Lectures Physics Colloquiums
  Rotation, Nut charge and Anti de Sitter space
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  Title slide
  (reveal title)
  The work I'm going to talk about has been carried out with Chris
  Hunter and Marika Taylor Robinson at Cambridge, and Don Page at
  Alberta.
  References
  It is described in the first three papers shown on the screen. Related
  work has been carried out by Chamblin, Emparan, Johnson, and Myers.
  However, they seemed a bit uncertain what reference background to use.
  I have also shown a reference to Dowker which is relevant.
  It has been known for quite a time, that black holes behave like they
  have entropy. The entropy is the area of the horizon, divided by 4 G,
  where G is Newton's constant.
  Black Hole Entropy
  The idea is that the Euclidean sections of black hole metrics are
  periodic in the imaginary time coordinate. Thus they represent black
  holes in equilibrium with thermal radiation. However there are
  problems with this interpretation.
  Problems with thermodynamic interpretation
  (first problem appear)
  First, one can not have thermal radiation in asymptotically flat
  space, all the way to infinity, because the energy density would curve
  the space, and make it an expanding or collapsing Friedmann universe.
  Thus, if you want a static situation, you have to resort to the
  dubious Gedanken experiment, of putting the black hole in a box. But
  you don't find black hole proof boxes, advertised on the Internet.
  (second problem appear)
  The second difficulty with black holes in equilibrium with thermal
  radiation is that black holes have negative specific heat. In many
  cases, when they absorb energy, they get larger and colder. This
  reduces the radiation they give off, and so they absorb faster than
  they radiate, and the equilibrium is unstable. This is closely related
  to the fact that the Euclidean metric has a negative mode. Thus it
  seems that asymptotically flat Euclidean black holes, describe the
  decay of hot flat space, rather than a black hole in equilibrium with
  thermal radiation.
  (third problem appear)
  The third difficulty with the idea of equilibrium is that if the black
  hole is rotating, the thermal radiation should be co-rotating with it.
  But far away from the black hole, the radiation would be co-rotating
  faster than light, which is impossible. Thus, again one has to use the
  artificial expedient, of a box of finite size.
  Way back in pre-history, Don page and I, realized one could avoid the
  first two difficulties, if one considered black holes in anti de
  Sitter space, rather than asymptotically-flat space. In anti de Sitter
  space, the gravitational potential increases as one goes to infinity.
  This red shifts the thermal radiation, and means that it has finite
  energy. Thus anti de Sitter space can exist at finite temperature,
  without collapsing. In a sense, the gravitational potential in anti de
  Sitter space, acts like a confining box.
  Anti de Sitter space can also help with the second problem, that the
  equilibrium between black holes and thermal radiation, will be
  unstable. Small black holes in anti de Sitter space, have negative
  specific heat, like in asymptotically flat space, and are unstable.
  But black holes larger than the curvature radius of anti de Sitter
  space, have positive specific heat, and are presumably stable.
  At the time, Don page and I, did not think about rotating black holes.
  But I recently came back to the problem, along with Chris Hunter, and
  Marika Taylor Robinson. We realized that thermal radiation in anti de
  Sitter space could co-rotate with up to some limiting angular
  velocity, without having to travel faster than light. Thus anti de
  Sitter boundary conditions, can solve all three problems, in the
  interpretation of Euclidean black holes, as equilibria of black holes,
  with thermal radiation.
  Anti de Sitter black holes may not seem of much interest, because we
  can be fairly sure, that the universe is not asymptotically anti de
  Sitter. However, they seem worth studying, both for the reasons I have
  just given, and because of the Maldacena conjecture, relating
  asymptotically anti de Sitter spaces, to conformal field theories on
  their boundary. I shall report on two pieces of work in relation to
  this conjecture. One is a study of rotating black holes in anti de
  Sitter space. We have found Kerr anti de Sitter metrics in four and
  five dimensions. As they approach the critical angular velocity in
  anti de Sitter space, their entropy, as measured by the horizon area,
  diverges. We compare this entropy, with that of a conformal field
  theory on the boundary of anti de Sitter space. This also diverges at
  the critical angular velocity, when the rotational velocity,
  approaches the speed of light. We show that the two divergences are
  similar.
