The curve of compactified 6Dgauge theories and integrable systems

We analyze the Seiberg-Witten curve of the six-dimensional N=(1,1) gaugetheory compactified on a torus to four dimensions. The effective theory in fourdimensions is a deformation of the N=2* theory. The curve is naturallyholomorphically embedding in a slanted four-torus--actually an abeliansurface--a set-up that is natural in Witten's M-theory construction of N=2theories. We then show that the curve can be interpreted as the spectral curveof an integrable system which generalizes the N-body elliptic Calogero-Moserand Ruijsenaars-Schneider systems in that both the positions and momenta takevalues in compact spaces. It turns out that the resulting system is not simplydoubly elliptic, rather the positions and momenta, as two-vectors, take valuesin the ambient abelian surface. We analyze the two-body system in some detail.The system we uncover provides a concrete realization of a Beauville-Mukaisystem based on an abelian surface rather than a K3 surface.Comment: 22 pages, JHEP3, 4 figures, improved readility of figures, added reference