One-instanton predictions of Seiberg–Witten curves for product groups

Abstract

One-instanton predictions for the prepotential are obtained from the Seiberg-Witten curve for the Coulomb branch of N = 2 supersymmetric gauge theory for the product group m n=1 SU(N n ) with a massless matter hypermultiplet in the bifundamental representation (N n ,N n+1 ) of SU(N n ) × SU(N n+1 ) for n = 1 to m − 1, together with N 0 and N m+1 matter hypermultiplets in the fundamental representations of SU(N 1 ) and SU(N m ) respectively. The derivation uses a generalization of the systematic perturbation expansion about a hyperelliptic curve developed by us in earlier work.

Two antisymmetric hypermultiplets in N = 2SU(N) gauge theory: Seiberg-Witten curve and M-theory interpretation

“…Using methods developed in refs. [12]- [15] for a systematic instanton expansion based on a perturbative expansion beginning with a hyperelliptic approximation to a SW curve, we were able to "reverse engineer" a quartic curve which reproduces F 1−inst . The leading order terms in L are unique, and there are strong constraints on the subleading terms.…”

“…It is clear from this Table, however, that the renormalized order parameters are the natural variables for this problem, as emphasized in refs. [3] and [12]- [15]. …”

“…We expect at least a cubic curve for two antisymmetric hypermultiplets. The curve for a theory with a product of m factors of SU(N ) with matter in bifundamental representations [10,8] is of order m + 1, but we found [15] that to compute the dual order parameters for any of the factor groups to one-instanton accuracy, it suffices to use a quartic approximation to the full curve. (In all cases, the quadratic approximation is sufficient to compute the one-loop prepotential.…”

“…For a gauge theory based on a product of m factors of SU(N ) with hypermultiplets in bifundamental representations, the Riemann surface is described by an (m + 1)th order curve [10,8]. In a series of papers [12]- [15], we have developed a systematic approximation scheme to compute the instanton expansion of the prepotential for non-hyperelliptic curves of cubic and higher order. This allows one to test the predictions of M-theory and geometric engineering for field theory, thereby increasing our confidence in the validity of these string-theoretic methods.…”

“…In section 3, we then "reverse engineer" a Seiberg-Witten curve for this theory from the prepotential, using methods we have developed [12]- [15] for computing the instanton expansion for non-hyperelliptic curves. The quartic curve that we derive has the correct limiting behavior as the mass of either of the antisymmetric hypermultiplets goes to infinity.…”