Inflation Panel, Cosmic Controversies Conference“Tour of the Swampland”
November 6, 2019
Hirosi Ooguri (Caltech & Kavli IPMU)
PowerPoint file accompanying the remarks is attached at the end of this text.

Swampland Conditions
In quantum field theory without gravitational degrees of freedom, it is generally
believed that we can write down any low energy effective theory and expect thatit has a consistent ultraviolet completion. The only requirement is consistency
that one can check in low energy. For example, all gauge anomalies must be
cancelled. This is what the Wilsonian view of the renormalization group tells us.
However, the situation may be different in theories with gravitational degrees offreedom. Indeed, there are non-trivial constraints that are not obvious in lowenergy.
One of such constraints says that quantum gravity does not have global
symmetry. We can easily write down a low energy effective theory with global
symmetry, but it cannot be exact symmetry in full quantum gravity. Any
accidental global symmetry in low energy has to be either broken or a gauge
symmetry in disguise. Daniel Harlow and I have recently completed a proof of thisstatement in the context of the AdS/CFT correspondence. We have also proventhe completeness hypothesis, which states that all possible charges with respect
to long range gauge symmetry must be realized.The logic we used to prove thesetheorems does not rely on string theory. We only assumed the AdS/CFTcorrespondence, which we believe should be applicable to any consistent
quantum theory of gravity.
In 2005, Cumrun Vafa proposed the swampland as a set of effective theories
that seem consistent in low energy, but cannot be derived as low energy

approximation to consistent quantum gravity. Swampland conditions delineate
the border between the swampland and the landscape. Over the last 14 years,
several swampland conditions have been proposed with various degrees of rigor.
No Global Symmetry, Weak Gravity Conjecture, and Distance Conjecture
As I mentioned, the absence of global symmetry and the completeness of gauge
representations have become theorems by my work with Harlow. The weakgravity conjecture and the distance conjecture strengthen and quantify thesetheorems as I am going to discuss now.
The weak gravity conjecture states that, in any effective theory described by theEinstein gravity coupled to Maxwell field, if it has a consistent ultra-violet
completion, there must be a particle with charge Q and mass m such that mmeasured in the Planck units is less than or equal to the absolute value of Q.
String theory has no continuous free parameters. All parameters can vary
locally. They are expectation values of scalar fields. The distance conjectureCumrun Vafa and I proposed in 2006 is about properties of the moduli space ofsuch scalar fields, with the metric defined by their kinetic terms. The distanceconjecture states that: The moduli space is non-compact. Pick any point X, one can go infinite
distance away from it.
 It also states that compared to the theory at X, the theory at Y in distancemuch greater than the Planck scale has a tower of light particles withexponentially small masses, namely masses suppressed by the exponential ofthe distance between X and Y measured in the Planck units.
 The low energy effective theory defined at X breaks down at Y because ofthese new light degrees of freedom.
Though the weak gravity conjecture and the distance conjecture have not been
proven rigorously, over the last decade, these conjectures have successfully
passed highly non-trivial tests with a large variety of Calabi-Yau compactifications.
Logical connections between these conjectures and other conjectures such as thecosmic censorship hypothesis have been found. We believe that theseconjectures are applicable to any consistent quantum gravity, not restricted tostring theory. So far, tests have only been done in string theory since it is the only
theory in which reliable calculations can be done.
These swampland conditions have stronger implications on low energy effective
theories. As I mentioned, the no global symmetry theorem does not tell us howsymmetry should be broken or gauged. The completeness theorem predicts
magnetic monopoles, but it does not tell us how heavy these monopoles are. The
weak gravity conjecture and the distance conjecture strengthen and quantify
these statements.
Let me give you one example. Daniel Harlow and I have shown that the noglobal symmetry theorem applies to spontaneously broken symmetry also. For
example, if there is a low energy effective theory with a massless scalar field witha flat potential, the theory has a symmetry which shifts the value of the scalar
field by a constant. This symmetry is spontaneously broken in three or more
spacetime dimensions, and the scalar field is a Goldson field. The no global
symmetry theorem prohibits such symmetry even if it is spontaneously broken.
Therefore, the shift symmetry must be violated. My distance conjecture with
Cumrun quantifies how it is violated. If the value of the scalar field shits much
more than the Planck scale, the low energy effective theory must break down by atower of exponentially light particles.
Axion Monodromy
The only potential counter-example to the distance conjecture I am aware of isthe axion monodromy model proposed in 2008 by Liam McAllister, Eva Silversteinand Alexander Westphal. Therefore, I would like to take this opportunity toexplain why it does not make a counter-example.
To make sure that what I am going to tell you is fair to those who worked on theaxion monodromy, I discussed it with Westphal, who is a co-inventor of themodel. He and I worked on this part of my presentation together, and heapproved of the remarks that I am going to read now
Over the past decade, several papers have appeared, pointing out that transPlanckian excursions in the axion monodromy may become difficult once one
takes into account moduli stabilization. In fact, recent studies based on some

