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# Cambridge Journal of Mathematics

## Volume 9 (2021)

### Number 3

### Geometric flows for the Type IIA string

Pages: 693 – 807

DOI: https://dx.doi.org/10.4310/CJM.2021.v9.n3.a3

#### Authors

#### Abstract

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