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Jian Zhou

Publications and source records attributed to Jian Zhou.

390 records · Page 22Linked to original sources
arXiv1999

Homological perturbation theory and mirror symmetry

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 such structures on the cohomology. We formulate the $A_{\infty}$ algebraic mirror symmetry as the identification of the $A_{\infty}$ algebras together with their canonical deformations constructed this way on different manifolds.

math.DG↗
arXiv1999

On quasi-isomorphic DGBV algebras

One of the methods to obtain Frobenius manifold structures is via DGBV (differential Gerstenhaber-Batalin-Vilkovisky) algebra construction. An important problem is how to identify Frobenius manifold structures constructed from two different DGBV algebras. For DGBV algebras with suitable conditions, we show the functorial property of a construction of deformations of the multiplicative structures of their cohomology. In particular, we show that quasi-isomorphic DGBV algebras yield identifiable Frobenius manifold structures.

math.DG↗
arXiv1999

Formal Frobenius manifold structure on equivariant cohomology

For a closed Kähler manifold with a Hamiltonian action of a connected compact Lie group by holomorphic isometries, we construct a formal Frobenius manifold structure on the equivariant cohomology by exploiting a natural DGBV algebra structure on the Cartan model.

math.DG↗
arXiv1998

Degenerate Chern-Weil Theory and Equivariant Cohomology

We develop a Chern-Weil theory for compact Lie group action whose generic stabilizers are finite in the framework of equivariant cohomology. This provides a method of changing an equivariant closed form within its cohomological class to a form more suitable to yield localization results. This work is motivated by our work on reproving wall crossing formulas in Seiberg-Witten theory, where the Lie group is the circle. As applications, we derive two localization formulas of Kalkman type for G = SU(2) or SO(3)-actions on compact manifolds with boundary. One of the formulas is then used to yield a very simple proof of a localization formula due to Jeffrey-Kirwan in the case of G = SU(2) or SO(3).

math.DG↗
arXiv1998

On Quantum de Rham Cohomology

We define quantum exterior product wedge_h and quantum exterior differential d_h on Poisson manifolds, of which symplectic manifolds are an important class of examples. Quantum de Rham cohomology is defined as the cohomology of d_h. We also define quantum Dolbeault cohomology. Quantum hard Lefschetz theorem is proved. We also define a version of quantum integral, and prove the quantum Stokes theorem. By the trick of replacing d by d_h and wedge by wedge_h in the usual definitions, we define many quantum analogues of important objects in differential geometry, e.g. quantum curvature. The quantum characteristic classes are then studied along the lines of classical Chern-Weil theory, i.e., they can be represented by expressions of quantum curvature. Quantum equivariant de Rham cohomology is defined in a similar fashion. Calculations are done for some examples, which show that quantum de Rham cohomology is different from the quantum cohomology defined using pseudo-holomorphic curves.

math.DG↗
arXiv1998

Identification of Two Frobenius Manifolds In Mirror Symmetry

We identify two Frobenius manifolds obtained from two different differential Gerstenhaber-Batalin-Vilkovisky algebras on a compact Kaehler manifold. One is constructed on the Dolbeault cohomology, and the other on the de Rham cohomology. Our result can be considered as a generalization of the identification of the Dolbeault cohomology ring with the complexified de Rham cohomology ring on a Kaehler manifold.

math.DG↗
arXiv1998

On Quantum de Rham Cohomology Theory

We define quantum exterior product wedge_h and quantum exterior differential d_h on Poisson manifolds (of which symplectic manifolds are an important class of examples). Quantum de Rham cohomology, which is a deformation quantization of de Rham cohomology, is defined as the cohomology of d_h. We also define quantum Dolbeault cohomology. A version of quantum integral on symplectic manifolds is considered and the correspoding quantum Stokes theorem is proved. We also derive quantum hard Lefschetz theorem. By replacing d by d_h and wedge by wedge_h in the usual definitions, we define many quantum analogues of important objects in differential geometry, e.g. quantum curvature. The quantum characteristic classes are then studied along the lines of classical Chern-Weil theory. Quantum equivariant de Rham cohomology is defined in the similar fashion.

math.DG↗
arXiv1996

A Frenkel-Kontorova Model with Two Spring Constants: New Phases and Phase Transitions

A discrete Frenkel-Kontorova model with two alternate spring constants $k_{1}$ and $k_{2}$ is studied. It is found that this model has many surprising behaviours. The continuum version of this model is different from the sine-Gordon equation, which is the continuum version of the standard FK model. More interestingly, it has an unpinned commensurate phase which is translationally invariant. A phase transition takes place on a lattice point in the parameter space. We have also predicted another interesting phase transition in which an unlocked commensurate phase breaking translational invariance will take place as the strength of the substrate potential is increased.

cond-mat↗
arXiv1996

Effects of a Magnetic Field on a Modulated Phase

The effects of a magnetic field on a modulated phase is studied. A modulated phase is found to have two critical fields $H_{1}$ and $H_{2}$. For a large enough magnetic field. $H_1$ can be approximated by a linear law. As a result, the minimum magnetic field needed to destroy a modulated phase is a constant. The minimum magnetic field also greatly depends on the order of a commensurate. A very high order commensurate phase and an incommensurate phase are impossible to survive a magnetic field. The behaviour of a magnetoelastic chain in a magnetic field is can be described by a harmless Devil's staircase. The inverse temperature is found to play a role similar to that of a special magnetic field. The deeper physics underlying these new phenomena is the breaking of the left-right symmetry of a phase diagram. As a result a controllable path to a modulated phase is found.

cond-mat↗