Formal loops II: A local Riemann-Roch theorem for determinantal gerbes

Details Formal loops II: A local Riemann-Roch theorem for determinantal gerbes Formal loops II: A local Riemann-Roch theorem for determinantal gerbes

2005-09-27

arxiv.org/abs/math/0509646v1

Mathematics

Algebraic Geometry

25 pages

Scientific paper

If V is a bundle of Tate vector spaces over a base B, its determinantal gerbe has a class C_1(V) in the second cohomology group of the sheaf of invertible functions which can be seen as the Deligne cohomology H^3(B, Z(2)). An example of such a "Tate bundle" can be obtained from a finite rank vector bundle E on the product of B and a punctured formal disk. Our main result identifies the corresponding C_1(V) with the cohomological direct image of ch_2(E), the second Chern character of E. It can be seen as a "local" version of the Riemann-Roch-Grothendieck theorem for a family of curves. This theorem explains the results of Gorbounov, Malikov and Schechtman relating ch_2 of the tangent bundle of an algebraic variety X to the existence of a sheaf of chiral differential operators. To be precise, it implies that the determinantal anomaly of the formal loop space of X is the transgression of ch_2(TX).

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