nLab positive energy representation

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.

Properties

Relation to quantum group representations

See this MO discussion.

Related concepts

• quantization of loop groups

• conformal block

• equivariant elliptic cohomology

• twisted ad-equivariant K-theory

• twisted ad-equivariant Tate K-theory

• Loop Groups and Twisted K-Theory

References

General

The standard textbook on loop groups is

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

A review talk is

• Graeme Segal, Loop groups (pdf)

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)

Relation to ad-equivariant K-theory

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:

• Daniel S. Freed, The Verlinde algebra is twisted equivariant K-theory, Turk. J. Math 25 (2001) 159-167 (arXiv:math/0101038)

• Daniel S. Freed, Michael Hopkins, Constantin Teleman,

  1. Loop Groups and Twisted K-Theory I,

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

     (arXiv:0711.1906, doi:10.1112/jtopol/jtr019)

  2. Loop Groups and Twisted K-Theory II,

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

     (arXiv:0511232, doi:10.1090/S0894-0347-2013-00761-4)

  3. Loop Groups and Twisted K-Theory III,

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

     (arXiv:math/0312155, doi:10.4007/annals.2011.174.2.5)

• Daniel S. Freed, Constantin Teleman,

  Dirac families for loop groups as matrix factorizations,

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

  (arxiv/1409.6051, doi:10.1016/j.crma.2015.02.011)