The Cohomology of Solvmanifold SYZ Mirrors

Abstract

This paper investigates the geometric and cohomological properties of non-Kähler SYZ mirror symmetry for dual torus fibrations over solvmanifolds in the sense of Lau, Tseng and Yau. We are mainly concerned with three questions: \textbf{(a)} How the Lau-Tseng-Yau notion of non-Kähler SYZ is related to the mapping of supersymmetric branes between symplectic and complex sides; \textbf{(b)} Finding explicit non-Kähler SYZ mirror pairs determined purely by Lie-theoretic data; \textbf{(c)} better understand the cohomological correspondence in the Lau-Tseng-Yau framework (given by a Fourier-Mukai transform), especially concerning the role of Tseng-Yau cohomology. We prove that the Fourier-Mukai transform introduced by Lau-Tseng-Yau exchanges type-A supersymmetric cycles, which are given by special Lagrangian sections equipped with flat $\mathrm{U}(1)$ connections, with type-B cycles, corresponding to line bundles whose connections satisfy the deformed Hermitian-Yang-Mills (dHYM) equation. We provide pure Lie-theoretic criteria for the existence of non-Kähler SYZ mirror pairs whose base manifolds are solvmanifolds. Applying these criteria, we construct new explicit families of mirror pairs from almost abelian and generalized Heisenberg Lie groups, and provide a complete classification of such pairs arising from nilpotent Lie groups. To contextualize the role of the Tseng-Yau cohomology, we link it to noncommutative geometry. We introduce the Tseng-Yau and Bott-Chern mirror bicomplexes. We show that (some of) their enclosed cohomologies reduce to the primitive Tseng-Yau and Bott-Chern cohomologies and that for basic forms they are isomorphic under the Fourier-Mukai transform. As a last contribution, we discuss how to explicitly compute the Tseng-Yau and the Bott-Chern cohomology for the non-Kähler SYZ mirror pairs constructed here.