$G_2$-instantons, associative submanifolds, and Fueter sections

Walpuski, Thomas

doi:10.48550/arxiv.1205.5350

Cited by 9 publications

(12 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We hope that the current work will be a first step towards the construction of singular G 2 -instantons on twisted connected sums. G 2 -instantons play a central role in Donaldson and Thomas' vision of gauge theory in higher dimensions [9], and understanding singularities and their formation is an important part of making their ideas rigorous; see, e.g., [37,36,15].…”

Section: Resultsmentioning

confidence: 99%

Hermitian Yang–Mills metrics on reflexive sheaves over asymptotically cylindrical Kähler manifolds

Jacob

Walpuski

2018

Communications in Partial Differential Equations

Self Cite

View full text Add to dashboard Cite

We prove an analogue of the Donaldson-Uhlenbeck-Yau theorem for asymptotically cylindrical Kähler manifolds: If E is a reflexive sheaf over an ACyl Kähler manifold, which is asymptotic to a µ-stable holomorphic vector bundle, then it admits an asymptotically translationinvariant projectively Hermitian Yang-Mills metrics (with curvature in L 2 loc across the singular set). Our proof combines the analytic continuity method of Uhlenbeck and Yau [35] with the geometric regularization scheme introduced by Bando and Siu [3]. 1 metrics on holomorphic vector bundles which are flat at infinity [2]. Ni and Ren [27] proved that a holomorphic vector bundle over a complete non-compact Kähler manifold with a spectral gap admits a PHYM metric if and only if it admits a metric whose failure to be PHYM is in L p for p > 1 (using an argument similar to Donaldson's solution of the Dirichlet problem for the PHYM equation [8]). Ni [26] showed that the same conclusion holds, for example, if the Kähler manifold satisfies a L 2 Sobolev inequality and p ∈ [1, n/2), or if it is non-parabolic (i.e., admits a positive Green's function) and p = 1. Main resultIn this article we concentrate on the asymptotically cylindrical case, and in view of the applications we have in mind we work with reflexive sheaves (not just holomorphic vector bundles).Theorem 1.1. Let V be an asymptotically cylindrical (ACyl) Kähler manifold with asymptotic cross-section D. Let E D be a µ-stable vector bundle over D, and E a reflexive sheaf asymptotic to E D .In this situation there exists an asymptotically translation-invariant Hermitian metric H on E which satisfies the projective Hermitian Yang-Mills (PHYM) equation). Remark 1.3. A PHYM metric H on E is Hermitian Yang-Mills (HYM) if and only if the induced metric h on det E is HYM, that is, iΛF h = tr(iΛF H ) rk Eis constant. Every asymptotically translationinvariant line bundle over an ACyl Kähler manifold has a HYM metric; however, this metric will typically not be asymptotically translation invariant. See Section 2.3 for a detailed discussion.Remark 1.4. The definition of asymptotically cylindrical Kähler manifolds we work with is given in Definition 2.1; it includes being asymptotically fibred.Remark 1.5. The question of the existence of HYM metrics on holomorphic bundles (with trivial determinant) over ACyl Calabi-Yau 3-folds was studied earlier by Sá Earp [29] (using the Yang-Mills heat flow). Our result improves on his in that we consider general ACyl Kähler manifolds and handle reflexive sheaves; moreover, we give a complete proof of the exponential decay to a PHYM metric over D (which is crucial for applications).Remark 1.6. In dimension four, there is prior work on the relation between ASD instantons and holomorphic vector bundles over cylindrical manifolds by Guo [12] and Owens [28]. Examples and applicationsThere are plenty of examples of ACyl Kähler manifolds and reflexive sheaves on them. Given any smooth projective variety Z containing a smooth divisor D and fibred by |D|, V := Z \ D can be given t...

show abstract

Section: Resultsmentioning

confidence: 99%

Hermitian Yang–Mills metrics on reflexive sheaves over asymptotically cylindrical Kähler manifolds

Jacob

Walpuski

2018

Communications in Partial Differential Equations

Self Cite

View full text Add to dashboard Cite

show abstract

“…All of these equations have potentially important applications. For example, Haydys and Walpuski [13] [14], conjecture a fundamental relation between the multi-spinor Seiberg-Witten equations on 3-manifolds and the spaces of G 2 -instantons on certain 7-dimensional manifolds (also see [7]). Meanwhile, Witten has conjectured [15] that spaces of solutions of the Kapustin-Witten equations can be used to compute the Jones polynomial for knots in S 3 .…”

Section: Introduction and Main Theoremmentioning

confidence: 99%

Index theorem for $\mathbb{Z}/2$-harmonic spinors

Takahashi

2018

Mathematical Research Letters

View full text Add to dashboard Cite

Let M denote a compact 3-manifold. The author proved in [8] that there exists a Kuranishi structure for the moduli space of pairs consisting of a Riemannian metric on M and a non-zero Z/2-harmonic spinor subject to certain natural regularity assumptions. This paper proves that the virtual dimension of Z/2-harmonic spinors for a generic metric is equal to zero. The paper also computes the virtual dimension of certain Z/2-harmonic spinors on 4-manifolds using an index theorem developed by Jochen Bruning and Robert Seeley and, independently, Fangyun Yang.

show abstract

“…Advances have been made both from the physics [9,10,11,12,13,14] and the mathematics [15,1,16,17,18,19,20,21], and involves aspects such as a string-worldsheet interpretation [22,23,24], a better understanding of exceptional gauge bundles and instantons [25,26,27,28,29,30,31], in addition to non-perturbative aspects such as M 2 instantons and their relation to exceptional enumerative geometry [32,33,34]. On the physics side such new insights often rely on dualities with other string theories and F-theory [35,36,37,38], in addition to dimensional reductions [39,40,41,42,43].…”

mentioning

confidence: 99%

New $G_2$-conifolds in $M$-theory and their Field Theory Interpretation

Acharya¹,

Foscolo²,

Najjar³

et al. 2020

Preprint

View full text Add to dashboard Cite

A recent theorem of Foscolo-Haskins-Nordström [1] which constructs complete G 2 -holonomy orbifolds from circle bundles over Calabi-Yau cones can be utilised to construct and investigate a large class of generalisations of the M -theory flop transition. We see that in many cases a UV perturbative gauge theory appears to have an infrared dual described by a smooth G 2 -holonomy background in M -theory. Various physical checks of this proposal are carried out affirmatively.

show abstract

$G_2$-instantons, associative submanifolds, and Fueter sections

Cited by 9 publications

References 0 publications

Hermitian Yang–Mills metrics on reflexive sheaves over asymptotically cylindrical Kähler manifolds

Hermitian Yang–Mills metrics on reflexive sheaves over asymptotically cylindrical Kähler manifolds

Index theorem for $\mathbb{Z}/2$-harmonic spinors

New $G_2$-conifolds in $M$-theory and their Field Theory Interpretation

Contact Info

Product

Resources

About