Fluxes Vacua and Tadpoles meet Landau-Ginzburg and Fermat Katrin Beckera Eduardo Gonzalob Johannes Walcherc and Timm Wraseb aGeorge P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy

2025-04-27 0 0 587.32KB 37 页 10玖币

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Fluxes, Vacua, and Tadpoles meet Landau-Ginzburg and Fermat

Katrin Beckera, Eduardo Gonzalob, Johannes Walcherc, and Timm Wraseb

aGeorge P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy

Texas A&M University, College Station, TX 77843, U.S.A.

bDepartment of Physics

Lehigh University, Bethlehem, PA 18018, U.S.A.

cMathematical Institute and Institute for Theoretical Physics

Ruprecht-Karls-Universit¨at Heidelberg, 69221 Heidelberg, Germany

Abstract

Type IIB ﬂux vacua based on Landau-Ginzburg models without K¨ahler deformations

provide fully-controlled insights into the non-geometric and strongly-coupled string

landscape. We show here that supersymmetric ﬂux conﬁgurations at the Fermat point

of the 19model, which were found long-time ago to saturate the orientifold tadpole,

leave a number of massless ﬁelds, which however are not all ﬂat directions of the super-

potential at higher order. More generally, the rank of the Hessian of the superpotential

is compatible with a suitably formulated tadpole conjecture for all ﬂuxes that we found.

Moreover, we describe new inﬁnite families of supersymmetric 4d N= 1 Minkowski

and AdS vacua and confront them with several other swampland conjectures.

October 2022

arXiv:2210.03706v1 [hep-th] 7 Oct 2022

Contents

1 Introduction 2

2 Moduli stabilization in non-geometric backgrounds 5

2.1 Orientifoldsandﬂuxes ........................... 5

2.2 Non-renormalization theorems . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Conditions for supersymmetric vacua . . . . . . . . . . . . . . . . . . . 7

2.4 Higher-order derivatives of the superpotential . . . . . . . . . . . . . . 9

3 Around the tadpole conjecture 11

3.1 Hodge theoretic formulation . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 The rank of the mass matrix for Minkowski solutions . . . . . . . . . . 15

3.3 Stabilization at higher order . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Minkowski solutions at ﬁxed coupling 18

4.1 Massless ﬁelds in old solutions . . . . . . . . . . . . . . . . . . . . . . . 18

4.2 Symmetries and new solutions . . . . . . . . . . . . . . . . . . . . . . . 19

4.3 Constraint on the G-ﬂux.......................... 21

5 New inﬁnite families 22

5.1 Two inﬁnite families of Minkowski vacua . . . . . . . . . . . . . . . . . 23

5.2 Implications for the landscape and the swampland . . . . . . . . . . . . 24

5.3 AdSvacua.................................. 25

6 Conclusions 28

A LG integrals 30

A.1 Single variable integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 31

A.2 Taylorcoeﬃcients.............................. 31

B The G-ﬂux for an AdS solution 32

1 Introduction

Moduli stabilization by ﬂuxes has been a cornerstone of realistic string models since the

advent of the string landscape. Early investigations of the GKP construction [1] such

as refs. [2, 3, 4, 5] appeared to conﬁrm the expectation that a generic ﬂux will stabilize

all complex structure moduli of Calabi-Yau manifolds in either type IIB or F-theory

compactiﬁcations. Based on more recent studies, however, it was argued that in models

with a large number of complex structure moduli it should not be possible to stabilize

all of them using ﬂuxes [6, 7, 8]. The basic idea, known as the tadpole conjecture, is

that the ﬂux contribution to the D3-brane tadpole scales linearly with the number of

stabilized moduli, with a proportionality constant that leads to an eﬀective bound in

many popular situations. This argument is part of the swampland program (reviewed

for example in [9, 10]), whose goal it is to determine what the low-energy eﬀective ﬁeld

theories are that can arise from a full-ﬂedged theory of quantum gravity like string

theory.

The relation between the size of the tadpole and the number of stabilized moduli

can easily be tested (and hence falsiﬁed) in many examples. Moreover, with a some-

what more precise deﬁnition of the “number of stabilized moduli”, the conjecture can

be stated essentially in classical Hodge theory, and could thus conceivably be proven

rigorously independent of complicated or unknown perturbative or non-perturbative

quantum corrections. Recent work along this line has provided evidence for the con-

jecture in the large complex structure limits [11, 12, 13, 14]. A scenario including

a putative counterexample was presented in [15]. However, this counterexample was

more recently challenged in [16].

The main aim of this paper is to shed light on the competition between the sta-

bilization of moduli and the size of the D3-brane tadpole in the deep interior of the

moduli space of type IIB ﬂux compactiﬁcation (see [17] for related work in F-theory).

Our investigation is based on a Landau-Ginzburg orbifold describing a non-geometric

compactiﬁcation with h1,1= 0, that was ﬁrst studied with this purpose some 16 years

ago in [18, 19]. It was shown there that while the model is intrinsically non-geometric,

the standard Hodge theoretic formulas for the ﬂux superpotential and tadpole continue

to apply, based on the holomorphic nature of the supersymmetric locus and thanks to

powerful non-renormalization theorems for the superpotential. Moreover, while the lat-

tice of supersymmetric ﬂuxes at the Fermat point has such a large rank that brute force

numerical searches for “short” ﬂux vectors compatible with tadpole cancellation are

prohibitively expensive, some explicit ﬂuxes were found that lead to supersymmetric

Minkowski and AdS vacua that are under full control despite an O(1) string coupling.

