Light Strings and Strong Coupling in F-theory Max Wiesner Center of Mathematical Sciences and Applications Harvard University 20 Garden Street

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Light Strings and Strong Coupling in F-theory

Max Wiesner

Center of Mathematical Sciences and Applications, Harvard University, 20 Garden Street,

Cambridge, MA 02138, USA

Jeﬀerson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA

Abstract

We consider 4d N= 1 theories arising from F-theory compactiﬁcations on elliptically-

ﬁbered Calabi–Yau four-folds and investigate the non-perturbative structure of their scalar

ﬁeld space beyond the large volume/large complex structure regime. We focus on regimes

where the F-theory ﬁeld space eﬀectively reduces to the deformation space of the world-

sheet theory of a critical string obtained from a wrapped D3-brane. In case that this

critical string is a heterotic string with a simple GLSM description, we identify new

strong coupling singularities in the interior of the F-theory ﬁeld space. Whereas from

the perturbative perspective these singularities manifest themselves through a breakdown

of the perturbative α0-expansion, the dual GLSM perspective reveals that at the non-

perturbative level these singularities correspond to loci in ﬁeld space along which the

worldsheet theory of the critical D3-brane string breaks down and a 7-brane gauge theory

becomes strongly coupled due to quantum eﬀects. Therefore these singularities signal a

transition to a strong coupling phase in the F-theory ﬁeld space which can be shown to

arise due to the failure of the F-theory ﬁeld space to factorize between complex structure

and K¨ahler sector at the quantum level. Such singularities are hence a feature of a genuine

N= 1 theory without a direct counterpart in N= 2 theories in 4d. By relating our setup

to recent studies of global string solutions associated to axionic strings we argue that

the D3-brane string dual to the perturbative heterotic string leaves the spectrum of BPS

strings when traversing into the strong coupling phase. The absence of the perturbative,

critical heterotic string then provides a physical explanation for the breakdown of the

perturbative expansion and the obstruction of certain classical inﬁnite distance limits in

accordance with the Emergent String Conjecture.

mwiesner at cmsa.fas.harvard.edu

arXiv:2210.14238v2 [hep-th] 21 Nov 2022

Contents

1 Introduction 1

2 Light string limits in F-theory 4

2.1 Generalsetup..................................... 4

2.2 Comparison to N=2ﬁeldtheorylimits ...................... 7

2.3 Generalstrategy ................................... 10

3 Dual Description 11

3.1 Heterotic string with standard embedding . . . . . . . . . . . . . . . . . . . . . 12

3.2 Deformations of the tangent bundle . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Strong Coupling Singularities in F-theory 24

4.1 Perturbative corrections to F-theory ﬁeld space . . . . . . . . . . . . . . . . . . 24

4.2 F-theory singularity structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.3 Singularities and the string spectrum . . . . . . . . . . . . . . . . . . . . . . . . 36

5 Conclusions 45

A Essentials of heterotic/F-theory duality 47

B Review of GLSMs 51

1 Introduction

In the quest to uncover the fundamental nature of quantum gravity, string theory provides an

ideal testing ground to identify general principles that are believed to be valid in any theory of

quantum gravity. For that reason, string theory plays a key role in the so-called Swampland

program, initiated in [1], that aims to ﬁnd criteria that any eﬀective theory needs to satisfy in

order to arise as a low-energy approximation to quantum gravity. In that context, perturbative

string theories provide the most striking evidence e.g. for the Distance Conjecture [2] or the

Weak Gravity Conjecture [3] (cf. [4–7] for reviews). However, in order to have computational

control over the string theory one typically requires the string to be weakly coupled, the eﬀective

theory to preserve a large amount of supersymmetry and the vevs of the scalar ﬁelds to be tuned

to asymptotic regions in the scalar ﬁeld space where the eﬀective theory can be described

by a string compactiﬁcation on some geometric background. Unfortunately when aiming to

understand the full nature of quantum gravity these requirements pose severe limitations. To

obtain a more complete picture, one should hence also turn for instance to eﬀective theories with

at most N= 1 supersymmetry in four dimensions which are not realized in strict asymptotic

regions of the scalar ﬁeld space in order to avoid the existence of an inﬁnite tower of light states

in these regimes as required by the Distance Conjecture.