  The other piece of work, is a study of gravitational entropy, in a
  more general setting. The quarter area law, holds for black holes or
  black branes in any dimension, d, that have a horizon, which is a d
  minus 2 dimensional fixed point set, of a U1 isometry group. However
  Chris Hunter and I, have recently shown that entropy can be associated
  with a more general class of space-times. In these, the U1 isometry
  group can have fixed points on surfaces of any even co-dimension, and
  the space-time need not be asymptotically flat, or asymptotically anti
  de Sitter. In these more general class, the entropy is not just a
  quarter the area, of the d minus two fixed point set.
  Among the more general class of space-times for which entropy can be
  defined, an interesting case is those with Nut charge. Nut charge can
  be defined in four dimensions, and can be regarded as a magnetic type
  of mass.
  Solutions with nut charge are not asymptotically flat in the usual
  sense. Instead, they are said to be asymptotically locally flat, or
  ALF. In the Euclidean regime, in which I shall be working, the
  difference can be described as follows. An asymptotically flat metric,
  like Euclidean Schwarzschild, has a boundary at infinity, that is a
  two-sphere of radius r, cross a circle, whose radius is asymptotically
  constant.
  To get finite values for the action and Hamiltonian, one subtracts the
  values for flat space, periodically identified. In asymptotically
  locally flat metrics, on the other hand, the boundary at infinity, is
  an S1 bundle over S2. These bundles are labeled by their first Chern
  number, which is proportional to the Nut charge. If the first Chern
  number is zero, the boundary is the product, S2 cross S1, and the
  metric is asymptotically flat. However, if the first Chern number is
  k, the boundary is a squashed three sphere, with mod k points
  identified around the S1 fibers.
  Such asymptotically locally flat metrics, can not be matched to flat
  space at infinity, to give a finite action and Hamiltonian, despite a
  number of papers that claim it can be done. The best that one can do,
  is match to the self-dual multi Taub nut solutions. These can be
  regarded as defining the vacuums for ALF metrics.
  In the self-dual Taub Nut solution, the U1 isometry group, has a zero
  dimensional fixed point set at the center, called a nut. However, the
  same ALF boundary conditions, admit another Euclidean solution, called
  the Taub bolt metric, in which the nut is replaced by a two
  dimensional bolt. The interesting feature, is that according to the
  new definition of entropy, the entropy of Taub bolt, is not equal to a
  quarter the area of the bolt, in Planck units. The reason is that
  there is a contribution to the entropy from the Misner string, the
  gravitational counterpart to a Dirac string for a gauge field.
  The fact that black hole entropy is proportional to the area of the
  horizon has led people to try and identify the microstates, with
  states on the horizon. After years of failure, success seemed to come
  in 1996, with the paper of Strominger and Vafa, which connected the
  entropy of certain black holes, with a system of D-branes. With
  hindsight, this can now be seen as an example of a duality, between a
  gravitational theory in asymptotically anti de Sitter space, and a
  conformal field theory on its boundary.
  It would be interesting if similar dualities could be found for
  solutions with Nut charge, so that one could verify that the
  contribution of the Misner string was reflected in the entropy of a
  conformal field theory. This would be particularly significant for
  solutions like Taub bolt, which don't have a spin structure. It would
  show whether the duality between anti de Sitter space, and conformal
  field theories on its boundary, depends on super symmetry. In fact, I
  will present evidence, that the duality requires super symmetry.
  To investigate the effect of Nut charge, we have found a family of
  Taub bolt anti de Sitter solutions. These Euclidean metrics are
  characterized by an integer, k, and a positive parameter, s. The
  boundary at large distances is an S1 bundle over S2, with first Chern
  number, k. If k=0, the boundary is a product, S1 cross S2, and the
  space is asymptotically anti de Sitter, in the usual sense. But if k
  is not zero, they are what may be called, asymptotically locally anti
  de Sitter, or ALADS.
  The boundary is a squashed three sphere, with k points identified
  around the U1 direction. This is just like asymptotically locally
  flat, or ALF metrics. But unlike the ALF case, the squashing of the
  three-sphere, tends to a finite limit, as one approaches infinity.
  This means that the boundary has a well-defined conformal structure.
  One can then ask whether the partition function and entropy, of a
  conformal field theory on the boundary, is related to the action and
  entropy, of these asymptotically locally anti de Sitter solutions.
  To make the ADS, CFT correspondence well posed, we have to specify the
  reference backgrounds, with respect to which the actions and
  Hamiltonians are defined. For Kerr anti de Sitter, the reference
  background is just identified anti de Sitter space. However, as in the
  asymptotically locally flat case, a squashed three sphere, can not be
  imbedded in Euclidean anti de Sitter. One therefore can not use it as
  a reference background, to make the action and Hamiltonian finite.
