We consider toroidal asymmetric orbifolds of the heterotic string preserving all 16 supercharges, developing a general formalism to study components of the moduli space characterized by rank reduction of the gauge group. In particular we construct six- and four-dimensional heterotic islands with no massless moduli other than the dilaton. The formalism involves the Leech lattice, its automorphisms and their corresponding invariant and normal, or coinvariant, sublattices.

Argentina

Gerardo Aldazabal

Eduardo Andrés

Venezuela

Germany

Anamaría Font

Italy

Kumar Narain

United Kingdom

Ida Zadeh

Asymmetric orbifolds, rank reduction and heterotic islands

We first describe the low energy dynamics of ten dimensional heterotic supergravity compactified on the smooth, flat 3-manifold   <math> <mi>T</mi> </math>  $$ \mathbbm{T} $$  3/ℤ2, without supersymmetry, and explain how it arises from flat heterotic gauge fields. The semi-classical theory has both Coulomb and Higgs branches of non-supersymmetric vacua. We then give an exact worldsheet description as asymmetric orbifolds of   <math> <mi>T</mi> </math>  $$ \mathbbm{T} $$  3, where the orbifold generator involves a Nikulin non-symplectic involution θ of the even self-dual lattice Γ(19,3). Along the way we briefly compare our findings with M-theory on K3/θ. Our construction gives a novel CFT description of the semi-classical field theory moduli space. In particular, the Wilson line parameters in the lattice I ⊂ Γ(19,3) of signature (19 − s, 1) which is invariant under θ, and in its orthogonal complement N , correspond respectively to Coulomb and Higgs branch moduli. There is a rich pattern of transitions amongst Higgs and Coulomb branches which we describe using the worldsheet theory.

Bobby Acharya

Heterotic strings on   <math> <mi>T</mi> </math>  $$ \mathbbm{T} $$  3/ℤ2, Nikulin involutions and M-theory

We conjecture that in a consistent supergravity theory with non-vanishing gravitino mass, the limit m 3/2 → 0 is at infinite distance. In particular one can write M tower ~   <math> <msubsup> <mi>m</mi> <mrow> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mrow> <mi>δ</mi> </msubsup> </math>  $$ {m}_{3/2}^{\delta } $$  so that as the gravitino mass goes to zero, a tower of KK states as well as emergent strings becomes tensionless. This conjecture may be motivated from the Weak Gravity Conjecture as applied to strings and membranes and implies in turn the AdS Distance Conjecture. We test this proposal in classical 4d type IIA orientifold vacua in which one obtains a range of values   <math> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </math>  $$ \frac{1}{3} $$  ≤ δ ≤ 1. The parameter δ is related to the scale decoupling exponent in AdS vacua and to the α exponent in the Swampland Distance Conjecture for the type IIA complex structure. We present a general analysis of the gravitino mass in the limits of moduli space in terms of limiting Mixed Hodge Structures and study in some detail the case of two-moduli F-theory settings. Moreover, we obtain general lower bounds δ ≥   <math> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> <mo>,</mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </math>  $$ \frac{1}{3},\frac{1}{4} $$  for Calabi-Yau threefolds and fourfolds, respectively. The conjecture has important phenomenological implications. In particular we argue that low-energy supersymmetry of order 1 TeV is only obtained if there is a tower of KK states at an intermediate scale, of order 108 GeV. One also has an upper bound for the Hubble constant upon inflation H ≲   <math> <msubsup> <mi>m</mi> <mrow> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mrow> <mi>δ</mi> </msubsup> <msubsup> <mi>M</mi> <mi>P</mi> <mfenced> <mrow> <mn>1</mn> <mo>−</mo> <mi>δ</mi> </mrow> </mfenced> </msubsup> </math>  $$ {m}_{3/2}^{\delta }{M}_{\mathrm{P}}^{\left(1-\delta \right)} $$ .

