Communications in Analysis and Geometry

Volume 26 (2018)

Number 4

Anomaly flows

Pages: 955 – 1008



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)


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.


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

Work supported in part by the National Science Foundation Grants DMS-12-66033 and DMS-1605968.

Received 27 May 2017

Published 6 September 2018