semi-explicit examples have demonstrated that back-reaction there isalready important at sub-Planckian distances.
Typically, the models contain both axion modes and radial modes. As we goaround the axion directions, back-reaction to the potential starts to move theradial modes and eventually destabilize them. The radial modes thenstart running away toward the large volume limit, shutting off the monodromy
excursion of the axion modes. At the same time, the runaway of the radial modes
generates a tower of light Kaluza-Klein modes. This behavior is consistent with thedistance conjecture.
To circumvent this problem, the F-term axion monodromy inflation wasproposed by several groups in 2014. However, papers by Irene Valenzuela and hercollaborators in 2016 and 17, following upon an earlier paper by Baume and Palti
in 2016, showed that the F-term monodromy also suffers from control issues, if ahierarchy of mass scales cannot be engineered. A similar problem for models with
D7 branes was pointed out last year by Liam McAllister, who is a co-inventor of
the axion monodromy. In the paper posted last December, McAllister and hisstudent reported that moving D7 branes induce the D3 brane charge and thatthe resulting back-reaction may preclude large field inflation due to what they calla dramatic effect on the superpotential even in one period of the axion field
The distance conjecture is an asymptotic statement and is applicable for a field
range parametrically larger than the Planck scale. So far, some models of axion
monodromy seem to avoid the instability within a field range of a few times thePlanck scale. If that can be confirmed in fully explicit constructions, it would
remain useful as an ingredient of certain inflation scenarios. Such a model would
not be a counter-example to the distance conjecture since there is an order one
factor leeway in the field range as for when the exponential behavior begins toset in.
Neutrino Mass and Dark Energy
Let me turn to other low energy implications of swampland conditions. 3 years
ago, Cumrun Vafa and I pointed out that the weak gravity conjecture will havefurther implications on low energy physics, if we strengthen it slightly by statingthat the equality holds only for BPS particles in theories with unbroken
supersymmetry.
We found that this strengthened version of the weak gravity conjecture would
put restrictions the neutrinos in the Standard Model. Our observation has been
examined more carefully by Ibanez, Martin-Lozano, and Valezuela, and byHamada and Shiu, leading to the relation between neutrino masses and thedensity of the dark energy of the universe. The fourth power of neutrino masses
must be bounded above by the dark energy density, under some assumption on
physics beyond the Standard Model of Particle Physics.
De Sitter Space
Given that a variety of constraints on low energy effective theories of consistent
quantum gravity have been proposed and some have been proven, it seems
reasonable to ask if there are constraints on the accelerated expansion of theuniverse. In the spirit of the title of this panel, let me pose the following question:“Can one derive a low energy effective theory from a known consistent quantum
theory of gravity that accommodates a stable or quasi-stable de Sitter space as its
solution that explains either the observed accelerated expansion of the current
universe or the proposed inflationary period of the early universe?”
Cumrun Vafa and I, together with our students Georges Obied and Lev
Spodyneiko, asked such a question and came up with a conjecture stating that, ifa low energy effective theory contains scalar fields, either the gradient of theirpotential is bounded below or the minimum of its Hessian eigenvalues is bounded
above. This conjecture as well as the distance conjecture does not exclude theinflation scenario, but they can put non-trivial constraints on inflation models.
Since there have been proposals on top-down constructions of quasi-stable deSitter spaces from string theory, it would be appropriate to discuss them at thispoint.
KKLT scenario proposed by Kachru, Kallosh, Linde, and Trivedi 16 years ago is
one of the best efforts to derive quasi-stable de Sitter solutions top-down fromstring theory. It has also been the most scrutinized proposal. KKLT starts with ananti-de Sitter solution, and then turns on supersymmetry breaking to lift thepotential, aiming at creating a local minimum in the potential with a small positive
vacuum energy. Several steps in this scenario have been questioned and theirvalidities are currently being debated at various fronts.
Dine-Seiberg Problem and Compactness of Internal Space
Rather than going into technical details of these on-going debates about KKLT,
let me point out two fundamental difficulties one encounters in any attempt at
top-down construction of this type from string theory. One is the Dine-Seiberg
problem and another is compactness of the internal Calabi-Yau manifold. Let mediscuss each of them.
String theory has no free parameter, and any parameters of a low energy
effective theory is an expectation values of some scalar field in high energy. KKLT
requires that all scalar fields be stabilized and their expectation values be fixed.
This means that, if they succeed in what they want to do, there is no adjustableparameter in their model. Already in 1988, Dine and Seiberg pointed out that it isdifficult, if not impossible, to find a quasi-stable state in the domain that allows aweak coupling expansion with respect the string coupling constant
When flux compactifications were found to be abundant in the early 2000, it
was hoped that, since there are so many of them, it may be possible to find aseries of flux compactification, guaranteeing smallness of the coupling constantby using fluxes as control parameters. Indeed, there are a large number ofparametrically controlled flux compactifications which give rise to the landscapeof anti-de Sitter spaces. Unfortunately, this idea has not been successfully
implemented in constructing de Sitter spaces. Since a controlled approximation
has not been possible, it has been difficult to check claims made about such
constructions in reliable manners.
Let me turn to issues with compactness of Calabi-Yau manifolds. In
mathematics, problems dealing with compact geometric objects are often moredifficult than those with non-compact counterparts. For example, no one has
been able to construct a Ricci-flat Kaehler metric explicitly on any smoothcompact Calabi-Yau manifold, even though its existence is guaranteed by Yau’s
proof of the Calabi conjecture. In contrast, it is very easy construct a Ricc-flat

