Abstract
In [F. Ferrari, hep-th/0211069], the author has discussed the quantum parameter space of the N=1 super Yang–Mills theory with one adjoint Higgs field Φ, tree-level superpotential Wtree=mΦ2/2+gΦ3/3, and gauge group U(N). In particular, full details were worked out for U(2) and U(3). By discussing higher rank gauge groups like U(4), for which the classical parameter space has a large number of disconnected components, we show that the phenomena discussed in [F. Ferrari, hep-th/0211069] are generic. It turns out that the quantum space is connected. The classical components are related in the quantum theory either through standard singularities with massless monopoles or by branch cuts without going through any singularity. The branching points associated with the branch cuts correspond to new strong coupling singularities, which are not associated with vanishing cycles in the geometry, and at which glueballs can become massless. The transitions discussed recently by Cachazo, Seiberg, and Witten are special instances of those phenomena.
Highlights
Introduction and review ofU(2) and U(3)In a recent paper [1], the powerful technology in the calculation of exact quantum effective superpotentials [2,3,4,5] was used for the first time to derive new physics in N = 1 supersymmetric U(N) gauge theories
The basic object considered in [1] is the quantum space of parameters Mq. This space is reminiscent of the quantum moduli space of theories with a larger number of supersymmetries
The most fundamental difference is that no massless scalar is associated with the motion on Mq
Summary
The calculations relevant to the ten sheets |k, 0; 4, 0 , |0, k; 0, 4 , and |k, k; 2, 2 were already performed in [1]. The eight solutions correspond to the eight sheets, and the singularities with massless glueballs correspond to critical values of λ for which two roots of R8 coincide. As expected, this occurs precisely for the values (2.13). The last step in the calculation of Mq consists of studying possible phase transitions with massless monopoles relating the eight Coulomb sheets discussed above to the confining sheets. From this we deduce the classical polynomial for u, analogous to (2.2), Q10,cl(u) = (u − 1)2(u − 2)3(u − 3)3(u − 4)2 This shows that the ten sheets are related to each other through branch cuts.
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