One-instanton test of a Seiberg-Witten curve from M-theory: the antisymmetric representation of SU(N)

Abstract

One-instanton predictions are obtained from the Seiberg-Witten curve derived from M-theory by Landsteiner and Lopez for the Coulomb branch of N = 2 supersymmetric SU(N) gauge theory with a matter hypermultiplet in the antisymmetric representation. Since this cubic curve describes a Riemann surface that is non-hyperelliptic, a systematic perturbation expansion about a hyperelliptic curve is developed, with a comparable expansion for the Seiberg-Witten differential. Calculation of the period integrals of the SW differential by the method of residues of D'Hoker, Krichever, and Phong enables us to compute the prepotential explicitly to one-instanton order.It is shown that the one-instanton predictions for SU(2), SU(3), and SU(4) agree with previously available results. For SU(N), N ≥ 5, our analysis provides explicit predictions of a curve derived from M-theory at the one-instanton level in field theory.

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]. …”

“…In particular, for SU(N ), the one-instanton prepotential has the form [3,13,14] 8πi one hypermultiplet in the antisymmetric representation and N f hypermultiplets in the defining representation, it is given by [12,14] 8πi…”

“…We begin by changing variables 4) to recast the curve into a form suitable for the hyperelliptic expansion [12]-[15]…”

“…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.…”