SO(N) superpotential, Seiberg–Witten curves and loop equations

Abstract

We consider the exact superpotential of N = 1 super Yang-Mills theory with gauge group SO(N ) and arbitrary tree-level polynomial superpotential of one adjoint Higgs field. A field-theoretic derivation of the glueball superpotential is given, based on factorization of the N = 2 Seiberg-Witten curve. Following the conjecture of Dijkgraaf and Vafa, the result is matched with the corresponding SO(N ) matrix model prediction. The verification involves an explicit solution of the first non-trivial loop equation, relating the spherical free energy to that of the non-orientable surfaces with topology RP 2 .

Effective SO superpotential for N=1 theory with Nf fundamental matter

“…The effective superpotential of SU(N c ) theory has been obtained in [24]. Comparison of the effective superpotentials of these two theories provides the following equivalence: 20) which agrees with the relation obtained in [14]. We shall now address the fundamental matter contribution F χ=1 to the effective potential.…”

“…Following the arguments in [24], for the effective superpotential evaluation we can still look at a point in the quantum moduli space of N = 2 pure gauge theory where r = [N c /2] (rank of SO(N c )) monopoles become massless [14]. This corresponds to the point where the Seiberg-Witten curve factorizes completely.…”

“…We are interested in the complete factorization of the Seiberg-Witten curve. The moduli that factorizes the Seiberg-Witten curve are given by [14] …”

Phases of SO(Nc) gauge theories with flavors

“…Due to the mass dependent term containing p i dependent term, we can not simplify the second term (A.20) (as we go to the contour integral around x again, when we consider massive flavors) further like as (A.23) for the massless case. 32 One can multiply z 2r both in the numerator and denominator and then the orders of the characteristic polynomial are changed into 2N c .…”

Geometric dual and matrix theory for SO/Sp gauge theories

“…The matrix models for the SO/Sp gauge groups have been proposed in [58,63,64] 5 The partition function of the matrix model is given by…”