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.The U 2 -bundle E` has Det E` = Det E and c2 E` =c2 E , `.The work of Feehan and Leness then proceeds by constructing the links ofthe strata Masd and Ms1 in the top Uhlenbeck level and their intersection withthe geometric representatives of the cohomology classes: H X; C ! H2, BE ; C :Q QIf there were no reducibles in the lower strata, this would be enough to re-cover the relation given in 12.3.Unfortunately, the lower strata also containreducibles, hence a detailed analysis of the intersection of the geometric repre-sentatives with the lower strata is necessary.The authors consider tubular neighbourhood of the lower strata de ned bythe gluing maps.This requires a very careful and technically demanding analysisof the relevant gluing theorems 10 , 11.It should be mentioned that the approach of Feehan and Leness has led toa rigorous mathematical proof of some results derived from the physical theoryof S-duality, besides the original goal of establishing the Witten conjecture.Anexample is the proof of a conjecture of Marino, Moore, and Peradze 19 , derived~by Feehan, Kronheimer, Leness, and Mrowka 6.In 19 the authors considerthe Seiberg Witten invariants of a smooth oriented 4-manifold X with b+ 1,2assembled in the expressionX1w w2 +c1 L w2SWX h = ,1 Ns X eh c1 L h; X i:s2S XHere L is the determinant line bundle of the Spinc-structure as we discussedin Part I, and h varies in H2 X; IR and w is any integral lift of the StiefelWhitney class w2 X.They introduce the notion of superconformal simpletype" to denote a class of compact oriented smooth 4-manifolds with b+ 12175wand of Seiberg Witten simple type, such that SWX h has a zero at h = 0 oforder at leastc X , 3;where1c X = , 7 X +11 X :4In 20 it is then shown that all known 4-manifolds with b+ 1 are of2superconformal simple type: in fact, it is shown that the superconformal simpletype property is preserved under blowup, bre sum along embedded tori, knotsurgery, and generalised log transforms.Moreover, it is shown that compactcomplex surfaces with b+ 1 are of superconformal simple type.This leads to2the following conjecture 20.Conjecture 12.6 All compact oriented smooth 4-manifolds with b+ 1 are of2superconformal simple type.Using the analysis of PU 2 -monopoles of Feehan and Leness, the conjectureis reduced in 6 to the technical hypothesis that reducibles which appear in thelower levels of the Uhlenbeck compacti cation of the moduli space of PU 2 -monopoles do not contribute any non-trivial Seiberg Witten invariants.This ispossible under the assumption that the 4-manifold is of Seiberg Witten simpletype and is abundant, that is, the intersection form restricted to the orthogonalcomplement of the basic classes contains a hyperbolic sublattice.The latter isa condition which guarantees that the index of the twisted Dirac operator 70is positive, Ind DA 0.This is a technical condition that is needed in orderto apply the results of 9.176References1 M.F.Atiyah, Duality and quantum eld theory, Topics in symplectic 4-manifolds Irvine, CA, 1996 , 1 7, First Int.Press Lect.Ser., I, Internat.Press, 1998.2 A.Bilal, Duality in N=2 SUSY SU 2 Yang-Mills Theory: A pedagogicalintroduction to the work of Seiberg and Witten, preprint hep-th 9601007.3 R.Bott, On some recent interactions between Mathematics and Physics,Canad.Math.Bull.28 N.2 1985 129-164.4 S.Bradlow, O.Garc ia-Prada, Non-abelian monopoles and vortices, Geome-try and physics Aarhus, 1995 , 567 589, Lecture Notes in Pure and Appl.Math., 184, Dekker, New York, 1997.5 R.Dijkgraaf, Fields, Strings, and Duality, Sym quantiques LesetriesHouches, 1995 , 3 147, North-Holland, 1998.6 P.M.N.Feehan, P.B.Kronheimer, T.G.Leness, T.S.Mrowka,PU 2 monopoles and a conjecture of Marino, Moore, and Peradze,math.DG 9812125.7 P.M.N.Feehan, T.G.Leness, PU 2 monopoles and relations between four-manifold invariants, preprint, dg-ga 9709022.8 P.M.N.Feehan, T.G.Leness, PU 2 Monopoles, I: Regularity, UhlenbeckCompactness, and Transversality.preprint, dg-ga 9710032.9 P.M.N.Feehan, T.G.Leness, PU 2 monopoles, II: Highest-level sin-gularities and relations between four-manifold invariants, preprint, dg-ga 9712005.10 P.M.N.Feehan, T.G.Leness, PU 2 monopoles, III: Existence of gluingand obstruction maps, preprint.11 P.M.N.Feehan, T.G.Leness, PU 2 monopoles, IV: Surjectivity of gluingmaps, preprint.12 P.M.N.Feehan, Generic metrics, irreducible rank-one PU 2 monopoles,and transversality, preprint, math.DG 9809001.13 P.M.N.Feehan, T.G.Leness, Donaldson invariants and wall-crossing for-mulas.I: Continuity of gluing maps, preprint, math.DG 9812060.14 P.M.N [ Pobierz całość w formacie PDF ]