Localization of Chern-Simons type invariants of Riemannian foliations

Goertsches, Oliver; Nozawa, Hiraku; Toeben, Dirk

doi:10.48550/arxiv.1508.07973

Cited by 2 publications

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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%

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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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“…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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show abstract

“…From ξ, we can construct for α eq = α 4 + α 2 + α 0 its primitive α eq = −(d − ι x ) ξα 0 + ξα 2 + ξdξα 0 away from the loci x r. Then one can use the Stokes theorem to reduce the calculation to some local computation around the loci x r. What is behind all this is that the Stokes theorem is still valid for the basic forms thanks to dκ being basic. For details as well as the typical scenario where our setting arises, see [46].…”

Section: A3 a Bottom Up Constructionmentioning

confidence: 99%

Transversally Elliptic Complex and Cohomological Field Theory

Festuccia,

Qiu,

Winding

et al. 2019

Preprint

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This work is a continuation of our previous paper arXiv:1812.06473 where we have constructed N = 2 supersymmetric Yang-Mills theory on 4D manifolds with a Killing vector field with isolated fixed points. In this work we expand on the mathematical aspects of the theory, with a particular focus on its nature as a cohomological field theory. The well-known Donaldson-Witten theory is a twisted version of N = 2 SYM and can also be constructed using the Atiyah-Jeffrey construction [1]. This theory is concerned with the moduli space of anti-self-dual gauge connections, with a deformation theory controlled by an elliptic complex. More generally, supersymmetry requires considering configurations that look like either instantons or anti-isntantons around fixed points, which we call flipping instantons. The flipping instantons of our 4D N = 2 theory are derived from the 5D contact instantons. The novelty is that their deformation theory is controlled by a transversally elliptic complex, which we demonstrate here. We repeat the Atiyah-Jeffrey construction in the equivariant setting and arrive at the Lagrangian (an equivariant Euler class in the relevant field space) that was also obtained from our previous work arXiv:1812.06473. We show that the transversal ellipticity of the deformation complex is crucial for the non-degeneracy of the Lagrangian and the calculability of the theory. Our construction is valid on a large class of quasi toric 4 manifolds.

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Localization of Chern-Simons type invariants of Riemannian foliations

Cited by 2 publications

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

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

Transversally Elliptic Complex and Cohomological Field Theory

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