arXiv research

Study compares thimbles to Morse theory on Lie theory models.

Study Hodge theory for Landau-Ginzburg models on Calabi-Yau manifolds.

A model for B-type Landau-Ginzburg theories rigorously defined.

This study compares Landau-Ginzburg theory to Khovanov homology.

New models for B-type topological theories using complex functions.

Study simplifies Landau-Ginzburg models on Stein manifolds.

Explains complex adjoint orbits in Lie theory and geometry.

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

Maps Kähler cones to moduli spaces of stable manifolds.

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

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

Study of B-type LG models on open Riemann surfaces.

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

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

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

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

Study cohomologies of complex bundles and sections.

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

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

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…

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 …

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

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

New groups and manifolds from Weyl groups, with Frobenius structures.

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

Defines monopole Floer homology for 3-manifolds with toroidal boundaries.

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…

It is known that knot homologies admit a physical description as spaces of open BPS states. We study operators and algebras acting on these spaces. This leads to a very rich story, which involves wall crossing phenomena, algebras of closed BPS states acting on spaces of open BPS states, and deformations of Landau-Ginzb…

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…

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…

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…

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…

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 …

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…

Study special complexified Kähler forms in mirror symmetry.

Novel gauge-theoretic Floer homologies defined from 5d N=2 theory, linking 4, 3, and 2-manifolds.

We show how to construct measures on Banach manifolds associated to supersymmetric quantum field theories. These measures are mathematically well-defined objects inspired by the formal path integrals appearing in the physics literature on quantum field theory. We give three concrete examples of our construction. The fi…

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…

Lectures detail field theory dynamics and exact WKB analysis.

The goal of this article is twofold. First, we find a natural home for the double affine Hecke algebras (DAHA) in the physics of BPS states. Second, we introduce new invariants of torus knots and links called "hyperpolynomials" that address the "problem of negative coefficients" often encountered in DAHA-based approach…

For each sphere with three orbifold points, we construct an algorithm to compute the open Gromov-Witten potential, which serves as the quantum-corrected Landau-Ginzburg mirror and is an infinite series in general. This gives the first class of general-type geometries whose full potentials can be computed. As a conseque…

Novel gauge-theoretic Floer homologies defined for 7, 6, and 5-manifolds.

Constructs exotic Lagrangian tori in Grassmannians using cluster algebra.

This is an introduction to some of the analytic (or integrable systems) aspects of quantum cohomology which have attracted much attention during the last few years. The small quantum cohomology algebra, regarded as an example of a Frobenius manifold, is described in the original naive manner, without going into the tec…

We conjecture the existence of four independent gradings in the colored HOMFLY homology. We describe these gradings explicitly for the rectangular colored homology of torus knots and make qualitative predictions of various interesting structures and symmetries in the colored homology of general knots. We also give a si…

This paper is focused on geometric aspects of two particular types of finite-variable reductions in the dispersionless Toda hierarchy. The reductions are formulated in terms of "Landau-Ginzburg potentials" that play the role of reduced Lax functions. One of them is a generalization of Dubrovin and Zhang's trigonometric…

Study Witten deformation on noncompact manifolds with bounded geometry.

The paper proves a precise SYZ conjecture for toric Calabi-Yau manifolds with singular fibers.