Cauchy-Riemann Operators, Self—Duality, and the Spectral Flow

Dostoglou, Stamatis; Salamon, Dietmar

doi:10.1007/978-3-0348-9110-3_17

Cited by 14 publications

(20 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…While the general outline of the proof of Theorem A is analogous to the proof of the Atiyah-Floer conjecture in [DS1,DS2] there are several new ingredients in the present paper.…”

Section: Outline Of the Proof Of Theorem Amentioning

confidence: 97%

“…This refers to the linear and quadratic estimates needed in the proof of Theorem B (Sections 4 and 5). Secondly, we extend the estimates of [DS1] for the (1, p, ε)-norms to the (2, p, ε)-norms (see Lemmata 4.5-4.8 and Proposition 5.1). For the linear estimates the extension to the higher derivatives is quite subtle because of the ε-dependent norms.…”

Section: Outline Of the Proof Of Theorem Amentioning

confidence: 99%

See 1 more Smart Citation

Gromov--Witten invariants of symplectic quotients and adiabatic limits

Gaio¹,

Salamon²

2005

Journal of Symplectic Geometry

Self Cite

127

View full text Add to dashboard Cite

We study pseudoholomorphic curves in symplectic quotients as adiabatic limits of solutions of a system of nonlinear first order elliptic partial differential equations in the ambient symplectic manifold. The symplectic manifold carries a Hamiltonian group action. The equations involve the Cauchy-Riemann operator over a Riemann surface, twisted by a connection, and couple the curvature of the connection with the moment map. Our main theorem asserts that the genus zero invariants of Hamiltonian group actions defined by these equations are related to the genus zero Gromov-Witten invariants of the symplectic quotient (in the monotone case) via a natural ring homomorphism from the equivariant cohomology of the ambient space to the quantum cohomology of the quotient.

show abstract

“…While the general outline of the proof of Theorem A is analogous to the proof of the Atiyah-Floer conjecture in [DS1,DS2] there are several new ingredients in the present paper.…”

Section: Outline Of the Proof Of Theorem Amentioning

confidence: 97%

Section: Outline Of the Proof Of Theorem Amentioning

confidence: 99%

Gromov--Witten invariants of symplectic quotients and adiabatic limits

Gaio¹,

Salamon²

2005

Journal of Symplectic Geometry

Self Cite

127

View full text Add to dashboard Cite

show abstract

“…(See for example [24], [31].) Conversely, any solution of (6), (7), and (8) has finite energy. Given any two fixed points…”

Section: Floer Homology For Symplectic Fixed Pointsmentioning

confidence: 99%

“…These paths are in one-to-one correspondence with the fixed points of φ 1 = ψ −1 1 φ. Now let u : R 2 → M be any solution of (11) and (7) and define…”

Section: Floer Homology For Symplectic Fixed Pointsmentioning

confidence: 99%

See 1 more Smart Citation

Instanton homology and symplectic fixed points

Dostoglou¹,

Salamon²

1994

Symplectic Geometry

Self Cite

View full text Add to dashboard Cite

* This research has been partially supported by the SERC. * (M ; Q). A special case is where the bundle Q = P f is the mapping cylinder of a nontrivial SO(3)-bundle π : P → Σ over a Riemann surface Σ for an automorphism f : P → P. The underlying 3-manifold is the mapping cylinder M = Σ h of Σ for the diffeomorphism h : Σ → Σ induced by f. Then the flat connections on P f correspond naturally to the fixed points of the symplectomorphism φ f : M(P) → M(P)

show abstract

Elliptic Yang–Mills flow theory

Janner¹,

Swoboda

2015

Mathematische Nachrichten

View full text Add to dashboard Cite

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.

show abstract

Cauchy-Riemann Operators, Self—Duality, and the Spectral Flow

Cited by 14 publications

References 22 publications

Gromov--Witten invariants of symplectic quotients and adiabatic limits

Gromov--Witten invariants of symplectic quotients and adiabatic limits

Instanton homology and symplectic fixed points

Elliptic Yang–Mills flow theory

Contact Info

Product

Resources

About