Exotic matter on singular divisors in F-theory

TL;DR

This work develops a non-UFD geometric framework for realizing exotic matter representations in F-theory by leveraging the normalization of the intrinsic local function ring on singular divisors. It provides explicit Weierstrass tunings for SU$(N)$ with 2-index symmetric matter and SU$(2)$ with 3-index symmetric matter, associated with double and triple point singularities, and demonstrates how discriminant cancellations arise from non-UFD structure. The authors connect these geometric constructions to 6D supergravity spectra via anomaly constraints, identify swampland examples, and describe matter transitions that interpolate between anomaly-equivalent models through superconformal points. The results suggest that, within F-theory, the only genuinely exotic representations tied to a single simple gauge factor are the $2$-index symmetric of SU$(N)$ and the $3$-index symmetric of SU$(2)$, with higher representations either disallowed by geometry or residing in the swampland. The work thus advances the landscape–swampland program by clarifying when and how non-Tate, non-UFD tunings can realize exotic matter and by mapping how geometric constraints govern low-energy spectra.

Abstract

We analyze exotic matter representations that arise on singular seven-brane configurations in F-theory. We develop a general framework for analyzing such representations, and work out explicit descriptions for models with matter in the 2-index and 3-index symmetric representations of SU($N$) and SU(2) respectively, associated with double and triple point singularities in the seven-brane locus. These matter representations are associated with Weierstrass models whose discriminants vanish to high order thanks to nontrivial cancellations possible only in the presence of a non-UFD algebraic structure. This structure can be described using the normalization of the ring of intrinsic local functions on a singular divisor. We consider the connection between geometric constraints on singular curves and corresponding constraints on the low-energy spectrum of 6D theories, identifying some new examples of apparent "swampland" theories that cannot be realized in F-theory but have no apparent low-energy inconsistency.

Figures (5)

• Figure 1: Schematic illustration of the transition for $\mathrm{SU}(4)$. The curve represents (a slice of) $h$, the curve along which the $\mathrm{SU}(4)$ singularity is tuned. Points represent codimension-two loci contributing charged matter; the labels give the $\mathrm{SU}(4)$ representations associated with particular codimension-two loci. (a) Initially, there are several codimension-two loci supporting charged matter. We assume that a double point has already been tuned on the curve, with $\mathbf{15}$ matter localized at the point. (b) The Weierstrass model is tuned so that two $\mathbf{6}$ loci move to the double point. (c) When the $\mathbf{6}$ loci reach the double point, the singularity type at the double point worsens, giving an SCFT. (d) Deformations in the Weierstrass model remove the SCFT by pulling away two $\mathbf{6}$ loci. The charged matter at the double point is now $\mathbf{10}+\mathbf{6}$, rather than the initial $\mathbf{15}$.
• Figure 2: Allowed embeddings of Dynkin diagrams corresponding to Kodaira singularities at codimension two points for (a) the two-index symmetric representation of SU(3), (b) the three-index symmetric representation of SU(2). Solid circles represent the Dynkin diagrams of the gauge group and the open circle represents the matter fields.
• Figure 3: Disallowed embeddings of Dynkin diagrams corresponding to Kodaira singularities at codimension two points for (a) the three-index symmetric representation of SU(3), (b) the four-index symmetric representation of SU(2). Solid circles represent the Dynkin diagrams of the gauge group and the open circle represents the matter fields. Circle with a cross represents the extra node of the extended Dynkin diagram.
• Figure 4: Exceptional curves for the $A_3\times A_3$ resolution. Circles denote the exceptional curves, with $x$'s marking the intersections. Arrows indicate how exceptional curves are exchanged under $\phi_0\rightarrow-\phi_0$ and $\beta\rightarrow-\beta$. Colors indicate which $C$ and $\tilde{C}$ curves are identified for the case with symmetric matter.
• Figure 5: Embedding of $A_3 \times A_3 \rightarrow A_7$ at a double point. Black dots represent exceptional curves for the $A_7$ singularity, with the lines between them denoting intersections between the exceptional curves. Colored lines indicate the combinations of $\gamma$ curves corresponding to the $A_3\times A_3$ exceptional curves. Colors indicate which $C$ and $\tilde{C}$ curves are identified for the case with symmetric matter.