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

TL;DR

The paper develops a comprehensive bridge between multiscale $N=2$ SUSY gauge theories and finite-dimensional integrable systems, casting Seiberg-Witten data in terms of Lax pairs, spectral curves, and monopole dynamics. It shows how brane constructions in Type IIB and Type IIA/M-theory realize the same integrable structures as spin chains (twisted $SL(2)$ XXX and higher-rank magnets) and spin Calogero systems, with KK/D0 degrees of freedom playing the role of nonperturbative regulators and expanding the BPS spectrum. A central theme is the dual Lax representations (2×2 vs n×n) connected by brane dualities, and the monopole–spin chain correspondence that unifies Nahm constructions with finite-dimensional spin systems. The work also introduces multiscale setups via multiple $\Lambda_i$ scales and discusses hidden symmetries, including embeddings into 2d integrable hierarchies and Yangian-type structures, highlighting how nonperturbative stringy degrees of freedom organize the low-energy dynamics. Overall, the paper provides a detailed, multifaceted framework tying gauge theory vacua, integrable models, and brane configurations into a cohesive picture with potential for extensions to elliptic models and higher-dimensional theories.

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 $Λ_{QCD}$. 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.

Figures (5)

• Figure 1: Various aspects (columns) of the same models (rows)
• Figure 2: Brane picture, corresponding to the $Sl(3)$ magnet
• Figure 3: Newton polygon $\Delta$
• Figure 4: Different representations of fundamental matter first by inserting sixbranes, and then transformation of them into semi-infinite fourbranes
• Figure 5: Two looks at the Toda chain: explanation of the embedding Toda $\hookrightarrow$ Nahm