Document Type

Article

Journal/Book Title/Conference

Rocky Mountain Journal of Mathematics

Volume

50

Issue

1

Publisher

Rocky Mountain Mathematics Consortium

Publication Date

8-22-2019

First Page

181

Last Page

212

Abstract

We survey the Hirzebruch signature theorem as a special case of the Atiyah–Singer index theorem. The family version of the Atiyah–Singer index theorem in the form of the Riemann–Roch–Grothendieck–Quillen (RRGQ) formula is then applied to the complexified signature operators varying along the universal family of elliptic curves. The RRGQ formula allows us to determine a generalized cohomology class on the base of the elliptic fibration that is known in physics as (a measure of) the local and global anomaly. Combining several anomalous operators allows us to cancel the local anomaly on a Jacobian elliptic surface, a construction that is based on the construction of the Poincaré line bundle over an elliptic surface.

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