Multiple Bernoulli series, an Euler-MacLaurin formula, and Wall crossings

Boysal, Arzu; Vergne, Michèle

doi:10.5802/aif.2696

Cited by 8 publications

(14 citation statements)

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“…[20]). In these examples, the twisted DuistermaatHeckman distributions are essentially Bernoulli series, and the formula (1) coincides with the decomposition formula in [8]. (These examples are closely related to Witten's formulas for intersection pairings on moduli spaces of flat connections on Riemann surfaces.)…”

Section: Introductionmentioning

confidence: 82%

“…In this example the contribution from 0 vanishes, and the remaining terms in (1) are constant multiples (±1 relative to suitably normalized Lebesgue measure) of indicator functions for half-spaces (see Section 5 for further discussion). The formula (1) is related to the decomposition formula for Bernoulli series in [8]. 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.…”

Section: Introductionmentioning

confidence: 99%

“…(These examples are closely related to Witten's formulas for intersection pairings on moduli spaces of flat connections on Riemann surfaces.) It is possible to derive (1) from the decomposition formula of [8] and the Duistermaat-Heckman formula for group-valued moment maps [2]; further details will appear in the author's PhD thesis. We adopt the cobordism approach here, as this approach leads quickly to the basic result (Theorem 4.1).…”

Section: Introductionmentioning

confidence: 99%

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Norm-square localization for Hamiltonian LG-spaces

Loizides

2017

Journal of Geometry and Physics

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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...

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Section: Introductionmentioning

confidence: 82%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Norm-square localization for Hamiltonian LG-spaces

Loizides

2017

Journal of Geometry and Physics

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“…In this case, s = 1, p = 0, q = 1, f = 3, |Z(SU(3))| = 3, hence, for g = 2, 2 s(2g−1) (f q) g−1 |Z(G)|(−1) (g−1)|Φ| = −3 2 . vol(SU(3), g = 2)(a) = −9 γ∈Preg e 2iπ a,γ Hα∈Φ (2iπ H α , γ ) 3 = −9 n 1 =0,n 2 =0,n 1 +n 2 =0 e 2iπ(n 1 a 1 +n 2 a 2 ) (2iπn 1 ) 3 (2iπn 2 ) 3 (2iπ(n 1 + n 2 )) 3 and we obtain vol(SU(3), g = 2)(a) = 1/40320(a 2 − 2a 1 )(a 2 − 1 + a 1 )(−1 + 2a 2 − a 1 )P 1 , a 1 ≤ a 2 1/40320(a 2 + 1 − 2a 1 )(2a 2 − a 1 )(a 2 − 1 + a 1 )P 2 , a 1 ≥ a 2…”

Section: Various Examples Of Volume Calculationsmentioning

confidence: 99%

Multiple Bernoulli series and volumes of moduli spaces of flat bundles over surfaces

Baldoni

Boysal

Vergne

2015

Journal of Symbolic Computation

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Section: Introductionmentioning

confidence: 99%

Norm-square localization for Hamiltonian $LG$-spaces

Loizides

2016

Preprint

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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.

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Multiple Bernoulli series, an Euler-MacLaurin formula, and Wall crossings

Cited by 8 publications

References 9 publications

Norm-square localization for Hamiltonian LG-spaces

Norm-square localization for Hamiltonian LG-spaces

Multiple Bernoulli series and volumes of moduli spaces of flat bundles over surfaces

Norm-square localization for Hamiltonian $LG$-spaces

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