Flux-complete extension of the Lau–Tseng–Yau non-Kähler SYZ mirror correspondence: H-flux, non-invariant forms, and the passage beyond the semi-flat regime

Lau, Tseng and Yau (Comm. Math. Phys. 340 (2015) 145–170) gave the semi-flat Strominger–Yau–Zaslow (SYZ) mirror correspondence a precise non-Kähler form: the Fourier–Mukai transform on differential forms exchanges supersymmetric Type IIA and Type IIB SU(n)-structure systems carrying Ramond–Ramond (RR) flux, together with their flux source terms and the associated $d+d^\Lambda$/Bott–Chern cohomologies. The theorem is proved under three hypotheses: only RR flux is present (no NS-NS H-flux, no B-field complexification of the symplectic form), the forms are torus-invariant, and the Lagrangian torus fibration is semi-flat (no singular fibers, hence no instanton corrections). The open problem is to remove these restrictions in a mathematically complete way: formulate and prove the mirror correspondence for Type II supersymmetric systems carrying both RR and H-flux (allowing generalized-complex or, if necessary, gerbe-theoretic/noncommutative target structures, with a criterion for when the mirror is geometric), establish the operator and cohomology correspondence for general non-invariant forms, and ultimately extend the correspondence across singular fibers with instanton corrections, resting on an understanding of the special real polarizations that must be fixed in order to define higher-dimensional Type IIA systems. No resolution of these directions is known in the literature to date.

Background

Mirror symmetry predicts a duality between a symplectic manifold (the A-side) and a complex manifold (the B-side) that exchanges symplectic and complex structures. The Strominger–Yau–Zaslow (SYZ) conjecture explains this geometrically: mirror Calabi–Yau manifolds should carry dual special Lagrangian torus fibrations over a common base (special Lagrangian tori are Lagrangian submanifolds — the symplectic form restricts to zero on them — on which the real part of the ambient holomorphic volume form restricts to a constant-phase volume form), and the mirror map should be fiberwise T-duality, exchanging each torus fiber with its dual torus. On the open locus where the fibers are nonsingular — the semi-flat regime — this duality is mathematically clean: the dual of a Lagrangian torus bundle carries a canonical complex structure, and the duality is implemented by the Fourier–Mukai transform, which on differential forms is a fiberwise Fourier series exchanging fiber directions of one side with base directions of the other (Leung–Yau–Zaslow). Supersymmetric string backgrounds with background flux are generically non-Kähler, which motivated Lau, Tseng and Yau (LTY) to build a non-Kähler version of semi-flat SYZ. An $SU(n)$-structure on a real $2n$-manifold is a pair $(\omega,\Omega)$, where $\omega$ is a nondegenerate real $(1,1)$-form defining a Hermitian metric and $\Omega$ is a decomposable complex $(n,0)$-form defining an almost complex structure; no closedness is required. A Type IIB system is a complex manifold with holomorphic volume form $\Omega$ and a balanced metric, i.e. $d\Omega=0$ and $d(\omega^{n-1})=0$ — the balanced condition is the natural non-Kähler replacement for the Kähler condition $d\omega=0$. A Type IIA system is a symplectic manifold with $d\omega=0$ and $d(\mathrm{Re}\,\Omega)=0$; in complex dimension three this is the familiar symplectic half-flat condition, but in higher dimensions the correct definition requires a special real polarization: an integrable Lagrangian distribution $\mathcal{P}$ (an involutive subbundle of half rank on which $\omega$ vanishes) such that $\Omega$ restricts to each leaf as a real-valued form of constant phase. Unlike a complex structure, which automatically provides the splitting into $(1,0)$ and $(0,1)$ directions, a symplectic structure does not come with a real polarization, and this choice is an essential, poorly understood input. In type II string theory there are two kinds of background flux: Ramond–Ramond (RR) flux, an inhomogeneous differential form appearing as source terms $\rho_A$, $\rho_B$ in the supersymmetry equations, and NS-NS (Neveu–Schwarz) H-flux, a closed three-form $H=dB$ built from the two-form $B$-field. LTY treated systems with RR flux and proved their central theorem: under SYZ plus Fourier–Mukai transform, a semi-flat supersymmetric Type IIA $SU(n)$-system maps to a Type IIB system, with the flux sources corresponding to each other. On the cohomological level, the relevant invariants are Bott–Chern cohomology $H_{BC}=\ker\partial\cap\ker\bar\partial/\mathrm{im}\,\partial\bar\partial$ on the complex side and the Tseng–Yau symplectic cohomology $H_{d+d^\Lambda}=\ker d\cap\ker d^\Lambda/\mathrm{im}\,dd^\Lambda$ on the symplectic side, where $d^\Lambda=[d,\Lambda]$ and $\Lambda$ is the adjoint of wedging with $\omega$; LTY proved that, for torus-invariant forms, the Fourier–Mukai transform induces an isomorphism $H^{p,q}_{d+d^\Lambda}\cong H^{n-p,q}_{BC}$ between mirror systems — a non-Kähler analogue of Hodge symmetry. Three restrictions of the theorem are explicit in the paper. First, the Fourier–Mukai machinery is applied only to torus-invariant forms; a footnote states that the $B$-field should be included and $\omega$ complexified for the mirror map to be an isomorphism, but this is omitted. Second, everything happens in the semi-flat regime, away from the singular fibers where quantum (instanton) corrections — contributions from holomorphic disks and curves — are expected. Third, in dimensions above three the Type IIA system depends on the choice of a special real polarization, whose global existence and uniqueness is not understood. The H-flux side is understood only in adjacent frameworks: Fidanza–Minasian–Tomasiello realized mirror symmetry for SU(3)-structure manifolds with NS fluxes through chains of T-dualities; Bouwknegt–Evslin–Mathai's topological T-duality shows that H-flux can change the topology of the dual torus bundle and is encoded in twisted K-theory (K-theory whose cocycle condition is deformed by the three-form $H$); Cavalcanti–Gualtieri established T-duality for generalized complex structures (the geometry on $TX\oplus T^*X$ of Hitchin and Gualtieri, whose Courant bracket is twisted by $H$) under isotropy conditions on the fibers. None of these provides an LTY-type, form-level Fourier–Mukai correspondence with H-flux, and in general the T-dual of a torus fibration with non-isotropic H-flux is not expected to be an ordinary manifold at all (a nongeometric T-fold background, i.e. one glued together by T-duality transformations rather than by coordinate charts). Physically, the six-dimensional systems come from type II supersymmetry; the higher-dimensional analogues introduced by LTY resemble the SU(4)-structure supergravity systems of Prins–Tsimpis and the generalized-geometry formulation of two-dimensional vacua by Rosa. Concrete examples come from nilmanifolds and solvmanifolds — compact quotients of nilpotent or solvable Lie groups, which are never Kähler in the non-abelian cases — such as the Iwasawa manifold.