Spain

Alberto Castellano

France

Alvaro Herráez

Luis Ibáñez

A gravitino distance conjecture

Compactifications of the heterotic string on special T  d /ℤ2 orbifolds realize a landscape of string models with 16 supercharges and a gauge group on the left-moving sector of reduced rank d + 8. The momenta of untwisted and twisted states span a lattice known as the Mikhailov lattice II(d), which is not self-dual for d &gt; 1. By using computer algorithms which exploit the properties of lattice embeddings, we perform a systematic exploration of the moduli space for d ≤ 2, and give a list of maximally enhanced points where the U(1) d+8 enhances to a rank d + 8 non-Abelian gauge group. For d = 1, these groups are simply-laced and simply-connected, and in fact can be obtained from the Dynkin diagram of E10. For d = 2 there are also symplectic and doubly-connected groups. For the latter we find the precise form of their fundamental groups from embeddings of lattices into the dual of II(2). Our results easily generalize to d &gt; 2.

Bernardo Fraiman

Mariana Graña

Carmen Núñez

Héctor Freitas

Exploring the landscape of CHL strings on T  d

Superstring/M-theory compactified on compact Ricci flat manifolds have recently been conjectured to exhibit instabilities whenever the metrics do not have special holonomy. We use worldsheet conformal field theory to investigate instabilities of Type II superstring theories on compact, Ricci flat, spin 3-manifolds including a worldsheet description of their spin structures. The instabilities are signalled by the appearance of stringy tachyons at small radius and a negative (1-loop) vacuum energy density at large radius. We briefly discuss the extension to higher dimensions.

Stringy tachyonic instabilities of non-supersymmetric Ricci flat backgrounds

Compactifications of the heterotic string on T  d  are the simplest, yet rich enough playgrounds to uncover swampland ideas: the U(1) d+16 left-moving gauge symmetry gets enhanced at special points in moduli space only to certain groups. We state criteria, based on lattice embedding techniques, to establish whether a gauge group is realized or not. For generic d, we further show how to obtain the moduli that lead to a given gauge group by modifying the method of deleting nodes in the extended Dynkin diagram of the Narain lattice II1,17. More general algorithms to explore the moduli space are also developed. For d = 1 and 2 we list all the maximally enhanced gauge groups, moduli, and other relevant information about the embedding in II d,d+16. In agreement with the duality between heterotic on T 2 and F-theory on K3, all possible gauge groups on T 2 match all possible ADE types of singular fibers of elliptic K3 surfaces. We also present a simple method to transform the moduli under the duality group, and we build the map that relates the charge lattices and moduli of the compactification of the E8 × E8 and Spin(32)/ℤ2 heterotic theories.

Exploring the landscape of heterotic strings on T  d

We study the separation of AdS and Kaluza-Klein (KK) scales in type II 4d AdS orientifold vacua. We first address this problem in toroidal/orbifold type IIA vacua with metric fluxes, corresponding to compactifications in twisted tori, both from the 4d and 10d points of view. We show how the naive application of the effective 4d theory leads to results which violate the AdS distance conjecture, in a class of   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 supersymmetric models which have a 10d lifting to a compactification on S 3 × S 3. We show how using KK scales properly modified by the compact metric leads to no separationof scales with   <math> <msubsup> <mi>M</mi> <mi>KK</mi> <mn>2</mn> </msubsup> <mo>=</mo> <mi>c</mi> <mfenced> <mi>Λ</mi> </mfenced> <mo>,</mo> </math>  $$ {M}_{\mathrm{KK}}^2=\mathfrak{c}\left|\Lambda \right|, $$  with c a numerical constant independent of fluxes. This applies with no need to keep non-leading fluxes fixed. We also consider a class of IIB models with non-geometric fluxes in which the effective field theory analysis seems to lead to a naive separation of scales and a violation of the AdS distance conjecture. It has a T-dual which again may be understood as a 10d type IIA theory compactified on S 3 × S 3. In this geometric dual one again observes that the strong AdS distance conjecture is obeyed with   <math> <msubsup> <mi>M</mi> <mi>KK</mi> <mn>2</mn> </msubsup> <mo>=</mo> <mi>c</mi> <mo>′</mo> <mfenced> <mi>Λ</mi> </mfenced> <mo>,</mo> </math>  $$ {M}_{\mathrm{KK}}^2=\mathfrak{c}^{\prime}\left|\Lambda \right|, $$  if one takes into account the curvature in the internal space. These findings seem to suggest that all toroidal/orbifold models with fluxes in this class obey   <math> <msubsup> <mi>M</mi> <mi>KK</mi> <mn>2</mn> </msubsup> <mo>=</mo> <mi>c</mi> <mfenced> <mi>Λ</mi> </mfenced> </math>  $$ {M}_{\mathrm{KK}}^2=\mathfrak{c}\left|\Lambda \right| $$  with   <math> <mi>c</mi> </math>  $$ \mathfrak{c} $$  a flux-independent numerical constant.

