop space of a Riemannian manifold is infinite-dimensional, it is a non-trivial problem to make sense of the top degree component of a differential form on it. In this paper, we show that a formula from finite dimensions generalizes to assign a sensible top degree component to certain composite forms, obtained by wedging with the exponential (in the exterior algebra) of the canonical 2-form on the loop space. The result is a section on the Pfaffian line bundle on the loop space. We then identify this with a section of the line bundle obtained by transgression of the spin lifting gerbe. These results are a crucial ingredient for defining the fermionic part of the supersymmetric path integral on the loop space.
Let G be a split, simple, simply connected, algebraic group over Q. The degree 4, weight 2 motivic cohomology group of the classifying space BG of G is identified with Z. We construct cocycles representing the generator of this group, known as the second universal motivic Chern class. If G = SL(m), there is a canonical cocycle, defined by the first author (1993). For any group G, we define a collection of cocycles parametrised by cluster coordinate systems on the space of G-orbits on the cube of the principal affine space G/U. Cocycles for different clusters are related by explicit coboundaries, constructed using cluster transformations relating the clusters. The cocycle has three components. The construction of the last one is canonical and elementary; it does not use clusters, and provides a canonical cocycle for the motivic generator of the degree 3 cohomology class of the complex manifold G(C). However to lift this component to the whole cocycle we need cluster coordinates: the construction of the first two components uses crucially the cluster structure of the moduli spaces A(G,S) related to the moduli space of G-local systems on S. In retrospect, it partially explains why the cluster coordinates on the space A(G,S) should exist. This construction has numerous applications, including an explicit construction of the universal extension of the group G by K_2, the line bundle on Bun(G) generating its Picard group, Kac-Moody groups, etc. Another application is an explicit combinatorial construction of the second motivic Chern class of a G-bundle. It is a motivic analog of the work of Gabrielov-Gelfand-Losik (1974), for any G.
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