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From the signature theorem to anomaly cancellation. (English) Zbl 1440.58008

In the paper under review, the authors consider 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 leds 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. They cancel the local anomaly on a Jacobian elliptic surface by combining several anomalous operators, a construction that is based on the construction of the Poincaré line bundle over an elliptic surface.

MSC:

58J20 Index theory and related fixed-point theorems on manifolds
14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations
14J28 \(K3\) surfaces and Enriques surfaces
81T50 Anomalies in quantum field theory
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