arXiv research

This paper deforms complex tori and their mirrors using gerbes.

Extends noncommutative deformations of holomorphic line bundles on complex tori and their mirror partners.

The paper explores a B-field transform of complex structures on complex tori.

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 s…

Mathematical framework for brane quantization using SYZ mirror symmetry.

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…

Identifies special Lagrangian submanifolds in non-Kähler Calabi-Yau manifolds.

We propose a new approach to the Mirror Symmetry Conjecture in a form suitable to possibly non-Kähler compact complex manifolds whose canonical bundle is trivial. We apply our methods by proving that the Iwasawa manifold $X$, a well-known non-Kähler compact complex manifold of dimension $3$, is its own mirror dual to t…

Study properties of holomorphic $p$-contact manifolds, including non-Kähler hyperbolicity and deformations.

New MD algorithms using Tempesta logarithms for machine learning.

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 o…

Via considerations of symplectic reduction, monodromy, mirror symmetry and Chern-Simons functionals, a conjecture is proposed on the existence of special Lagrangians in the hamiltonian deformation class of a given Lagrangian submanifold of a Calabi-Yau manifold. It involves a stability condition for graded Lagrangians,…

Defines a volume functional for Hermitian connections on manifolds, proving its properties and connections.

The paper studies deformations of Hermitian Yang-Mills and Donaldson-Thomas connections on $G_2$-manifolds.

The paper examines properties of deformed Donaldson-Thomas connections on G2-manifolds.

In this article we discuss the geometry of moduli spaces of (1) flat bundles over special Lagrangian submanifolds and (2) deformed Hermitian-Yang-Mills bundles over complex submanifolds in Calabi-Yau manifolds. These moduli spaces reflect the geometry of the Calabi-Yau itself like a mirror. Strominger, Yau and Zaslow c…

Machine learning models predict DM performance for AO systems.

Paper proves solvability condition for complex equation on special submanifolds.

We provide an introduction to the mathematics and physics of the deformed Hermitian-Yang-Mills equation, a fully nonlinear geometric PDE on Kahler manifolds which plays an important role in mirror symmetry. We discuss the physical origin of the equation, and some recent progress towards its solution. In dimension 3 we …

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 Solom…

We exhibit a transformation taking special Lagrangian submanifolds of a Calabi-Yau together with local systems to vector bundles over the mirror manifold with connections obeying deformed Hermitian-Yang-Mills equations. That is, the transformation relates supersymmetric A- and B-cycles. In this paper, we assume that th…

We study the connection between topological strings and contact homology recently proposed in the context of knot invariants. In particular, we establish the proposed relation between the Gromov-Witten disk amplitudes of a Lagrangian associated to a knot and augmentations of its contact homology algebra. This also impl…

We present the extended Kuranishi space for Kodaira surface as a non-trivial example to Kontsevich and Barannikov's extended deformation theory. We provide a non-trivial example of Hertling-Manin's weak Frobenius manifold. In addition, we find that Kodaira surface is its own mirror image. Our computation is done in the…

We study generalized complex structures on K3 surfaces, in the sense of Hitchin. For each real parameter t between one and infinity we exhibit two families of generalized K3 surfaces, (M,cal{I}_{zeta}) and (M,cal{J}_{zeta}), parametrized by zeta in CP^1, which are Mukai dual for zeta=0 and infinity, amd mirror partners…

We construct from a real affine manifold with singularities (a tropical manifold) a degeneration of Calabi-Yau manifolds. This solves a fundamental problem in mirror symmetry. Furthermore, a striking feature of our approach is that it yields an explicit and canonical order-by-order description of the degeneration via f…

Given a six-dimensional symplectic manifold $(M, B)$, a nondegenerate, co-closed four-form $C$ introduces a dual symplectic structure $\widetilde{B} = *C$ independent of $B$ via the Hodge duality $*$. We show that the doubling of symplectic structures due to the Hodge duality results in two independent classes of nonc…

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 a…

The paper explores non-Kähler SYZ mirrors for solvmanifolds, proving cohomological properties and constructing new mirror pairs.

