Matter in transition

TL;DR

This work identifies and analyzes a new class of matter transitions in 6D (and potentially 4D) supergravity theories in which the gauge group remains fixed while the matter content changes, with the transition points passing through strongly coupled superconformal fixed points. The authors develop a dual perspective in F-theory and heterotic string theory, using Weierstrass-model tunings beyond Tate form to realize nonstandard representations (notably SU($N$) with three-index antisymmetric matter) and to describe the exchange between different anomaly-equivalent matter contents. They provide detailed analyses for SU($6$), SU($7$), and SU($8$) blocks, including Higgsing deformations, product groups, and the heterotic bundle decompositions, and they map these transitions across heterotic/F-theory duality through stable degeneration and spectral-cover data. The paper also extends the discussion to other groups (Sp, SO, SU($3$) with symmetric matter) and outlines implications for 4D theories and the geometry of transition manifolds. Overall, the results give a concrete framework for understanding nonperturbative matter transitions, highlight the central role of SCFT points, and offer a path toward broader classes of string-compactification transitions with potential phenomenological relevance.

Abstract

We explore a novel type of transition in certain 6D and 4D quantum field theories, in which the matter content of the theory changes while the gauge group and other parts of the spectrum remain invariant. Such transitions can occur, for example, for SU(6) and SU(7) gauge groups, where matter fields in a three-index antisymmetric representation and the fundamental representation are exchanged in the transition for matter in the two-index antisymmetric representation. These matter transitions are realized by passing through superconformal theories at the transition point. We explore these transitions in dual F-theory and heterotic descriptions, where a number of novel features arise. For example, in the heterotic description the relevant 6D SU(7) theories are described by bundles on K3 surfaces where the geometry of the K3 is constrained in addition to the bundle structure. On the F-theory side, non-standard representations such as the three-index antisymmetric representation of SU(N) require Weierstrass models that cannot be realized from the standard SU(N) Tate form. We also briefly describe some other situations, with groups such as Sp(3), SO(12), and SU(3), where analogous matter transitions can occur between different representations. For SU(3), in particular, we find a matter transition between adjoint matter and matter in the symmetric representation, giving an explicit Weierstrass model for the F-theory description of the symmetric representation that complements another recent analogous construction.

Figures (3)

• Figure 1: $\rm SU(7)$ transition point when blown up. Here, the compactification base was taken to be $\mathbb{F}_n$, while the original $\rm SU(7)$ gauge group was tuned on $\tilde{S}$. The blow-up procedure introduces three exceptional curves shown in red, one of which has an $I_2$ singularity. The $I_2$ singularity indicates there is a strongly coupled $\rm SU(2)$ at the transition point.
• Figure 2: Higgsing chain involving $\rm SU(6)$, $\rm Sp(3)$, $\rm SU(3)\times\rm SU(3)$, and $\rm SU(3)$ along with associated Dynkin diagrams. Dotted lines indicate nodes exchanged under monodromy. The Higgsing chain involves two types of F-theory deformations that either introduce monodromy or remove the central node in the Dynkin diagrams. The $\rm SU(3)$ singularity occurs when both types of deformations are performed.
• Figure 3: The matter transitions studied in this paper can be seen as arising along one-parameter families of theories, with the transition point at $\hat{\varepsilon} = 0$, and field theories with the same gauge group but distinct matter representation content at $\hat{\varepsilon} > 0$ and $\hat{\varepsilon} < 0$. For these matter transitions in 6D theories, the transition point at $\hat{\varepsilon} = 0$ is a superconformal field theory, from which an additional tensor branch generally extends, as well as separate branches for each of the field theories with distinct matter contents, though the tensor branch is incidental to the matter transition. The matter representations shown are for the simplest (SU(6)) matter transition.