Flat connections in three-manifolds and classical Chern–Simons invariant

Guadagnini, Enore; Mathieu, Philippe; Thuillier, Frank

doi:10.1016/j.nuclphysb.2017.10.021

Cited by 4 publications

(5 citation statements)

References 35 publications

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“…Accordingly, there is no obvious normal-ization of the functional integrals defining these partition functions. Nevertheless, using a Heegaard splitting of M, the fact that the lagrangian is a DB class allows to rewrite classical CS invariants -define as the CS action evaluated on flat connections -as the sum of two surface terms, one typically representing colored intersections and the other being a Wess-Zumino term [15]. The colored intersections term is the only one surviving in the U(1) case and gives rise to the correct expression of the classical U(1) invariant.…”

Section: Introductionmentioning

confidence: 99%

3D topological models and Heegaard splitting. I. Partition function

Thuillier

2019

Journal of Mathematical Physics

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Section: Introductionmentioning

confidence: 99%

3D topological models and Heegaard splitting. I. Partition function

Thuillier

2019

Journal of Mathematical Physics

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“…For r = 2 these have been studied in detail in [32], considering the Heegaard splitting of L 3 (p, ±1)…”

Section: 33)mentioning

confidence: 99%

“…We will show in section §4, following [32], how this affects the evaluation of a Chern-Simons term on L 3 (p, ±1). Moreover, both for r = 2 and r = 3, when we will study the partition function for a vector multiplet on L 3 (p, ±1) and L 5 (p, ±1), we will have to sum over flat connections differing by their wrapping along the Hopf fiber.…”

Section: 33)mentioning

confidence: 99%

“…Starting from the the classical piece on L 3 (p, ±1) we need to evaluate the Chern-Simons term on the Abelian flat connections (2.35). Following [32] one finds:…”

Section: Perturbative Partition Function On Cpmentioning

confidence: 99%

“…The function f (σ, m) can be determined generalizing the approach of [32] to higher dimensional lens spaces and we leave it for a future study. In the large p limit, keeping pg 2 Y M constant 17 , the classical piece reduces to that computed on CP 2 .…”

Section: Perturbative Partition Function On Cpmentioning

confidence: 99%

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SYM on Quotients of Spheres and Complex Projective Spaces

Lundin,

Ruggeri

2021

Preprint

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We introduce a generic procedure to reduce a supersymmetric Yang-Mills (SYM) theory along the Hopf fiber of squashed S 2r−1 with U (1) r isometry, down to the CP r−1 base. This amounts to fixing a Killing vector v generating a U (1) ⊂ U (1) r rotation and dimensionally reducing either along v or along another direction contained in U (1) r . To perform such reduction we introduce a Z p quotient freely acting along one of the two fibers. For fixed p the resulting manifolds S 2r−1 /Z p ≡ L 2r−1 (p, ±1) are a higher dimensional generalization of lens spaces. In the large p limit the fiber shrinks and effectively we find theories living on the base manifold.Starting from N = 2 SYM on S 3 and N = 1 SYM on S 5 we compute the partition functions on L 2r−1 (p, ±1) and, in the large p limit, on CP r−1 , respectively for r = 2 and r = 3. We show how the reductions along the two inequivalent fibers give rise to two distinct theories on the base.Reducing along v gives an equivariant version of Donaldson-Witten theory while the other choice leads to a supersymmetric theory closely related to Pestun's theory on S 4 . We use our technique to reproduce known results for r = 2 and we provide new results for r = 3. In particular we show how, at large p, the sum over fluxes on CP 2 arises from a sum over flat connections on L 5 (p, ±1). Finally, for r = 3, we also comment on the factorization of perturbative partition functions on non simply connected manifolds.

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SYM on quotients of spheres and complex projective spaces

Lundin

Ruggeri

2022

J. High Energ. Phys.

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We introduce a generic procedure to reduce a supersymmetric Yang-Mills (SYM) theory along the Hopf fiber of squashed S2r−1 with U(1)r isometry, down to the ℂℙr−1 base. This amounts to fixing a Killing vector v generating a U(1) ⊂ U(1)r rotation and dimensionally reducing either along v or along another direction contained in U(1)r. To perform such reduction we introduce a ℤp quotient freely acting along one of the two fibers. For fixed p the resulting manifolds S2r−1/ℤp ≡ L2r−1(p, ±1) are a higher dimensional generalization of lens spaces. In the large p limit the fiber shrinks and effectively we find theories living on the base manifold. Starting from $$ \mathcal{N} $$ N = 2 SYM on S3 and $$ \mathcal{N} $$ N = 1 SYM on S5 we compute the perturbative partition functions on L2r−1(p, ±1) and, in the large p limit, on ℂℙr−1, respectively for r = 2 and r = 3. We show how the reductions along the two inequivalent fibers give rise to two distinct theories on the base. Reducing along v gives an equivariant version of Donaldson-Witten theory while the other choice leads to a supersymmetric theory closely related to Pestun’s theory on S4. We use our technique to reproduce known results for r = 2 and we provide new results for r = 3. In particular we show how, at large p, the sum over fluxes on ℂℙ2 arises from a sum over flat connections on L5(p, ±1). Finally, for r = 3, we also comment on the factorization of perturbative partition functions on non simply connected manifolds.

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Flat connections in three-manifolds and classical Chern–Simons invariant

Cited by 4 publications

References 35 publications

3D topological models and Heegaard splitting. I. Partition function

3D topological models and Heegaard splitting. I. Partition function

SYM on Quotients of Spheres and Complex Projective Spaces

SYM on quotients of spheres and complex projective spaces

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