Ivancevic_Applied-Diff-Geom

770 Applied Differential Geometry: A Modern IntroductionN = 2 theory with one massive hypermultiplet was fully established bythe authors in [D’Hoker and Phong (1998a); D’Hoker and Phong (1998b);D’Hoker and Phong (1998c)], where it was shown that:(i) The resulting effective prepotential F (and thus the low energy effectiveaction) reproduces correctly the logarithmic singularities predictedby perturbation theory.(ii) F satisfies a renormalization group type equation which determinesexplicitly and efficiently instanton contributions to any order.(iii) The prepotential in the limit of large hypermultiplet mass m (as wellas large gauge scalar expectation value and small gauge coupling) correctlyreproduces the prepotentials for N = 2 super–YM theory with any numberof hypermultiplets in the fundamental representation of the gauge group.The N = 2 theory for arbitrary gauge algebra g and with one massivehypermultiplet in the adjoint representation was one such outstandingcase when g ≠ SU(N). Actually, as discussed previously, upon takingsuitable limits, this theory contains a very large number of modelswith smaller hypermultiplet representations R, and in this sense has auniversal aspect. It appeared difficult to generalize directly the Donagi–Witten construction of Hitchin systems to arbitrary g, and it was thusnatural to seek this generalization directly amongst the elliptic CM integrablesystems. It has been known now for a long time, thanks to thework of Olshanetsky and Perelomov [Olshanetsky and Perelomov (1976);Olshanetsky and Perelomov (1981)], that CM–systems can be defined forany simple Lie algebra. Olshanetsky and Perelomov also showed that theCM–systems for classical Lie algebras were integrable, although the existenceof a spectral curve (or, a Lax pair with a spectral parameter) aswell as the case of exceptional Lie algebras remained open. Thus severalimmediate questions are:(i) Does the elliptic CM–system for general Lie algebra g admit a Laxpair with spectral parameter?(ii) Does it correspond to the N = 2 supersymmetric gauge theory withgauge algebra g and a hypermultiplet in the adjoint representation?(ii) Can this correspondence be verified in the limiting cases when themass m tends to 0 with the theory acquiring N = 4 supersymmetry andwhen m → ∞, with the hypermultiplet decoupling in part to smaller representationsof g?According to [D’Hoker and Phong (1998a); D’Hoker and Phong (1998b);D’Hoker and Phong (1998c)], the answers to these questions can be statedsuccinctly as follows: