We introduce the notion of symplectic flatness for connections and fiber bundles over symplectic manifolds. Given an $A_\infty$-algebra, we present a flatness condition that enables the twisting of the differential complex associated with the $A_\infty$-algebra. The symplectic flatness condition arises from twisting the $A_\infty$-algebra of differential forms constructed by Tsai, Tseng and Yau. When the symplectic manifold is equipped with a compatible metric, the symplectic flat connections represent a special subclass of Yang-Mills connections. We further study the cohomologies of the twisted differential complex and give a simple vanishing theorem for them.

This paper studies a notion of enumerative invariants for stable A-branes, and discusses its relation to invariants defined by spectral and exponential networks. A natural definition of stable A-branes and their counts is provided by the string theoretic origin of the topological A-model. This is the Witten index of the supersymmetric quantum mechanics of a single D3 brane supported on a special Lagrangian in a Calabi-Yau threefold. Geometrically, this is closely related to the Euler characteristic of the A-brane moduli space. Using the natural torus action on this moduli space, we reduce the computation of its Euler characteristic to a count of fixed points via equivariant localization. Studying the A-branes that correspond to fixed points, we make contact with definitions of spectral and exponential networks. We find agreement between the counts defined via the Witten index, and the BPS invariants defined by networks. By extension, our definition also matches with Donaldson-Thomas invariants of B-branes related by homological mirror symmetry.

No abstract uploaded!

We prove a version of equivariant split generation of Fukaya category when a symplectic manifold admits a free action of a finite group <i>G</i>. Combining this with some generalizations of Seidel's algebraic frameworks from [35], we obtain new cases of homological mirror symmetry for some symplectic tori with non-split symplectic forms, which we call <i>special isogenous tori</i>. This extends the work of AbouzaidSmith [2]. We also show that derived Fukaya categories are complete invariants of special isogenous tori.

Mirror symmetry conjecture identifies the complex geometry of a Calabi $ Yau manifold with the symplectic geometry of its mirror Calabi $ Yau man $ ifold. Using the SYZ mirror transform, we argue that (i) the mirror of an elliptic Calabi $ Yau manifold admits a twin Lagrangian fibration structure and (ii) the mirror of the Fourier $ Mukai transform for dual elliptic fibra $ tions is a symplectic Fourier $ Mukai transform for dual twin Lagrangian fibrations, which is essentially an identity transformation in this case.

In this paper we study the birational geometry of HyperKaehler manifolds by combining the method of minimal model program and the traditional approach of symplectic geometry.

We study the CalabiYau equation on symplectic manifolds. We show that Donaldson's conjecture on estimates for this equation in terms of a taming symplectic form can be reduced to an integral estimate of a scalar potential function. Under a positive curvature condition, we show that the conjecture holds.

We give ChernWeil definitions of the Maslov indices of bundle pairs over a Riemann surface \Sigma with boundary, which consists of symplectic vector bundle on \Sigma and a Lagrangian subbundle on \Sigma as well as its generalization for transversely intersecting Lagrangian boundary conditions. We discuss their properties and relations to the known topological definitions. As a main application, we extend Maslov index to the case with orbifold interior singularities, via curvature integral, and find also an analogous topological definition in these cases.

We give Chern-Weil definitions of the Maslov indices of bundle pairs over a Riemann surface with boundary, which consists of symplectic vector bundle on and a Lagrangian subbundle on\partial as well as its generalization for transversely intersecting Lagrangian boundary conditions. We discuss their properties and relations to the known topological definitions. As a main application, we extend Maslov index to the case with orbifold interior singularites, via curvature integral, and find also an analogous topological definition in these cases.

Lagrangian Floer homology in a general case has been constructed by Fukaya, Oh, Ohta and Ono, where they construct an A-algebra or an A-bimodule from Lagrangian submanifolds. They developed obstruction and deformation theories of the Lagrangian Floer homology theory. But for obstructed Lagrangian submanifolds, the standard Lagrangian Floer homology cannot be defined. We explore several well-known homology theories on these A-objects, which are Hochschild and cyclic homology for an A-objects and ChevalleyEilenberg or cyclic ChevalleyEilenberg homology for their underlying L-objects. We show that these homology theories are well-defined and invariant even in the obstructed cases. Due to the existence of m 0, the standard homological algebra does not work and we develop analogous homological algebra over Novikov fields. We provide computations of these homology