We invoke probe brane arguments to classify the asymptotic behavior of general five-dimensional supergravity theories with eight supercharges near infinite distance boundaries of the vector multiplet moduli space. Imposing consistency of supergravity strings we derive several constraints on the Chern-Simons couplings entering the prepotential, including their non-negativity. This establishes a classification of infinite distance limits analogous to those for theories obtained as Calabi-Yau compactifications, but without having to assume a geometric or string theoretic origin. All infinite distance limits are found to be either vector or tensor limits, depending on the nature of the gauge potential becoming weakly coupled at the fastest rate. In particular, we prove uniqueness results for the asymptotically leading gauge fields. The asymptotic physics along these limits is in perfect agreement with the predictions of the Emergent String Conjecture and hence serves as bottom-up evidence for the latter. Our findings imply that every consistent five-dimensional $$\mathcal{N}$$ = 1 supergravity with a non-compact vector multiplet moduli space either descends from six dimensions or contains a stringy subsector.

Germany

Lukas Kaufmann

Stefano Lanza

Timo Weigand

Asymptotics of 5d supergravity theories and the emergent string conjecture

<p>The dark dimension scenario, predicting one extra mesoscopic dimension in the micron range, has emerged by applying various swampland principles to the dark energy. In this note we find that realizing the grand unification of gauge forces is highly constraining in this context. Without actually constructing any grand unified theory (GUT) models, we argue that the mere assumption of grand unification of forces in this scenario, together with the experimental bounds on massive replicas of the Standard Model gauge bosons, predicts an upper bound for the GUT scale, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>M</mi><mi>GUT</mi></msub><mo>≲</mo><msup><mn>10</mn><mn>16</mn></msup><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>. Combined with the experimental bound on the proton lifetime, this predicts that the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>X</mi></math> gauge boson mediating proton decay is a 5D solitonic string of Planckian tension stretched across a length scale <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>L</mi><mo>∼</mo><mo stretchy="false">(</mo><mn>1</mn><mi>–</mi><mn>10</mn><mtext> </mtext><mtext> </mtext><mi>TeV</mi><msup><mo stretchy="false">)</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup></math> ending on gauge branes of the same diameter <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo>∼</mo><mi>L</mi></math>. This leads to a mass of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>M</mi><mi>X</mi></msub><mo>∼</mo><msup><mn>10</mn><mn>15</mn></msup><mi>–</mi><msup><mn>10</mn><mn>16</mn></msup><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>. In particular assuming grand unification in the dark dimension scenario results in a tower of Kaluza-Klein excitations of Standard Model gauge bosons on the gauge branes in the 1–10 TeV range. This suggests that the diameter/separation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>L</mi></math> of the gauge branes correlates with both the weak scale <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo>∼</mo><mn>1</mn><mo>/</mo><mi>L</mi></math> near a TeV <i>and</i> the GUT scale <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo>∼</mo><msubsup><mi>M</mi><mn>5</mn><mn>2</mn></msubsup><mi>L</mi></math> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>10</mn><mn>16</mn></msup><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>.</p>

United States

Jonathan J. Heckman

Cumrun Vafa

China

Fengjun Xu

Dark dimension and the grand unification of forces

We interpret infinite-distance limits in the complex structure moduli space of F-theory compactifications to six dimensions in the light of general ideas in quantum gravity. The limits we focus on arise from non-minimal singularities in the elliptic fiber over curves in a Hirzebruch surface base, which do not admit a crepant resolution. Such degenerations take place along infinite directions in the non-perturbative brane moduli space in F-theory. A blow-up procedure, detailed generally in Part I of this project [1], gives rise to an internal space consisting of a union of log Calabi-Yau threefolds glued together along their boundaries. We geometrically classify the resulting configurations for genus-zero single infinite-distance limits. Special emphasis is put on the structure of singular fibers in codimension zero and one. As our main result, we interpret the central fiber of these degenerations as endpoints of a decompactification limit with six-dimensional defects. The conclusions rely on an adiabatic limit to gain information on the asymptotically massless states from the structure of vanishing cycles. We also compare our analysis to the heterotic dual description where available. Our findings are in agreement with general expectations from quantum gravity and provide further evidence for the Emergent String Conjecture.

