Non-existence and uniqueness results for boundary value problems for Yang-Mills connections

Isobe, Takeshi

doi:10.1090/s0002-9939-97-03804-5

Cited by 3 publications

(2 citation statements)

References 17 publications

(22 reference statements)

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“…Thus the upper bound B ε limits the number of disjoint balls in the set {B n (y n,m )} . Choose a maximal disjoint set B n (y n,mj ) J j=1 of balls in {B n (y n,m )}, and consider the set B * n (y n,mj ) J j=1 of balls centered at the points y n,mj but having radius 3 n . Then we have {yn,m} B n (y n,m ) ⊂ J j=1 B * n (y n,mj ).…”

Section: Solving the Yang-mills Dirichlet Problemmentioning

confidence: 99%

“…However, while still an open question, there exist partial results toward establishing uniqueness of a minimizer for given initial data in the compact case. In [3], Isobe has shown that for flat boundary values, the Dirichlet problem on a star-shaped bounded domain in R n can only have a flat solution. Non-uniqueness results are proven by Isobe and Marini [4] for Yang-Mills connections in bundles over B 4 , but the solutions are topologically distinct, belonging to differing Chern classes.…”

Section: The Euclidean Yang-mills Hamilton-jacobi Functionalmentioning

confidence: 99%

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Gauge-invariant ground state for canonically quantized Yang-Mills theory

Maitra

2008

Preprint

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We use Hamilton-Jacobi theory to construct a gauge-invariant zeroenergy candidate ground state for canonically quantized Yang-Mills theory with a "nonlinear normal" factor ordering, generalizing an analogous ordering introduced by Moncrief and Ryan for problems with finitely many degrees of freedom. Invariance under spatial rotations and translations is immediate; boost invariance remains under investigation. The motivation is to find a model for constructing a candidate ground state in general relativity, canonically quantized a la the Ashtekar variables. We seek to avoid replicating some of the more troublesome features of the Kodama state, inherited from the Chern-Simons state.

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Section: Solving the Yang-mills Dirichlet Problemmentioning

confidence: 99%

Section: The Euclidean Yang-mills Hamilton-jacobi Functionalmentioning

confidence: 99%

Gauge-invariant ground state for canonically quantized Yang-Mills theory

Maitra

2008

Preprint

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A singular radial connection over $$\mathbb B^5$$ B 5 minimizing the Yang-Mills energy

Petrache

2014

Calc. Var.

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Small coupling limit and multiple solutions to the Dirichlet problem for Yang-Mills connections in four dimensions. I

Isobe

Marini

2012

Journal of Mathematical Physics

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In this paper, the third of this series, we prove that the spaces A * ,p k (A 0 ; q) and B * ,p 0,k (A 0 ; q) which contain L p k -approximate solutions to the Dirichlet problem for the ǫ-Yang Mills equations on a four dimensional disk B 4 , carry a natural manifold structure (more precisely a natural structure of Banach bundle), for p(k + 1) > 4. All results apply also if B 4 is replaced by a general compact manifold with boundary, and SU (2) is replaced by any compact Lie group. We also construct bases for the tangent space to the space of approximate solutions, thus showing that this space is 8-dimensional for ǫ sufficiently small, and prove some technical results used in Parts I and II for the proof of the existence of multiple solution and, in particular, non-minimal ones, for this non-compact variational problem.where, ι : ∂B 4 → B 4 is the inclusion, the symbol ∼ stands for gauge equivalence via a gauge transformation that extends smoothly to the interior, and d * A ǫ := * d * + * [A, * ·] ǫ := * d * +ǫ * [A, * ·],where * is the Hodge star operator with respect to the flat metric on R 4 .An absolute minimum, say A ǫ , for the Yang-Mills functional is known to exist by [6]. Moreover,in [3], it is shown that the space of connections with boundary value A 0 , denoted by A(A 0 ), has countable connected components, i.e., A(A 0 ) = ∞ j=−∞ A j (A 0 ), where A j (A 0 ) is the space of connections with relative 2nd Chern number with respect to A ǫ equal to j, and that there * Tokyo Institute of Technology;

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Non-existence and uniqueness results for boundary value problems for Yang-Mills connections

Cited by 3 publications

References 17 publications

Gauge-invariant ground state for canonically quantized Yang-Mills theory

Gauge-invariant ground state for canonically quantized Yang-Mills theory

A singular radial connection over $$\mathbb B^5$$ B 5 minimizing the Yang-Mills energy

Small coupling limit and multiple solutions to the Dirichlet problem for Yang-Mills connections in four dimensions. I

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