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

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 RP2.