On the Geometry of Super Yang-Mills Theories: Phases and Irreducible Polynomials

Abstract

We study the algebraic and geometric structures that underly the space of vacua of N=1 super Yang-Mills theories at the non-perturbative level. Chiral operators are shown to satisfy polynomial equations over appropriate rings, and the phase structure of the theory can be elegantly described by the factorization of these polynomials into irreducible pieces. In particular, this idea yields a powerful method to analyse the possible smooth interpolations between different classical limits in the gauge theory. As an application in U(Nc) theories, we provide a simple and completely general proof of the fact that confining and Higgs vacua are in the same phase when fundamental flavors are present, by finding an irreducible polynomial equation satisfied by the glueball operator. We also derive the full phase diagram for the theory with one adjoint when Nc is less than or equal to 7 using computational algebraic geometry programs.

arXiv comment87 pages; v2: typos and eq. (4.44) corrected

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Cited by in corpus (6)

• A Simple Introduction to Grobner Basis Methods in String Phenomenology
• Cutting and gluing with running couplings in $\mathcal{N}=2$ QCD
• Confining vacua in SQCD, the Konishi anomaly and the Dijkgraaf-Vafa superpotential
• Permutations of Massive Vacua
• Galois symmetries in Super Yang-Mills Theories
• Flavors in the microscopic approach to N=1 gauge theories