Gerbes on complex reductive Lie groups

Abstract

We construct a gerbe over a complex reductive Lie group G attached to an invariant bilinear form on a maximal diagonalizable subalgebra which is Weyl group invariant and satisfies a parity condition. By restriction to a maximal compact subgroup K, one then gets a gerbe over K. For a simply-connected group, the parity condition is the same used by Pressley and Segal; in general, it was introduced by Deligne and the author. The gerbe is defined by geometric methods, using the so-called Grothendieck manifold. It is equivariant under the conjugation action of G; its restriction to a semisimple orbit is not always trivial. The paper starts with a discussion of gerbe data (in the sense of Chatterjee and Hitchin) and of gerbes as geometric objects (sheaves of groupoids); the relation between the two approaches is presented. There is an Appendix on equivariant gerbes, discussed from both points of view.

33 pages; plain tex

References in corpus (4)

• Lectures on Special Lagrangian Submanifolds
• Positive Energy Representations of the Loop Groups of Non Simply Connected Lie Groups
• Equivariant K-Theory of Simply Connected Lie Groups
• Group-valued equivariant localization

Cited by in corpus (16)

• Unoriented WZW Models and Holonomy of Bundle Gerbes
• Abelian and non-Abelian branes in WZW models and gerbes
• Basic gerbe over non simply connected compact groups
• Differentiable Stacks and Gerbes
• The basic gerbe over a compact simple Lie group
• The basic bundle gerbe on unitary groups
• Topological Quantum Field Theories from Compact Lie Groups
• Equivariant characteristic forms in the Cartan model and Borel equivariant cohomology
• Basic equivariant gerbes on non-simply connected compact simple Lie groups
• Gerbes in classical Chern-Simons theory
• Reduction of strongly equivariant bundle gerbes with connection and curving
• Anomalies in gauge theory and gerbes over quotient stacks
• Equivariant gerbes over compact simple Lie groups
• Equivariant smooth Deligne cohomology
• Relationship between equivariant gerbes and gerbes over the quotient space
• K-twisted K-theory for SU(N)