Ane algebras at innite distance limits in the Heterotic String Veronica Collazuol Mariana Gra na

2025-04-30 0 0 2.35MB 53 页 10玖币
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Affine algebras at infinite distance limits
in the Heterotic String
Veronica Collazuol, Mariana Gra˜na,
Alvaro Herr´aez and H´ector Parra De Freitas
Institut de Physique Th´eorique,
Universit´e Paris Saclay, CEA, CNRS,
Orme des Merisiers, F-91191 Gif sur Yvette, France
veronica.collazuol, mariana.grana, alvaro.herraezescudero, hector.parradefreitas
@ ipht.fr
Abstract
We analyze the boundaries of the moduli spaces of compactifications of the heterotic string
on Td, making particular emphasis on d= 2 and its F-theory dual. We compute the OPE
algebras as we approach all the infinite distance limits that correspond to (possibly partial)
decompactification limits in some dual frame. When decompactifying kdirections, we find
infinite towers of states becoming light that enhance the algebra arising at a given point in the
moduli space of the Td−kcompactification to its k-loop version, where the central extensions
are given by the kKK vectors. For T2compactifications, we reproduce all the affine algebras
that arise in the F-theory dual, and show all the towers explicitly, including some that are not
manifest in the F-theory counterparts. Furthermore, we construct the affine SO(32) algebra
arising in the full decompactification limit, both in the heterotic and in the F-theory sides,
showing that not only affine algebras of exceptional type arise in the latter.
arXiv:2210.13471v1 [hep-th] 24 Oct 2022
Contents
1 Introduction 1
2 Decompactification limits of the Heterotic String 2
2.1 Decompactification limit of the E8×E8Heterotic String on S1......... 4
2.2 Decompactification limit of the SO(32) Heterotic String on S1......... 5
3 Decompactifications of the Heterotic String on tori 5
3.1 Decompactification limits of the Heterotic String on T2............. 6
3.2 Decompactification limits of the Heterotic String on Td............. 11
4 Review of string junctions 13
4.1 Basic concepts on String Junctions . . . . . . . . . . . . . . . . . . . . . . . . 14
4.2 TheJunctionLattice................................ 16
4.3 String Junctions and (Affine) Lie Algebras . . . . . . . . . . . . . . . . . . . . 18
5 Heterotic-F theory duality at infinite distance 22
5.1 ( b
E9⊕b
E9)/∼.................................... 23
5.2 ( b
E8−n⊕b
E8⊕b
An+1)/∼............................... 25
5.3 b
b
D16 .......................................... 26
6 Conclusions 28
A Details on the OPEs 29
A.1 T2decompactifications ............................... 29
A.2 Tddecompactification ............................... 38
B Decompactification limits in F-theory 42
B.1 ReviewofKulikovmodels ............................. 42
B.2 Realisation of double loop D16 inF-theory .................... 44
B.3 Heterotic/F-theory duality map . . . . . . . . . . . . . . . . . . . . . . . . . . 48
1 Introduction
In the recent years, there has been an increasing interest in the so-called Swampland pro-
gram [1] (reviewed in [2–6]), which aims at drawing the boundaries between the low energy
effective theories than can be consistently coupled to Quantum Gravity and those that cannot.
These boundaries are typically formulated as a series of conjectures that constrain the low
energy effective theories, whose arguments range from general black hole physics to explicit
String Theory constructions. One of the best established conjectures at the moment is the
Swampland Distance Conjecture [7], which predicts the presence of an infinite tower of states
becoming exponentially light with the distance as one moves far away in the moduli space of
the theory. The nature of the possible massless towers has been analysed in [8], where it was
conjectured that it must always come from either a KK tower (or dual versions), or from the
excitations of an emergent critical string becoming tensionless (this is known as the Emergent
String Conjecture).
On the other hand, massless vectors at a given point in moduli space are a signature of a
larger gauge group, and all massless states arrange into representations of this group. Thus,
exploring specific points (or more generally, hypersufaces) in moduli space allows to examine
the symmetry enhancements associated to this phenomenon. In particular, a detailed know-
ledge of all the possible enhancements is essential to determine whether all consistent low
energy vacua in Quantum Gravity can be obtained from String Theory (i.e. string universal-
ity). Progress along this direction has been recently made by systematically studying toroidal
compactifications of the heterotic/CHL string [9–12] and F-theory on K3 surfaces [13], as
well as by applying general Swampland arguments to supergravity vacua [14–17] . In this
context, a natural question arises, namely what is the interplay between these symmetry en-
hancements and infinite distance boundaries of the moduli space, specially in relation to the
Distance Conjecture. Such points have been recently studied from this perspective in the
circle compactification of the heterotic string [18] and in 8d theories realised from F-theory
on K3 surfaces [19–21], where it was shown that in these limits one gets an infinite tower of
massless states that realize an affine algebra. Particularly, in the decompactification limit of