From string theory, the notion of deformed Hermitian Yang-Mills connections has been introduced by Mariño, Minasian, Moore and Strominger. After that, Leung, Yau and Zaslow proved that it naturally appears as mirror objects of special Lagrangian submanifolds via Fourier-Mukai transform between dual torus fibrations. In…

In this paper we investigate the properties of series of vacua in the string theory landscape. In particular, we study minima to the flux potential in type IIB compactifications on the mirror quintic. Using geometric transitions, we embed its one dimensional complex structure moduli space in that of another Calabi-Yau …

Wedge product on deRham complex of a Riemannian manifold $M$ can be pulled back to $H^*(M)$ via explicit homotopy, constructed using Green's operator, to give higher product structures. We prove Fukaya's conjecture which suggests that Witten deformation of these higher product structures have semiclassical limits as op…

Study special Lagrangian sections in Calabi-Yau threefolds, showing stability conditions imply isomorphism to special Lagrangians.

Paper confirms conjecture for projective manifolds in supercritical phase.

Study adiabatic limits of calibrated submanifolds in Riemannian geometry.

According to our previous results, the conjugacy class of the involution induced by the complex conjugation in the homology of a real non-singular cubic fourfold determines the fourfold up to projective equivalence and deformation. Here, we show how to eliminate the projective equivalence and to obtain a pure deformati…

The mean curvature flow is an evolution process under which a submanifold deforms in the direction of its mean curvature vector. The hypersurface case has been much studied since the eighties. Recently, several theorems on regularity, global existence and convergence of the flow in various ambient spaces and codimensio…

Three-dimensional N=2 superconformal field theories are constructed by compactifying M5-branes on three-manifolds. In the infrared the branes recombine, and the physics is captured by a single M5-brane on a branched cover of the original ultraviolet geometry. The branch locus is a tangle, a one-dimensional knotted subm…

Geometrically, Legendrian surfaces related by surgery have related skein-valued cluster spaces.

This article studies the geometry of moduli spaces of G2-manifolds, associative cycles, coassociative cycles and deformed Donaldson-Thomas bundles. We introduce natural symmetric cubic tensors and differential forms on these moduli spaces. They correspond to Yukawa couplings and correlation functions in M-theory. We ex…

This paper considers the natural geometric structure on the moduli space of deformations of a compact special Lagrangian submanifold $L^n$ of a Calabi-Yau manifold. From the work of McLean this is a smooth manifold with a natural $L^2$ metric. It is shown that the metric is induced from a local Lagrangian immersion int…

We report some minimal surfaces that can be seen as copies of a triply periodic minimal surface (TPMS) related by reflections in parallel mirrors. We call them minimal twin surfaces for the resemblance with twin crystal. Brakke's Surface Evolver is employed to construct twinnings of various classical TPMS, including Sc…

The geometric approach to diffeomorphic image registration known as "large deformation by diffeomorphic metric mapping" (LDDMM) is based on a left action of diffeomorphisms on images, and a right-invariant metric on a diffeomorphism group, usually defined using a reproducing kernel. We explore the use of left-invariant…

Proves stability condition for Lagrangian sections in toric weak Fano manifolds.

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 re…

This is the first of two papers devoted to showing how the rich algebraic formalism of Eliashberg-Givental-Hofer's symplectic field theory (SFT) can be used to define higher algebraic structures on the symplectic cohomology of open symplectic manifolds. Using the SFT of Hamiltonian mapping tori we show how to define a …

In this article we study Weinstein structures endowed with a Lefschetz fibration in terms of the Legendrian front projection. First we provide a systematic recipe for translating from a Weinstein Lefschetz bifibration to a Legendrian handlebody. Then we present several applications of this technique to symplectic topol…

Study of deformed Hermitian Yang-Mills equations with variable Kähler metrics.

Study of geometric properties of almost calibrated forms on Kähler manifolds.