Peng Zhou

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cheatsheets:symplectic-contact-manifold

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cheatsheets:symplectic-contact-manifold [2022/12/12 04:43] – pzhoucheatsheets:symplectic-contact-manifold [2023/06/25 15:53] (current) – external edit 127.0.0.1
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 ====== Symplectic and Contact manifolds ====== ====== Symplectic and Contact manifolds ======
  
 +Let's follow the reference of [[https://arxiv.org/pdf/0803.2455.pdf | EES]] (Ekholm-Etnyre-Sabloff). The standard contact form on $\R^3$ is $dz - ydx$.
  
 ===== Example of $\C$ ===== ===== Example of $\C$ =====
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 Given an immersed Lagrangian $L$ in $W$ (no wrapping turned on) equipped with a potential, we can do the lift $L_+$ to $M$. Then, we have a free algebra generated by the double point in $L$, or Reeb chord ending in $L_+$. **It is hugely non-commutative**. The only thing makes geometric sense is the differential, whose input is a positive chord, and outputs are many negative chord.  Given an immersed Lagrangian $L$ in $W$ (no wrapping turned on) equipped with a potential, we can do the lift $L_+$ to $M$. Then, we have a free algebra generated by the double point in $L$, or Reeb chord ending in $L_+$. **It is hugely non-commutative**. The only thing makes geometric sense is the differential, whose input is a positive chord, and outputs are many negative chord. 
  
-So [[https://hal.archives-ouvertes.fr/hal-01873728/file/generation-v2.pdf | CRGG ]] (Definition 3.4)'s sign convention for the punctures is correct after all. +So [[https://hal.archives-ouvertes.fr/hal-01873728/file/generation-v2.pdf | CDGG ]] (Definition 3.4)'s sign convention for the punctures is correct after all. 
  
 ==== Linearized version ==== ==== Linearized version ====
 Fix a comm ring $F$, and DGA $A$ over $F$. Suppose we have augmentation from a DGA $\epsilon: A \to F$.  Fix a comm ring $F$, and DGA $A$ over $F$. Suppose we have augmentation from a DGA $\epsilon: A \to F$. 
  
-Consider two Legendrians $L_0^+, L_1^+$, with DGA $A_0, A_1$. Consider the set of self-Reeb chords $D_1, D_2$, and $C$ the Reeb chords from $L_1^+$ to $L_0^+$. +Consider two Legendrians $L_0^+, L_1^+$, with DGA $A_0, A_1$. Consider the set of self-Reeb chords $D_1, D_2$, and $C$ the Reeb chords from $L_1^+$ to $L_0^+$. We build a total immersed Lagrangian $L = L_0 \cup L_1$.  
 + 
 +Then, given the augmentation data, we have $LCC(L_0^+, L_1^+)$ is an $F$-module freely generated by the wrong way Reeb chord from $L_1$ to $L_0$. Then differential counting 'jagged bigon', with one edge going around $L_0$, and the other going around $L_1$, and whenever we meet a double point along the boundary
  
  
cheatsheets/symplectic-contact-manifold.1670820228.txt.gz · Last modified: 2023/06/25 15:53 (external edit)