arXiv research

Study compares thimbles to Morse theory on Lie theory models.

New construction of Fukaya-Seidel categories using complex gradient flow equation.

Given a smooth projective toric variety X, we construct an A-infinity category of Lagrangians with boundary on a level set of the Landau-Ginzburg mirror of X. We prove that this category is quasi-equivalent to the DG category of line bundles on X. This establishes part of the Homological Mirror Conjecture for toric var…

Study Witten deformation on noncompact manifolds with bounded geometry.

We summarize the main results of our investigation of B-type topological Landau-Ginzburg models whose target is an arbitrary open Riemann surface. Such a Riemann surface need not be affine algebraic and in particular it may have infinite genus or an infinite number of Freudenthal ends. Under mild conditions on the Land…

The paper proves an isomorphism between $tt^*$ structures of Landau-Ginzburg and Calabi-Yau models.

Let X be a non-compact Calabi-Yau manifold and f be a holomorphic function on X with compact critical locus. We introduce the notion of f-twisted Sobolev spaces for the pair (X,f) and prove the corresponding Hodge-to-de Rham degeneration property via L2-Hodge theoretical methods when f satisfies an asymptotic condition…

Generalizes Landau-Ginzburg mirrors for Frobenius manifolds in Dynkin type A.

We propose a family of differential models for B-type open-closed topological Landau-Ginzburg theories defined by a pair $(X,W)$, where $X$ is any non-compact Calabi-Yau manifold and $W$ is any holomorphic complex-valued function defined on $X$ whose critical set is compact. The models are constructed at cochain level …

We study N=2 nonlinear two dimensional sigma models with boundaries and their massive generalizations (the Landau-Ginzburg models). These models are defined over either Kahler or bihermitian target space manifolds. We determine the most general local N=2 superconformal boundary conditions (D-branes) for these sigma mod…

Establishes correspondence between Calabi-Yau and Landau-Ginzburg structures.

Maps Kähler cones to moduli spaces of stable manifolds.

The paper constructs monopole Floer homology for specific 3-manifolds and surfaces.

We describe a mathematically rigorous differential model for B-type open-closed topological Landau-Ginzburg theories defined by a pair $(X,W)$, where $X$ is a non-compact Kählerian manifold with holomorphically trivial canonical line bundle and $W$ is a complex-valued holomorphic function defined on $X$ and whose criti…

Study connects mirror symmetry invariants to K-stability for toric manifolds.

New insights into mirror symmetry via Monge-Ampère domains and pre-Frobenius manifolds.

The study connects K-stability and large complex structure limits in mirror symmetry.

In this note we make an attempt to compare a cohomological theory of Hilbert spaces of ground states in the ${\cal N}=(2,2)$ 2d Landau-Ginzburg theory in models describing link embeddings in ${\mathbb{R}}^3$ to Khovanov and Khovanov-Rozansky homologies. To confirm the equivalence we exploit the invariance of Hilbert sp…

The paper proves mirror symmetry for del Pezzo surfaces and computes related structures.

Novel $A_{\infty}$-categories derived from gauge theories for manifold homologies.

The abstract proves a conjecture about geometric structures in Calabi-Yau orbifolds.

We consider the bulk algebra and topological D-brane category arising from the differential model of the open-closed B-type topological Landau-Ginzburg theory defined by a pair $(X,W)$, where $X$ is a non-compact Calabi-Yau manifold and $W$ has compact critical set. When $X$ is a Stein manifold (but not restricted to b…

Let V be a holomorphic bundle over a complex manifold M, and s be a holomorphic section of V. We study different types of cohomology associated to the Koszul complex induced by s. When M is complete, these cohomologies are isomorphic to each other and have self duality.

