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Homological algebra and moduli spaces in topological field theories

Kenji Fukaya

Abstract

This is a survey of various types of Floer theories (both in symplectic geometry and gauge theory) and relations among them.

Homological algebra and moduli spaces in topological field theories

Abstract

This is a survey of various types of Floer theories (both in symplectic geometry and gauge theory) and relations among them.
Paper Structure (13 sections, 15 theorems, 126 equations, 12 figures)

This paper contains 13 sections, 15 theorems, 126 equations, 12 figures.

Table of Contents

  1. Numerical invariants: Donaldson and Gromov.
  2. Floer homology.
  3. Topological field theory.
  4. Lagrangian Floer theory.
  5. $A_{\infty}$ algebras and $A_{\infty}$ categories.
  6. Homological Mirror symmetry.
  7. Elliptic curves and tori
  8. Generator of the Category 1: Decomposition of the monodromy.
  9. Generator of the Category 2: Open-closed map.
  10. Family Floer homology.
  11. Matrix factorizations and weak bounding cochains.
  12. Seidel's directed $A_{\infty}$ category.
  13. Lagrangian correspondence and Gauge theory.

Key Result

Theorem 1.1

(Gr) If there exists a map $u$ from $\overset{\circ}{D^4}(r)$ (the open $r$-ball in $\mathbb C{$ C$}^2$) to $D^2(1) \times \mathbb C{$ C$}$ such that $u^*\omega = \omega$ (where $\omega$ is the standard symplectic form) then $r<1$.

Figures (12)

  • Figure 1: Stasheff 2-gon
  • Figure 2: Universal cover of an elliptic curve and its three Lagrangians
  • Figure 3: Dehn twist
  • Figure 4: A pseudo-holomorphic map to $-X \times X$.
  • Figure 5: A pseudo-holomorphic map to $X$.
  • ...and 7 more figures

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 2.1
  • Corollary 2.2
  • Theorem 2.3
  • Proposition 3.1
  • Conjecture 3.2
  • Theorem 3.3
  • Remark 3.4
  • Theorem 3.5
  • Theorem 4.1
  • ...and 24 more