On the integrable geometry of soliton equations and $N=2$ supersymmetric gauge theories

Abstract

We provide a unified construction of the symplectic forms which arise in the solution of both N=2 supersymmetric Yang-Mills theories and soliton equations. Their phase spaces are Jacobian-type bundles over the leaves of a foliation in a universal configuration space. On one hand, imbedded into finite-gap solutions of soliton equations, these symplectic forms assume explicit expressions in terms of the auxiliary Lax pair, expressions which generalize the well-known Gardner-Faddeev-Zakharov bracket for KdV to a vast class of 2D integrable models; on the other hand, they determine completely the effective Lagrangian and BPS spectrum when the leaves are identified with the moduli space of vacua of an N=2 supersymmetric gauge theory. For SU(N c ) with N f ≤ N c + 1 flavors, the spectral curves we obtain this way agree with the ones derived by Hanany and Oz and others from physical considerations.

Vector Bundles and Lax Equations on Algebraic Curves

“…Moreover, a'priori it is not clear, why all the systems constructed above are Hamiltonian. In this section we show that the general algebraic approach to the Hamiltonian theory of the Lax equations proposed in [9,10] and developed in [11] is evenly applicable to the Lax equations on the Riemann surfaces.…”

“…where d i is the order of zero dz at P i (compare with the definition of the universal configuration space in [9]). Note, that although the functions (4.8) are multivalued, their common level sets are leaves of a well-defined foliation on L D .…”

“…Moreover, a'priori it's not clear, why the Lax equations are Hamiltonian. In Section 4 we clarify this problem using the approach to the Hamiltonian theory of soliton equations proposed in [9,10,11]. It turns out that for D = K the universal two-form which is expressed in terms of the Lax matrix and its eigenvectors coincides with canonical symplectic structure on the cotangent bundle T * (M).…”

Integrable chains on algebraic curves

“…In this section we apply the general algebraic approach to the Hamiltonian theory of the Lax equations proposed in [8,9], and developed in [13], to the Lax equations for periodic chains on the algebraic curves. As it was mentioned in the introduction, this approach is based on the existence of two universal two-forms on a space of meromorphic matrix-function.…”

“…The alternative approach to the Hamiltonian theory of the soliton equations was developed in [8,9]. It is based on the existence of some universal two-form defined on a space of meromorphic matrix-functions.…”