232 research outputs found
Schr\"odinger equations, deformation theory and -geometry
This is the first of a series of papers to construct the deformation theory
of the form Schr\"odinger equation, which is related to a section-bundle system
, where is a noncompact complete K\"ahler manifold with
bounded geometry and is a holomorphic function defined on .
This work is also the first step attempting to understand the whole
Landau-Ginzburg B-model including the higher genus invariants. Our work is
mainly based on the pioneer work of Cecotti, Cecotti and Vafa
\cite{Ce1,Ce2,CV}.Comment: Deformation theory, Landau-Ginzburg B model, 114 page
Torsion type invariants of singularities
Inspired by the LG/CY correspondence, we study the local index theory of the
Schr\"odinger operator associated to a singularity defined on
by a quasi-homogeneous polynomial . Under some mild assumption on , we
show that the small time heat kernel expansion of the corresponding
Schr\"odinger operator exists and is a series of fractional powers of time .
Then we prove a local index formula which expresses the Milnor number of by
a Gaussian type integral. Furthermore, the heat kernel expansion provides
spectral invariants of . Especially, we define torsion type invariants
associated to a singularity. These spectral invariants provide a new direction
to study the singularity
Novikov-Morse theory for dynamical systems
The present paper contains an interpretation and generalization of Novikov's
theory of Morse type inequalities for 1-forms in terms of Conley's theory for
dynamical systems.Comment: 63 page
A twisted -Neumann problem and Toeplitz -tuples from singularity theory
A twisted -Neumann problem associated to a singularity
is established. By constructing the connection to the
Koszul complex for toeplitz -tuples on Bergman spaces
, we can solve this -Neumann problem. Moreover, we
can compute the cohomology of the holomorphic Koszul complex
explicitlyComment: 20 page
Conley Index Theory and Novikov-Morse Theory
We derive general Novikov-Morse type inequalities in a Conley type framework
for flows carrying cocycles, therefore generalizing our results in [FJ2]
derived for integral cocycle. The condition of carrying a cocycle expresses the
nontriviality of integrals of that cocycle on flow lines. Gradient-like flows
are distinguished from general flows carrying a cocycle by boundedness
conditions on these integrals.Comment: 35pages, 6 figure
The Moduli Space in the Gauged Linear Sigma Model
This is a survey article for the mathematical theory of Witten's Gauged
Linear Sigma Model, as developed recently by the authors. Instead of developing
the theory in the most general setting, in this paper we focus on the
description of the moduli.Comment: Credit for support added. arXiv admin note: substantial text overlap
with arXiv:1506.0210
Martin points on open manifolds of non-positive curvature
The Martin boundary of a Cartan-Hadamard manifold describes a fine geometric
structure at infinity, which is a sub-space of positive harmonic functions. We
describe conditions which ensure that some points of the sphere at infinity
belong to the Martin boundary as well. In the case of the universal cover of a
compact manifold with Ballmann rank one, we show that Martin points are generic
and of full harmonic measure. The result of this paper provides a partial
answer to an open problem of S. T. Yau
A Mathematical Theory of the Gauged Linear Sigma Model
We construct a mathematical theory of Witten's Gauged Linear Sigma Model
(GLSM). Our theory applies to a wide range of examples, including many cases
with non-Abelian gauge group. Both the Gromov-Witten theory of a Calabi-Yau
complete intersection X and the Landau-Ginzburg dual (FJRW-theory) of X can be
expressed as gauged linear sigma models. Furthermore, the
Landau-Ginzburg/Calabi-Yau correspondence can be interpreted as a variation of
the moment map or a deformation of GIT in the GLSM. This paper focuses
primarily on the algebraic theory, while a companion article will treat the
analytic theory.Comment: Minor correction
Harmonic Hopf Constructions Between Spheres II
We prove that a necessary condition for the existence of the remaining
problem in the harmonic Hopf construction is also sufficient. We also give some
topological applications based on our result.Comment: 9 page
The Witten equation and its virtual fundamental cycle
We study a system of nonlinear elliptic PDEs associated with a
quasi-homogeneous polynomial. These equations were proposed by Witten as the
replacement for the Cauchy-Riemann equation in the singularity
(Landau-Ginzburg) setting. We introduce a perturbation to the equation and
construct a virtual cycle for the moduli space of its solutions. Then, we study
the wall-crossing of the deformation of the virtual cycle under perturbation
and match it to classical Picard-Lefschetz theory. An extended virtual cycle is
obtained for the original equation. Finally, we prove that the extended virtual
cycle satisfies a set of axioms similar to those of Gromov-Witten theory and
r-spin theory.Comment: Major revision. Additional axioms proved and additional details
provided over previous versio
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