Localization of Chern–Simons type invariants of Riemannian foliations

Goertsches, Oliver; Nozawa, Hiraku; Töben, Dirk

doi:10.1007/s11856-017-1608-6

Cited by 13 publications

(7 citation statements)

References 48 publications

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“…We note that for this result, it is sufficient to assume that all G-fixed points have a closed Reeb orbit, an assumption that is weaker than assuming 0 to be a regular value of Ψ and that is automatically satisfied for total spaces in the Boothby-Wang fibration. This theorem is closely related to results obtained in [Töb14,GNT17].…”

supporting

confidence: 77%

“…Then the (j = 0)-summand tends to (−1) n , the others vanish (cf. [GNT17]). Now, let us consider the special case of the odd sphere M = S 3 ⊂ C 2 with Sasakian structure determined by the weight (w, 1) with w > 0 irrational.…”

Section: Weighted Sasakian Structures On Odd Spheresmentioning

confidence: 99%

“…We now define equivariant basic cohomology and give its basic properties. See [GNT12,Töb14,GNT17,Cas17] for related constructions. We suppose now that a torus G acts on M, preserving the contact form.…”

Section: Equivariant Basic Cohomologymentioning

confidence: 99%

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Localization for $K$-contact manifolds

Casselmann

Fisher²

2019

Journal of Symplectic Geometry

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We prove an analogue of the Atiyah-Bott-Berline-Vergne localization formula in the setting of equivariant basic cohomology of K-contact manifolds. As a consequence, we deduce analogues of Witten's nonabelian localization and the Jeffrey-Kirwan residue formula, which relate equivariant basic integrals on a contact manifold M to basic integrals on the contact quotient M 0 := µ −1 (0)/G, where µ denotes the contact moment map for the action of a torus G. In the special case that M → N is an equivariant Boothby-Wang fibration, our formulae reduce to the usual ones for the symplectic manifold N . ContentsRemark 1.5. In §5.1 we explain in detail how Theorems 1.1 and 1.4 may be used to deduce the analogous theorems for symplectic manifolds that occur as M/F in the case that R induces a free S 1 -action. In this sense, these theorems provide a strict generalization of their symplectic analogues, at least in the case of an integral symplectic form and a Hamiltonian group action that lifts to the S 1 -bundle in the Boothby-Wang fibration [BW58].Remark 1.6. The first named author has obtained a surjectivity result for the basic contact Kirwan map [Cas17]. Since basic cohomology satisfies Poincaré duality (see Lemma 2.3), Theorem 1.4 in principle provides a method to compute the kernel of theProposition 4.3. The distribution Q η (y) may be represented by a piecewise polynomial function.As in the proof of Lemma 4.2, the previous equation becomesRecall that the local normal form of the moment map is given by µ(p, z) = z. Then, for sufficiently small y, δ(−µ − y) is supported away from S h and it follows that ∆ = 0. This means that, for sufficiently small y, Q η (y) = 1 (2π) s/2 g µ −1 (0)×B h α ∧ π * q * η 0 ∧ e idα+i −µ−y,φ dφ = (2π) s/2 µ −1 (0)×B h α ∧ π * q * η 0 ∧ e idα δ(−µ − y)

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confidence: 77%

Section: Weighted Sasakian Structures On Odd Spheresmentioning

confidence: 99%

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Localization for $K$-contact manifolds

Casselmann

Fisher²

2019

Journal of Symplectic Geometry

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“…This formula is derived using localisation techniques on K-contact manifolds in [57]. The sum is over the corners of ∆ µ (see section 3 for notations).…”

Section: Comparison With Flat Space Resultsmentioning

confidence: 99%

“…To match with[57] it is useful to note that (ι X κ) 2 is the 0-form component of the equivariantly completed form (dκ) 2 .…”

mentioning

confidence: 99%

N = 2 $$ \mathcal{N}=2 $$ supersymmetric gauge theory on connected sums of S 2 × S 2

Festuccia¹,

Qiu²,

Winding³

et al. 2017

J. High Energ. Phys.

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Abstract:We construct 4D N = 2 theories on an infinite family of 4D toric manifolds with the topology of connected sums of S 2 × S 2 . These theories are constructed through the dimensional reduction along a non-trivial U(1)-fiber of 5D theories on toric SasakiEinstein manifolds. We discuss the conditions under which such reductions can be carried out and give a partial classification result of the resulting 4D manifolds. We calculate the partition functions of these 4D theories and they involve both instanton and anti-instanton contributions, thus generalizing Pestun's famous result on S 4 .

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Equivariant basic cohomology under deformations

Caramello

Töben

2021

Math. Z.

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Localization of Chern–Simons type invariants of Riemannian foliations

Cited by 13 publications

References 48 publications

Localization for $K$-contact manifolds

Localization for $K$-contact manifolds

N = 2 $$ \mathcal{N}=2 $$ supersymmetric gauge theory on connected sums of S 2 × S 2

Equivariant basic cohomology under deformations

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