Multiscale N=2 SUSY field theories, integrable systems and their stringy/brane origin -- I

Abstract

We discuss supersymmetric Yang-Mills theories with the multiple scales in the brane language. The issue concerns N=2 SUSY gauge theories with massive fundamental matter including the UV finite case of $n_{f}=2n_c$, theories involving products of SU(n) gauge groups with bifundamental matter, and the systems with several parameters similar to $\Lambda_{QCD}$. We argue that the proper integrable systems are, accordingly, twisted XXX SL(2) spin chain, $SL(p)$ magnets and degenerations of the spin Calogero system. The issue of symmetries underlying integrable systems is addressed. Relations with the monopole systems are specially discussed. Brane pictures behind all these integrable structures in the IIB and M theory are suggested. We argue that degrees of freedom in integrable systems are related to KK excitations in M theory or D-particles in the IIA string theory, which substitute the infinite number of instantons in the field theory. This implies the presence of more BPS states in the low-energy sector.