[Paper Review] Twisted conjugacy classes, coadjoint orbits of loop groups and D-branes in the WZW-model

S. Mohrdieck, Robert Wendt|ArXiv.org|Mar 10, 2003

Algebraic structures and combinatorial models16 references9 citations

TL;DR

This paper establishes a geometric correspondence between integral twisted and untwisted conjugacy classes in compact, simply connected Lie groups and irreducible highest weight representations of the corresponding twisted and untwisted affine Lie algebras. By translating the integrality condition on conjugacy classes into a cohomological condition on coadjoint orbits of loop groups, the authors use geometric quantization to show that such classes naturally parametrize these representations, generalizing the classical Borel-Weil-Bott correspondence to the twisted setting.

ABSTRACT

We show that untwisted respectively twisted conjugacy classes of a compact and simply connected Lie group which satisfy a certain integrality condition correspond naturally to irreducible highest weight representations of the corresponding affine Lie algebra. Along the way, review the classification of twisted conjugacy classes of a simply connected compact Lie group $G$ and give a description of their stabilizers in terms of the Dynkin diagram of the corresponding twisted affine Lie algebra.

Motivation & Objective

• To generalize the correspondence between integral conjugacy classes and highest weight representations of affine Lie algebras to the case of twisted conjugacy classes.
• To provide a geometric explanation for this correspondence using coadjoint orbits of twisted loop groups and the orbit method.
• To classify twisted conjugacy classes in terms of convex polytopes and twisted affine Dynkin diagrams.
• To describe the stabilizers of twisted conjugacy classes via sub-diagrams of twisted affine Dynkin diagrams.
• To establish a precise integrality condition on conjugacy classes that ensures quantization via relative cohomology and leads to unitary representations.

Proposed method

• The integrality condition is defined via the relative cohomology class of a 3-cocycle involving the canonical 3-form η on G and a 2-form ω on the conjugacy class with dω = ι*η.
• The authors use transgression to lift the relative 3-cocycle from G to the loop group L(G), connecting it to coadjoint orbits of the twisted loop group LGτ.
• The classification of twisted conjugacy classes is achieved by identifying them with the fundamental domain of the twisted affine Weyl group, which is a convex polytope in a Euclidean space.
• Stabilizers of twisted conjugacy classes are shown to correspond to subgroups whose Dynkin diagrams are sub-diagrams of the twisted affine Dynkin diagram.
• Homotopy groups of fiber spaces over conjugacy classes are computed using long exact sequences, proving simply connectedness of fibers to enable cycle filling in coadjoint orbits.
• The correspondence to unitary representations is established via geometric quantization, where integral coadjoint orbits of LGτ yield irreducible highest weight representations of the affine Lie algebra.

Experimental results

Research questions

• RQ1 How can twisted conjugacy classes in a compact, simply connected Lie group be classified geometrically, and how do they relate to the structure of twisted affine Lie algebras?
• RQ2 What is the precise integrality condition on conjugacy classes that ensures their quantization via geometric quantization?
• RQ3 How do the stabilizers of twisted conjugacy classes relate to the root systems of the corresponding twisted affine Lie algebras?
• RQ4 Can the correspondence between integral conjugacy classes and highest weight representations be extended from the untwisted to the twisted case using coadjoint orbits of loop groups?
• RQ5 What role does the relative cohomology class of the 3-cocycle (η, ω) play in determining which conjugacy classes yield unitary representations?

Key findings

• Integral twisted conjugacy classes in a compact, simply connected Lie group G correspond bijectively to irreducible highest weight representations of the corresponding twisted affine Lie algebra.
• The set of twisted conjugacy classes is parametrized by a convex polytope in a Euclidean space, which is the fundamental domain of the twisted affine Weyl group.
• The stabilizer of a twisted conjugacy class is a connected subgroup of G whose Dynkin diagram is a sub-diagram of the twisted affine Dynkin diagram associated to the automorphism τ.
• The fundamental group of each stabilizer is trivial, and the fibers of the evaluation map from the loop group to the conjugacy class are simply connected, enabling the construction of closed 2-cycles in coadjoint orbits.
• The integrality condition on conjugacy classes—requiring the relative 3-cocycle a(η, ω) to define an integral class in H³(G, C)—is equivalent to the level a being an integer and ensures that the associated coadjoint orbit of the loop group admits a pre-quantum line bundle.
• The geometric quantization of coadjoint orbits of the twisted loop group LGτ yields irreducible highest weight representations of the affine Lie algebra, generalizing the untwisted Borel-Weil-Bott correspondence.