Perturbed geodesics on the moduli space of flat connections and Yang–Mills theory

Janner, Remi

doi:10.1007/s00209-012-1025-9

Cited by 2 publications

(8 citation statements)

References 17 publications

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“…Furthermore, he showed that the Morse homologies, defined from the perturbed parabolic Yang–Mills gradient flow on

\hat{P} \to Σ \times S^{1}

and the heat flow on

{scriptM}^{γ} (P)

, are isomorphic, provided that ε is small enough and that an energy bound b is chosen (cf. , ). Hence we have

H M_{*} (L^{b} M^{γ} (P), Z_{2}) ≅ H M_{*} (A^{ɛ, b} ((), \hat{P}) / G_{0} ((), \hat{P}), Z_{2}),

where

{scriptL}^{b} {scriptM}^{γ} (P) \subseteq L {scriptM}^{γ} (P)

and

{scriptA}^{ɛ, b} (\overset{P}{true}) \subseteq A (\overset{P}{true})

, respectively, denote the subsets with energies bounded from above by b .…”

Section: Three‐dimensional Product Manifoldsmentioning

confidence: 97%

“…In fact, by results of the first author (cf. ), there is a bijection between the perturbed Yang–Mills connections on the bundle

\overset{P}{true}

and the perturbed geodesics on

{scriptM}^{γ} (P)

. Furthermore, he showed that the Morse homologies, defined from the perturbed parabolic Yang–Mills gradient flow on

\hat{P} \to Σ \times S^{1}

and the heat flow on

{scriptM}^{γ} (P)

, are isomorphic, provided that ε is small enough and that an energy bound b is chosen (cf.…”

Section: Three‐dimensional Product Manifoldsmentioning

confidence: 99%

“…Furthermore, apart from the two‐dimensional case, the Morse–Smale transversality is not yet proven and thus the homology

H M_{*} (A^{ɛ, b} ((), \hat{P}) / G_{0} ((), \hat{P}), Z_{2})

is presently not defined in the general case. One notable exception are products

Y = Σ \times S^{1}

as studied by the first author in , . He shows that in this case the gauge equivalence classes of gradient flow lines are in bijective correspondence with the gauge equivalence classes of heat gradient flow lines on

L {scriptM}^{γ} (P)

provided that ε is small enough and thus, the homology

H M_{*} (A^{ɛ, b} ((), \hat{P}) / G_{0} ((), \hat{P}), Z_{2})

is well‐defined.…”

Section: Three‐dimensional Product Manifoldsmentioning

confidence: 99%

“…The linearized flow equation then splits into an operator of mixed parabolic and elliptic type (cf. [4], [12], [23]). Short-time existence of the thus augmented equation is shown in [15].…”

Section: Introductionmentioning

confidence: 99%

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Elliptic Yang–Mills flow theory

Janner¹,

Swoboda

2015

Mathematische Nachrichten

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We lay the foundations of a Morse homology on the space of connections A(P) on a principal G‐bundle over a compact manifold Y, based on a newly defined gauge‐invariant functional scriptJ on A(P). While the critical points of scriptJ correspond to Yang–Mills connections on P, its L2‐gradient gives rise to a novel system of elliptic equations. This contrasts previous approaches to a study of the Yang–Mills functional via a parabolic gradient flow. We carry out the analytical details of our programme in the case of a compact two‐dimensional base manifold Y. We furthermore discuss its relation to the well‐developed parabolic Morse homology over closed surfaces. Finally, an application of our elliptic theory is given to three‐dimensional product manifolds Y=Σ×S1.

