PARTITION FUNCTIONS AND ELLIPTIC GENERA FROM SUPERGRAVITY

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PARTITION FUNCTIONS AND ELLIPTIC GENERA FROM
SUPERGRAVITY
Niels Bohr Institute Summer Workshop August 1,
2006

• Finn Larsen
• University of Michigan

Refs P. Kraus and FL hep-th/0506176,
hep-th/0508218, hep-th/0607138
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INTRODUCTION

• Modern interpretation of internal structure of
  black holes
• Here ZCFT is the partition function of the brane
  constituents that form the black holes.
• Agreement of leading asymptotics (for large
  charges) amounts to a microscopic explanation of
  the black hole entropy.
• Subleading corrections on black hole side higher
  derivative corrections to geometry.
• Subleading corrections on CFT side central
  charge and/or level of the CFT not necessarily
  large.

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SIGNIFICANCE OF AdS/CFT

• All examples where this strategy has been
  successful involve black holes with near horizon
  geometry of the form AdS3xSpxX.
• Therefore, it is natural to to interpret the
  agreement ZBH ZCFT as a special case of the
  AdS3/CFT2 correspondence.
• In this framework the agreements at leading order
  are automatic, due to symmetries (in particular,
  trace anomalies guarantee agreement of central
  charges asymptotically).
• Corrections that are higher order in the charge
  (though still saddle point) also agree due to
  symmetries (in particular gravitational anomalies
  guarantee agreement of exact central charges).
• This talk develop AdS/CFT point of view in more
  detail, to go beyond saddle point approximation.
  The particulars are motivated by the Farey-tale.

Dijkgraaf, Maldacena, Moore, Verlinde
P. Kraus and FL hep-th/0506176, hep-th/0508218
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ROLE OF CHERN-SIMONS THEORY

• Effective theory on AdS3 has massless fields and
  numerous massive fields which we also want to
  take into account.
• Massless fields gravity, gauge fields from
  R-symmetries and also from other currents, and
  scalars.
• The action for the gauge field is dominated at
  long distances by a Chern-Simons term. Important
  task develop Chern-Simons theory using AdS/CFT
  reasoning.
• Note we want to allow any other terms (starting
  with Maxwell but higher derivative terms too).
  The asymptotic dominance of Chern-Simons theory
  will explain why many results are robust.

P. Kraus and FL hep-th/0607138
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ASIDE ON TOPOLOGICAL STRINGS

• It has further been conjectured that the CFT on
  the brane is related to topological string
  theory
• Precise statement! The elliptic genus on both
  sides so these are robust indices.
• Confusions the measure, the ensemble..
• Limitations supersymmetric black holes only.
• And why is the index related to the black hole
  (for example, Walds entropy formula does not
  apply to the index.
• Therefore want to develop robust aspects of the
  higher derivative story.

Ooguri, Strominger, Vafa
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REVIEW SPECTRAL FLOW

• Consider a 2d CFT with 4 SUSYs in the holomorphic
  sector the bosonic part of the algebra is
• The algebra permits the spectral flow
  automorphism
• For the elliptic genus
• This amounts to the transformation

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ASIDE MODULAR INVARIANCE

• Consider a boson coupled to a scalar field.
  Partition function
• A simple change of variables verifies modular
  invariance
• The partition function
• corresponds to a Hamiltonian that differs by a
  term proportional to A2. Therefore
• Consequently the modular transformation of Z is
  non-trivial it is given by that of the
  exponential prefactor.

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SUMMARY TRANFORMATION PROPERTIES

• Quite generally, spectral flow is manifest in
  Hamiltonian formalism.
• In other words, it is just a change of variables
  in the partition function or in the elliptic
  genus.
• Modular invariance, on the other hand is manifest
  in the path integral formalism where it is just a
  change of variables.
• The Lagrangian and the Hamiltonian are related to
  each other by a nontrivial pre-factor.
• This accounts for non-trivial modular
  properties/spectral flow in the
  Hamiltonian/Lagrangian formalism.