Rafael Álvarez-García

South Korea

Seung-Joo Lee

Non-minimal elliptic threefolds at infinite distance II: asymptotic physics

We study infinite-distance limits in the complex structure moduli space of elliptic Calabi-Yau threefolds. In F-theory compactifications to six dimensions, such limits include infinite-distance trajectories in the non-perturbative open string moduli space. The limits are described as degenerations of elliptic threefolds whose central elements exhibit non-minimal elliptic fibers, in the Kodaira sense, over curves on the base. We show how these non-crepant singularities can be removed by a systematic sequence of blow-ups of the base, leading to a union of log Calabi-Yau spaces glued together along their boundaries. We identify criteria for the blow-ups to give rise to open chains or more complicated trees of components and analyse the blow-up geometry. While our results are general and applicable to all non-minimal degenerations of Calabi-Yau threefolds in codimension one, we exemplify them in particular for elliptic threefolds over Hirzebruch surface base spaces. We also explain how to extract the gauge algebra for F-theory probing such reducible asymptotic geometries. This analysis is the basis for a detailed F-theory interpretation of the associated infinite-distance limits that will be provided in a companion paper [1].

Non-minimal elliptic threefolds at infinite distance. Part I. Log Calabi-Yau resolutions

We examine the minimal constraints imposed by the Weak Gravity Conjecture (WGC) on the particle spectrum of a quantum gravity theory. Towers of super-extremal states have previously been argued to be required for consistency of the WGC under circle reduction. At the same time, there exist classes of theories where no tower of super-extremal particle states below the black hole threshold has been established with current techniques. We resolve this tension by arguing for the existence of a minimal radius for circle reductions of generic quantum gravity theories. Below this threshold, the notion of a circle compactification breaks down, bypassing the need for a tower of super-extremal states to satisfy the WGC after circle reduction. Based on this we propose that if a theory satisfies the WGC at the particle level below the black hole threshold, these states are sufficient for consistency under dimensional reduction, even in absence of a tower of super-extremal particles. Apart from general arguments, we provide independent evidence for this main result in F-, M- and string theory compactifications. According to the Emergent String Conjecture the only exception to the appearance of a minimal radius arises in asymptotically weak-coupling limits for heterotic strings, which aligns with the appearance of a weakly coupled super-extremal tower of particle states. This observation motivates a Minimal Weak Gravity Conjecture which states that towers of super-extremal particles occur if and only if they are required by consistency of the WGC under dimensional reduction.

Cesar Cota

Alessandro Mininno

Max Wiesner

The minimal weak gravity conjecture

The tower Weak Gravity Conjecture predicts infinitely many super-extremal states along every ray in the charge lattice of a consistent quantum gravity theory. We show this far-reaching claim in five-dimensional compactifications of M-theory on Calabi-Yau 3-folds for gauge groups with a weak coupling limit. We first characterize the possible weak coupling limits, building on an earlier classification of infinite distance limits in the Kähler moduli space of M-theory compactifications. We find that weakly coupled gauge groups are associated to curves on the compactification space contained in generic fibers or in fibers degenerating at finite distance in their moduli space. These always admit an interpretation as a Kaluza-Klein or winding U(1) in a dual frame or as part of a dual perturbative heterotic gauge group, in agreement with the Emergent String Conjecture. Using the connection between Donaldson-Thomas invariants and Noether-Lefschetz theory, we then show that every ray in the associated charge lattice either supports a tower of BPS states or of non-BPS states, and prove that these satisfy the super-extremality condition, at least in the weak coupling regime.

