Norm-square localization for Hamiltonian LG-spaces

Abstract

Abstract. We prove a formula for twisted Duistermaat-Heckman distributions associated to a Hamiltonian LG-space. The terms of the formula are localized at the critical points of the norm-square of the moment map, and can be computed in cross-sections. Our main tools are the theory of quasi-Hamiltonian G-spaces, as well as the Hamiltonian cobordism approach to norm-square localization introduced recently by Harada and Karshon. IntroductionLet M be a Hamiltonian G-space with moment map φ : M → g * , and equip g * with an inner product. The function φ 2 : M → R has been studied extensively. An early paper of AtiyahBott [4] studied an infinite dimensional example, the Yang-Mills functional on the space of connections on a compact Riemann surface. In the setting where M and G are compact, Kirwan [17] extended techniques of Morse theory to φ 2 , using this to prove Kirwan surjectivity. Closer to the subject of this paper are the early results of Witten [25], who studied certain integrals on g * × M , and found that they localize to the critical set of φ 2 , with the dominant contribution coming from the 0-level set. In [21,22], Paradan proved a norm-square localization formula for twisted Duistermaat-Heckman measures, in the general setting where M can be non-compact, but φ is proper. Another approach to these norm-square localization formulas was developed by Woodward [26], and more recently an approach based on Hamiltonian cobordism techniques was introduced by Harada and Karshon [14]. In their approach, M is found to be cobordant (in a suitable sense) to a small open neighbourhood of the critical set, once the moment map on the neighbourhood has been suitably polarized and completed. A norm-square localization formula for twisted Duistermaat-Heckman measures then follows from Stokes' theorem. We give an overview of the Harada-Karshon Theorem in Section 2 and Appendix B.Many well-known results on compact Hamiltonian G-spaces have parallels for proper Hamiltonian LG-spaces (LG is the loop group). Examples include the cross-section theorem, the convexity theorem, and Duistermaat-Heckman formulas, cf. [20,1,3]. The objects of study in this paper are twisted Duistermaat-Heckman (DH) distributions for Hamiltonian LG-spaces that carry information about cohomology pairings on symplectic quotients. For example, the (untwisted) DH distribution that we study is a signed measure on t (the Lie algebra of a maximal torus T ⊂ G) that gives volumes of symplectic quotients.Let Ψ : M → Lg * be a proper Hamiltonian LG-space. We prove a 'norm-square localization' formula expressing a twisted DH distribution m on t as a sum of contributions:where B indexes components of the critical set of Ψ 2 , and W = N G (T )/T is the Weyl group. The sum is infinite but locally finite, in the sense that the supports of only finitely many terms intersect each bounded set. The terms of (1) consist of a central contribution (from the critical value 0), and correction terms supported in half-spaces not containing the origin. Using the terminology of H...

Quantization of Hamiltonian loop group spaces

“…The latter motivated us to define the 'quantization' of (M, L) as this particular element of R k (G). In a related paper [27], the first author showed that this definition agrees with that of E. Meinrenken [38] based on quasi-Hamiltonian spaces and twisted K-homology.…”

“…If ξ, ζ ∈ Lg ⊥ µ ⊆ Lg, the second term in (21) does not contribute, while the first term is given by (20). We conclude that the given metric on ker(T q) z = Lg differs from the metric (20) only on a finite-dimensional subspace.…”

“…In the case of a compact Hamiltonian G-space, an analogous formula was proved by Paradan [31]. In the loop group setting, the norm-square of the moment map has been used to prove a Kirwan surjectivity theorem [7], and to study (twisted) Duistermaat-Heckman distributions [20].…”

“…Its moment polytope is a triangle, with vertices the mid-points of the edges of the alcove. See [23] or [20] for a description of this space. Proof.…”

Hamiltonian loop group spaces and a theorem of Teleman and Woodward

“…The first type is a non-abelian localization formula (Theorem 3.19), which shows that the index is expressible as a sum of contributions localized near the components of the critical set of the norm-square of the moment map µ M : M → Lg * . There is a large literature on non-abelian localization in various forms, for example [39,65,61,72] amongst many others; in the more specific context of Hamiltonian loop group spaces references include [24,76,43,48,44]. We prove non-abelian localization by adapting a technique of Bismut-Lebeau [21, Chapter IX] to analyse the resolvents of a 1-parameter family of operators in the limit as the parameter goes to infinity.…”