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.The sections of E is given by the Seiberg Witten equations with a gauge xing condition+s A; = FA , ; ; d A , A0 ; DA ; 66with ; = eiej ; ei ^ ej.The linearisation Ds at a point A; in Zs is the Fredholm operatorDs : 1 X; iIR , X; W+ ! 2+ X; iIR 0 X; iIR , X; W,given by81d+ , Im eiej ; ei ^ ej2Ds j A; ; = d , i ;:DA + i :This can be rewritten in terms of the long deformation complexG T0 ! 0 ! 1 , X; W+ ! 2+ , X; W, ! 0;where the operator G is the linearisation of the action of the gauge group G f =,idf; if and T is the linearisation of the section s regarded as a map fromA to E,1T j A; ; = d+ , Im eiej ; ei ^ ej; DA + i :2Thus we obtain Ds = T + G.In the case of the long deformation complex we have H0 C = 0 thein nitesimal action of the gauge group in an injective operator whenever isnot identically zero.We also have H1 C = Ker T =Im G Ker Ds and=H2 C = Coker T Coker Ds.Thus we get the exact same information.=In particular notice that the H2 C is an obstruction to Zs being a smoothand cut out transversely.Under the assumption that the four-manifold hasb+ X 1 it is possible to perturb the section by adding a self dual 2-form2and make H2 C trivial, as we discussed previously.The zero set Zs is theSeiberg Witten moduli space, which in this case is a compact smooth manifoldof dimension given by the index of the operator Ds,2c1 L , 2 +3Ind Ds = :4147 Another example is the dimensional reduction of the Seiberg Witten theory~on three-manifolds.Let Y be a closed oriented three-manifold.Let W be thespinor bundle associated to a Spinc-structure.If Y has b1 Y 1, then we canconsider the in nite dimensional manifold A of U 1 connections and non-trivialspinors, and the quotient X by the action of the gauge group.The bundle E~has bre 1 Y; iIR 0 Y; iIR , Y; W and the section s is given bys A; = FA , ; ; d A , A0 ; @A ; 67where ; is the 1-form given in local coordinates as ei ; ei and @A~is the Dirac operator on W twisted with the connection A.The linearisation Ds determines the short deformation complexDs~ ~0 ! 0 Y 1 Y , Y; W ! 0 Y 1 Y , Y; W ! 0;with8d +2Im ; , dfDs j A; f; ; = ,@A , i + if:d , if ; :~The space 0 Y 1 Y , Y; W is the tangent space TX at the pointA; of Zs.Since we are assuming b1 Y 1, we can guarantee that undera suitable perturbation of the section by a 1-form the zero set will not containpoints with trivial.The more general case can be worked out in the equivariant setup 27 , byconsidering X = A=Gb where Gb is the group of based gauge transformations,~i.e.those maps that act as the identity on a preferred bre of W.In this casethe framed con guration space X is a manifold, even though the action of thefull gauge group is not free.11.3.1 Atiyah-Je rey descriptionSome of the references available on the Mathai-Quillen formalism in SeibergWitten gauge theory are 25 , 10.A di erent construction that uses the BRSTmodel of equivariant cohomology see 19 , 12 , 13 can be found for instancein 16.Consider the case of perturbed Seiberg Witten equations on four-manifolds.The section s A; given in 66 satis esj s A; j2= S A; ;where S A; is the Seiberg Witten functional de ned in 11.We can identify the various terms of the Euler class, as given in proposition11.8 following the case of Donaldson theory analysed in 4 : this has been donein 25 , 10.148 Note that, according to 17 , we have variables dy = ; , with 2 1 X^and 2 , X; W+ , that represent a basis of forms on A; and the variable fwhich counts the gauge directions": f 2 0 X satis es eif 2 G.Similarly qand h are in the Lie algebra, i.e.in 0 X.w = ; 2 2+ X , X; W,is the variable along the bre.The term ds, therefore, is just the linearisation Ds of the Seiberg Wittenequations given in lemma 4.2:1ds A; ; = DA + i ; d+ , Im eiej ; ei ^ ej :2Moreover, it is clear that the operator is the map G of the complex 17that describes the in nitesimal action of the gauge group
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