On scale separation in type II AdS flux vacua

The Swampland Distance Conjecture states that at infinite distance in the scalar moduli space an infinite tower of particles become exponentially massless. We study this issue in the context of 4d type IIA and type IIB Calabi-Yau compactifications. We find that for large moduli not only towers of particles but also domain walls and strings become tensionless. We study in detail the case of type IIA and IIB 𝒩 = 1 CY orientifolds and show how for infinite Kähler and/or complex structure moduli towers of domain walls and strings become tensionless, depending on the particular direction in moduli space. For the type IIA case we construct the monodromy orbits of domain walls in detail. We study the structure of mass scales in these limits and find that these towers may occur at the same scale as the fundamental string scale or the KK scale making sometimes difficult an effective field theory description. The structure of IIA and IIB towers are consistent with mirror symmetry, as long as towers of exotic domain walls associated to non-geometric fluxes also appear. We briefly discuss the issue of emergence within this context and the possible implications for 4d vacua.

The Swampland Distance Conjecture and towers of tensionless branes

We study a class of 6d  <math><mi>N</mi></math> $$ \mathcal{N} $$ = (1, 0) non-geometric vacua of the Spin(32)/ℤ 2 heterotic string which can be understood as fibrations of genus-two curves over a complex one-dimensional base. The 6d  <math><mi>N</mi></math> $$ \mathcal{N} $$ = (1, 0) theories living on the defects that arise when the genus-two fiber degenerates at a point of the base are analyzed by dualizing to F-theory on elliptic K3-fibered non-compact Calabi-Yau threefolds. We consider all possible degenerations of genus-two curves and systematically attempt to resolve the singularities of the dual threefolds. As in the analogous non-geometric vacua of the E 8 × E 8 heterotic string, we find that many of the resulting dual threefolds contain singularities which do not admit a crepant resolution. When the singularities can be resolved crepantly, we determine the emerging effective theories which turn out to be little string theories at a generic point on their tensor branch. We also observe a form of duality in which theories living on distinct defects are the same.

Christoph Mayrhofer

Non-geometric vacua of the Spin(32)/ℤ 2 heterotic string and little string theories

We study the (1, 0) six-dimensional SCFTs living on defects of non-geometric heterotic backgrounds (T-fects) preserving a E 7 × E 8 subgroup of E 8 × E 8 . These configurations can be dualized explicitly to F-theory on elliptic K3-fibered non-compact Calabi-Yau threefolds. We find that the majority of the resulting dual threefolds contain non-resolvable singularities. In those cases in which we can resolve the singularities we explicitly determine the SCFTs living on the defect. We find a form of duality in which distinct defects are described by the same IR fixed point. For instance, we find that a subclass of non-geometric defects are described by the SCFT arising from small heterotic instantons on ADE singularities.