Matter and singularities

TL;DR

This work analyzes matter representations arising from codimension-two singularities in F-theory for $SU(N)$ gauge groups, connecting local singularity geometry to global Weierstrass models. It develops a systematic framework to realize a wide range of representations via rank-one enhancements, incomplete/completed resolutions, and singular brane loci, and demonstrates consistency with 6D anomaly constraints by constructing global models on $oldsymbol{ ext{P}^2}$ and $oldsymbol{ ext{P}^3}$. It uncovers novel phase transitions between matter sectors without changing the gauge group and identifies a mild constraint on certain 4D supergravity theories arising from compactifications on $oldsymbol{ ext{P}^3}$, linking geometry, topology, and low-energy physics. The results underline a close correspondence between geometric singularity structures and the spectrum of charged matter in both 6D and 4D, and chart a course for broader classifications and phenomenological implications of F-theory vacua.

Abstract

We analyze the structure of matter representations arising from codimension two singularities in F-theory, focusing on gauge groups SU(N). We give a detailed local description of the geometry associated with several types of singularities and the associated matter representations. We also construct global F-theory models for 6D and 4D theories containing these matter representations. The codimension two singularities encountered include examples where the apparent Kodaira singularity type does not need to be completely resolved to produce a smooth Calabi-Yau, examples with rank enhancement by more than one, and examples where the 7-brane configuration is singular. We identify novel phase transitions, in some of which the gauge group remains fixed but the singularity type and associated matter content change along a continuous family of theories. Global analysis of 6D theories on P^2 with 7-branes wrapped on curves of small degree reproduces the range of 6D supergravity theories identified through anomaly cancellation and other consistency conditions. Analogous 4D models are constructed through global F-theory compactifications on P^3, and have a similar pattern of SU(N) matter content. This leads to a constraint on the matter content of a limited class of 4D supergravity theories containing SU(N) as a local factor of the gauge group.

Figures (5)

• Figure 1: Embedding of $A_3 \rightarrow D_4$ singularity encoded in eq. (\ref{['eq:a3d4']}). Curves in $D_4$ are depicted in black solid lines, while $A_3$ curves are in colored dashed lines. Two different methods are used to depict the same embedding. (a) depicts each curve as a line, with intersections associated with crossings, as in much mathematical literature. (b) depicts $D_4$ curves in Dynkin diagram notation, with nodes for curves and lines for intersection, and depicts $A_3$ curves as colored dashed curves depicting $\mathbb{P}^1$'s at generic $s$ and limit as $s \rightarrow 0$, with intersections denoted by "x"'s. There are four possible embeddings depending upon choices for codimension two resolutions. Choice depicted has $\tau_+ = 1, \tau_-= 0$, according to notation in Section \ref{['sec:a3-appendix']} of Appendix, so for example $C_1^+ \rightarrow \delta_1 + \delta_2^+$ as $s \rightarrow 0$.
• Figure 2: Embedding $A_5 \rightarrow E_6$ with incomplete resolution of $E_6$ singularity in threefold.
• Figure 3: Embedding $A_5 \rightarrow E_6$ with complete resolution of $E_6$ singularity in threefold.
• Figure 4: A schematic depiction of the decomposition of the adjoint of $E_6$ under the action of $A_5$. The action of $A_5$ is taken to be in the horizontal direction. The root $\epsilon_4$ is perpendicular to all roots of $A_5$. In the incompletely resolved $E_6$, a projection is taken in the $\epsilon_4$ direction that combines the two 20's of $A_5$ into a single half hypermultiplet.
• Figure 5: Embedding of $A_3 \rightarrow A_7$ at an ordinary double point singularity, giving a two-index symmetric representation as well as antisymmetric representation ( ${\tiny\yng(2)} + {\tiny\yng(1,1)}$ ).