Kaehler metric explicitly on a non-compact Calabi-Yau manifold. One of thereasons is that, in compact cases, we cannot hide problems by pushing themaway toward the infinity; rather we have to confront global constraints
These global constraints in mathematics mirror the swampland conditions inphysics. The connection is that the Newton coupling in a low energy theory
vanishes in the limit when the volume of the Calabi-Yau manifold become infinite.
In the infinite volume limit, the gravity decouples and the constraints disappear,
taking us back to the traditional Wilsonian view of the renormalization group andeffective field theories, where we can write down any low energy effective theory
and expect that it has a consistent ultraviolet completion. For a compact internal
space giving rise to a non-zero Newton coupling, we have to confront global
issues over the compact manifold, and swampland conditions emerge.
At the Strings 2019 conference in Brussels in June, Liam McAllister gave an
excellent review talk in support of KKLT, responding to a variety of issues raised
about the scenario. Leaving aside the question on whether his assessments areoverly optimistic or not, he agreed at the end of his talk that there has been no
explicit compactification that unifies all necessary components of KKLT. Given that
KKLT is a proposal for a top down construction from string theory, it is desirable
to have at least one concrete model to show how everything works out.
ClosingOver the next decade, a variety of astrophysical experiments are planned forprecision tests of the inflation. They include the Japanese satellite project
LiteBIRD to measure the primordial B-mode polarization. As the Director of theKavli IPMU, I proposed this project to the Master Plan 2020 of the Japanese
Science Council. Just a few months ago, we heard a wonderful news that JAXA,
the Japanese Aerospace Exploration Agency, approved the launch of LiteBIRD in 8years. We should improve our theoretical tools and strive to make more preciseand sharper statements about low energy effective theories of string theory andtheir implications on the current and early universes before the next astrophysicalexperiment verifies or falsifies the inflation in the early universe or nails theequations of state of the dark energy in the current universe.

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