Cambridge Journal of Mathematics

Volume 9 (2021)

Number 3

Geometric flows for the Type IIA string

Pages: 693 – 807



Teng Fei (Department of Mathematics & Computer Science, Rutgers University, Newark, New Jersey, U.S.A.)

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

Sebastien Picard (Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada)

Xiangwen Zhang (Department of Mathematics, University of California, Irvine, Calif., U.S.A.)


A geometric flow on $6$-dimensional symplectic manifolds is introduced which is motivated by supersymmetric compactifications of the Type IIA string. The underlying structure turns out to be $\mathrm{SU}(3)$ holonomy, but with respect to the projected Levi–Civita connection of an almost-Hermitian structure. The short-time existence is established, and new identities for the Nijenhuis tensor are found which are crucial for Shi-type estimates. The integrable case can be completely solved, giving an alternative proof of Yau’s theorem on Ricci-flat Kähler metrics. In the non-integrable case, models are worked out which suggest that the flow should lead to optimal almost-complex structures compatible with the given symplectic form.

The second-named author is supported in part by the National Science Foundation Grant DMS-1855947.

The fourth-named author is supported in part by the National Science Foundation Grant DMS-1809582.

Received 22 February 2021

Published 7 December 2021