Ivancevic_Applied-Diff-Geom

782 Applied Differential Geometry: A Modern IntroductionF (1) =q2πiF (2) = q28πi∑S i (a i ), (4.289)⎡i⎣ ∑ iS i (a i )∂ 2 i S i (a i ) + 4 ∑ i≠j⎤S i (a i )S j (a j )(a i − a j ) 2 − S i(a i )S j (a j )⎦(a i − a j − m) 2 .The perturbative corrections to the prepotential of (4.288) indeed preciselyagree with the predictions of asymptotic freedom. The formulas (4.289) forthe instanton corrections F (1) and F (2) are new, as they have not yet beencomputed by direct field theory methods. The moduli k i , 1 ≤ i ≤ N, of thegauge theory are evidently integrals of motion of the system. To identifythese integrals of motion, denote by S be any subset of {1, · · · , N}, andlet S ∗ = {1, · · · , N} \ S, ℘(S) = ℘(x i − x j ) when S = {i, j}. Let also p Sdenote the subset of momenta p i with i ∈ S.4. For any K, 0 ≤ K ≤ N, let σ K (k 1 , · · · , k N ) = σ K (k) be the K−thsymmetric polynomial of (k 1 , · · · , k N ), defined byH(u) =N∑(−) K σ K (k)u N−K .K=0[K/2]∑σ K (k) = σ K (p) +l=1m 2l∑|S i ∩S j |=2δ ij1≤i,j≤lThen4.14.11.7 CM and SW Theory for General Lie Algebral∏σ K−2l (p (∪ li=1 S ∗) i) [℘(S i ) + η 1].ω 1Now, we consider the N = 2 supersymmetric gauge theory for a generalsimple gauge algebra g and a hypermultiplet of mass m in the adjointrepresentation. Then we have the following results [D’Hoker and Phong(1998d)].The SW curve of the theory is given by the spectral curveΓ = {(k, z) ∈ C × Σ; det(kI − L(z)) = 0}of the twisted elliptic CM–system associated to the Lie algebra g. The SWdifferential dλ is given by dλ = kdz.The function R(k, z) = det(kI − L(z)) is polynomial in k and meromorphicin z. The spectral curve Γ is invariant under the Weyl group of g.It depends on n complex moduli, which can be thought of as independentintegrals of motion of the CM–system.The differential dλ = kdz is meromorphic on Γ, with simple poles.i=1