# Wednesday, August 25

- Setup: $G\actson M$ for $M$ a symplectic manifold and $G$ a Lie group acting by symplectomorphisms.
	- Canonical pairing $\lieg \tensor \lieg\dual \to \RR$, written $\inner{\xi_1}{\xi_2}$.
	- Exponential map: $\exp: \lieg\to G$
	- Elements $g\in \lieg$ induce a vector field $s(g)$, where at points $p$ it's given by $s(g)(p) = \dd{}{t}\ro{}{t=0} \exp(t g)\actson p$, using the $G$ action.
	- Take the contraction $\iota_{s(g)} \omega$ of the symplectic form
		-  Contraction: given $X$ a vector field, cook up $\iota_X: \Omega^p \to \Omega^{p-1}$. 
		-  For $\alpha \in \Omega^p$ a $p\dash$form, $\iota_X\alpha (X_1, \cdots, X_{p-1}) \da \alpha(X, X_1, \cdots, X_{p-1})$.
	-  Moment map: a map $\mu:M\to\lieg\dual$ where for all $\xi\in \lieg$, one defines $d(\inner{\mu}{\xi}) = \iota_{s(\xi)} \omega$
	-  Symplectic reduction:  take a moment map $\mu:M\to\lieg\dual$, then if $0\in\lieg\dual$ is a regular value where $G$ acts freely on $\mu\inv(0)$, then $M/G$ inherits a symplectic structure from $(M, \omega)$.
	-  Question: what does $\Omega^1(\Sigma, \lieg)$ mean?

	![](attachments/Pasted%20image%2020210825233011.png)
	
	This is the space of connections on the trivial bundle $\Sigma \cross G$, which carries an infinite dimensional symplectic form $a\tensor b\mapsto \int_\Sigma \tr(a\wedge b)$. The gauge group $\tilde G \da \Map(\Sigma, G)$ acts by conjugation.
	Also have $\Lie(\tilde G) \cong \Omega^0(\Sigma, \lieg)$.
	
	- Can use this to put a symplectic structure on the space of flat connection modulo gauge equivalence by sending a connection to its curvature $A\mapsto dA + {1\over 2}[A\wedge A]$.


Uses in Goldman

- Action of $\mcg\actson \liea$ is Hamiltonian with moment map $\liea \mapsto A^2(S; \Ad P) \cong A^0(S; \Ad P)\dual$ assigning connections to curvature.


# Results

![](attachments/Pasted%20image%2020210825232723.png)

![](attachments/Pasted%20image%2020210825232833.png)

![](attachments/Pasted%20image%2020210825232920.png)