The asymptotic weak gravity conjecture in M-theory

We propose quantum gravitational constraints on effective four-dimensional theories with   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 supersymmetry. These Swampland constraints arise by demanding consistency of the worldsheet theory of a class of axionic, or EFT, strings whose existence follows from the Completeness Conjecture of quantum gravity. Modulo certain assumptions, we derive positivity bounds and quantization conditions for the axionic couplings to the gauge and gravitational sector at the two- and four-derivative level, respectively. We furthermore obtain general bounds on the rank of the gauge sector in terms of the gravitational couplings to the axions. We exemplify how these bounds rule out otherwise consistent effective supergravity theories as theories of quantum gravity. Our derivations of the quantum gravity bounds are tested and further motivated in concrete string theoretic settings. In particular, this leads to a sharper version of the bound on the gauge group rank in F-theory on elliptic four-folds with a smooth base, which improves the known geometrical Kodaira bounds. We furthermore provide a detailed derivation of the EFT string constraints in heterotic string compactifications including higher derivative corrections to the effective action and apply the bounds to M-theory compactifications on G 2 manifolds.

Italy

Luca Martucci

Nicolò Risso

Quantum gravity bounds on   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 effective theories in four dimensions

We investigate the asymptotic Tower Weak Gravity Conjecture in weak coupling limits of open string theories with minimal supersymmetry in four dimensions, focusing for definiteness on gauge theories realized on 7-branes in F-theory. Contrary to expectations, we find that not all weak coupling limits contain an obvious candidate for a tower of states marginally satisfying the super-extremality bound. The weak coupling limits are classified geometrically in the framework of EFT string limits and their generalizations. We find three different classes of weak coupling limits, whose physics is characterized by the ratio of the magnetic weak gravity scale and the species scale. The four-dimensional Tower Weak Gravity Conjecture is satisfied by the (non-BPS) excitations of the weakly coupled EFT string only in emergent string limits, where the EFT string can be identified with a critical (heterotic) string. All other weak coupling limits lead to a decompactification either to an in general strongly coupled gauge theory coupled to gravity or to a defect gauge theory decoupling from the gravitational bulk, in agreement with the absence of an obvious candidate for a marginally super-extremal tower of states.

The asymptotic Weak Gravity Conjecture for open strings

Motivated by the Swampland Distance and the Emergent String Conjecture of Quantum Gravity, we analyse the infinite distance degenerations in the complex structure moduli space of elliptic K3 surfaces. All complex degenerations of K3 surfaces are known to be classified according to their associated Kulikov models of Type I (finite distance), Type II or Type III (infinite distance). For elliptic K3 surfaces, we characterise the underlying Weierstrass models in detail. Similarly to the known two classes of Type II Kulikov models for elliptic K3 surfaces we find that the Weierstrass models of the more elusive Type III Kulikov models can be brought into two canonical forms. We furthermore show that all infinite distance limits are related to degenerations of Weierstrass models with non-minimal singularities in codimension one or to models with degenerating generic fibers as in the Sen limit. We explicitly work out the general structure of blowups and base changes required to remove the non-minimal singularities. These results form the basis for a classification of the infinite distance limits of elliptic K3 surfaces as probed by F-theory in the companion paper [1]. The Type III limits, in particular, are (partial) decompactification limits as signalled by an emergent affine enhancement of the symmetry algebra.

Elliptic K3 surfaces at infinite complex structure and their refined Kulikov models

We investigate infinite distance limits in the complex structure moduli space of F-theory compactified on K3 to eight dimensions. While this is among the simplest possible arenas to test ideas about the Swampland Distance Conjecture, it is nevertheless non-trivial enough to improve our understanding of the physics for these limiting geometries, including phenomena of emergence. It also provides a perspective on infinite distance limits from the viewpoint of open strings. The paper has two quite independent themes. In the main part we show that all degenerations of elliptic K3 surfaces at infinite distance as analysed in the companion paper [1] can be interpreted as (partial) decompactification or emergent string limits in F-theory, in agreement with the Emergent String Conjecture. We present a unified geometric picture of the possible towers of states that can become light and illustrate our general claims via the connection between Kulikov models of degenerating K3 surfaces and the dual heterotic string. As an application we classify the possible maximal non-abelian Lie algebras and their Kac-Moody and loop extensions that can arise in the infinite distance limits. In the second part we discuss the infinite distance behaviour of certain exact quartic gauge couplings. We encounter a tension with the hypothesis that effective couplings should be fully generated by integrating out massive states. We show that by appropriately renormalizing the string coupling, at least partial emergence can be achieved.