Ivancevic_Applied-Diff-Geom

780 Applied Differential Geometry: A Modern Introduction(ii) The gauge coupling g and the θ−angle are related to the modulusof the torus Σ = C/(2ω 1 Z + 2ω 2 Z) byτ = ω 2ω 1= θ2π + 4πig 2 ;(iii) The SW curve Γ is the spectral curve of the elliptic CM model,defined byΓ = {(k, z) ∈ C × Σ, det(kI − L(z)) = 0}and the SW 1–orm is dλ = k dz. Γ is invariant under the Weyl group ofSU(N).(iv) Using the Lax equation ˙L = [L, M], it is clear that the spectral curveis independent of time, and can be dependent only upon the constants ofmotion of the CM–system, of which there are only N. These integrals ofmotion may be viewed as parametrized by the quantum moduli of the SWsystem.(v) Finally, dλ = kdz is meromorphic, with a simple pole on each of theN sheets above the point z = 0 on the base torus. The residue at each ofthese poles is proportional to m, as required by the general set–up of SWtheory.Four Fundamental Theorems1. The spectral curve equation det(kI − L(z)) = 0 is equivalent to( 1ϑ 1 (z − m ∂ )2ω 1 ∂k )|τ H(k) = 0,where H(k) is a monic polynomial in k of degree N, whose zeros (or equivalentlywhose coefficients) correspond to the moduli of the gauge theory. IfH(k) = ∏ Ni=1 (k − k i), then∮limq→012πiA ikdz = k i − 1 2 m.Here, ϑ 1 is the Jacobi ϑ−function, which admits a simple series expansionin powers of the instanton factor q = e 2πiτ , so that the curve equation mayalso be rewritten as a series expansion∑(−) n q 1 2 n(n−1) e nz H(k − n · m) = 0, (4.286)n∈Z