6D SCFTs and Gravity

TL;DR

This work develops a geometric framework for coupling 6D SCFTs to gravity within F-theory, showing that on bases with orbifold singularities, the 6D anomaly polynomial is captured entirely by the intersection theory of fractional divisors, with anomaly coefficients identified as elements of the orbifold homology. A lattice refinement is required: the fractional divisor data alone would violate charge quantization, but extra light states from the SCFT, living on the resolved base, restore a unimodular string-charge lattice. The authors illustrate the construction with a detailed study of a compact model with base $\mathbb{P}^2/\mathbb{Z}_3$, where three $A_2$ SCFTs sit at fixed points and can be brought into gauge theories by tuning moduli; anomaly cancellation proceeds via Green–Schwarz terms that involve both the conventional tensor sector and the SCFT sector. Overall, the paper provides a consistent picture in which SCFTs contribute fractional anomaly data that are precisely balanced by refined string-charge lattices arising from the resolved geometry, offering a path toward understanding gravity-coupled 6D SCFTs and potential extensions to 4D vacua.

Abstract

We study how to couple a 6D superconformal field theory (SCFT) to gravity. In F-theory, the models in question are obtained working on the supersymmetric background R^{5,1} x B where B is the base of a compact elliptically fibered Calabi-Yau threefold in which two-cycles have contracted to zero size. When the base has orbifold singularities, we find that the anomaly polynomial of the 6D SCFTs can be understood purely in terms of the intersection theory of fractional divisors: the anomaly coefficient vectors are identified with elements of the orbifold homology. This also explains why in certain cases, the SCFT can appear to contribute a "fraction of a hypermultiplet" to the anomaly polynomial. Quantization of the lattice of string charges also predicts the existence of additional light states beyond those captured by such fractional divisors. This amounts to a refinement to the lattice of divisors in the resolved geometry. We illustrate these general considerations with explicit examples, focusing on the case of F-theory on an elliptic Calabi-Yau threefold with base P^2 / Z_3.

Figures (5)

• Figure 1: The tensor branch of the SCFT $\mathcal{T}_{p} (N, N + p k)$. On the left, $\mathcal{T}_{p} (N, N + p k)$ is localized at the $A_{p-1}$ locus which two flavor branes, each of type $I_N$ and $I_{N+pk}$, pass through. The $A_{p-1}$ singularity in the base is resolved on the right, by introducing the resolution divisors $D_m$, hence moving on to the tensor branch of the theory. A singular fiber of type $I_{N+mk}$ fibers over the divisor $D_m$. The effective gauge group of the tensor branch theory can thus be identified as $\prod_{m=1}^{p-1} SU(N+mk)$.
• Figure 2: $B = \mathbb{P}^2 / \mathbb{Z}_3$ and its resolution $\hat{B}$. $\hat{B}$ is a $dP_6$. There are six resolution divisors $D_{xy,a}$ that resolve the three $A_2$ singularities of $B$. The divisors $D_x$ of $B$ are mapped to divisors $\hat{D}_x$. Each pair of adjacent divisors in the diagram have intersection number $1$.
• Figure 3: A schematic picture of ${\widetilde{\sigma}}$ as it becomes reducible in ${\widetilde{B}}$. The upper diagrams depict the locus of the divisor ${\widetilde{\sigma}}$ (bold curves) on ${\widetilde{B}}$, while the lower diagrams depict its projection, ${\sigma}$ (also bold curves) on $B$. The orbifold $B$ is depicted as a cone, while the dotted lines on ${\widetilde{B}}$ are used to show the fundamental domain of ${\widetilde{B}}$ under the orbifold action. When ${\widetilde{\sigma}}$ is irreducible (left), its projection is a smooth divisor on $B$. Meanwhile, when ${\widetilde{\sigma}}$ become reducible (right), it factors into three copies of divisors related by the $\mathbb{Z}_3$ action. Upon projection to $B$, ${\sigma}$ develops a double-point.
• Figure 4: A diagram depicting the resolution of the $A_2$ singularity at $U=V=0$ when the gauge brane ${\sigma}$ carrying an $I_N$ singularity passes through. The two resolution divisors $D_{UV,1}$ and $D_{UV,2}$ each have a $I_N$ singularity along them.
• Figure 5: A schematic diagram of the configuration of divisors on $B$ (left) and $\hat{B}$ (right). The singularity type of the elliptic fiber over each divisor is indicated. On the left, $D_x$ are the fractional divisors, while the dotted line represents $\mathcal{F}$, the residual divisor of the discriminant of the elliptic fibration. The points where the fundamentals matter of $G_x$ lie are represented by points where $\mathcal{F}$ meets $D_x$ transversally, and the points where antisymmetrics lie are represented by the points where $\mathcal{F}$ meets $D_x$ tangentially. Each pair of divisors $D_x$ and $D_y$ meet at a single orbifold point, where superconformal matter jointly charged under $G_x \times G_y$ lie. On the right, $\hat{D}_x$ are integral divisors on $\hat{B}$ obtained by resolving $D_x$. The theory now has only ordinary matter. In particular, there exist bifundamental matter at the intersection loci of adjacent divisors.