We consider Landau-Ginzburg (LG) models with boundary conditions preserving A-type N=2 supersymmetry. We show the equivalence of a linear class of boundary conditions in the LG model to a particular class of boundary states in the corresponding CFT by an explicit computation of the open-string Witten index in the LG mo…

Let $L$ be a closed, orientable, monotone Lagrangian 3-manifold of a symplectic manifold $(M, ω)$, for which there exists a local system such that the corresponding Lagrangian quantum homology vanishes. We show that its cohomology ring satisfies a certain dichotomy, which depends only on the parity of the first Betti n…

We construct a spectral sequence that converges to the cohomology of the chiral de Rham complex over a Calabi-Yau hypersurface and whose first term is a vertex algebra closely related to the Landau-Ginburg orbifold. As an application, we prove an explicit orbifold formula for the elliptic genus of Calabi-Yau hypersurfa…

We describe all smooth solutions of the two-function tt*-Toda equations (a version of the tt* equations, or equations for harmonic maps into SL(n,R)/SO(n)) in terms of (i) asymptotic data, (ii) holomorphic data, and (iii) monodromy data. This allows us to find all solutions with integral Stokes data. These include solu…

To construct mirror symmetric Landau-Ginzburg models, P.Berglund, T.Hübsch and M.Henningson considered a pair $(f,G)$ consisting of an invertible polynomial $f$ and an abelian group $G$ of its symmetries together with a dual pair $(\widetilde{f}, \widetilde{G})$. Here we study the reduced orbifold Euler characteristics…

Study special complexified Kähler forms in mirror symmetry.

5D gauge theories are dual to 3D and 2D models via Floer homologies.

This expository article is an introduction to the adjoint orbits of complex semisimple groups, primarily in the algebro-geometric and Lie-theoretic contexts, and with a pronounced emphasis on the properties of semisimple and nilpotent orbits. It is intended to build a foundation for more specialized settings in which a…

The mirror of a projective toric manifold $X_Σ$ is given by a Landau-Ginzburg model $(Y,W)$. We introduce a class of Lagrangian submanifolds in $(Y,W)$ and show that, under the SYZ mirror transformation, they can be transformed to torus-invariant hermitian metrics on holomorphic line bundles over $X_Σ$. Through this ge…

This is the first of a series of papers to construct the deformation theory of the form Schrödinger equation, which is related to a section-bundle system $(M,g,f)$, where $(M,g)$ is a noncompact complete Kähler manifold with bounded geometry and $f$ is a holomorphic function defined on $M$. This work is also the first …

We present a new class of extended affine Weyl groups $\widetilde{W}^{(k,k+1)}(A_l)$ for $1\leq k <l$ and obtain an analogue of Chevalley-type theorem for their invariants. We further show the existence of Frobenius manifold structures on the orbit spaces of $\widetilde{W}^{(k,k+1)}(A_l)$ and also construct Landau--Gin…

Morse neural networks improve uncertainty quantification and detection.

Research resolves sign conventions in Floer theory for Morse-Bott case.

Counterexample disproves Borde-Sorkin conjecture on causal continuity of Morse spacetimes.

New method extends discrete Morse theory to simplicial complexes.

For the root systems of type $B_l, C_l$ and $D_l$, we generalize the result of \cite{DZ1998} by showing the existence of Frobenius manifold structures on the orbit spaces of the extended affine Weyl groups that correspond to any vertex of the Dynkin diagram instead of a particular choice made in \cite{DZ1998}. It also …

Extends Morse-Novikov Homology to include differential graded coefficients and fibration structures.

Paper constructs continuous families of topological Morse functions.

New proof for discrete Morse theory using combinatorial construction.

Characterizes geodesics on spheres with Morse index bounds and inequalities.

Study continuation maps for Morse fundamental group properties.

For Morse-Smale pairs on a smooth, closed manifold the Morse-Smale-Witten chain complex can be defined. The associated Morse homology is isomorphic to the singular homology of the manifold and yields the classical Morse relations for Morse functions. A similar approach can be used to define homological invariants for i…

The paper introduces Morse theory for Lie groupoids and proves inequalities.

Discrete Morse functions induce shellings with critical tiles corresponding to function's critical faces.

Unified Morse-Bott-Smale chain complex, resolves well-definedness issue.