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.Then the solutionA t ; t is of nite energy.In fact, the total variation of the Chern-Simons-Dirac functional isZ1dC Aa ; a , C Ab; b = , C A t ; t dtdt,1Z Z1 1d= , rC A t ; t ; A t ; t dt = jrC A t ; t j2 dt;L2dt,1 ,1by equations 18 and 19.The interesting point is the fact that the nite energy condition 40 is alsosu cient to provide the existence of asymptotic values.This implication is moresubtle and it relies upon the proof of some estimates for solutions on cylinders23 24.Suppose given a nite energy solution A t ; t of 18 and 19 on Y IR.The solution A t ; t is irreducible if it is an irreducible solution of thefour-dimensional Seiberg Witten equations, that is, if the spinor t is notidentically vanishing.Proposition 7.6 Let A t ; t be an irreducible nite energy solution of 18and 19 on Y IR.Then there exist critical points a = Aa ; a and b = Ab ; b~in Mc Y; s , such that we havelim A t ; t = Aa; a ;t!,1lim A t ; t = Ab; b ;t!+179 in the L2 topology.Moreover, if Aa; a is an irreducible critical point, a =0,61then we have exponential decay2k A t ; t , Aa; a k Ce, j tj ;L2 Y ftg1for all t ,T0.The rate of decay is controlled by the eigenvalues of theHessian at the critical point: minfj ajg, where f ag is the set of non-zeroeigenvalues of the Hessian H Aa ; a.We have a similar decay as t ! +1 ifAb; b is an irreducible critical point.A proof of the existence of asymptotic values and an estimate of the rateof decay can be derived from the Lojasiewicz inequalities 34.These estimatethe distance from a critical point in terms of the L2-norm of the gradient -nite energy implies nite length".The argument depends on the fact that theChern-Simons-Dirac functional C is a real analytic function on the con gura-tion space of connections and sections.Since the rate of the exponential decay is controlled by the smallest non-trivial absolute value of the eigenvalues of the Hessian HAa ; a , we can choose aunique for all critical points.Thus, given two irreducible critical points a = Aa; a and b = Ab; b in~Mc Y; s , we consider all solutions A t ; t of the ow equations with limitslim A t ; t = Aa; a and lim A t ; t = Ab ; b ;t!,1 t!+1and such thatk A t ; t , Aa; a k Ce tL21for t 2 ,1; ,T0 andk A t ; t , Ab; b k Ce, tL21for t 2 T0; 1.In the next section we describe how to t these solutions into a con gurationspace with a free action of a gauge group, so that we can nally introduce themoduli spaces of ow lines.7.4 Flow lines: moduli spacesThere are other technical issues that need to be addressed in order to de nemoduli spaces of ow lines.The reader may refer to 21 , and to 23 for theanalogous problems treated within the context of Donaldson theory.So far we have considered the ow equations 18 and 19 for A t ; t ina temporal gauge.With this notation, we can consider the action of the groupof time independent gauge transformations on the set of solutions.Notice,however, that, in order to de ne virtual tangent spaces to the moduli spaces of80 ow lines and set up the corresponding Fredholm analysis, we need to considerslices of the gauge action.This corresponds to the analysis worked out in 23for the case of Donaldson Floer theory.The slice at an element A; is de ned by the condition G ; = 0.A;It is easy to verify that the ow A t ; t is tangent to the slice at A t ; titself.However, if we need to consider positive dimensional moduli spaces thiswill be the case especially in the context of the equivariant Floer theory discussedlater in the chapter , we need to consider solutions in the slice at a xed elementA0; 0.A solution A t ; t of 18 and 19 is then no longer in the sliceat A0; 0 , unless time dependent gauge transformations are allowed, but thesewill break the temporal gauge condition.This problem can be overcome by replacing the temporal gauge conditionwith the condition of standard form introduced in 23.This allows timedependent gauge transformations.One can then add a correction term A;in the ow equations in order to make the ow tangent to a xed slice of thegauge action at the point A0; 0.The linearisation of the equations containsthe extra term that linearises A; ,@D A t ; t ; = + A t ; t + :@tAnisotropic Sobolev norms L2 can be introduced on the spaces of connec-k;mtions and sections and gauge transformations, as analysed in 23.The linearisa-tion is a compact operator with respect to these norms
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