# Sasha's Talk (Monday, November 29)

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Topic: Segal-Sagawara construction.
Define $\mathrm{Witt} = \Lie(\Diff^+ S^1)$, regarded as polynomial vector fields on $S^1$.
$H^2(\mathrm{Witt}; \CC) = \CC$, so there is a 1-dimensional space of central extensions, with a distinguished one: the Virasoro algebra.
There is a SES $0 \to\CC\mathrm{charge} \to \mathrm{Vir} \to \mathrm{Witt}\to 0$, and for $LG\da C^\infty(S^1, G)$, a SES $0\to S^1 \to \tilde{LG} \to LG\to 0$.
Here $\mathrm{charge}$ is some distinguished central element.
Does the Virasoro group act on this extension?
Not quite, but almost -- pass to Lie algebras to get $0\to \CC\to \tilde{L\lieg}\to \lieg \to 0$.
Theorem: for $\rho: \tilde{L\lieg} \to \Endo_\CC(V)$ an admissible representation, there is a representation $\rho': \mathrm{Vir}\to \Endo_\CC(V)$.
Note $L\lieg \da \lieg \tensor_\CC \CC[t, t\inv]$.
Write $X_i \gens{m} \da X_i\tensor m$.
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Admissible representations: for all $v\in V$ and $X\in\lieg$, there exists an $m$ such that $\rho(X\gens{m})(v) = 0$.
Define the Casimir element $\sum_i X_i X^i \in Z(\mcu(\lieg))$.
Levels: level $\ell$ if $\mathrm{charge}$ acts by $\ell \cdot \id$.
Critical level: $\ell \neq \cdots$ some constant (roughly the dual Coxeter number), avoid this $\ell$ for the reps in the theorem statement.
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