Seiberg-Witten theory, monopole spectral curves and affine Toda solitons

Abstract

Using Seiberg-Witten theory it is known that the dynamics of N = 2 supersymmetric SU (n) Yang-Mills theory is determined by a Riemann surface Σ n . In particular the mass formula for BPS states is given by the periods of a special differential on Σ n . In this note we point out that the surface Σ n can be obtained from the quotient of a symmetric n-monopole spectral curve by its symmetry group. Known results about the Seiberg-Witten curves then implies that these monopoles are related to the A (1) n−1 Toda lattice. We make this relation explicit via the ADHMN construction. Furthermore, in the simplest case, that of two SU (2) monopoles, we find that the general two monopole solution is generated by an affine Toda soliton solution of the imaginary coupled theory.

Three-dimensional gauge theories and monopoles

“…One way to construct this space is to consider its simply connected double cover and then mod by the cyclic symmetry of two monopoles -Z 2 . We will now describe a procedure due to Sutcliffe [16] which relates the spectral curves to the curves associated with d = 4 super Yang-Mills theory. We recall the spectral curve for two monopoles…”

Two integrable systems related to hyperbolic monopoles

“…It seems likely that this elliptic k = 2 solution corresponds to hyperbolic 2-monopoles with gauge group SU(2), via the Braam-Austin construction. More generally, for k > 2 one may speculate that discrete-Toda solutions correspond to hyperbolic k-monopoles with C k cyclic symmetry, since this is what happens for Euclidean monopoles (Sutcliffe 1996).…”

Branes and integrability in the N = 2 SUSY YM theory

“…Following [11] we can identify the world volume theory of strings as coming from 10d SYM theory via the dimensional reduction. Moreover there is one to one correspondence between the equation defining the ground state in the SUSY σ model and the Nahm equations describing the monopole moduli space.Finally to get the Toda equation in the brane terms we use the observation [12] that the spectral curve for SL(N,C) Toda system coincides with the one for the cyclic N-monopoles. The spectral curve for the cyclic N-monopole follows from the generic spectral curve…”