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., 1=2 1=2This explains why one gets det Q instead of det Q = Pf Q.In order to generalise this argument to the in nite dimensional context, theproblem is reformulated in terms of a functional F with non-degenerate minima.The approximation of the partition function in this case can be taken to be,1=2the well de ned mathematical object det Q , where Q is a positive ellipticoperator the Hessian of the functional F at a minimum and the determinantis the Ray-Singer determinant 24 , 25.If the coupling constant is large thisapproximation method no longer works and the partition function is in generalno longer computable.12.1.4 The u-planeIn the case of N = 2 supersymmetry, the auxiliary elds that are introducedcan be described as two independent variables of the type of the w used inthe de nition of the fermionic integral, and a eld which is a section of theadjoint bundle of E.The classical potential can be written as a function Vand as mentioned before the supersymmetry imposes that in the vacuum stateV = 0.This allows for certain symmetries of the vacuum.This means thatthe vacuum state is not an isolated point but there is some parametrisation ofa certain manifold of possible vacuum states.In our case the parameter thatclassi es inequivalent vacua is Tr 2.This is better said by introducing a variable u = hTr 2 i which is theexpectation value with respect to the partition function Z of Tr 2.Theexpectation value of the eld is proportional to a variable a, h i a.In the1classical limit, that is, when the coupling is weak, one has the relation u a2.2In the strong coupling range the relation is more complicated.169In terms of the parameter a one has the corresponding modulusa 4 ia = + :2 g2 a, 1The symmetry of the action under the transformation 7! can be formallydescribed in terms of a Legendre transformation over a potential called prepo-tential in the Physics literature F.In fact, a dual variable aD is introduced bythe relation@F aaD = ; 69@aand a dual eld D is de ned by h D i aD.Here dual" is intended inanalogy to coordinates and moments in classical mechanics that are related by, 1a Legendre transform similar to 69.The transformation 7! exchangesthe action Z with a dual action ZD where the eld is replaced with D anda with aD.This exchanges weak and strong coupling.The reason why this can be still thought of as electromagnetic duality is thatone thinks of the purely electric or purely magnetic charge as quantities qe = neaand qm = nmaD, for a pair of integers ne; nm.One can also consider stateswhich are called dyons in the literature that have both electric and magneticcharge q = nea + nmaD.The group SL 2; Z acts by mixing the electric andZthe magnetic chargene a b ne7! :nm c d nmIf one wants to express the variables a and aD as functions of the parameterthat determines the vacuum state, a u and aD u , one gets two multivaluedfunctions, de ned for u 2 C with branch cuts.In particular one can compute theImonodromy at the branch points 2.One point is certainly the one at in nity,where the weak coupling range is attained.In this case the prepotential takes thei a2form F a a2 ln and as u 7! e2 iu one has a 7! ,a and aD 7! ,aD +2a.2 2Thus the monodromy at u = 1 is,1 2M1 = :0 ,1An argument depending on the factorisation of the matrix M1 in SL 2; ZZshows that there are other two branching points.Up to the choice of a normal-ising constant these can be taken to be u = 1.As u ! 1 the strong couplingrange is attained.The corresponding monodromies 2 are1 0M1 =,2 1170and,1 2M, 1 = :,2 3The physical interpretation of the eigenvalues of the monodromy matricesleads to interpreting these branch points as the vacua at which a magneticmonopole when u =1 or a 1; ,1 -dyon when u = ,1 become massless.12.1.5 Elliptic curvesGiven the data obtained above by physical arguments, namely the puncturedsphere CP1 ,f1; 1; ,1g or u-plane and the prescribed monodromies at theIpunctures, it is possible to proceed with a rigorous construction.The mon-odromies obtained above span a subgroup , 2 in SL 2; Z.The u-plane isZequivalent to the quotient of the upper half plane with respect to the group, 2.This gives the moduli of the family of elliptic curvesy2 = x2 , 1 x , uthat becomes singular at the points u = 1
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