the heterotic string on the circle the KK tower combines with the E8×E8or Spin(32)/Z2
gauge group to give rise to the affine version of their algebras, respectively, with the U(1) of
the circle direction playing the role of imaginary root.
In this paper, we study the boundaries of the moduli space of toroidal compactifications of
the heterotic string, which correspond to (duals of) different decompactification limits. Despite
being a very simple playground, the toroidally-compactified heterotic theory proves to be rich
enough to describe all the possible decompactifications and the related affine algebras that are
realised in the corresponding moduli space. We show that when decompactifying kdimensions
from 10 −dto a point in the 10 −d+kdimensional theory with gauge algebra AL×U(1)d−k
R,
we obtain the k-loop version of AL×U(1)d−k
R. The central extension for the left/right moving
algebra is the holomorphic/antiholomorphic part of the current associated to the kdirections
that are decompactified.
1
We show that for T2compactifications, the heterotic results match those obtained in the
dual F theory on K3, in the context of Kulikov models and K3 complex structure degener-
ations [19,20]. The heterotic framework allows however for a clearer characterisation of the
decompactification limits and corresponding algebras in all the corners of moduli space, and
moreover can be (straightforwardly) generalised to lower dimensions.
The paper is organised as follows. After setting our conventions and reviewing the affin-
isation process in the decompactification of the heterotic theory on S1in Section 2, we discuss
in Section 3its generalisations, first to T2and then to a generic Tdcompactification, showing
the explicit world-sheet realisation of the algebras in Appendix A. In Section 4we review the
string junction picture, which we explicitly use in Section 5in order to make contact with the
F-theory dual description of [19,20], reviewed in Appendix B. We also build the explicit real-
isation of the Kulikov Weierstrass model for b
b
D16, which from the heterotic dual is predicted to
be, together with ( b
E9⊕b
E9)/∼, the only algebra in the full decompactification limit. Finally,
we summarise and make some concluding remarks in Section 6.
2 Decompactification limits of the Heterotic String
To set our notation, in this section we summarize the main features of heterotic compactific-
ation on a ddimensional torus Td, and review the behaviour of the theory on S1when going
to infinite distance in the moduli space.
Let Tdbe associated with a lattice Λ generated by the vectors ei, i = 1, ..., d, and dual
lattice ˜
Λ generated by the dual vectors e∗i. The background of the theory compactified on
this space is specified by1
•the metric on the torus Gij =ea
iδabeb
j, a = 1, ..., d ,
•the internal B-field Bij and
•the Wilson lines AI
i,I= 1, ..., 16 .
Restricting the worldsheet action to the bosonic sector and setting α0= 1, we have
S=−1
4πZd2σhηαβGMN ∂αXM∂βXN+1
2δIJ ∂αXI∂βXJ
+αβBMN ∂αXM∂βXN+AI
i∂αYi∂βXIi.
(2.1)
In the following we split the 10 dimensional index M= 0, ..., 9 as M= (µ, i), with µ= 0, ...9−d
and i= 10 −d, ..., 9. We write XM= (Xµ, Y i) with Yithe compact spacetime bosons.
The heterotic states are characterized by the winding and momentum numbers associated
to the Tddirections, respectively wi, ni∈Z, and by the heterotic momenta πIalong the 16
1One should also include the dilaton, but we are taking it to be fixed (and small).
2
dimensional heterotic torus. These can be arranged in a charge vector
Z= (πI, wi, ni).(2.2)
The vector πlies either in the root lattice of E8×E8, denoted Γ8×Γ8, or in the weight lattice
of Spin(32)/Z2, denoted Γ16. Moreover, the states have the following left (L) and right (R)
internal momenta along the Tddirections,
pR,i =1
√2(ni−Eijwj−π·Ai),
pL,i =1
√2(ni+ (2Gij −Eij)wj−π·Ai),
(2.3)
where
Eij =Gij +1
2Ai·Aj+Bij ,(2.4)
and along the heterotic torus,
pI=πI+AI
iwi.(2.5)
The vector p= (pR,ie∗i
a;pL,ie∗i
a, pI) = (pR,a;pL,a, pI)≡(pR;pL) lies in an even and self-dual
lattice Γd,d+16 with Lorentzian signature (−d,+d+16), as seen from the expression
p2=−p2
R+p2
L= 2niwi+|π|2∈2Z.(2.6)
The mass formula and the level-matching condition (LMC) for the NS sector of the spectrum
then read
M2=p2
L+p2
R+ 2N+˜
N−3
2,
0 = p2
L−p2
R+ 2N−˜
N−1
2,
(2.7)
where Nand ˜
Nare respectively the left and right-moving oscillator numbers. These formulas
determine the group of symmetries of the spectrum to be O(d, d + 16,Z), which we refer to
as the T-duality group.
In order to read the symmetry algebra at a given point in moduli space one should look at
the set of massless vectors. For generic values of the moduli, these are the U(1)d
R×U(1)16+d
L
vector bosons, characterized by N= 1,˜
N=1
2,pL= 0,pR= 0:
αµ
−1˜
ψi
−1
2|0iNS , αi
−1˜
ψµ
−1
2|0iNS , αI
−1˜
ψµ
−1
2|0iNS −→ (gµi ∓Bµi), AµI.(2.8)
Due to the moduli dependence of (2.7) it turns out that at fixed points of the T-duality group
there can be additional massless vectors with N= 0,¯
N=1
2,|pL|2= 2,pR= 0, of the form
˜
ψµ
−1
2|πα, wi, niiNS −→ Aα
µ(2.9)
3
摘要：

Anealgebrasatin nitedistancelimitsintheHeteroticStringVeronicaCollazuol,MarianaGra~na,AlvaroHerraezandHectorParraDeFreitasInstitutdePhysiqueTheorique,UniversiteParisSaclay,CEA,CNRS,OrmedesMerisiers,F-91191GifsurYvette,Franceveronica.collazuol,mariana.grana,alvaro.herraezescudero,hector.parradef...

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