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GRAVITATIONAL ACTION

• The gravitational part of the action is
• The boundary term is the Gibbons-Hawking term
  (cancelling second derivatives) and the
  counter-term (cancelling infrared divergences).
  Dots refer to higher orders in bulk. Those do not
  lead to additional infrared divergences.
• Asymptotically AdS3 spacetimes take the
  Fefferman-Graham form
• The stress tensor is defined as
• It can be written explicitly in terms of the
  departure from pure AdS3 as

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GAUGE FIELD ACTION

• We want to repeat the treatment of the
  gravitational field for the case of a gauge field
  with a Chern-Simons interaction.
• The leading term in the Fefferman-Graham
  expansion is a flat connection no matter the
  detailed bulk action
• The holomorphic part of the boundary gauge field
  is an independent field, not specified by
  boundary condition
• The condition that this be so determines the
  boundary term as
• Similar considerations apply to the
  anti-holomorphic gauge fields.

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BOUNDARY STRESS TENSOR FOR GAUGE FIELD

• The boundary term depends on the metric. So there
  is a universal contribution to the boundary
  stress tensor that takes the form
• The Aw enters the anti-holomorphic stress tensor
  through the gauge field Aw which is not an
  independent field, it is a function of Aw (and
  all this is mirrored for the anti-holomorphic
  fields with tildes).
• Important point these expressions for the stress
  tensor are exact, no matter the details of the
  bulk action.
• The dependence on higher derivative terms enter
  through the detailed relation between the
  propagating gauge fields Aw and the dependent
  gauge fields Aw.

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EXAMPLE NS vs R SECTOR

• Consider global AdS3
• The generators of the isometry group SL(2,R) x
  SL(2,R) vanish in this state so
• The R-vacuum has opposite periodicity on the
  fermions. We can generate these periodicities by
  performing a spectral flow that turns on the flat
  connection
• The boundary tensor receives a contribution from
  the flat connection which is precisely the value
  that is associated with the R-vacuum.

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U(1) CURRENTS

• We have discussed so far just the gauge fields
  associated with the R-charges.
• For N4 these are non-abelian (with gauge group
  SU(2) ) but we focus on a subgroup U(1) .
• There is an entirely analogous structure for
  other U(1) currents. Although such currents are
  not a part of the SUSY algebra there are charges
  associated with them, and they play an important
  role in applications to black holes.
• In particular, we can define spectral flow, and
  we can determine the universal form of the gauge
  field contribution to the stress tensor.
• The incorporation of other U(1) fields just
  involves the introduction of a an additional
  index on the gauge fields (which in fact we may
  want to suppress).

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ANOMALIES

• Variation of the full action (including the
  boundary terms) gives the universal form of the
  currents
• Taking derivatives (and remembering that the
  connection is flat) we find
• Thus the currents are not conserved. In fact,
  this is the standard form of the anomaly.
• Anomalies are thus taken into account
  automatically by the formalism the introduces
  boundary counterterms.

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THE EXACT ACTION

• We have determined the exact stress tensor and
  currents in the R-vacuum with a gauge field
  turned on
• They are related to variations of the action
  through
• Integrating (with no angular gauge field as
  boundary condition) we find the exact action
• This result takes all higher derivative terms
  into account.

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BLACK HOLE ENTROPY

• This was for the thermal vacuum because no
  angular part of the gauge field corresponds to
  angular loop contractible in bulk.
• Euklidean black holes have the temporal direction
  contractible in bulk. So a modular transformation
  gives the exact black hole action. It is
• The result is the saddlepoint contribution to the
  path integral. Changing to the Hamiltonian
  formulation and keeping just the saddle-point
  contribution we find
• Corresponding to the entropy
• This result is exact in that it takes higher
  derivatives into account. It is even applicable
  to non-extremal black holes. And it agrees with
  macroscopic considerations.
• However, we would like to go beyond the
  saddle-point approximation.