Due to the lack of computational control over the corrections to N= 1 theories in four

dimensions, the interior of the scalar ﬁeld space of such theories is relatively unexplored. Still,

one might hope to encounter interesting structures once moving away from the strict asymp-

totic/weak coupling limits. In this paper, we want to partially address this question for the

special case of F-theory compactiﬁcations on elliptically-ﬁbered Calabi–Yau four-folds. Such

setups give rise to four-dimensional eﬀective theories with N= 1 supersymmetry and thus

provide an interesting setting to uncover the structure of the N= 1 scalar ﬁeld spaces. The

asymptotic regions of the scalar ﬁeld space for these theories have recently been investigated

in the context of the ﬂux compactiﬁcations [8–12] as well as the Weak Gravity and Swampland

Distance/Emergent String Conjecture [13–17]. The latter conjecture [18] states that any inﬁnite

distance limit in a consistent theory of quantum gravity is either a limit in which a critical string

becomes tensionless and weakly coupled, or a limit in which the theory eﬀectively decompacti-

ﬁes. Among others this conjecture has been shown to hold in the K¨ahler sector of the F-theory

scalar ﬁeld space [13,15] and the strict diﬀerentiation between emergent string and decompac-

tiﬁcation limits has recently been stressed again in [17]. More precisely, in asymptotic limits in

the K¨ahler ﬁeld space of N= 1 four-dimensional F-theory compactiﬁcations [17] showed that

there cannot be any tower with mass below the quantum gravity cut-oﬀ, i.e. the Planck scale

or the species scale [19], that does not arise from KK modes of a higher-dimensional theory or

the excitations of a critical string. In particular any particle-like string excitations necessarily

arise from weakly coupled, genuinely four-dimensional strings obtained by wrapping D3-branes

on certain curves in the base of the elliptic CY four-fold which indeed can be shown [13,15,17]

to be always dual to critical type II or heterotic strings.

In this work we aim to investigate the interior of the F-theory scalar ﬁeld space, MF, away

from strict weak coupling points. More precisely, our goal is to uncover the physics in corners

of the scalar ﬁeld space of genuine N= 1 theories in 4d where the asymptotic, weakly-coupled

description breaks down. We refer to the loci in ﬁeld space where the asymptotic description

breaks down as the border of the asymptotic region. In this context, it has already previously

been noticed [15] that, for instance, certain regimes in ﬁeld space that classically look like an

asymptotic emergent string region are obstructed due to a breakdown of the perturbative α0-

expansion. One of our goals in this work is to revisit these obstructions and give a physical

explanation for the absence of emergent strings in these regions.

As mentioned previously, asymptotic regions in the scalar ﬁeld space of N= 1 theories

in 4d have the property that any tower of massive excitations with mass below the quantum

gravity cutoﬀ is either made up by KK modes or the excitations of a critical string [17]. In this

work, we want to exploit this property to ﬁnd the borders of these asymptotic regimes in MF.

For deﬁniteness, we exclusively focus on the case where the light, massive states arise from a

critical string. In the regimes of MFwhere this is the case, the full F-theory eﬀectively reduces

to a critical string theory. Such regimes are obtained in the case that a D3-brane wrapped on

a curve becomes classically lighter than any other stringy scale as the physics associated to

these other scales eﬀectively decouples. We are then left with a theory of a single string and its

excitations. Though this is similar in spirit to the emergent string limits, unlike for emergent

string limits we only require that the D3-brane string becomes light at the classical level and at

this point are agnostic about whether it remains light and weakly coupled also at the quantum

level. Still, the beneﬁt of such regimes is that we are left with a residual scalar ﬁeld space which

can be identiﬁed with the deformation space of the string worldsheet theory.

