Calibrated manifolds and Gauge theory

Akbulut, Selman; Salur, Sema

doi:10.1515/crelle.2008.094

Cited by 16 publications

(27 citation statements)

References 10 publications

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“…In particular, one can define ν σ the normal bundle to exp σ (Y ), and F the Banach bundle over E with fiber F σ = W k−1,p (Y, ν σ ). It is clear that the operator F defined by (3) extends to a section F k,p of F over E. The proof of Theorem 2.1 shows that F k,p is smooth and the derivative of F in the direction of a vector field s ∈ T 0 E = W k,p (Y, ν) is computed by (1). Now, the operator D :…”

Section: The Operator D and The Deformation Problemmentioning

confidence: 99%

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Smooth Moduli Spaces of Associative Submanifolds

Gayet

2014

The Quarterly Journal of Mathematics

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Let M 7 be a smooth manifold equipped with a G 2 -structure φ, and Y 3 be a closed compact φ-associative submanifold. In [17], R. McLean proved that the moduli space M Y,φ of the φ-associative deformations of Y has vanishing virtual dimension. In this paper, we perturb φ into a G 2 -structure ψ in order to ensure the smoothness of M Y,ψ near Y . If Y is allowed to have a boundary moving in a fixed coassociative submanifold X, it was proved in [7] that the moduli space M Y,X of the associative deformations of Y with boundary in X has finite virtual dimension. We show here that a generic perturbation of the boundary condition X into X ′ gives the smoothness of M Y,X ′ . In another direction, we use Bochner's technique to prove a vanishing theorem that forces M Y or M Y,X to be smooth near Y . For every case, some explicit families of examples will be given. MSC 2000: 53C38 (35J55, 53C21, 58J32).

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Section: The Operator D and The Deformation Problemmentioning

confidence: 99%

“…Proof of Theorem 4.12 (1). Firstly, if M is equipped with a closed G 2 -structure φ, note that an associative submanifold Y with boundary in a coassociative X minimizes the volume in the relative homology class [Y ] ∈ H 3 (M, X, Z).…”

Section: Extensions From the Calabi-yau Worldmentioning

confidence: 99%

Smooth Moduli Spaces of Associative Submanifolds

Gayet

2014

The Quarterly Journal of Mathematics

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“…The abelian 3d Seiberg-Witten equations with n = 2 have appeared in relation to deformations of associative three-cycles in manifolds with (not necessarily torsion-free) G 2 -structure in [49][50][51], where the twisted harmonic spinor condition of [18] is supplemented with an additional Seiberg-Witten-like equation, which couples a U(1) gauge field to sections of the normal bundle (2.13), in order to make the space of deformations compact and zero-dimensional. In the context of G 2 -strings, a non-abelian version was shown to arise as the equations of motion of the world-volume theory of topological 3-branes wrapped on an associative three-cycle in a G 2 -manifold [52].…”

Section: Jhep07(2018)052mentioning

confidence: 99%

An $$ \mathcal{N}=1 $$ 3d-3d correspondence

Eckhard

Schäfer‐Nameki

Wong

2018

J. High Energ. Phys.

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M5-branes on an associative three-cycle M 3 in a G 2 -holonomy manifold give rise to a 3d N = 1 supersymmetric gauge theory, T N =1 [M 3 ]. We propose an N = 1 3d-3d correspondence, based on two observables of these theories: the Witten index and the S 3 -partition function. The Witten index of a 3d N = 1 theory T N =1 [M 3 ] is shown to be computed in terms of the partition function of a topological field theory, a super-BFmodel coupled to a spinorial hypermultiplet (BFH), on M 3 . The BFH-model localizes on solutions to a generalized set of 3d Seiberg-Witten equations on M 3 . Evidence to support this correspondence is provided in the abelian case, as well as in terms of a direct derivation of the topological field theory by twisted dimensional reduction of the 6d (2, 0) theory. We also consider a correspondence for the S 3 -partition function of the T N =1 [M 3 ] theories, by determining the dimensional reduction of the M5-brane theory on S 3 . The resulting topological theory is Chern-Simons-Dirac theory, for a gauge field and a twisted harmonic spinor on M 3 , whose equations of motion are the generalized 3d Seiberg-Witten equations. For generic G 2 -manifolds the theory reduces to real Chern-Simons theory, in which case we conjecture that the S 3 -partition function of T N =1 [M 3 ] is given by the Witten-ReshetikhinTuraev invariant of M 3 .

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“…References for this section are [1], [6], [18] and [20]. Recall that for each n ≥ 3, the Lie group SO(n) is connected, and it has a doublecovering map ι : Spin(n) → SO(n) where the Lie group Spin(n) is a compact, connected, simply-connected Lie group.…”

Section: G 2 -Structures On T * X × Rmentioning

confidence: 99%

A note on closed $G_2$-structures and $3$-manifolds

Cho¹,

Salur²,

Todd³

2014

Turk J Math

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This article shows that given any orientable 3 -manifold X , the 7 -manifold T * X × R admits a closed G2 -where Ω is a certain complex-valued 3 -form on T * X ; next, given any 2 -dimensional submanifold S of X , the conormal bundle N * S of S is a 3 -dimensional submanifold of T * X × R such that φ|N * S ≡ 0 .A corollary of the proof of this result is that N * S×R is a 4 -dimensional submanifold of T * X ×R such that φ| N * S×R ≡ 0 .

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Calibrated manifolds and Gauge theory

Cited by 16 publications

References 10 publications

Smooth Moduli Spaces of Associative Submanifolds

Smooth Moduli Spaces of Associative Submanifolds

An $$ \mathcal{N}=1 $$ 3d-3d correspondence

A note on closed $G_2$-structures and $3$-manifolds

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