On non-$L^2$ solutions to the Seiberg-Witten equations
We show that a previous paper of Freund describing a solution to the Seiberg-Witten 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
002 raM 51 1v521300/0ht-pe:hviXraOnnon-L2solutionstotheSeiberg–Wittenequations
by
ChristophAdam ,BrunoMuratori andCharlesNash
Institutf¨urTheoretischePhysik DepartmentofMathematicalPhysics,
Universit¨atKarlsruhe,NationalUniversityofIreland,
Germany.
Maynooth,Ireland.
Abstract:WeshowthatapreviouspaperofFreunddescribingasolutiontotheSeiberg–Wittenequationshasasignerrorrenderingitasolutiontoarelatedbutdi erentsetofequations.Thenon-L2natureofFreund’ssolutionisdiscussedandclari edandwealsoconstructawholeclassofsolutionstotheSeiberg–Wittenequations.
With§1.Introduction
theintroductionoftheSeiberg–Wittenequations[1]therecomeawealthofresultsonfourmanifoldtheoryandanewimprovedpointofviewonDonaldsontheorywithanAbeliangaugetheorysupplantinganon-Abelianone—cf.[2]forareview.
Animportantvanishingtheoremof[1],reminiscentoftheLichernowicz–Weitzenb¨ockvanishingtheorems,showsthattherearenonon-trivialsolutionstotheSeiberg–Wittenequationsonfourmanifoldswithnon-negativeRiemannianscalarcurvature.Howeveronecanhavenon-trivialsolutionswhicharesingularinsomeway—forexampleonecouldhaveanon-trivialsolutionwhichwasnotL2:in[3]Freunddescribessuchanon-L2totheSeiberg–WittenequationsonR4.Unfortunatelyasigndiscrepancyin[3]meansthattheexpressionsgiventhereobeyequationswhichdi erfromtheSeiberg–Wittenequationsinacertainsign.TheseotherequationsalsoadmitL2solutionsaswellasnon-L2onesandsoFreund’sequationsarefundamentallydi erentfromtheSeiberg–Wittenequations.
In§2wedescribetheSeiberg–Wittenequationsinafairlyexplicitmannersoastoexposenotationalconventionsandmattersofsigns.Insection§3wegivethedetailscon-cerningFreund’sworkandtheninsection§4wegiveanL2solutionofFreund’sequationsandaclassofsolutionstotheSeiberg–Wittenequations.§If2.MTheisanSeiberg–WittenorientedRiemannianequations
fourmanifoldwithmetricgijthenthedataweneedfortheSeiberg–WittenequationsareaU(1)connectionAionMIfFijisthecurvatureofAi,sothatitsself-dualpartF+
andalocalspinor eldij
isgivenbyF+M.
ij=1/2(Fij+(
√2