In the cases of interest for us, the light string is a critical string and we can thus invest-

igate the properties of the residual scalar ﬁeld space by studying the deformation space of a

critical string in 4d. By the emergent string conjecture the existence of the asymptotic region

and the presence of the perturbative excitations of this string are tightly related. In order to

identify the borders of the asymptotic region in ﬁeld space, the relevant question pertinent to

the analysis in this paper is whether the light, critical string remains weakly coupled in the

interior of the residual ﬁeld space also at the non-perturabtive quantum level. To answer this

question, in practice we restrict to the case that the critical string is a heterotic string whose

worldsheet theory allows for a description as the low-energy limit of a (2,2) (or (0,2) deforma-

tion thereof) supersymmetric Gauged Linear Sigma Model (GLSM). This case corresponds to

heterotic standard embedding and thus to a situation with space-time gauge theory E6×E8.

Though this is certainly a strong restriction, it enables us to explicitly study the FI-parameter

space of the GLSM as a proxy for the quantum K¨ahler deformation space of the heterotic string

(cf. [20] and [21,22] for the (0,2) case). Via heterotic/F-theory duality this then translates into

a description of the residual F-theory K¨ahler ﬁeld space in the light string limit.

In fact, the GLSM description allows us to identify a region in the residual scalar ﬁeld

space where the light string ceases to be weakly coupled. We arrive at this conclusion by

considering the singular loci in the GLSM FI-parameter space and, more precisely, identify the

principal component of the singular locus as being responsible for the light string failing to be

weakly coupled. This is due to the fact that in the heterotic theory with standard embedding,

the unbroken E8becomes strongly coupled along this locus in ﬁeld space. Furthermore, as

the correlators of the heterotic worldsheet theory become singular along this locus, also the

worldsheet theory of the perturbative heterotic string breaks down entirely. Using heterotic/F-

theory duality and employing the perturbative corrections to MFderived in [15, 23–25], we

translate the structure of the FI-parameter space into the structure of MF. Thereby we are able

to identify a strong coupling phase also in the latter. By studying the perturbative corrections

to MFit has already been noticed in [15] that certain limits, in which a critical string becomes

classically weakly-coupled, are obstructed since the perturbative α0-expansion breaks down in

F-theory. Our analysis shows that, at the non-perturbative level, this obstruction is due to the

strong coupling phase in MFwhose presence we infer via heterotic/F-theory duality and which

is closely linked to the failure of MFto factorize in K¨ahler and complex structure sectors at

the quantum level.

Going further, our analysis provides a physical explanation for why this obstruction/break-

down of the α0-expansion occurs. Though this eﬀect is a genuine property of a N= 1 theory

without a direct counterpart in N= 2 theories, we can still draw an analogy to a similar situ-

ation in the vector multiplet moduli space of CY threefold compactifcations of type II string

theory. More precisely our approach to study the interior of the F-theory scalar ﬁeld space in

regions where the full F-theory reduces to a string theory can be viewed as the N= 1 analogue

of studying point particle limits of N= 2 compactiﬁcations of type II string theory [26]. In

these setups the α0→0 limit of the type II moduli space can be identiﬁed with the Coulomb

branch of N= 2 SU(2) SYM theory with D-brane states playing the role of the W±-bosons.

The conifold singularity of the full type II vector multiplet moduli space can then be identi-

ﬁed with the singularities on the Coulomb branch at which the gauge theory becomes strongly

coupled. In our N= 1 version of this, the relevant BPS objects are not particles but 1

2-BPS

strings. Still, in a similar spirit, the theory associated to the 1

2-BPS string becomes strongly

coupled at the singularity in the ﬁeld space and we expect a strong coupling phase beyond

that point. In fact, following the approach of [27], we can also identify the singularity with a

non-critical, non-geometric string that becomes light along that locus replacing the D3-brane

string as the fundamental BPS object. Since the critical string fails to be weakly-coupled and

to be part of the BPS spectrum in the strong coupling phase, it also does not provide us with

a tower of perturbative string excitations. The absence of such a tower then implies that we

also reached the border of the asymptotic region in ﬁeld space and any attempt to extend the

asymptotic region into the strong coupling region should be obstructed. Hence, the failure of

the critical string to be part of the BPS-string spectrum provides a physical explanation why

certain classically allowed emergent string limits in MFare obstructed [15].

