Veronese geometry and the electroweak vacuum moduli space

Abstract

We explain the origin of the Veronese surface in the vacuum moduli space geometry of the MSSM electroweak sector. While this result appeared many years ago using techniques of computational algebraic geometry, it has never been demonstrated analytically. Here, we present an analytical derivation of the vacuum geometry of the electroweak theory by understanding how the F-and D-term relations lead to the Veronese surface. We moreover give a detailed description of this geometry, realising an extra branch as a zero-dimensional point when quadratic Higgs lifting deformations are incorporated into the superpotential. * hey@maths.ox.ac.uk † vishnu@neo.phys.wits.ac.za ‡ Cyril. Matti.1@city.ac.uk

Testing R-parity with geometry

“…The geometry, or geometries plural, defined solely by the GIOs corresponds to the ideal of their relations, or syzygies. 3 Such geometries serve, in some sense, as master spaces common to many of the outcomes we will study in this paper. These geometries may be thought of as the vacuum moduli spaces given by the non-vanishing GIOs when the F-term equations do not impose additional relations among themselves.…”

“…The vacuum of an N = 1 supersymmetric quantum field theory consists of field configurations that satisfy the F-term and D-term constraints. Recent advances in understanding the vacuum moduli space of the minimal supersymmetric standard model (MSSM) establish that non-trivial geometrical characteristics correlate to specific parameter choices of the theory such as the number of generations of matter fields, the number of pairs of Higgs doublets, and the vanishing of coupling constants [1,2,3,4]. These are the first hints that a bottom up strategy to studying low dimensional effective field theories might yield information about the structure of its higher energy completion [7].…”

“…While the F-term equations (3) simply correspond to a Jacobian and are straightforward to solve, it tends to be much more difficult to satisfy the D-term equations (4). Nonetheless [10,11] demonstrates that solving (3) and (4) is equivalent to an elimination problem (see also [3] for a summary). For every solution to the F-term equations, there exists a unique solution to the D-terms in the completion of the complexified orbit of the gauge group [12,13,14,15,16].…”

Standard model plethystics

“…The flat directions of the MSSM have been identified in [5]. Combining these directions of thought, a long programme was launched to study the vacuum geometry of the MSSM and its relatives [6][7][8][9][10][11]: under the guiding principle that "interesting geometry is coextensive with interesting physics", the ultimate goal is to use geometric and topological properties of VMS as a selection rule for operators in the Standard Model Lagrangian. Specifically, 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, then this special form should be regarded as a consequence of some unknown physics.…”

“…In such sense, we are placing very restrictive constraints on allowed physical processes that are mediated by certain operators. Already, many interesting features have been found, such as the VMS of the electro-weak sector being an affine cone over the classical Veronese surface, a structure ruined by addition of R-parity-violating operators, or the sensitive dependence of the geometry on the number of generations, or the appearance of Calabi-Yau varieties, etc [9][10][11]. Supersymmetry and the VMS thus provide us with a low energy window of how geometry can guide certain phenomenological questions.…”