Liouville vortex and φ4 kink solutions of the Seiberg–Witten equations

Nergiz, Serdar; Saçlıoğlu, Cihan

doi:10.1063/1.531628

Cited by 12 publications

(7 citation statements)

References 10 publications

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“…Hence any pair (M G , A G i ) with G given by 4.14 is a solution the Seiberg-Witten equations 3.6. In fact, these solutions resemble the two-dimensional solutions of the SeibergWitten equations that were discussed in [6]. Their solutions emerged as solutions to the same Liouville equation 4.14, however the coordinate space variable x + = x + iy appears rather than the target space variable χ used here.…”

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confidence: 86%

Non-L2 solutions to the Seiberg–Witten equations

Adam

Muratori

Nash

2000

Journal of Mathematical Physics

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Abstract:We show that a previous paper of Freund describing a solution to the SeibergWitten equations has a sign error rendering it a solution to a related but different set of equations. The non-L 2 nature of Freund's solution is discussed and clarified and we also construct a whole class of solutions to the Seiberg-Witten equations. An important vanishing theorem of [1], reminiscent of the Lichernowicz-Weitzenböck vanishing theorems, shows that there are no non-trivial solutions to the Seiberg-Witten equations on four manifolds with non-negative Riemannian scalar curvature. However one can have non-trivial solutions which are singular in some way-for example one could have a non-trivial solution which was not L 2 : in [3] Freund describes such a non-L 2 to the Seiberg-Witten equations on R 4 . Unfortunately a sign discrepancy in [3] means that the expressions given there obey equations which differ from the Seiberg-Witten equations in a certain sign. These other equations also admit L 2 solutions as well as non-L 2 ones and so Freund's equations are fundamentally different from the Seiberg-Witten equations.In § 2 we describe the Seiberg-Witten equations in a fairly explicit manner so as to expose notational conventions and matters of signs. In section § 3 we give the details concerning Freund's work and then in section § 4 we give an L 2 solution of Freund's equations and a class of solutions to the Seiberg-Witten equations. § 2. The Seiberg-Witten equations If M is an oriented Riemannian four manifold with metric g ij then the data we need for the Seiberg-Witten equations are a U (1) connection A i on M and a local spinor field M .If F ij is the curvature of A i , so that its self-dual part F + ij is given by F + ij = 1/2(F ij + ( √ g/2)ǫ ijkl F kl ), then the Seiberg-Witten equations are

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confidence: 86%

Non-L2 solutions to the Seiberg–Witten equations

Adam

Muratori

Nash

2000

Journal of Mathematical Physics

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“…The equations (4.36) with θ 11 = 2θ 12 = 0 and their explicit solutions were extensively discussed in the literature (see, e.g., [37,38,30]). Note that the choice of the projector P (2) 0 on H 2 ensures a finite charge 13 Note that these equations can also be obtained in the context of the perturbed SW equations if one chooses the perturbation proportional to P (2) 0 . Then there is no necessity to introduce an indefinite normed space.…”

Section: Solutions To the Unperturbed Sw Equationsmentioning

confidence: 99%

“…This assertion is also true for lower-dimensional reductions of the SW equations, i.e., these reductions also do not exhibit regular solutions on Ê n≤3 with a nonzero topological charge. Nevertheless, one may construct nontrivial non-L 2 solutions, as it has been done, e.g., in [12,13,14,15].…”

Section: Introductionmentioning

confidence: 99%

Seiberg–Witten monopole equations on noncommutative R4

Popov

Сергеев

Wolf

2003

Journal of Mathematical Physics

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It is well known that, due to vanishing theorems, there are no nontrivial finite action solutions to the Abelian Seiberg-Witten (SW) monopole equations on Euclidean four-dimensional space Ê 4 . We show that this is no longer true for the noncommutative version of these equations, i.e., on a noncommutative deformation Ê 4 θ of Ê 4 there exist smooth solutions to the SW equations having nonzero topological charge. We introduce action functionals for the noncommutative SW equations and construct explicit regular solutions. All our solutions have finite energy. We also suggest a possible interpretation of the obtained solutions as codimension four vortexlike solitons representing D(p − 4)-and D(p − 4)-branes in a Dp-Dp brane system in type II superstring theory.Ê 4 and fix our notation. In section 3 we introduce the noncommutatively deformed non-Abelian SW equations. We derive them from properly deformed U (2) self-duality type equations in eight dimensions [17] by a dimensional reduction to four dimensions (cf.[18]). The resulting U + (1) × U − (1), U + (1) and U − (1) noncommutative SW equations can also be produced from appropriate action functionals by using a Bogomolny type transformation. We point out that the U + (1)×U − (1), U + (1) and U − (1) noncommutative SW equations share the same commutative limit. In section 4 we present a number of regular solutions to the noncommutative SW equations and discuss their D-brane interpretation in a string theoretic context. In section 5 we conclude with a brief summary and open problems. Finally, in the Appendix we perform the Bogomolny type transformation for the noncommutative U + (1) × U − (1) SW action functional.2 SW monopole equations on Ê 4 SW action functional. In this paper we consider the SW equations on the Euclidean space Ê 4 , provided with the standard metric g = (δ µν ), where µ, ν, . . . = 1, . . . , 4. The (energy) functional E = E(A, Φ) for these equations has the form (cf., e.g., [19,20,21])(2.1) u(1)) is a connection one-form on Ê 4 with pure imaginary smooth coefficients and Φ ∈ C ∞ (Ê 4 , 2 ) is a Weyl spinor given by a smooth complex-valued vector function on Ê 4 . We denote by F + A ∈ Ω 2 + (Ê 4 , u(1)) the self-dual part of the curvature F A of A and by D A the covariant

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“…(31) of Ref. [34] (near the singularity) which is a solution of the dimensionally reduced monopole equations of Ref. [27].…”

Section: )mentioning

confidence: 99%

The BF formalism for QCD and quark confinement

Fucito¹,

Martellini²,

Zeni³

1997

Nuclear Physics B

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Using the BF version of pure Yang-Mills, it is possible to find a covariant representation of the 't Hooft magnetic flux operator. In this framework, 't Hooft's pioneering work on confinement finds an explicit realization in the continuum. Employing the Abelian projection gauge we compute the expectation value of the magnetic variable and find the expected perimeter law. We also check the area law behaviour for the Wilson loop average and compute the string tension which turns out to be of the right order of magnitude.

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Liouville vortex and φ4 kink solutions of the Seiberg–Witten equations

Cited by 12 publications

References 10 publications

Non-L2 solutions to the Seiberg–Witten equations

Non-L2 solutions to the Seiberg–Witten equations

Seiberg–Witten monopole equations on noncommutative R4

The BF formalism for QCD and quark confinement

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