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“…Furthermore, he showed that the Morse homologies, defined from the perturbed parabolic Yang–Mills gradient flow on

\hat{P} \to Σ \times S^{1}

and the heat flow on

{scriptM}^{γ} (P)

, are isomorphic, provided that ε is small enough and that an energy bound b is chosen (cf. , ). Hence we have

H M_{*} (L^{b} M^{γ} (P), Z_{2}) ≅ H M_{*} (A^{ɛ, b} ((), \hat{P}) / G_{0} ((), \hat{P}), Z_{2}),

where

{scriptL}^{b} {scriptM}^{γ} (P) \subseteq L {scriptM}^{γ} (P)

and

{scriptA}^{ɛ, b} (\overset{P}{true}) \subseteq A (\overset{P}{true})

, respectively, denote the subsets with energies bounded from above by b .…”

Section: Three‐dimensional Product Manifoldsmentioning

confidence: 97%

“…In fact, by results of the first author (cf. ), there is a bijection between the perturbed Yang–Mills connections on the bundle

\overset{P}{true}

and the perturbed geodesics on

{scriptM}^{γ} (P)

. Furthermore, he showed that the Morse homologies, defined from the perturbed parabolic Yang–Mills gradient flow on

\hat{P} \to Σ \times S^{1}

and the heat flow on

{scriptM}^{γ} (P)

, are isomorphic, provided that ε is small enough and that an energy bound b is chosen (cf.…”

Section: Three‐dimensional Product Manifoldsmentioning

confidence: 99%

“…Furthermore, apart from the two‐dimensional case, the Morse–Smale transversality is not yet proven and thus the homology

H M_{*} (A^{ɛ, b} ((), \hat{P}) / G_{0} ((), \hat{P}), Z_{2})

is presently not defined in the general case. One notable exception are products

Y = Σ \times S^{1}

L {scriptM}^{γ} (P)

provided that ε is small enough and thus, the homology

H M_{*} (A^{ɛ, b} ((), \hat{P}) / G_{0} ((), \hat{P}), Z_{2})

is well‐defined.…”

Section: Three‐dimensional Product Manifoldsmentioning

confidence: 99%

“…The linearized flow equation then splits into an operator of mixed parabolic and elliptic type (cf. [4], [12], [23]). Short-time existence of the thus augmented equation is shown in [15].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Elliptic Yang–Mills flow theory

Janner¹,

Swoboda

2015

Mathematische Nachrichten

Self Cite

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show abstract

“…where denotes the standard Liouville form on T M . This construction turns out to be quite flexible and has been applied by several authors to various situations, both in symplectic geometry and in gauge theory (see [4,6,12,14,28,32,38,39,44]).…”

Section: Introductionmentioning

confidence: 99%

The Role of the Legendre Transform in the Study of the Floer Complex of Cotangent Bundles

Abbondandolo

Schwarz

2014

Comm Pure Appl Math

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Consider a classical Hamiltonian H on the cotangent bundle T * M of a closed orientable manifold M , and let L : T M → R be its Legendre-dual Lagrangian. In a previous paper we constructed an isomorphism Φ from the Morse complex of the Lagrangian action functional which is associated to L to the Floer complex which is determined by H. In this paper we give an explicit construction of a homotopy inverse Ψ of Φ. Contrary to other previously defined maps going in the same direction, Ψ is an isomorphism at the chain level and preserves the action filtration. Its definition is based on counting Floer trajectories on the negative half-cylinder which on the boundary satisfy "half" of the Hamilton equations. Albeit not of Lagrangian type, such a boundary condition defines Fredholm operators with good compactness properties. We also present a heuristic argument which, independently on any Fredholm and compactness analysis, explains why the spaces of maps which are used in the definition of Φ and Ψ are the natural ones. The Legendre transform plays a crucial role both in our rigorous and in our heuristic arguments. We treat with some detail the delicate issue of orientations and show that the homology of the Floer complex is isomorphic to the singular homology of the loop space of M with a system of local coefficients, which is defined by the pull-back of the second Stiefel-Whitney class of T M on 2-tori in M .

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Perturbed geodesics on the moduli space of flat connections and Yang–Mills theory

Cited by 2 publications

References 17 publications

Elliptic Yang–Mills flow theory

Elliptic Yang–Mills flow theory

The Role of the Legendre Transform in the Study of the Floer Complex of Cotangent Bundles

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