Y. He

V. Jejjala

B. D. Nelson

 H. Schenck

M. Stillman

The rich and intricate vacuum geometry of the Minimal Supersymmetric Standard Model—a complex manifold—is characterised for the first time.

The vacuum moduli space of the supersymmetric Standard Model, a central geometric object in mathematical physics, encodes all possible vacuum states of the theory. Despite decades of study, its full geometry was not known. In this work, the vacuum moduli space is described explicitly as an algebraic variety for the first time, with three irreducible components of dimensions 1, 15 and 29, structured by Segre geometry. 

Vacuum structure of a quantum field theory is a crucial property. In theories with extended symmetries, such as supersymmetric gauge theories, the vacuum is typically a continuous manifold, called the vacuum moduli space, parametrized by the expectation values of scalar fields. Starting from the R-parity preserving superpotential at renormalizable order, we use Gröbner bases to determine the explicit structure, as an algebraic variety, of the vacuum geometry of the minimal supersymmetric extension of the Standard Model. Gröbner bases have doubly exponential computational complexity (for this case, 721023 operations); we exploit symmetry and multigrading to render the computation tractable. This geometry has three irreducible components of complex dimensions 1, 15, and 29, each being a so-called rational variety. The defining equations of the components express the solutions to F-terms and D-terms in terms of the gauge invariant operators and are interpreted in terms of classical geometric constructions.

Standard Model vacuum

Vacuum geometry of the standard model