Varieties of four-dimensional gauge theories

TL;DR

The paper recasts the problem of anomaly-free chiral representations in four-dimensional gauge theories as a problem in algebraic geometry for $\mathfrak{su}_n$ with $n\ge5$: anomaly-free irreps correspond to rational points on a cubic projective variety $V_n$, whose real ordered points form a region diffeomorphic to $\mathbb{R}^{n-3}$ and whose rational points are dense. By introducing Okubo coordinates $\{\sigma_i\}$ and exploiting the cubic constraint $\sum_i \sigma_i^3=0$ together with $\sum_i \sigma_i=0$, the authors develop a Secants-based, group-symmetric method to generate infinite families of anomaly-free chiral irreps, unify the $n=5$ and general $n$ cases, and establish density and topological structure of the ordered region. The Clebsch diagonal surface $V_5$ serves as a concrete showcase: it is rational, its rational lines yield a practical parameterization, and elliptic-curve analysis bounds exceptional rational points, while comparisons with prior brute-force approaches highlight the method’s efficiency and discoverability. The work further extends to reducible representations, outlining how the polynomial anomaly conditions generalize and how reducible constructions may yield phenomenologically relevant models, albeit with greater arithmetic complexity. Overall, the paper provides a geometric, scalable framework for enumerating anomaly-free representations in four-dimensional gauge theories and reveals a rich landscape of possibilities, especially for $n\ge5$.

Abstract

We use algebraic geometry to study the anomaly-free representations of an arbitrary gauge Lie algebra for 4-dimensional spacetime fermions. For irreducible representations, the problem reduces to studying the Lie algebras $\mathfrak{su}_n$ for $n\geq 3$. We show that there exist equivalence classes of such representations that are in bijection with the rational points on a projective variety that are dense in a region of the underlying real variety diffeomorphic to $\mathbb{R}^{n-3}$. It follows that the chiral ones overwhelm the non-chiral ones for $n \geq 5$. We present an efficient algorithm to find explicit anomaly-free irreducible representations and discuss the generalization to reducible representations.

Figures (5)

• Figure 1: Rational points on an affine patch of the Clebsch diagonal surface, with coordinates $x_{1,2,3}$ as defined in \ref{['eq:xi']}. Each red point corresponds to an infinite class of chiral anomaly-free irreps of $\mathfrak{su}_5$. The points in orange correspond to non-chiral irreps. The blue points are rational points in the $S_5$ orbits of red ones, so can be used in our algorithm to find anomaly-free irreps, while the black points are not in such orbits, and cannot be used. The green point corresponds to the trivial representation.
• Figure 2: Close-up view of the region of interest in \ref{['fig:figclebschscanv7']}.
• Figure 3: An affine patch of the Clebsch diagonal surface (in blue), along with the palindromic region (in orange) and the remaining 11 rational lines (in black). The co-ordinates $x_{1,2,3}$ are defined in \ref{['eq:xi']}.
• Figure 4: Elliptic curves on the Clebsch diagonal cubic surface corresponding to the unorderable region for $\mathfrak{su}_5$. There are 10 real curves (each shown in a different colour). The orange curve is the curve $C(\mathbb{R})$ discussed in the main text. Each curve contains 6 rational points and collectively they contain 25 points, of which 16 are visible (in black) in this affine patch. Four of the points in $C(\mathbb{Q}) \cong \mathbb{Z}/6$ discussed in the main text are visible, labelled as powers of the generator (the generator itself is at infinity).
• Figure 5: An affine patch of the Clebsch diagonal cubic surface (in blue). The weakly ordered region on $\mathbb{R} P^3$ is shown as the yellow tetrahedron. Secants between the two edges shown in black intersect the surface exactly once and define a homeomorphism between the weakly-ordered region on the surface (in red) and a square. This descends to a diffeomorphism between the ordered region on the surface (corresponding to anomaly-free irreps of $\mathfrak{su}_5$) and $\mathbb{R}^2$.