We analyse divergences of the scalar curvature of the vector multiplet moduli space of type IIA string theory compactified on a Calabi-Yau , along infinite-distance large volume limits. Extending previous results, we classify the origin of the divergence along trajectories which implement decompactifications to F-theory on and/or emergent heterotic string limits. In all cases, the curvature divergence can be traced back to a 4d rigid field theory that decouples from gravity along the limit. This can be quantified via the asymptotic relation ~ (Λ Λ ) , with Λ ≡ and Λ the species scale. In the UV, the 4d rigid field theory becomes a higher-dimensional, strongly-coupled rigid theory that also decouples from gravity. The nature of this UV theory is encoded in the exponent , and it either corresponds to a 5d SCFT, 6d SCFT or a Little String Theory.

Spain

Fernando Marchesano

Luca Melotti

United States

Max Wiesner

Asymptotic curvature divergences and non-gravitational theories

We study how tiny Dirac neutrino masses consistent with experimental constraints can arise in string theory SM-like vacua. We use as a laboratory 4d $$ \mathcal{N} $$ = 1 type IIA Calabi-Yau orientifold compactifications, and in particular recent results on Yukawa couplings at infinite field-space distance. In this regime we find Dirac neutrino masses of the form ≃ ⟨ ⟩, with the gauge coupling of the massive U(1) under which the right-handed neutrinos are charged, and which should be in the range ≃ 10 14 − 10 12 to reproduce neutrino data. The neutrino mass suppression occurs because the right-handed neutrino kinetic term behaves as ≃ 1 $$ {g}_{\nu}^2 $$ . At the same time a tower of -like states appears with characteristic scale ≃ $$ {g}_{\nu}^2{M}_{\textrm{P}} $$ ≃ 0 1 − 500 eV, in agreement with Swampland expectations. Two large hidden dimensions only felt by the sector arise at the same scale, while the string scale is around ≃ ≃ 10 − 700 TeV. Some phenomenological implications and model building challenges are described. We also describe the difficulties in obtaining appropriate tiny neutrino Yukawas in the cases of a single or more than two large dimensions. Thus the case with two large dimensions seems to be quite unique.

Gonzalo Casas

Luis Ibáñez

On small Dirac neutrino masses in string theory

We study limits of vanishing Yukawa couplings of 4d chiral matter fields in Quantum Gravity, using as a laboratory type IIA orientifolds with D6-branes. In these theories chiral fermions arise at brane intersections, where an infinite tower of charged particles dubbed gonions are localised. We show that in the limit Y → 0 some of these towers become asymptotically massless, while at the same time the kinetic term of some chiral fields becomes singular and at least two extra dimensions decompactify. For limits parametrised by a large complex structure saxion u, Yukawa couplings have a behaviour of the form Y ~ 1/u  r , with r some positive rational number. Moreover, in this limit some of the gauge couplings associated to the Yukawa vanish. The lightest gonion scales are of order m gon ~ g  s  M P with s &gt; 1, verifying the magnetic WGC with room to spare and with no need of its tower/sublattice versions. We also show how this behaviour can be understood in the context of the emergence of kinetic terms in Quantum Gravity. All these results may be very relevant for phenomenology, given the fact that some of the Yukawa couplings in the Standard Model are very small.

Yukawa couplings at infinite distance and swampland towers in chiral theories

We analyse the scalar curvature of the vector multiplet moduli space   <math> <msubsup> <mi>M</mi> <mi>X</mi> <mi>VM</mi> </msubsup> </math>  $$ {\mathcal{M}}_X^{\textrm{VM}} $$  of type IIA string theory compactified on a Calabi-Yau manifold X. While the volume of   <math> <msubsup> <mi>M</mi> <mi>X</mi> <mi>VM</mi> </msubsup> </math>  $$ {\mathcal{M}}_X^{\textrm{VM}} $$  is known to be finite, cases have been found where the scalar curvature diverges positively along trajectories of infinite distance. We classify the asymptotic behaviour of the scalar curvature for all large volume limits within   <math> <msubsup> <mi>M</mi> <mi>X</mi> <mi>VM</mi> </msubsup> </math>  $$ {\mathcal{M}}_X^{\textrm{VM}} $$ , for any choice of X, and provide the source of the divergence both in geometric and physical terms. Geometrically, there are effective divisors whose volumes do not vary along the limit. Physically, the EFT subsector associated to such divisors is decoupled from gravity along the limit, and defines a rigid   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 field theory with a non-vanishing moduli space curvature R rigid. We propose that the relation between scalar curvature divergences and field theories decoupled from gravity is a common trait of moduli spaces compatible with quantum gravity.

