Phases of 5d SCFTs from M-/F-theory on Non-Flat Fibrations

TL;DR

The work develops a framework in which non-flat resolutions of non-minimal elliptic singularities in Calabi–Yau threefolds encode 5d SCFT data directly through M-/F-theory duality. By resolving fiber non-minimalities while keeping the base fixed, surface components arise that realize 5d gauge sectors and mass deformations corresponding to circle reductions of 6d conformal matter. This construction naturally embeds the 5d enhanced flavor symmetry as a subgroup of the 6d flavor symmetry, read off from the global fiber geometry, and clarifies how flop transitions connect different 5d phases. The authors demonstrate the program in toric tops, obtaining rank-1, rank-2, and rank-4 theories from 6d E-string and $(E_8,SU(2))$, $(E_8,SU(3))$ conformal matter, and show precise matches between geometric prepotentials and field-theory expectations, including CS levels and theta angles. Overall, the approach offers a concrete geometric route toward classifying 5d SCFTs via their 6d ancestors and suggests a path toward a systematic catalog of non-flat fiber resolutions.

Abstract

We initiate the systematic investigation of non-flat resolutions of non-minimal singularities in elliptically fibered Calabi--Yau threefolds. Compactification of M-theory on these geometries provides an alternative approach to studying phases of five-dimensional superconformal field theories (5d SCFTs). We argue that such resolutions capture non-trivial holonomies in the circle reduction of the 6d conformal matter theory that is the F-theory interpretation of the singular fibration. As these holonomies become mass deformations in the 5d theory, non-flat resolutions furnish a novel method in the attempt to classify 5d SCFTs through 6d SCFTs on a circle. A particularly pleasant aspect of this proposal is the explicit embedding of the 5d SCFT's enhanced flavor group inside that of the parent 6d SCFT, which can be read off from the geometry. We demonstrate these features in toric examples which realize 5d theories up to rank four.

Figures (10)

• Figure 1: A cartoon illustrating the splitting of fibers after blowing up the non-minimal singularity over a point of the base. The gray squares represent the surface components of the non-flat fiber. In general, codimension one fibers can split into several curves which all (green) or only partly (red) lie inside the surfaces. They can also remain irreducible, and sit inside (blue) or outside (black) of any surface. By blowing down the surfaces to a point, the blue and green fibral curves are forced to shrink over the codimension one loci of the base. In the case depicted here, this would lead to an enhancement of an $SU(3)$ subgroup of the full $E_6$ flavor symmetry.
• Figure 2: A visualization of the $E_8$-top over $F_{10}$. The vertical direction is ${\rm x}_3$. At ${\rm x}_3 = 0$, the red points are the lattice vectors of $F_{10}$. For ${\rm x}_3 >0$, the blue points represent the vectors of the exceptional divisors of the $E_8$, as they sit on the edges of the top. The green points are facet interiors and correspond to non-flat surfaces. Different facets encode the local geometry at different non-flat fibers. The triangulation shown in the figures is arbitrary; for the rank one sector it defines the dP$_1$ phase.
• Figure 3: Toric diagram of $F_{10}$ with blow-ups on the edges. The vertices $(\vec{x}, \vec{y}, \vec{z})$ give rise to bi-, tri-, and rational sections, respectively, of the elliptic fibration. The $SU(2)$ singularity over $\{s_4 = 0\}$ is resolved by $j$, and the $SU(3)$ over $\{s_8 =0 \}$ is resolved by $f_1$ and $f_2$. The edges of length $l$ are labelled by ${\cal E}_l$.
• Figure 4: The full facet \ref{['fig:rank_1_facette']} of $\Diamond$ over the edge ${\cal E}_1$ (see figure \ref{['fig:F10_with_blow-up']}) containing one interior vector $\vec{r}$. Depending on the triangulation of top, which induces different triangulations \ref{['fig:rank_1_dP3']}--\ref{['fig:rank_1_P2']} of the facet, the non-flat geometry realizes a different weakly coupled phase of the rank one 5d SCFT. The similarity between this facet and the base polygon $F_{10}$ of $\Diamond$ is purely coincidental. The red lines indicate a projection of the corresponding toric diagram preserving the cones, and thus a ruling of the toric surface.
• Figure 5: The facet of the $E_8$ top over the edge ${\cal E}_2$, containing the non-flat surfaces $\{t_1\}$ and $\{t_2\}$ over $\{s_4\} \cap W$.