10.4 Loop groups and their central extensions

Loop groups are infinite-dimensional Lie groups of smooth maps from a circle to a finite-dimensional Lie group. They inherit properties from their underlying group and have applications in physics and mathematics.

Central extensions of loop groups create larger groups with additional structure. The most common is the Kac-Moody group, whose Lie algebra is an affine Lie algebra. These extensions are crucial in conformal field theory and integrable systems.

Loop Groups and Their Properties

Definition and Basic Structure

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• Loop groups are infinite-dimensional Lie groups consisting of smooth maps from the circle $S^1$ to a finite-dimensional Lie group $G$
• The group operation in a loop group is point-wise multiplication of the maps
  • $(fg)(\theta) = f(\theta)g(\theta)$ for all $\theta \in S^1$, where $f, g$ are elements of the loop group
• Loop groups inherit many properties from their underlying finite-dimensional Lie group
  • Connectedness
  • Compactness
  • Semisimplicity

Loop Algebras and Constructions

• The Lie algebra of a loop group, called a loop algebra, consists of smooth maps from $S^1$ to the Lie algebra of $G$
• Loop groups can be constructed from various finite-dimensional Lie groups
  • Unitary group $U(n)$
  • Special unitary group $SU(n)$
  • Orthogonal group $O(n)$
• Examples of loop groups include the loop group of $SU(2)$, denoted as $LSU(2)$, and the loop group of $U(1)$, denoted as $LU(1)$

Central Extensions of Loop Groups

Motivation and Definition

• Central extensions of loop groups are new loop groups that contain the original loop group as a quotient by a central subgroup
• Central extensions allow for the construction of new, larger loop groups with additional structure and properties
• The most common central extension of a loop group is the Kac-Moody group
  • Kac-Moody group is a central extension by the circle group $S^1$

Affine Lie Algebras and Cocycles

• The Lie algebra of a Kac-Moody group is an affine Lie algebra
  • Affine Lie algebra is a central extension of the loop algebra by a one-dimensional center
• Central extensions can be classified by the second cohomology group $H^2(G, S^1)$, where $G$ is the original loop group
• The construction of central extensions involves the use of cocycles
  • Cocycles are maps satisfying certain properties that define the extension
• An example of a cocycle is the Kac-Moody cocycle, which defines the central extension of a loop algebra to an affine Lie algebra

Loop Groups vs Affine Lie Algebras

Relationship and Correspondence

• Affine Lie algebras are infinite-dimensional Lie algebras that are central extensions of loop algebras
• The Kac-Moody group, which is a central extension of a loop group, has an affine Lie algebra as its Lie algebra
• The highest weight representations of affine Lie algebras correspond to the positive energy representations of loop groups

Representation Theory and Applications

• Affine Lie algebras have a rich representation theory that is closely related to the representation theory of loop groups
• Affine Lie algebras have important applications in various areas
  • Conformal field theory
  • Study of critical phenomena in statistical mechanics
• Examples of affine Lie algebras include the affine Kac-Moody algebras $\hat{su}(2)$ and $\hat{su}(3)$, which are central extensions of the loop algebras of $SU(2)$ and $SU(3)$, respectively

Applications of Loop Groups in Physics

Conformal Field Theory and String Theory

• Loop groups and their central extensions have numerous applications in mathematical physics
• In conformal field theory, loop groups and affine Lie algebras are used to construct the Wess-Zumino-Witten (WZW) model
  • WZW model describes the propagation of strings on group manifolds
• The representation theory of loop groups and affine Lie algebras plays a crucial role in the classification and study of conformal field theories
• In string theory, loop groups arise naturally in the description of closed strings propagating on group manifolds
  • Central extensions of loop groups are related to the anomalies that appear in the quantization of these strings

Integrable Systems

• Loop groups and affine Lie algebras also appear in the study of integrable systems
• Examples of integrable systems include
  • Korteweg-de Vries (KdV) equation
  • Sine-Gordon equation
• Loop groups and affine Lie algebras provide a framework for constructing and classifying solutions to these integrable systems
• The representation theory of loop groups and affine Lie algebras is used to construct soliton solutions and study the integrability of these systems