We propose a graph-based approach to 5d superconformal field theories (SCFTs) based on their realization as M-theory compactifications on singular elliptic Calabi-Yau threefolds. Field-theoretically, these 5d SCFTs descend from 6d $$ \mathcal{N} $$
 = (1, 0) SCFTs by circle compactification and mass deformations. We derive a description of these theories in terms of graphs, so-called Combined Fiber Diagrams, which encode salient features of the partially resolved Calabi-Yau geometry, and provides a combinatorial way of characterizing all 5d SCFTs that descend from a given 6d theory. Remarkably, these graphs manifestly capture strongly coupled data of the 5d SCFTs, such as the superconformal flavor symmetry, BPS states, and mass deformations. The capabilities of this approach are demonstrated by deriving all rank one and rank two 5d SCFTs. The full potential, how- ever, becomes apparent when applied to theories with higher rank. Starting with the higher rank conformal matter theories in 6d, we are led to the discovery of previously unknown flavor symmetry enhancements and new 5d SCFTs.

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Fibers add flavor. Part I. Classification of 5d SCFTs, flavor symmetries and BPS states

Following a recent proposal, we delineate a general procedure to classify 5d SCFTs via compactifications of 6d SCFTs on a circle (possibly with a twist by a discrete global symmetry). The path from 6d SCFTs to 5d SCFTs can be divided into two steps. The first step involves computing the Coulomb branch data of the 5d KK theory obtained by compactifying a 6d SCFT on a circle of finite radius. The second step involves computing the limit of the KK theory when the inverse radius along with some other mass parameters is sent to infinity. Under this RG flow, the KK theory reduces to a 5d SCFT. We illustrate these ideas in the case of untwisted compactifications of rank one 6d SCFTs that can be constructed in F-theory without frozen singularities. The data of the corresponding KK theory can be packaged in the geometry of a Calabi-Yau threefold that we explicitly compute for every case. The RG flows correspond to flopping a collection of curves in the threefold and we formulate a concrete set of criteria which can be used to determine which collection of curves can induce the relevant RG flows, and, in principle, to determine the Calabi-Yau geometries describing the endpoints of these flows. We also comment on how to generalize our results to arbitrary rank.

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Classifying 5d SCFTs via 6d SCFTs: rank one

https://cds.cern.ch/record/2709400/files/1-s2.0-S0370157319303072-main.pdf

Data science applications to string theory

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.

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Phases of 5d SCFTs from M-/F-theory on non-flat fibrations

We study the topological properties of Calabi-Yau threefold hypersurfaces at large h1,1. We obtain two million threefolds X by triangulating polytopes from the Kreuzer-Skarke list, including all polytopes with 240 ≤ h1,1 ≤ 491. We show that the Kahler cone of X is very narrow at large h1,1, and as a consequence, control of the α′ expansion in string compactifications on X is correlated with the presence of ultralight axions. If every effective curve has volume ≥ 1 in string units, then the typical volumes of irreducible effective curves and divisors, and of X itself, scale as (h1,1)p, with 3 ≲ p ≲ 7 depending on the type of cycle in question. Instantons from branes wrapping these cycles are thus highly suppressed.

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The Kreuzer-Skarke axiverse

We systematically analyze the fibration structure of toric hypersurface Calabi-Yau threefolds with large and small Hodge numbers. We show that there are only four such Calabi-Yau threefolds with h1,1 ≥ 140 or h2,1 ≥ 140 that do not have manifest elliptic or genus one fibers arising from a fibration of the associated 4D polytope. There is a genus one fibration whenever either Hodge number is 150 or greater, and an elliptic fibration when either Hodge number is 228 or greater. We find that for small h1,1 the fraction of polytopes in the KS database that do not have a genus one or elliptic fibration drops exponentially as h1,1 increases. We also consider the different toric fiber types that arise in the polytopes of elliptic Calabi-Yau threefolds.