Problem Statement

Extend the Lau–Tseng–Yau Fourier–Mukai mirror correspondence for supersymmetric Type IIA/IIB $SU(n)$-structure systems with flux so that none of its three hypotheses remains, in the following layered form, where each layer supports the same overarching objective — a flux-complete mirror correspondence for non-Kähler supersymmetric systems. (1) H-flux on equal footing with RR flux: formulate the Type IIA and Type IIB supersymmetric systems carrying both RR flux and NS-NS H-flux, and prove that a SYZ/Fourier–Mukai-type transform exchanges the two systems together with their flux source terms. The formulation may pass to generalized-complex, gerbe-theoretic, or noncommutative target structures, but must then state precisely under which conditions on the flux the mirror remains an ordinary manifold and justify when it does not. (2) General forms: remove the torus-invariance hypothesis — prove the operator correspondence and the cohomology isomorphism between $H_{d+d^\Lambda}$-type (or its H-flux-twisted replacement) and Bott–Chern-type cohomologies for general differential forms, which is what the deformation theory of the systems requires. (3) Beyond semi-flat: extend the correspondence across the singular fibers of the Lagrangian torus fibrations, incorporating the instanton (quantum) corrections, and understand the choice of the special real polarizations that must be fixed, since a symplectic structure does not come with a canonical one, in order to define the higher-dimensional Type IIA systems. (4) Physical identification: clarify which supersymmetric compactification backgrounds — which flux combinations, dimensions, and structure groups, including the $SU(4)$-structure systems of Prins–Tsimpis — are captured by the resulting mathematical structures. Layers (1) and (2) are set at the semi-flat level; layer (3) extends the same correspondence beyond the semi-flat regime; layer (4) states the physical identification of (1)–(3). A resolution of (1)–(2) consists of precise definitions and proofs of the structure-level, operator-level, and cohomology-level correspondences.

References

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