Lorenzo Paoloni

On the moduli space curvature at infinity

Massive type IIA flux compactifications of the form AdS4 × X 6, where X 6 admits a Calabi-Yau metric and O6-planes wrapping three-cycles, display families of vacua with parametric scale separation between the compactification scale and the AdS4 radius, generated by an overall rescaling of internal four-form fluxes. For toroidal orbifolds one can perform two T-dualities and map this background to an orientifold of massless type IIA compactified on an SU(3)-structure manifold with fluxes. Via a 4d EFT analysis, we generalise this last construction and embed it into new branches of supersymmetric and non-supersymmetric vacua with similar features. We apply our results to propose new infinite families of vacua based on elliptic fibrations with metric fluxes. Parametric scale separation is achieved by an asymmetric flux rescaling which, however, in general is not a simple symmetry of the 4d equations of motion. At this level of approximation the vacua are stable but, unlike in the Calabi-Yau case, they display a non-universal mass spectrum of light fields.

Rafael Carrasco

Thibaut Coudarchet

David Prieto

New families of scale separated vacua

Conventional wisdom dictates that ℤ N  factors in the integral cohomology group H  p (X  n , ℤ) of a compact manifold X  n  cannot be computed via smooth p-forms. We revisit this lore in light of the dimensional reduction of string theory on X  n , endowed with a G-structure metric that leads to a supersymmetric EFT. If massive p-form eigenmodes of the Laplacian enter the EFT, then torsion cycles coupling to them will have a non-trivial smeared delta form, that is an EFT long-wavelength description of p-form currents of the (n − p)-cycles of X  n . We conjecture that, whenever torsion cycles are calibrated, their linking number can be computed via their smeared delta forms. From the EFT viewpoint, a torsion factor in cohomology corresponds to a ℤ N  gauge symmetry realised by a Stückelberg-like action, and calibrated torsion cycles to BPS objects that source the massive fields involved in it.

Matteo Zatti

Torsion in cohomology and dimensional reduction

The analysis of type IIB flux vacua on warped Calabi-Yau orientifolds becomes considerably involved for a large number of complex structure fields. We however show that, for a quadratic flux superpotential, one can devise simplifying schemes which effectively reduce the large number of equations down to a few. This can be achieved by imposing the vanishing of certain flux quanta in the large complex structure regime, and then choosing the remaining quanta to respect the symmetries of the underlying prepotential. One can then implement an algorithm to find large families of flux vacua with a fixed flux tadpole, independently of the number of fields. We illustrate this approach in a Calabi-Yau manifold with 51 complex structure moduli, where several reduction schemes can be implemented in order to explicitly solve the vacuum equations for that sector. Our findings display a flux-tadpole-to-stabilized-moduli ratio that is marginally above the bound proposed by the Tadpole Conjecture, and we discuss several effects that would take us below such a bound.

Mikel Urkiola

Symmetric fluxes and small tadpoles

We revisit the Emergence Proposal in 4d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 vector multiplet sectors that arise from type II string Calabi-Yau compactifications, with emphasis on the role of axionic fundamental strings, or EFT strings. We focus on large-volume type IIA compactifications, where EFT strings arise from NS5-branes wrapping internal four-cycles, and consider a set of infinite-distance moduli-space limits that can be classified in terms of a scaling weight w = 1, 2, 3. It has been shown before how one-loop threshold effects of an infinite tower of BPS particles made up of D2/D0-branes generate the asymptotic behaviour of the gauge kinetic functions along limits with w = 3. We extend this result to w = 2 limits, by taking into account D2-brane multi-wrapping numbers. In w = 1 limits the leading tower involves EFT string oscillations, and one can reproduce the behaviour of both weakly and strongly-coupled U(1)’s independently on whether the EFT string is critical or not, by assuming that charged modes dominate the light spectrum.

EFT strings and emergence

We analyze the tree-level potential of type IIB flux compactifications in warped Calabi-Yau orientifolds, in regions of weak coupling and moderately large complex structure. In this regime, one may approximate the flux-induced superpotential W by a polynomial on the axio-dilaton and complex structure fields, and a significant fraction of vacua corresponds to a quadratic W. In this quadratic case, we argue that vacua fall into three classes, for which one can push the analytic description of their features. In particular, we provide analytic expressions for the vacuum expectation values and flux-induced masses of the axio-dilaton and complex structure fields in a large subclass of vacua, independently of the Calabi-Yau and the number of moduli. We show that supersymmetric vacua always contain flat directions, at least at this level of approximation. Our findings allow to generate vast ensembles of flux vacua in specific Calabi-Yau geometries, as we illustrate in a particular example.

Analytics of type IIB flux vacua and their mass spectra

We consider massive type IIA orientifold compactifications of the form AdS4 × X 6, where X 6 admits a Calabi-Yau metric and is threaded by background fluxes. From a 4d viewpoint, fluxes generate a potential whose vacua have been classified, including one   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 and three perturbatively stable   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 0 branches. We reproduce this result from a 10d viewpoint, by solving the type IIA equations at the same level of detail as previously done for the   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 branch. All solutions exhibit localised sources and parametric scale separation. We then analyse the non-perturbative stability of the   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 0 branches. We consider new 4d membranes, obtained from wrapping D8-branes on X 6 or D6-branes on its divisors, threaded by non-diluted worldvolume fluxes. Using them we show that all branches are compatible with the Weak Gravity Conjecture for membranes. In fact, most vacua satisfy the sharpened conjecture that predicts superextremal membranes in   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 0 settings, except for a subset whose non-perturbative stability remains an open problem.