Communications in Analysis and Geometry

Volume 26 (2018)

Number 4

Anomaly flows

Pages: 955 – 1008

DOI: https://dx.doi.org/10.4310/CAG.2018.v26.n4.a9

Authors

Duong H. Phong (Department of Mathematics, Columbia University, New York, N.Y., U.S.A.)

Sebastien Picard (Department of Mathematics, Columbia University, New York, N.Y., U.S.A.)

Xiangwen Zhang (Department of Mathematics, University of California at Irvine)

Abstract

The Anomaly flow is a flow which implements the Green–Schwarz anomaly cancellation mechanism originating from superstring theory, while preserving the conformally balanced condition of Hermitian metrics. There are several versions of the flow, depending on whether the gauge field also varies, or is assumed known. A distinctive feature of Anomaly flows is that, in $m$ dimensions, the flow of the Hermitian metric has to be inferred from the flow of its $(m-1)\textrm{-th}$ power $\omega^{m-1}$. We show how this can be done explicitly, and we work out the corresponding flows for the torsion and the curvature tensors. The results are applied to produce criteria for the long-time existence of the flow, in the simplest case of zero slope parameter.

Keywords

Green–Schwarz anomaly cancellation, quadratic terms in the curvature tensor, torsion constraints, flows, maximum principle

Received 27 May 2017

Published 6 September 2018