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ELLIPTIC GENUS HAMILTONIAN FORMALISM

• In order to do better we must consider not just
  the vacuum, but also more general bulk states
  with
• And their spectrally flowed versions with
• Their contribution to the elliptic genus is
• We take all states that do not correspond to
  black holes into account. Denote the degeneracy
  of such states by
• Then the polar part of the elliptic genus (only
  states not forming black holes) becomes

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ELLIPTIC GENUS SUPERGRAVITY

• The supergravity treatment uses path integral
  formalism to sum over configurations
• The charges of the sources in bulk give rise to
  nontrivial holonomies
• These are encoded in boundary conditions
• Again, we know exact values of the boundary
  stress tensor and currents, as function of gauge
  fields and complex structure. Integrating the
  variation of the action with these boundary
  conditions imposed we find
• Summing over all possible sources this in fact
  agrees with the result of the CFT analysis

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SUMMING OVER ORBITS

• So far we just computed the polar part include
  sources that are below threshold for black hole
  formation.
• Black holes are added back in when modular
  invariance (in spacetime) is restored, by summing
  over SL(2,Z) orbits. The transformed
  configurations contribute
• Transforming back to the hamiltonian formalism
  (remember the phase) we find the full result
  after sum over SL(2,Z) orbit as
• This is not right (the sum diverges) and instead
  we should sum over the Farey tale transformed
  versions (would be nice to understand why, from
  supergravity)

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SUPERGRAVITY FLUCTUATIONS

• We now need to sum over the actual spectrum of
  possible sources. In M-theory on some CY 3-fold
  the 5d effective theory has spectrum
• After sphere reduction to AdS3 the spectrum of
  chiral primaries is

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SUMMING OVER TOWERS
Gaiotto, Strominger, Yin

• The tower index (angular momentum on sphere) is
  l.
• But we can also include descendants of these by
  acting p times with L-1 .
• And by then we just have the single particle
  spectrum their can be m independent particles. A
  single bosonic tower now gives
• Taking the full spectrum into account we find

McMahon function
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SINGLETONS

• Singletons are formally pure gauge but actually
  correspond to single boundary degree of freedom.
• Including nL gauge fields and the diffeomorphism
  singletons too, and taking descendents into
  account, we have
• The singletons cancel many bulk modes in the
  complete elliptic genus

Removed by Farey tale transform?
Topological string in flat space
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NON-PERTUBATIVE EXCITATIONS

• This was just the part of the spectrum that came
  from light modes in M-theory (which may be heavy
  in AdS3).
• We also need to include modes coming from
  M2-branes wrapping various cycles, and from
  anti-M2-branes.
• These have been the focus of much recent effort
  and we have nothing to add in the case of generic
  CYs.
• For T6 there are no contributions to the elliptic
  genus.
• For K3 x T2 only the M2s wrapping the T2 matter,
  and these give rise to 24 Dedekind functions, as
  needed for duality with the heterotic string.

Gaiotto, Strominger, Yin Amsterdam
group DenefMoore
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SUMMARY

• The overall goal systematic computation of the
  elliptic genus for a black hole using
  supergravity
• Kay feature that makes this possible
  Chern-Simons theory dominates at long distances
  and this theory is topological. Complications
  such as higher order corrections, fluctuation
  determinants do not present problems.
• String input the precise spectrum of allowed
  nontrivial holonomies. This distinguishes
  different examples.
• Boundary terms play an central role. They are
  taken correctly into account using conventional
  AdS/CFT techniques.
• Remark we computed elliptic genus but most
  formulae hold for the partition function, it is
  just that the latter is not a protected quantity.

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SOME OPEN PROBLEMS

• The precise interpretation in supergravity of the
  Farey tale transform remains obscure. It seems
  related to the omission of center of mass modes
  (a standard feature of AdS/CFT) but some details
  do not fit exactly.
• What are the ultimate limitations on extracting
  the black hole partition function from the
  spacetime action? After all, many string theories
  have the same low energy spectrum but differ in
  details (proposed answer the whole thing can be
  done, we just need to interpret it right).
• What are the prospects for non-BPS black holes?
  The saddle point works exactly (including higher
  order corrections in the charges) but higher
  order corrections (prefactors to the
  exponentials) are not robust and seem impossibly
  difficult, unless some simplifying principle can
  be discovered.