[...]... cients These will be useful in discussing the approach to Seiberg Witten gauge theory through Quantum Field Theory and in the construction of the Seiberg WittenFloer homology The brief exposition presented here partly follows 4 , which we recommend as a very good reference for the role of equivariant cohomology in Quantum Field Theory The basic idea is the following: when a group G which we assume... Seiberg Witten theory on Kahler and symplectic manifolds is the following simple remark Remark 2.14 let V be a 4,dimensional real vector space endowed with a positive de nite scalar product ; Let J be an orthogonal complex structure on V , and let !v; w = v; Jw Then we have a decomposition 2 C = C!  2;0  0;2 : + I I Symplectic geometry plays a prominent role in Seiberg Witten gauge theory Computation... formula p L 2.6 holds true for the twisted Dirac operator on S + 2.4 Topology of the gauge group Recall that the gauge group of a G bundle is de ned as the group of self equivalences of the bundle, namely the group of smooth maps  :U !G  =g  g ; where the bundle is trivial over U and has transition functions g The gauge group is an in nite dimensional manifold It is made into a Banach manifold with... implicit function theorem is a crucial tool in order to show that, after a generic perturbation, the moduli spaces of solutions of elliptic equations modulo gauge symmetries are smooth manifolds In a later chapter, in order to construct a gauge theory of three-manifolds, we shall consider the Seiberg Witten equations on a tube X = Y  IR, where Y is a compact oriented three-manifold without boundary... norms will be recalled later in this section In the following k we always assume to work with L2 -gauge transformations, with k  3, since we k think of the con guration space of pairs A;  endowed with the L2,1 -norm k If the structure group is abelian, or if the G-bundle is topologically trivial, then the gauge group has a simpler description as G = MX; G, the space of maps from X to G We have the... space of sections of the full spinor bundle W = W +  W , , the Dirac operator of the form 0 D D 0 is formally self adjoint An essential tool in Spin geometry, which is very useful in Seiberg Witten gauge theory, is the Weitzenbock formula Theorem 2.6 The Dirac operator D satis es the Weitzenbock formula: D2 s = r r +  + ,i F s; 4 4 where r is the formal adjoint of the covariant derivative  is... building blocks" of smooth four-manifolds could be A good reference for the notions introduced in this section is 18 20 2.6 The index theorem We recall here very brie y some essential results of index theory A very readable reference is the book by R Boos and D.D Bleecker, 2 Recall that a bounded linear operator acting between Banach spaces is Fredholm if it has nite dimensional kernel and cokernel... trivial, then the gauge group has a simpler description as G = MX; G, the space of maps from X to G We have the following easy lemma Lemma 2.11 In the case G = U 1, the set of connected components of the gauge group G is H 1 X; Z  Z 17 2.5 Symplectic and Kahler Manifolds Recall that a manifold X is endowed with a symplectic structure if a closed 2 form ! is given on X such that it is non-degenerate,... with di erential dW 1 + 1 d computes the equivariant cohomology of X , H   X ; dW 1 + 1 d  HG X; IR: =  G 2.8 Sobolev norms It is well know since 20 and the subsequent 23 see also 8  that in gauge theories the appropriate way of endowing the in nite dimensional spaces of connections and sections with a manifold structure is by appropriate Sobolev norms We recall here a few basic notions Let... conversation a most enjoyable experience It is also dedicated to the memory of my mother, a successful architect and fashion designer, who rst taught me the love of mathematics as a form of art Matilde Marcolli Department of Mathematics, 2-275 Massachusetts Institute of Technology Cambridge, MA 02139, USA 11 Part I Seiberg Witten on four-manifolds You have the glow of a man who knows brahman! Tell me .