However, the exact content of the low-energy theory and the full set of supersymmetric

ﬂuxes remained unexplored at the time.

In this work, we will show ﬁrst of all that in the Minkowski vacua of the 19Landau-

Ginzburg model presented in [18] in fact only a small subset of the 63 complex structure

moduli (that survive the orientifold projection) obtain a mass as a consequence of the

ﬂux. Secondly, we will present a more complete list of supersymmetric ﬂuxes saturating

the tadpole and leading to 4d Minkowski vacua, and show that all of them contain a

substantial number of massless ﬁelds. Thirdly, based on the evaluation of the cubic

(and higher-order) terms in the superpotential, we show that not all of these massless

ﬁelds correspond to truly ﬂat directions, although we are not able to show that all

moduli are actually stabilized. Based on these insights, we present a mathematically

precise (if perhaps somewhat simpliﬁed) formulation of the tadpole conjecture that can

be tested non-trivially over the entire moduli space.

We then turn to other aspects of the swampland program, in which context the

compactiﬁcations of [18, 19] were revisited in the recent works [20, 21], focusing only

on the stabilization of the three bulk complex structure moduli (that are mirror dual

to the untwisted K¨ahler moduli in the mirror dual toroidal type IIA compactiﬁcation).

An intriguing result of [21] was the presence of an inﬁnite family of SUSY Minkowski

vacua. In this inﬁnite family a quantized ﬂux, which is unconstrained by the tadpole,

goes to inﬁnity. Here, we discover similar inﬁnite families of Minkowski vacua that

include all complex structure moduli of the model. One is then forced to accept that

an inﬁnite family of 4d Minkowski solutions is part of the string landscape. This may

sound contradictory to the standard lore that the landscape is ﬁnite, that is, that there

is a ﬁnite number of vacua (and corresponding EFTs) below a certain energy cutoﬀ

[22, 23]. We argue that our inﬁnite families of Minkowski vacua are consistent with

the ﬁniteness conjecture since we expect that for each family there is a tower of states

becoming light.

Additionally, we revisit AdS solutions in these settings. There we ﬁnd likewise new

inﬁnite families of AdS solutions. The existence of these solutions was known based on

a study of simple models that restrict to the bulk moduli [19, 20, 21]. Those families

are reminiscent of the DGKT [24, 25] SUSY AdS vacua which are included in the

mirror of these construction. Here we show that such solutions also exist when taking

all complex structure moduli into account. We present explicitly two examples that

have peculiar features that are relevant to the swampland program.

2 Moduli stabilization in non-geometric backgrounds

Following [18], in this paper we focus on orientifolds of the 19Landau-Ginzburg (LG)

model, with worldsheet superpotential

i=1

Φ3

i.(2.1)

We orbifold by the Z3symmetry Φi→ωΦiwhere ω=e2πi

3. It can be shown that

the model is the mirror dual of a rigid Calabi-Yau manifold (T6/Z3×Z3). A basis for

the (c, c) primary chiral superﬁelds of the untwisted sector is given by the monomials

ΦiΦjΦki6=j6=k6=i. There are 84 of them and they can be identiﬁed as complex

structure moduli. In the untwisted sector of a LG model there is no (a, c) sector,

but there could be K¨ahler moduli in the twisted sector. However, in the case of a

Z3orbifold one ﬁnds no non-trivial (a, c) primary superﬁelds, so there are no K¨ahler

moduli. Intuitively, the orbifold is ﬁxing the volume in string frame. Notice that this

breaks S-duality in our setup.

2.1 Orientifolds and ﬂuxes

The diﬀerent consistent orientifold projections were studied in [18]. Here we will focus

only on the ﬁrst orientifold considered in [18], which combines the worldsheet parity

operator with the operator g1:

g1: (Φ1,Φ2,Φ3, ..., Φ9)→ −(Φ2,Φ1,Φ3, ..., Φ9).(2.2)

This reduces the number of complex structure moduli down to 63: 7 coming from

Φ1Φ2Φi,i= 3,4, . . . 9, 7

2= 21 coming from (Φ1+ Φ2)ΦiΦjand 7

3= 35 coming from

ΦiΦjΦk.

Using results from [26], the authors of [18] calculated the Ramond-Ramond charge

of the crosscap state in the orientifold (2.2), and showed that it amounts to 12 units

of the one elementary B-brane in the model, which can naturally be addressed as a

“D3-brane”, keeping in mind that this is really an abuse of language because the model

is intrinsically non-geometric.

Similarly, the possible Ramond-Ramond and Neveu-Schwarz ﬂuxes, F3and H3, can

be studied by consistency and comparison with the A-branes in the Landau-Ginzburg

theory, which are the analogues of supersymmetric three-cycles in ordinary Calabi-Yau

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Fluxes,Vacua,andTadpolesmeetLandau-GinzburgandFermatKatrinBeckera,EduardoGonzalob,JohannesWalcherc,andTimmWrasebaGeorgeP.andCynthiaWoodsMitchellInstituteforFundamentalPhysicsandAstronomyTexasA&MUniversity,CollegeStation,TX77843,U.S.A.bDepartmentofPhysicsLehighUniversity,Bethlehem,PA18018,U.S.A.cMath...

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Fluxes Vacua and Tadpoles meet Landau-Ginzburg and Fermat Katrin Beckera Eduardo Gonzalob Johannes Walcherc and Timm Wraseb aGeorge P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy

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