# 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

### General

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)

On twisted ad-equivariant K-theory of compact Lie groups and the identification with the Verlinde ring of positive energy representations of their loop group:

1. J. Topology, 4 (2011), 737-789

2. J. Amer. Math. Soc. 26 (2013), 595-644

3. Annals of Mathematics, Volume 174 (2011) 947-1007

• Dirac families for loop groups as matrix factorizations,

Comptes Rendus Mathematique, Volume 353, Issue 5, May 2015, Pages 415-419

Last revised on November 1, 2020 at 22:58:30. See the history of this page for a list of all contributions to it.