arXiv research

The paper calculates dimensions of higher Landau levels on compact manifolds.

The paper constructs quantizations for symplectic manifolds with specific Laplacian properties.

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 …

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

Ginzburg-Landau fields are the solutions of the Ginzburg-Landau equations which depend on two positive parameters, $α$ and $β$. We give conditions on $α$ and $β$ for the existence of irreducible solutions of these equations. Our results hold for arbitrary compact, oriented, Riemannian 2-manifolds (for example, bounded …

The paper studies eigenvalues in gaps of the essential spectrum of a Bochner-Schrödinger operator.

New solutions found to Ginzburg-Landau equations on surfaces.

New solutions found for Ginzburg-Landau equations on complex manifolds.

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 shows only rotations can be approximated by Ginzburg-Landau critical points.

In this paper, we bring in General Landau-Lifshitz-Bloch equation and prove that it admits a local strong solution.

By using the geometric concept of PDEs with prescribed curvature representations, we show that the 1+2 dimensional Landau-Lifshitz equation is gauge equivalent to a 1+2 dimensional nonlinear Schrödinger-type system. From the nonlinear Schrödinger-type system, we construct blowing up $H^3({\bf R}^2)$-solutions to the 1+…

We establish a glueing theorem for the Ginzburg-Landau equations in dimension $n > 2$. To this end, we consider a nondegenerate minimal submanifold of codimension 2, and construct a one-parameter family of solutions to the Ginzburg-Landau equations such that the energy density concentrates near this submanifold. The pr…

Study compares thimbles to Morse theory on Lie theory models.

Researchers find entire solutions to magnetic Ginzburg-Landau equations in 4D.

Study local minimizers of Ginzburg-Landau functionals in high dimensions, showing energy measures converge to rectifiable measures.

In this paper, we consider the smooth map from a Riemannian manifold to the standard Euclidean space and the p-Ginzburg-Landau energy. Under suitable curvature conditions on the domain manifold, some Liouville type theorems are established by assuming either growth conditions of the p-Ginzburg-Landau energy or an asymp…

The Landau-Lifshitz equation is derived as the reduction of a geodesic flow on the group of maps into the rotation group. Passing the symmetries of spatial isotropy to the reduced space is an example of semidirect product reduction by stages.

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…

We analyze 2-dimensional Ginzburg-Landau vortices at critical coupling, and establish asymptotic formulas for the tangent vectors of the vortex moduli space using theorems of Taubes and Bradlow. We then compute the corresponding Berry curvature and holonomy in the large volume limit.

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

In this paper, a type of integrable evolution equation--the generalized Landau-Lifshitz equation into $S^n$ is considered. We deal with this equation from a geometric point of view by rewriting it in a geometric form. Through the geometric energy method, we show the global well-posedness of the corresponding Cauchy pro…

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…

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

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…

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…

In this short note, we show how the parallel adaptive Wang-Landau (PAWL) algorithm of Bornn et al. (2013) can be used to automate and improve simulated tempering algorithms. While Wang-Landau and other stochastic approximation methods have frequently been applied within the simulated tempering framework, this note demo…

Uniform small energy regularity for fractional geometric problems proved.

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

Constructs surfaces with conical singularities using variational methods.

We use min-max techniques to produce nontrivial solutions $u_ε:M\to \mathbb{R}^2$ of the Ginzburg-Landau equation $Δu_ε+\frac{1}{ε^2}(1-|u_ε|^2)u_ε=0$ on a given compact Riemannian manifold, whose energy grows like $|\logε|$ as $ε\to 0$. When the degree one cohomology $H^1_{dR}(M)=0$, we show that the energy of these s…

Study of critical points in Ginzburg-Landau approximation with stability results.

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…

Maps Kähler cones to moduli spaces of stable manifolds.

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

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…

In this paper, we consider a parabolic system from a bounded domain in a Euclidean space or a closed Riemannian manifold into a unit sphere in a compact Lie algebra $\mathfrak{g}$, which can be viewed as the extension of Landau-Lifshtiz (LL) equation and was proposed by V. Arnold. We follow the ideas taken from the wor…

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

Minimal submanifolds are found as energy concentration sets in variational problems.

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

Study bounds the index of minimal submanifolds using energy measures and Yang-Mills-Higgs equations.

In this paper, we illustrated one scenario to modify the Ivanenko-Landau-Kähler equation. Since Ivanenko and Landau introduced the equation in 1928, the equation has been regarded as having a certain role as a fermion in particular in the discrete Lattice. Also, although it correctly is formulated as an alternative cla…

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

The paper studies momentum-based minimization for Ginzburg-Landau on Euclidean spaces and graphs.

For Ginzburg-Landau vortices, energy quantization holds only when density is less than 2.

A global weak solution of the biharmonic wave map equation in the energy space for spherical targets is constructed. The equation is reformulated as a conservation law and solved by a suitable Ginzburg-Landau type approximation.

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