Learning Seminar: SYZ Mirror Symmetry

Introduction

After Kontsevich's famous proposal on Homological Mirror Symmetry [Kon94], the question of geometric construction of mirrors to a given variety has been an important question for mathematicians. One important proposal was made by Strominger-Yau-Zaslow [SYZ96], which stated that mirrors to a given Kähler manifold $X$ equipped with a Lagrangian torus fibration $X\to B$ should be a Kähler manifold $\check{X}\to B$ with a Lagrangian torus fibration over the same base $B$, and with "dual torus fiber".

This proposal has been successful firstly for elliptic curves whose corresponding Homological Mirror Symmetry was (partly) verified by Polishchuk-Zaslow [PZ98]. People quickly found finding a smooth Lagrangian torus fibration for any Kähler manifold $X$ is almost impossible: there can be toric singularities where the fibration has lower dimensional toric fibers, and there can be singular fibers. Cho and Oh [CO06] studied the effects of toric singularities for toric Fano varieties and showed that existence of toric singularities gives Landau-Ginzburg superpotentials $W$ of the mirror, via counting intersection numbers of holomorphic disks with the toric singularities. For toric varieties which admits canonical Lagrangian torus fibrations with only toric singularities, the corresponding Homological Mirror Symmetry statement was then proved by Fang-Liu-Treumann-Zaslow [FLTZ11] for smooth projective toric varieties and further generalized by Kuwagaki [Kuw20] to all toric stacks.

The simplest and the most generic case of singular fibers are non-degenerate singularities, similar to singular fibers of Lefschetz fibration, obtained by contracting a codimension $1$ subtorus to a point. This situation was studied by Auroux [Aur07] in the case when $X=\mathbb{C}^2\setminus\{xy=1\}$ and the Lagrangian torus fibration was constructed out of the Lefschetz fibration $\pi (x,y)=xy$.

In this case one can clearly see a wall-crossing phenomenon, where there's a single singular value for the torus fibration $X\to\mathbb{R}^2$ coming from singularities of the Lefschetz fibration, $\{xy=0\}\subseteq\mathbb{C}^2$, which corresponds to the wall $\{x=0\}\times\mathbb{R}\subseteq\mathbb{R}^2$. Auroux showed that for any $b=(b_1,b_2)$ with $b_1>1$ or $b_1<1$, disk countings computing mirror superpotential to $X\subseteq\mathbb{P}^2$ are different, so that the two moduli spaces of Lagrangian tori are glued together with non-trivial transition functions.

References

[Aur07] Auroux, D. (2007). Mirror symmetry and T-duality in the complement of an anticanonical divisor. Journal of Gökova Geometry Topology. GGT, 1, 51–91. Retrieved from http://www.ams.org/mathscinet-getitem?mr=2386535

[CO06] Cho, C.-H., & Oh, Y.-G. (2006). Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toric manifolds. Asian J. Math., 10(4), 773–814. doi: 10.4310/AJM.2006.v10.n4.a10

[FLTZ11] Fang, B., Liu, C.-C. M., Treumann, D., & Zaslow, E. (2011). A categorification of Morelli’s theorem. Invent. Math., 186(1), 79–114. doi: 10.1007/s00222-011-0315-x

[Kon94] Kontsevich, M. (1994). Homological Algebra of Mirror Symmetry. arXiv, alg-geom/9411018. Retrieved from https://arxiv.org/abs/alg-geom/9411018v1.

[Kuw20] Kuwagaki, T. (2020). The nonequivariant coherent-constructible correspondence for toric stacks. Duke Math. J., 169(11), 2125–2197. doi: 10.1215/00127094-2020-0011

[PZ98] Polishchuk, A., & Zaslow, E. (1998). Categorical mirror symmetry: The elliptic curve. Adv. Theor. Math. Phys., 2(2), 443–470. doi: 10.4310/ATMP.1998.v2.n2.a9

[SYZ96] Strominger, A., Yau, S. T., & Zaslow, E. (1996). Mirror symmetry is T-duality. Nuclear Physics B, 479(1-2), 243-259.