Symplectic field theory: an overview

Abstract

We summarize some of the main ideas and results around symplectic field theory, from its early inception up to recent and ongoing developments.

Figures (4)

• Figure 3.1: An asymptotically cylindrical pseudoholomorphic pair of pants and a stable pseudoholomorphic building to which it could a priori degenerate under the SFT compactness theorem.
• Figure 5.1: Two possible breakings of a cylinder in a symplectization into a two level building. The latter breaking prevents the naive differential which counts cylinders from squaring to zero, and this motivates the definition of the contact homology algebra.
• Figure 5.2: A schematic picture of the curves involved in the contact homology algebra, rational symplectic field theory, and full symplectic field theory.
• Figure 5.3: A schematic picture of the anchored curves involved after linearized the invariants of the contact manifold $Y$ induced by its Liouville filling $X$. Note that the invariants $C_{{\operatorname{CH}}_{{\operatorname{lin}}}}(X)$ and $C_{\mathcal{B}{{\operatorname{CH}}_{{\operatorname{lin}}}}}(X)$ have no natural unlinearized counterparts, whereas the remaining three invariants only differ by their unlinearized versions by a change of coordinates.

Theorems & Definitions (15)

• Remark 3.1
• Remark 3.2
• Remark 3.3
• Example 4.1
• Remark 5.2
• Remark 5.3
• Remark 5.4
• Theorem 6.1: viterbo1994properties
• proof : Proof of Theorem \ref{['thm:hyp']}
• Theorem 6.2: CM2