Extremal N=2 2D CFT

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Extremal N2 2D CFTs and Constraints of
Modularity
IAS, October 3, 2008

• Work done with
• M. Gaberdiel, S. Gukov, C. Keller and H. Ooguri
• arXiv0805.4216

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Outline

• 1. Two Motivations
• 2. Results
• 3. Extremal N0 and N1 theories
• 4. 2D N2 Theories Elliptic Genus Polarity
• 5. Counting Polar Terms
• 6. Search for Extremal N2 theories
• 7. Near Extremal N2 Theories
• 8. Possible Applications to the Landscape
• 9. Summary Concluding Remarks

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Motivation 1

• An outstanding question in theoretical physics
  is the existence of three-dimensional AdS pure
  quantum gravity.
• Witten proposed that it should be defined by a
  holographically dual extremal CFT.
• We do not know if such CFTs exist for general
  central charge c24 k, kgt 1.
• In AdS3 one can define OSp(p2)xOSp(q2)
  sugra dual to theories
• with (p,q) supersymmetry.
• What can we say about those?

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Motivation 2

• There are widely-accepted claims of the
  existence of a
• landscape of d4 N1 AdS solutions of string
  theory
• with all moduli fixed.
• The same techniques should apply to M-theory
• compactifications on say CY 4-folds to AdS3.
• Such backgrounds would be holographically dual
  to 2D CFT
• Does modularity of partition functions put any
  interesting
• constraints on the landscape?

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Result 1

• We give a natural definition of an extremal
  N(2,2) CFT
• And we then show that there are at most a finite
  number of exceptional examples

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Result -2
We present evidence for the following conjecture
Any N(2,2) CFT must contain a state of the form
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Result - 3

• The bound is nearly optimal
• There are candidate partition functions (elliptic
  genera) where all states with

are descendents of the vacuum.
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Extremal Conformal Field Theory
Definition An extremal conformal field theory
of level k is a CFT with c24k with a (weight
zero) modular partition function as close as
possible to the vacuum Virasoro character.
Not modular
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Reconstruction Theorem
Define the polar polynomial of Zk to be the sum
of terms with nonpositive powers of q.
The weight zero modular function Zk can be
uniquely reconstructed from its polar
polynomial
This is the step which will fail (almost always)
in the N(2,2) case.
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Wittens proposal for pure 3D quantum gravity
The holographic dual of pure AdS3 quantum
gravity is a left-right product of extremal
conformal field theories.
Justification Chern-Simons form of action
Polar terms Chiral edge states (Brown-Henneaux)
Nonpolar terms Black holes (c.f. Fareytail)
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What can we say about supergravity?
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N1 Theories
Witten already pointed out that there is an
analog of the j-function for the modular group
preserving a spin structure on the torus.
Therefore one can construct the analog of
extremal N1 partition functions for the NS and R
sectors. We will return to N1 later, but for
now let us focus on N2.
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Pure N(2,2) AdS3 supergravity
N(2,2) AdS3 supergravity can be written as a
Chern-Simons theory for OSp(22) x OSp(22)
A natural extension of Wittens conjecture is
that pure N(2,2) sugra is dual to an extremal
(2,2) SCFT
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Extremal N(2,2) SCFT
Define an extremal N(2,2) theory to be a
theory whose partition function is as close as
possible to the vacuum character
It is useful to parametrize c 6 m
This is neither spectral flow invariant, nor
modular invariant.
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Extremal N(2,2) SCFT II
Impose spectral flow by hand
So, more precisely, an extremal N(2,2) CFT is a
CFT with a modular and spectral flow invariant
partition function of the above form.
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What do we mean by nonpolar terms ?
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Cosmic Censorship Bound
Black holes with near horizon geometry
Must satisfy the cosmic censorship bound
Cvetic Larsen
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Polarity
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Elliptic Genus
Modular
Spectral flow invariant
(Assume m integer, U(1) charges integral.)
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Weak Jacobi Forms
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Do such weak Jacobi forms exist?
Extremal Elliptic Genus
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Polar Region and Polar Polynomial
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Reconstruction Theorem
Dijkgraaf, Maldacena, Moore, Verlinde Manschot
Moore
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Obstructions
However, some polar polynomials cannot be
extended to a full weak Jacobi form!
Does not converge. It must be regulated.
The regularization can spoil modular invariance
Knopp Niebur Manschot Moore
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Counting Weak Jacobi Forms
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Counting Polar Polynomials
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Digression on Number Theory
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Digression on Number Theory
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Extremal Polar Polynomial
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Search for the extremal elliptic genus
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Polarity-Ordered Basis of Vm
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Computer Search
Recall P(m)gtj(m) for mgt4. Is there magic?
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Not Much Magic
We find five exceptional solutions
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Finiteness Theorem
Theorem There is an M such that for mgt M an
extremal elliptic genus does not exist.
Difficult proof. Compare the elliptic genus at
The NS and R cusps to derive the constraint
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Numerical analysis strongly indicates that our
five exceptional solutions are indeed the only
ones.
All this suggests that there are at most a
finite number of pure N(2,2) supergravity theorie
s.
But.
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Near-Extremal Theories
Perhaps our definition of extremal was too
restrictive
Maybe there are quantum corrections to the cosmic
censorship bound
Define a b-extremal N2 CFT by only demanding
agreement of the polar terms with the vacuum
character for polarity less than or equal to - b
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b-extremal theories
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Number of equations Number of variables.
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Estimate b(m)
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Escape Hatch for Pure N2 Sugra?
suggests a loophole Perhaps there
are quantum corrections to the cosmic censorship
bound.
Perhaps pure N2 sugra exists, and P(m)-j(m)
polar quantum numbers in fact admit states
which are semiclassically described as black
holes (or other nondescendent geometries).
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Bound on Conformal Weight
Also implies an interesting bound on the
conformal weight of the first N2 primary.
There must be some polar-state which is NOT a
descendent and satisfies
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Bound on Conformal Weight-2
Conclusion The conjecture
implies that for any unitary N(2,2) theory there
must exist a state of the form
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Explicit Construction of Nearly Extremal Elliptic
Genera
We can explicitly construct elliptic genera so
that only descendents of the vacuum
contribute to the polar subregion
So our bound is close to optimal.
Gritsenko
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Flux Compactifications
(Discussions with T. Banks, F. Denef M.
Gaberdiel, C. Keller, J. Maldacena)
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Existence of a Hauptmodul K There are no
restrictions on the polar polynomial
However, given the polar piece, modularity does
make predictions about the degeneracies at hc/24
a, a1/2, 1,3/2,
Characterize the flux vacuum by c J
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Single Particle Spectrum
Planck scale
Kaluza-Klein mode
Complex modulus
Kahler modulus
Vacuum
Large c, moderate J Near extremal
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Supergravity Fock Space
Supplies polar polynomial
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Optimistically one could estimate the
degeneracies at h c/24 a, a1/2, 1, 3/2,
Modularity
Descendents

• Jltlt1 Contradiction with EFT
• Jgtgt1 No contradiction. But large
• degeneracy might possibly invalidate EFT

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Conclusion

• There are at most 9 extremal N(2,2) 2D CFTs.
• We have strong evidence for an upper bound
• on the weight h of the first N2 primary.
• It is plausible that pure N2 supergravity
• does not exist.
• It would be good to investigate more precisely
• quantum corrections to the CC bound.

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Conclusions 2
It is possible that there can be applications to
the landscape, but our original motivation was
naïve such applications will require further
input than just modularity of the partition
function.