Musings

Geometry of the MSSM

A very pretty paper, today, by Gray, He, Jejjala and Nelson. They study the algebraic geometry of the space of supersymmetric vacua of $\mathcal{N}=1$ supersymmetric gauge theories.

The obvious (and well-known) statement is that the solution to the F-flatness conditions is an affine algebraic variety, $\mathcal{F}\subset \mathbb{C}^n$. The ring of functions on this variety is $$F = \mathbb{C}[\phi_1,\dots,\phi_n]/\langle\partial_i W\rangle$$ where $\langle\partial_i W\rangle$ denotes the ideal generated by the derivatives of the superpotential. Imposing D-flatness corresponds to the symplectic reduction, or GIT quotient $$\mathcal{F}//G \simeq \mathcal{F}/G_{\mathbb{C}}$$

Their application of interest is the MSSM. Needless to say, finding a Gröbner basis for this complicated ideal in $\mathbb{C}^{49}$ is a somewhat formidable computational task. So they end up looking at various simplified versions of the problem.

Setting all the colour-triplet fields to zero (that is, keeping only $H$,$\tilde{H}$, $\tilde{E}_i$, $L_i$), and turning on the most general quartic terms in the superpotential compatible with R-parity, they find that the vacuum manifold for this truncated theory is an affine cone over the Veronese surface (the degree-2 embedding of $\mathbb{P}^2$ in $\mathbb{P}^5$)¹.

It’s not at all clear what the significance of this observation is, though they do find other examples where the same Veronese structure arises, provided one is willing to tune the superpotential in various ways.

What would be really nice is a characterization of supersymmetry-breaking which allows one to carry over at least some of the techniques of algebraic geometry (which apply so beautifully to the supersymmetric case). Which reminds me that I ought to think some more about some of the obvious extensions of my old paper with Varadarajan, which explores similar questions, using similar techniques.