Hamiltonian loop group actions and Verlinde factorization

Meinrenken, Eckhard; Woodward, Chris

doi:10.4310/jdg/1214424966

Cited by 65 publications

(98 citation statements)

References 40 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…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.…”

Section: Introductionmentioning

confidence: 99%

“…Indeed, a collection of examples of Hamiltonian loop group spaces are moduli spaces of flat connections on a compact Riemann surface having at least 1 boundary component, with moment map given by pullback of the connection to the boundary (cf. [20]). In these examples, the twisted DuistermaatHeckman distributions are essentially Bernoulli series, and the formula (1) coincides with the decomposition formula in [8].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Norm-square localization for Hamiltonian LG-spaces

Loizides

2017

Journal of Geometry and Physics

View full text Add to dashboard Cite

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...

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Norm-square localization for Hamiltonian LG-spaces

Loizides

2017

Journal of Geometry and Physics

View full text Add to dashboard Cite

show abstract

“…In this paper, we chose to investigate the symplectic structure of these representation spaces. This symplectic structure can be obtained and described in a variety of ways (see for instance [13,15,5,6,22]), each of which has its own advantages. The description given by Alekseev, Malkin and Meinrenken in [6] will prove particularly well-suited for our study of representations of π g,l .…”

Section: Introductionmentioning

confidence: 99%

Decomposable representations and Lagrangian submanifolds of moduli spaces associated to surface groups

Schaffhauser

2008

Math. Ann.

View full text Add to dashboard Cite

Abstract. The importance of explicit examples of Lagrangian submanifolds of moduli spaces is revealed by papers such as [9,25]: given a 3-manifold M with boundary ∂M = Σ, Dostoglou and Salamon use such examples to obtain a proof of the Atiyah-Floer conjecture relating the symplectic Floer homology of the representation space Hom(π 1 (Σ = ∂M ), U )/U (associated to an explicit pair of Lagrangian submanifolds of this representation space) and the instanton homology of the 3-manifold M . In the present paper, we construct a Lagrangian submanifold of the space of representations M g,l := Hom C (π g,l , U )/U of the fundamental group π g,l of a punctured Riemann surface Σ g,l into an arbitrary compact connected Lie group U . This Lagrangian submanifold is obtained as the fixed-point set of an anti-symplectic involutionβ defined on M g,l . We show that the involutionβ is induced by a form-reversing involution β defined on the quasi-Hamiltonian space (The fact thatβ has a non-empty fixed-point set is a consequence of the real convexity theorem for group-valued momentum maps proved in [28]. The notion of decomposable representation provides a geometric interpretation of the Lagrangian submanifold thus obtained.

show abstract

“…Biswas [11], Agnihotri-Woodward [1] and Belkale [7] [37], Corollary 4.13, Ab is a convex polytope of maximal dimension in ~b . We wish to find the defining inequalities for Ab.…”

mentioning

confidence: 99%

Parabolic bundles, products of conjugacy classes, and Gromov-Witten invariants

Teleman¹,

Woodward²

2003

Annales De L’institut Fourier

View full text Add to dashboard Cite

The set of conjugacy classes appearing in a product of conjugacy classes in a compact, 1-connected Lie group K can be identified with a convex polytope in the Weyl alcove [37]. In this paper we identify linear inequalities defining this polytope. Each inequality corresponds to a non-vanishing Gromov-Witten invariant for a generalized flag variety G/P , where G is the complexification of K and P is a maximal parabolic subgroup. This generalizes the results for SU (n) of Agnihotri and the second author [1] and Belkale [7] on the eigenvalues of a product of unitary matrices and quantum cohomology of Grassmannians.

show abstract

Hamiltonian loop group actions and Verlinde factorization

Cited by 65 publications

References 40 publications

Norm-square localization for Hamiltonian LG-spaces

Norm-square localization for Hamiltonian LG-spaces

Decomposable representations and Lagrangian submanifolds of moduli spaces associated to surface groups

Parabolic bundles, products of conjugacy classes, and Gromov-Witten invariants

Contact Info

Product

Resources

About