Towards Exotic Matter and Discrete Non-Abelian Symmetries in F-theory

TL;DR

This work develops a geometric framework in F-theory to realize exotic bifundamental matter in 6D through collisions of discriminant components with high tangency and T-brane deformations, connecting the local geometry to weakly coupled gauge theories. It provides explicit global constructions for the pair $({\mathbf{56}},{\mathbf{2}})$ in $\mathfrak{e}_7\times\mathfrak{su}_2$ and $({\mathbf{27}},{\mathbf{3}})$ in $\mathfrak{e}_6\times\mathfrak{su}_3$, demonstrating their heterotic orbifold duals and showing that the exotic states are delocalized across multiple collision points. By further higgsing these exotic bifundamentals, the paper shows routes to higher-dimensional representations and to discrete non-abelian symmetries, such as $Q_8$ and $A_4$, highlighting the potential richness of symmetry structures realizable in F-theory. The analysis emphasizes the importance of tensor and Higgs branches, anomaly matching, and global geometry, and identifies open questions about general tangency criteria, the multiplicity of exotic bifundamentals, and extensions to 4D chiral theories.

Abstract

We present a prescription in F-theory for realizing matter in "exotic" representations of product gauge groups. For 6D vacua, bifundamental hypermultiplets are engineered by starting at a singular point in moduli space which includes 6D superconformal field theories coupled to gravity. A deformation in Higgs branch moduli space takes us to a weakly coupled gauge theory description. In the corresponding elliptically fibered Calabi--Yau threefold, the minimal Weierstrass model parameters $(f,g,Δ)$ vanish at collisions of the discriminant at least to order $(4,6,12)$, but with sufficiently high order of tangency to ensure the existence of T-brane deformations to a weakly coupled gauge theory with exotic bifundamentals. We present explicit examples including bifundamental hypermultiplets of $\mathfrak{e}_7 \times \mathfrak{su}_2$ and $\mathfrak{e}_6 \times \mathfrak{su}_3$, each of which have dual heterotic orbifold descriptions. Geometrically, these matter fields are delocalized across multiple points of an F-theory geometry. Symmetry breaking with such representations can be used to produce high dimension representations of simple gauge groups such as the four-index symmetric representation of $\mathfrak{su}_2$ and the three-index symmetric representation of $\mathfrak{su}_3$, and after further higgsing can yield discrete non-abelian symmetries.

Figures (4)

• Figure 1: Depiction of the deformations of singular F-theory models which can generate an exotic bifundamental. The minimal Weierstrass model contains points where the multiplicity of vanishing for $(f,g,\Delta)$ is at least $(4,6,12)$. T-brane deformations lead to models with exotic bifundamentals, and tensor branch deformations lead to deformations of 6D SCFTs coupled to gravity.
• Figure 2: Blow-up of the base at one of the four points $u=p_4=0$, where the $\mathfrak{e}_7$ divisor $\{u\}$ and the $\mathfrak{su}_2$ divisor $\{\sigma \equiv d_0\,u + p_4^3 \,v\}$ intersect. It requires five $\mathbb{P}^1$'s $\{\epsilon_i\}$, carrying the indicated Kodaira fibers and associated gauge algebras, to resolve all non-minimal points (black dots). After the blow-ups the curve $\{u\}$ has self-intersection $-8$, and $\{\sigma\}$ has $0$. All of the exceptional curves have self-intersection $-2$, except for $\{\epsilon_5\}$ which has $-1$. All displayed matter multiplets are half-hypermultiplets, except the ${\bf 7}$ of $\mathfrak{g}_2$ on $\{\epsilon_3\}$, which is a full hypermultiplet spread across the three intersection points with the residual discriminant.
• Figure 3: Blowing-up the base at the three points $u=p_3=0$ where the $\mathfrak{e}_6$ divisor $\{u\}$ and the $\mathfrak{su}_3$ divisor $\{\sigma \equiv u + p_3^2 \,v\}$ intersect. It requires five $\mathbb{P}^1$'s $\{\epsilon_i\}$ to resolve all non-minimal points (black dots). After the blow-ups the curve $\{u\}$ has self-intersection $-6$, and $\{\sigma\}$ has $-3$. The sequence of curves $\epsilon_1 - \epsilon_2 - \epsilon_3$ has self-intersections $(-2, -3, -2)$ and have the gauge algebras $\mathfrak{su}_2 - \mathfrak{so}_7 - \mathfrak{su}_2$. At the two intersection points we have $\frac{1}{2} ( {\bf 8,2})$, where ${\bf 8}$ is the spinor representation of $\mathfrak{so}_7$.
• Figure 4: Depiction of breaking patterns associated with vevs for exotic bifundamentals. Here, we assume that the continuous symmetries are flavor symmetries so that no D-term constraints need to be satisfied. The final stage of the breaking patterns leads to a finite order non-abelian group. Here, $Q_8$ denotes the order eight quaternion group and $A_4$ the alternating group on four letters. In the case of the breaking pattern for $\mathfrak{e}_6 \times \mathfrak{su}_3$ to $\mathfrak{su}_3 \times A_4$, this also requires a vev for a scalar in the $(\mathbf{1},\mathbf{6})$ of $\mathfrak{su}_3 \times \mathfrak{su}_3$, which descends from the $(\mathbf{27} , \mathbf{1})$ of $\mathfrak{e}_6 \times \mathfrak{su}_3$.