Berline–Vergne

Berline–Vergne
Name	Berline–Vergne
Field	Mathematics
Introduced	1980s
Contributors	Nicole Berline, Michèle Vergne

Berline–Vergne is a mathematical result linking index theory, equivariant cohomology, and symplectic geometry through a localization formula for equivariant differential forms and the equivariant index of transversally elliptic operators. It provides an explicit expression for characters and indices in terms of contributions from fixed points or critical sets, bridging ideas from the Atiyah–Bott fixed-point theorem, the Atiyah–Singer index theorem, and the theory of Duistermaat–Heckman theorem.

Definition and Statement

The Berline–Vergne formula gives an equivariant localization formula: for a compact Lie group action on a compact manifold with an invariant elliptic operator or an equivariant differential form, the equivariant index or integral localizes to contributions indexed by components of the fixed point set or zeros of the moment map. In the setting of a symplectic manifold with a Hamiltonian circle action or a general compact torus action, the formula expresses the equivariant character as a sum over fixed components, analogous to the formulas in the Atiyah–Bott fixed-point theorem and the Witten nonabelian localization approach.

Historical Context and Development

The formula originated in the 1980s in work by Nicole Berline and Michèle Vergne as part of efforts to make the abstract Atiyah–Singer index theorem and the Atiyah–Bott fixed-point theorem computationally effective for equivariant situations. Their work built on earlier contributions by Michael Atiyah, Raoul Bott, Isadore Singer, and subsequent developments by Jeffrey-Kirwan and Edward Witten in localization in symplectic geometry and quantization. The Berline–Vergne localization complemented contemporary research by Victor Guillemin, Shlomo Sternberg, and Berndtsson on moment maps and equivariant cohomology, and it played a role in clarifying the relation between geometric quantization as in the Guillemin–Sternberg conjecture and analytical index theory.

Mathematical Framework and Key Concepts

The Berline–Vergne formula operates in the setting of a compact connected Lie group G acting smoothly on a compact oriented manifold M, often equipped with an invariant Riemannian metric, a G-equivariant vector bundle, and a G-invariant elliptic operator such as a Dirac operator or a Dolbeault operator. Key concepts include equivariant differential forms, the equivariant de Rham complex developed by Cartan (Élie Cartan), equivariant characteristic classes like the equivariant Chern character and equivariant Todd class, and the notion of fixed point components encoded by the Lefschetz fixed-point theorem machinery. In the symplectic context, the moment map apparatus of Kostant, Souriau, and Meyer interacts with the Duistermaat–Heckman measure and the theory of Hamiltonian torus actions studied by Frankel and Kirwan.

Proofs and Techniques

Proofs of the Berline–Vergne formula use heat kernel methods inspired by Atiyah–Bott and Patodi and rely on stationary phase or localization arguments reminiscent of Witten's deformation techniques. The analytic approach constructs equivariant heat kernels for transversally elliptic operators and analyzes asymptotic expansions, while the cohomological approach uses the equivariant Thom form, the Mathai–Quillen formalism, and the equivariant Chern–Weil theory developed by Chern and Weil. Deformation to the normal cone and an application of the Duistermaat–Heckman theorem or the Local Index Theorem often appear in modern expositions by Berline, Getzler, and Vergne.

Applications and Examples

The Berline–Vergne formula is applied to compute equivariant indices and characters for actions of circle groups and higher-dimensional tori on complex projective spaces such as CP^n, flag varieties like G/B for complex reductive Lie groups, and symplectic manifolds arising from coadjoint orbits such as those of SU(n), SO(n), and Sp(n). It underpins explicit formulas in representation theory, including multiplicity formulas related to the Weyl character formula, the Kostant multiplicity formula, and the Kostant–Dirac operator approach. In mathematical physics it informs computations in topological field theories studied by Witten and in localization calculations for partition functions explored by Pestun and Nekrasov.

Generalizations and Related Results

Generalizations include non-compact and equivariant index versions by Atiyah and Hirzebruch, transversally elliptic index theory for non-compact group actions developed by Atiyah–Singer extensions, and the equivariant Riemann–Roch theorem by Grothendieck-style approaches of Quillen and refinements by Paradan, Vergne, and Meinrenken. Related results encompass the Atiyah–Bott localization theorem in equivariant cohomology, the Jeffrey–Kirwan residue formula for symplectic quotients, the Guillemin–Sternberg "quantization commutes with reduction" framework, and analytic localization techniques in index theory by Getzler and Bismut.

Category:Index theory Category:Equivariant cohomology Category:Symplectic geometry