The curve of compactified 6D gauge theories and integrable systems

We analyze the Seiberg-Witten curve of the six-dimensional N = (1, 1) gauge theory compactifted on a torus to four dimensions. The effective theory in four dimensions is a deformation of the N = 2* theory. The curve is naturally holomorphically embedding in a slanted four-torus - actually an abelian surface - a set-up that is natural in Witten's M-theory construction of M - 2 theories. We then show that the curve can be interpreted as the spectral curve of an integrable system which generalizes the N-body elliptic Calogero-Moser and Ruijsenaars-Schneider systems in that both the positions and momenta take values in compact spaces. It turns out that the resulting system is not simply doubly elliptic, rather the positions and momenta, as two-vectors, take values in the ambient abelian surface. We analyze the two-body system in some detail. The system we uncover provides a concrete realization of a Beauville-Mukai system based on an abelian surface rather than a K3 surface.