On the perturbation lemma, and deformations

伪 Maurer-Cartan 扰动代数和伪扰动引理

Pseudo Maurer-Cartan perturbation algebra and pseudo perturbation lemma

本文提出伪 Maurer-Cartan 扰动代数及其结构结果，推导出伪扰动引理并证明其等价于普通扰动引理。

Ideal Perturbation Lemma

We explain the essence of perturbation problems. The key to understanding is the structure of chain homotopy equivalence -- the standard one must be replaced by a finer notion which we call a strong chain homotopy equivalence. We prove an Ideal Perturbation Lemma and show how both new and classical results follow from this ideal statement.

量子 $L_\infty$ 代数与同伦扰动引理

Quantum $L_\infty$ Algebras and the Homological Perturbation Lemma

本文通过同伦扰动引理构造量子 $L_\infty$ 代数的极小模型，证明其由 Feynman 图展开给出并计算有效作用量，同时建立了原代数与有效代数间的同伦。

Algebraic Morse theory and homological perturbation theory

We show that the main result of algebraic Morse theory can be obtained as a consequence of the perturbation lemma of Brown and Gugenheim.

The formal Kuranishi parametrization via the universal homological perturbation theory solution of the deformation equation

Using homological perturbation theory, we develop a formal version of the miniversal deformation associated with a deformation problem controlled by a differential graded Lie algebra over a field of characteristic zero. Our approach includes a formal version of the Kuranishi method in the theory of deformations of complex manifolds.

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.