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On the prevalence of elliptic and genus one fibrations among toric hypersurface Calabi-Yau threefolds

We compare the sets of Calabi-Yau threefolds with large Hodge numbers that are constructed using toric hypersurface methods with those can be constructed as elliptic fibrations using Weierstrass model techniques motivated by F-theory. There is a close correspondence between the structure of “tops” in the toric polytope construction and Tate form tunings of Weierstrass models for elliptic fibrations. We find that all of the Hodge number pairs (h1,1, h2,1) with h1,1 or h2,1 ≥ 240 that are associated with threefolds in the Kreuzer-Skarke database can be realized explicitly by generic or tuned Weierstrass/Tate models for elliptic fibrations over complex base surfaces. This includes a relatively small number of somewhat exotic constructions, including elliptic fibrations over non-toric bases, models with new Tate tunings that can give rise to exotic matter in the 6D F-theory picture, tunings of gauge groups over non-toric curves, tunings with very large Hodge number shifts and associated nonabelian gauge groups, and tuned Mordell-Weil sections associated with U(1) factors in the corresponding 6D theory.

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Comparing elliptic and toric hypersurface Calabi-Yau threefolds at large Hodge numbers

We compare the sets of Calabi-Yau threefolds with large Hodge numbers that are constructed using toric hypersurface methods with those can be constructed as elliptic fibrations using Weierstrass model techniques motivated by F-theory. There is a close correspondence between the structure of "tops" in the toric polytope construction and Tate form tunings of Weierstrass models for elliptic fibrations. We find that all of the Hodge number pairs ($h^{1, 1},h^{2, 1}$) with $h^{1,1}$ or $h^{2, 1}\geq 240$ that are associated with threefolds in the Kreuzer-Skarke database can be realized explicitly by generic or tuned Weierstrass/Tate models for elliptic fibrations over complex base surfaces. This includes a relatively small number of somewhat exotic constructions, including elliptic fibrations over non-toric bases, models with new Tate tunings that can give rise to exotic matter in the 6D F-theory picture, tunings of gauge groups over non-toric curves, tunings with very large Hodge number shifts and associated nonabelian gauge groups, and tuned Mordell-Weil sections associated with U(1) factors in the corresponding 6D theory.

/pdf/comparing-elliptic-and-toric-hypersurface-calabi-yau-1l4gi5rcss.pdf

We find that for many Calabi-Yau threefolds with elliptic or genus one fibrations mirror symmetry factorizes between the fiber and the base of the fibration. In the simplest examples, the generic CY elliptic fibration over any toric base surface B that supports an elliptic Calabi-Yau threefold has a mirror that is an elliptic fibration over a dual toric base surface $$ \tilde{B} $$
 that is related through toric geometry to the line bundle −6KB. The Kreuzer-Skarke database includes all these examples and gives a wide range of other more complicated constructions where mirror symmetry also factorizes. Since recent evidence suggests that most Calabi-Yau threefolds are elliptic or genus one fibered, this points to a new way of understanding mirror symmetry that may apply to a large fraction of smooth Calabi-Yau threefolds. The factorization structure identified here can also apply for CalabiYau manifolds of higher dimension.

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Mirror symmetry and elliptic Calabi-Yau manifolds

We find through a systematic analysis that all but 29,223 of the 473.8 million 4D reflexive polytopes found by Kreuzer and Skarke have a 2D reflexive subpolytope. Such a subpolytope is generally associated with the presence of an elliptic or genus one fibration in the corresponding birational equivalence class of Calabi-Yau threefolds. This extends the growing body of evidence that most Calabi-Yau threefolds have an elliptically fibered phase.

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Yu-Chien Huang

Papers

On the prevalence of elliptic and genus one fibrations among toric hypersurface Calabi-Yau threefolds

Comparing elliptic and toric hypersurface Calabi-Yau threefolds at large Hodge numbers

Comparing elliptic and toric hypersurface Calabi-Yau threefolds at large Hodge numbers

Mirror symmetry and elliptic Calabi-Yau manifolds

Fibration structure in toric hypersurface Calabi-Yau threefolds