Abstract
We study the vacuum geometry prescribed by the gauge invariant operators of the minimal supersymmetric standard model via the plethystic program. This is achieved by using several tricks to perform the highly computationally challenging Molien-Weyl integral, from which we extract the Hilbert series, encoding the invariants of the geometry at all degrees. The fully refined Hilbert series is presented as the explicit sum of 1422 rational functions. We found a good choice of weights to unrefine the Hilbert series into a rational function of a single variable, from which we can read off the dimension and the degree of the vacuum moduli space of the minimal supersymmetric standard model gauge invariants. All data in Mathematica format are also presented.
Highlights
AND SUMMARYThe Standard Model of particle theory containing specific gauge interactions is expected to have more structures when extended to energies above 1–10 TeV, where supersymmetry might be incorporated
We study the vacuum geometry prescribed by the gauge invariant operators of the minimal supersymmetric standard model via the plethystic program
We found a good choice of weights to unrefine the Hilbert series into a rational function of a single variable, from which we can read off the dimension and the degree of the vacuum moduli space of the minimal supersymmetric standard model gauge invariants
Summary
The Standard Model of particle theory containing specific gauge interactions is expected to have more structures when extended to energies above 1–10 TeV, where supersymmetry might be incorporated. If the VMS were to be found to have some special form in the mathematical sense, which (1) cannot be explained in terms of symmetries relating the relevant degree of freedom in the low energy effective field theory; and (2) is very unlikely to have occurred by chance, this special form should be regarded as a consequence of some unknown physics In this setting, we take special to mean nontrivial properties of algebraic geometry, such as exhibited by interesting topological invariants or emergence of special holonomy. The case of nontrivial superpotential W ≠ 0 is left for future work
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