This paper is structured as follows: In section 2 we introduce the general setup and review

basic properties of the scalar ﬁeld space of N= 1 F-theory compactiﬁcations to 4d. We further

introduce the relevant string theory limits of F-theory and draw the analogy to the ﬁeld theory

limits of N= 2 string theory compactiﬁcations. In section 3 we then take a closer look at

the deformation space of the worldsheet theory of the light string to which the F-theory scalar

ﬁeld space reduces in the string theory limit. In particular, we identify candidates for strong

coupling singularities in the associated GLSM FI-parameter space that can be responsible for

obstructions to the classical light string limits. Based on heterotic/F-theory duality, we show

in section 4 that these strong coupling singularities also arise in the F-theory scalar ﬁeld space.

We further discuss the physical interpretation of the F-theory strong coupling phases in terms

of the 1

2-BPS string spectrum. We present our conclusions in section 5. The appendices A and

B provide some background information about heterotic/F-theory duality and GLSMs.

2 Light string limits in F-theory

In this section, we set the stage for our analysis in the rest of the paper. In particular, we

review certain light string limits in F-theory that play a central role throughout our analysis.

Such light string limits have been investigated previously in [13, 15]. Here, we would like to

revisit such limits in order to get insights into the non-perturbative structure of the F-theory

scalar ﬁeld space, MF. In section 2.1 we start by giving some background on the F-theory

scalar ﬁeld space and introduce the classical light string regimes pertinent to the analysis in this

paper. In section 2.2 we then discuss the analogy between these string theory limits in four-

dimensional N= 1 compactiﬁcations of F-theory and ﬁeld theory limits in four-dimensional

N= 2 compactiﬁcations of type II string theory. This analogy then serves as a guide to the

general strategy to analyze the properties of the F-theory scalar ﬁeld space in such string theory

limits which we summarize in Section 2.3.

2.1 General setup

In this work, we are primarily interested in F-theory compactiﬁcations on elliptically-ﬁbered

Calabi–Yau four-folds. The resulting eﬀective four-dimensional theory has N= 1 supersym-

metry and its eﬀective action has been derived in detail in [28]. Let us review the central

aspects: The scalar ﬁelds of this eﬀective theory are part of chiral multiplets and can be as-

sociated to the complex structure deformations of the CY four-fold X4and the (complexiﬁed)

K¨ahler deformations of the base, B3, of the elliptic ﬁbration π:X4→B3. In the limit of large

volume and large complex structure, the scalar ﬁeld space eﬀectively factorizes as

chiral −→ MF

c.s.× MF

cK .(2.1)

The ﬁrst factor corresponds to the complex structure deformations of X4whereas the second

factor is spanned by the complexiﬁed K¨ahler deformations parametrized by

Si=1

2ZDi

J∧J+ZDi

C4.(2.2)

Here Di,i= 1, . . . , h1,1(B3), are generators of the cone of eﬀective divisors on B3,Jis the K¨ahler

form on B3and C4the type IIB RR four-form. The classical factorization (2.1) translates into

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LightStringsandStrongCouplinginF-theoryMaxWiesnerCenterofMathematicalSciencesandApplications,HarvardUniversity,20GardenStreet,Cambridge,MA02138,USAJeersonPhysicalLaboratory,HarvardUniversity,Cambridge,MA02138,USAAbstractWeconsider4dN=1theoriesarisingfromF-theorycompacticationsonelliptically-bered...

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