# nLab positive energy representation

Contents

### Context

#### Representation theory

representation theory

geometric representation theory

# Contents

## Definition

Let $G$ be a compact Lie group and write $L G$ for its loop group. See there for details and notation.

Write

$t_\theta \colon L G \to L G$

for the automorphism which rotates loops by an angle $\theta$.

The corresponding semidirect product group we write $L G \rtimes S^1$

###### Definition

Let $V$ be a topological vector space. A linear representation

$S^1 \to Aut(V)$

of the circle group is called positive if $\exp(i \theta)$ acts by $\exp(i A \theta)$ where $A \in End(V)$ is a linear operator with positive spectrum.

A linear representation

$\rho : L G \to Aut(V)$

is said to have positive energy or to be a positive energy representation if it extends to a representation of the semidirect product group $L G \rtimes S^1$ such that the restriction to $S^1$ is positive.

## References

The standard textbook on loop groups is

• Andrew Pressley, Graeme Segal, Loop groups Oxford University Press (1988)

A review talk is

Discussion in the context of string theory (the Witten genus) is in

• Kefeng Liu, section 2.2 of On modular invariance and rigidity theorems, J. Differential Geom. Volume 41, Number 2 (1995), 247-514 (EUCLID, pdf)

Last revised on October 17, 2019 at 08:41:44. See the history of this page for a list of all contributions to it.