We describe some recent development on the theory of formal Frobenius manifolds via a construction from differential Gerstenhaber-Batalin-Vilkovisk (DGBV) algebras and formulate a version of mirror symmetry conjecture: the extended deformation problems of the complex structure and the Poisson structure are described by two DGBV algebras; mirror symmetry is interpreted in term of the invariance of the formal Frobenius manifold structures under quasi-isomorphism.
One of the methods to obtain Frobenius manifold structures is via DGBV (differential Gerstenhaber-Batalin-Vilkovisky) algebra construction. An important problem is how to identify Frobenius manifold structures constructed from two different DGBV algebras. For DGBV algebras with suitable conditions, we show the functorial property of a construction of deformations of the multiplicative structures of their cohomology. In particular, we show that quasi-isomorphic DGBV algebras yield identifiable Frobenius manifold structures.
For a closed Kähler manifold with a Hamiltonian action of a connected compact Lie group by holomorphic isometries, we construct a formal Frobenius manifold structure on the equivariant cohomology by exploiting a natural DGBV algebra structure on the Cartan model.
The goal of this article is to provide an explicit algorithmic construction of formal $F$-manifold structures, formal Frobenius manifold structures, and higher residue pairings on the primitive middle-dimensional cohomology $\mathbb{H}$ of a smooth projective Calabi-Yau complete intersection variety $X$ defined by homogeneous polynomials $G_1(\underline x), \dots, G_k(\underline x)$. Our main method is to analyze a certain dGBV (differential Gerstenhaber-Batalin-Vilkovisky) algebra $\mathcal{A}$ obtained from the twisted de Rham complex which computes $\mathbb{H}$. More explicitly, we introduc…
We explain how deformation theories of geometric objects such as complex structures, Poisson structures and holomorphic bundle structures lead to differential Gerstenhaber or Poisson algebras. We use homological perturbation theory to obtain $A_{\infty}$ algebra structures and some canonically defined deformations of such structures on the cohomology. We formulate the $A_{\infty}$ algebraic mirror symmetry as the identification of the $A_{\infty}$ algebras together with their canonical deformations constructed this way on different manifolds.
We give a vertex algebra proof of the Berglund-Hübsch duality of nondegenerate invertible potentials. We suggest a way to unify it with the Batyrev-Borisov duality of reflexive Gorenstein cones.
A complex symplectic structure on a Lie algebra $\lie h$ is an integrable complex structure $J$ with a closed non-degenerate $(2,0)$-form. It is determined by $J$ and the real part $Ω$ of the $(2,0)$-form. Suppose that $\lie h$ is a semi-direct product $\lie g\ltimes V$, and both $\lie g$ and $V$ are Lagrangian with respect to $Ω$ and totally real with respect to $J$. This note shows that $\lie g\ltimes V$ is its own weak mirror image in the sense that the associated differential Gerstenhaber algebras controlling the extended deformations of $Ω$ and $J$ are isomorphic. The geometry of $(Ω, J)$…
The existence of a flat torsion-free connection, or left symmetric algebra structure on a Lie algebra g gives rise to a canonically defined complex structure on g+g and a symplectic structure on g+g^*. We verify that the associated differential Gerstenhaber algebras controlling the deformation theories of the complex and symplectic form are isomorphic. This provides a class of examples of "weak mirror symmetry" as suggested by Merkulov. For nilpotent algebras in dimension 4 and 6 the isomorphism classes of the semi-direct products g+g and g+g^* are listed. A one-parameter family of inequivalen…
We construct a differential Gerstenhaber-Batalin-Vilkovisky algebra from Dolbeault complex of any close Kaehler manifold, and a Frobenius manifold structure on Dolbeault cohomology.
This article provides a complete description of the differential Gerstenhaber algebras of all nilpotent complex structures on any real six-dimensional nilpotent algebra. As an application, we classify all pseudo-Kählerian complex structures on six-dimensional nilpotent algebras such that the differential Gerstenhaber algebra of its complex structure is quasi-isomorphic to that of its symplectic structure. In a weak sense of mirror symmetry, it is a classification of pseudo-Kähler structures on six-dimensional nilpotent algebras whose mirror images are themselves.
We identify two Frobenius manifolds obtained from two different differential Gerstenhaber-Batalin-Vilkovisky algebras on a compact Kaehler manifold. One is constructed on the Dolbeault cohomology, and the other on the de Rham cohomology. Our result can be considered as a generalization of the identification of the Dolbeault cohomology ring with the complexified de Rham cohomology ring on a Kaehler manifold.
The paper is devoted to the comparison of the Fukaya category (it is responcible for the A-side of mirror symmetry) with the category of holonomic modules over the quantized algebra of functions on the same symplectic manifold. We conjecture that these categories become $A_{\infty}$-equivalent after a twist by a kind of integral transformation.
We study the deformed Hermitian-Yang-Mills (dHYM) equation, which is mirror to the special Lagrangian equation, from the variational point of view via an infinite dimensional GIT problem mirror to Thomas' GIT picture for special Lagrangians. This gives rise to infinite dimensional manifold $\mathcal{H}$ mirror to Solomon's space of positive Lagrangians. In the hypercritical phase case we prove the existence of smooth approximate geodesics, and weak geodesics with $C^{1,α}$ regularity. This is accomplished by proving sharp with respect to scale estimates for the Lagrangian phase operator on col…
This is a write-up of the author's talk in the conference "Algebraic Geometry in East Asia 2016" held at the University of Tokyo in January 2016. We give a survey on a series of papers of the author and his collaborators Daniel Pomerleano and Kazushi Ueda where we show how Strominger-Yau-Zaslow (SYZ) transforms can be applied to understand the geometry of Kontsevich's homological mirror symmetry (HMS) conjecture for certain local Calabi-Yau manifolds.
We develop a unifed theory to study geometry of manifolds with different holonomy groups. They are classified by (1) real, complex, quaternion or octonion number they are defined over and (2) being special or not. Specialty is an orientation with respect to the corresponding normed algebra A. For example, special Riemannian A-manifolds are oriented Riemannian, Calabi-Yau, Hyperkahler and G_2-manifolds respectively. For vector bundles over such manifolds, we introduce (special) A-connections. They include holomorphic, Hermitian Yang-Mills, Anti-Self-Dual and Donaldson-Thomas connections. Simila…
The Chinese University of Hong Kong（香港中文大学）；The Institute of Mathematical Sciences and Department of Mathematics, The Chinese University of Hong Kong（香港中文大学数学研究所及数学系）；Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology（南方科技大学量子科学工程研究院）；Department of Mathematics, National Cheng Kung University（国立成功大学数学系）；Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo Institutes for Advanced Study, The University of Tokyo（东京大学宇宙线研究所 Kavli 宇宙物理数学研究所（WPI））
We study the moduli spaces of flat SL(r)- and PGL(r)-connections, or equivalently, Higgs bundles, on an algebraic curve. These spaces are noncompact Calabi-Yau orbifolds; we show that they can be regarded as mirror partners in two different senses. First, they satisfy the requirements laid down by Strominger-Yau-Zaslow (SYZ), in a suitably general sense involving a B-field or flat unitary gerbe. To show this, we use their hyperkahler structures and Hitchin's integrable systems. Second, their Hodge numbers, again in a suitably general sense, are equal. These spaces provide significant evidence …
