Gravitational Positive Energy Theorems
from Infromation Inequalities
Hirosi Ooguri
Walter Burke Institute for Theoretical Physics, Caltech
Kavli IPMU, University of Tokyo

Quantum Matter, Spacetime and Information
Yukawa Institute for Theoretical Physics, Kyoto, 13 ‐ 17 June 2016

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Swampland Question
Given an effective theory of gravity, how canone judge whether it is realized as a low energy
appropximation to a consistent quantum theory
with ultra‐violet completion, such as string theory?

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Constraints on Symmetry

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Conjectures:☆ There are no global symmetry.
☆ All continuous gauge symmetries are compact.
☆ The spectrum of electric and magnetic charges
forms a complete set consistent with the Diracquantization condition.

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Conjectures:☆ There are no global symmetry.
☆ All continuous gauge symmetries are compact.
☆ The spectrum of electric and magnetic charges
forms a complete set consistent with the Diracquantization condition.

Holographic understanding:
Harlow, arXiv: 1510.07911
Harlow + H.O., to appear
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More Conjectures:
Gravity is the weakest force in Nature.
Arkani‐Hamed, Motl, Nicolis + Vafa, hep‐th/0601001

Every symmetry is gauged.
With a gauge field, there is always a particle
whose mass is smaller than its charge in the Planck unit.
☆ Black holes at the Reissner‐Nordstrom boundare unstable unless protected by supersymmetry.
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Constraints
on Moduli Space

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Conjectures:☆ The moduli space is non‐compact, complete, andhas finite volume.
☆ If we move a large distance T from a reference point,a tower of light particles emerges with mass of the orderexp( ‐ a T ) for some a. The number of such light particles
becomes infinite at T tends to the infinity.
☆ There is no non‐trivial one‐cycle with minimal length
within a given homotopy class in the moduli space.
as formulated by Vafa + H.O., arXiv:0605264

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Conjectures:☆ The moduli space is non‐compact, complete, andhas finite volume.
☆ If we move a large distance T from a reference point,a tower of light particles emerges with mass of the orderexp( ‐ a T ) for some a. The number of such light particles
becomes infinite at T tends to the infinity.
☆ There is no non‐trivial one‐cycle with minimal length
within a given homotopy class in the moduli space.
as formulated by Vafa + H.O., arXiv:0605264

These moduli space constraints have been provenfor theories with N=3 or higher supersymmetry.
Cecotti, "Supersymmetric Field Theories," section 4.9.1

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Constraints onCalabi‐Yau Topology

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Modular invariance constraints

Keller + H.O., arXiv: 1209.4649

conformal dimensions

Below this curve, there arealways non‐chiral operators.

Density grows linearly in the Hodge number.
total Hodge
number
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Recent experimental data on Calabi‐Yau 3 and 4 folds

Taylor + Wang,
arXiv: 1510.04978, 1511.03209

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Holographic Constraints

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Suppose there is a low energy effective field theory
whose gravity solutions asymptote to the anti‐deSitter space at the infinity.

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Suppose there is a low energy effective field theory
whose gravity solutions asymptote to the anti‐deSitter space at the infinity.
Holography of Quantum Gravity:
Consistent quantum gravity in AdS
is equivalent to a conformal field
theory on the boundary.
AdS/CFT Correspondence
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Question: What does consistency of the conformalfield theory mean for the gravity theory?

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Gravity theory in (d+1)‐dim AdS

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Gravity theory in (d+1)‐dim AdS
is equivalent to d‐dim CFT.

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Gravity theory in (d+1)‐dim AdS
is equivalent to d‐dim CFT.
A

Entanglement Density Matrix
For any statein CFT,
choose a spacelike region A.
☆ The trace is on the Hilbert spaceover the complement of A.
☆ It is an operator acting on theHilbert space over A.
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Entanglement Density Matrix ρA
Entanglement Entropy S

S measures the amount of entanglement
between the region A and its complement.
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Entanglement Entropy S

AWhen the bulk gravity theory is
described with smooth geometry,the entanglement entropy S is
proportional to the area of the minimum
surface ending of the boundary of A.

Ryu‐Takayanagi (2006)
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Entanglement Entropy
satisfies inequalities:☆ Some inequalities are satisfied
both by any CFT and by AdS gravity.
☆ Some inequalities are satisfied
by any CFT but not always by AdS gravity.
☆ Some inequalities are satisfied
by any AdS gravity but not always by CFT.
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AdSCFT

Monogamyof mutual
information:

Positivity/monotonicityof relative entropy
Strong subadditivity:
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AdS

CFT

Holographic
Entropy Inequalities

CFT states with gravitational duals haveinteresting entanglement properties.
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Entropy Inequalities(Classical) Shannon Entropy:
There are infinite number of independent entropy inequalities
for more than 3 regions.
⇒ Asymptotic performance for information processing tasks
Matus (2007)(Quantum) von Neumann Entropy:
For more than 3 regions, the complete set of independent inequalities
is not known.
⇒ Numerical evidences that the number is infinite.

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For holographic states:☆ Finite algorithm to classify all inequalities.
☆ There are finitely many independent
inequalities for a fixed number of regions.
☆ Complete classification for 2, 3, 4 regions.
☆ A new family of inequalities for 5 and moreregions.
Bao, Nezami, Stoica, Sully, Walter + H.O., arXiv:1505.07839
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Holographic Entropy Cone
Entanglement entropies for n regions
make a vector indimensions.

Entropy vectors of holographic states populate
inside of a convex rational polyhedral cone.
The number of independent inequalities is finite for each n.
Are these implied by the gap conditions on CFT?
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AdS

CFT

Hologrpahic
entropy inequalities
Positivity/monotonicityof relative entropy
Strong subadditivity:
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AdS

CFT

Hologrpahic
entropy inequalities
Positivity ofquasi‐local energy
Strong subadditivity:
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AdS

Delineate the Swampland bypositive energy conditions.

CFT

Positivity ofquasi‐local energy

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Positivity of Quasi‐Local Energy

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Energy and Entropy
based on formalism
developed by Wald & collaborators

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Relative Entropy

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A

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A

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A

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A low energy effective theory of a consistentultraviolet complete quantum theory of gravity
must satisfy the positive energy conditions implied by
the positivity and monotonicity of the relative entropy.
How strong are these conditions?
Which low energy theories are ruled out by them?
Do they follow from the bulk reconstruction argument?

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Swampland Question:
How to characterize an effective gravity theory that can emerge
in a low energy approximation to a consistent quantum theory,
such as string theory.

Constraints on Symmetry
Constraints on Moduli SpaceConstraints on Calabi‐Yau Topology
Positive Energy Theorems
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