  Instead, one has to use Taub Nut anti de Sitter, which is a limiting
  case of our family. If mod k is greater than one, there is an orbifold
  singularity in the reference backgrounds, but not in the Taub bolt
  anti de Sitter solutions. These orbifold singularities in the
  backgrounds could be resolved, by replacing a small neighbourhood of
  the nut, by an ALE metric. We shall take it, that the orbifold
  singularities are harmless.
  Another issue that has to be resolved, is what conformal field theory
  to use, on the boundary of the anti de Sitter space. For five
  dimensional Kerr anti de Sitter space, there are good reasons to
  believe the boundary theory is large N Yang Mills. But for
  four-dimensional Kerr anti de Sitter, or Taub bolt anti de Sitter, we
  are on shakier ground. On the three dimensional boundaries of four
  dimensional anti de Sitter spaces, Yang Mills theory is not
  conformally invariant.
  The folklore is that one takes the infrared fixed point, of
  three-dimensional Yang Mills, but no one knows what this is. The best
  we can do, is calculate the determinants of free fields on the
  squashed three sphere, and see if they have the same dependence on the
  squashing, as the action. Note that as the boundary is odd
  dimensional, there is no conformal anomaly. The determinant of a
  conformally invariant operator, will just be a function of the
  squashing. We can then interpret the squashing, as the inverse
  temperature, and get the number of degrees of freedom, from a
  comparison with the entropy of ordinary black holes, in four
  dimensional anti de Sitter.
  I now turn to the question, of how one can define the entropy, of a
  space-time. A thermodynamic ensemble, is a collection of systems,
  whose charges are constrained by La-grange multipliers.
  Partition function
  One such charge, is the energy or mass, with the Lagrange multiplier
  being the inverse temperature, beta. But one can also constrain the
  angular momentum, and gauge charges. The partition function for the
  ensemble, is the sum over all states, of e to the minus, La-grange
  multipliers, times associated charges.
  Thus it can be written as, trace of e to the minus Q. Here Q is the
  operator that generates a Euclidean time translation, beta, a
  rotation, delta phi, and a gauge transformation, alpha. In other
  words, Q is the Hamiltonian operator, for a lapse that is beta at
  infinity, and a shift that is a rotation through delta phi. This means
  that the partition function can be represented by a Euclidean path
  integral.
  The path integral is over all metrics which at infinity, are periodic
  under the combination of a Euclidean time translation, beta, a
  rotation through delta phi, and a gauge rotation, alpha. The lowest
  order contributions to the path integral for the partition function
  will come from Euclidean solutions with a U1 isometry, that agree with
  the periodic boundary conditions at infinity.
  The Hamiltonian in general relativity or supergravity, can be written
  as a volume integral over a surface of constant tau, plus surface
  integrals over its boundaries.
  Gravitational Hamiltonian
  The volume integral vanishes by the constraint equations. Thus the
  numerical value of the Hamiltonian, comes entirely from the surface
  terms. The action can be related to the Hamiltonian in the usual way.
  Because the metric has a time translation isometry, all dotted
  quantities vanish. Thus the action is just beta times the Hamiltonian.
  If the solution can be foliated by a family of surfaces, that agree
  with Euclidean time at infinity, the only surface terms will be at
  infinity.
  Family of time surfaces
  In this case, a solution can be identified under any time translation,
  rotation, or gauge transformation at infinity.
  This means that the action will be linear in beta, delta phi, and
  alpha. If one takes such a linear action, for the partition function,
  and applies the standard thermodynamic relations, one finds the
  entropy is zero.
  The situation is very different however, if the solution can't be
  foliated by surfaces of constant tau, where tau is the parameter of
  the U1 isometry group, which agrees with the periodic identification
  at infinity.
  Breakdown of foliation
  The break down of foliation can occur in two ways. The first is at
  fixed points of the U1 isometry group. These occur on surfaces of even
  co-dimension. Fixed-point sets of co-dimension-two play a special
  role. I shall refer to them as bolts. Examples include the horizons of
  non-extreme black holes and p-branes, but there can be more
  complicated cases, like Taub bolt.
  The other way the foliation by surfaces of constant tau, can break
  down, is if there are what are called, Misner strings.
  Kaluza Klein metric
  To explain what they are, write the metric in the Kaluza Klein form,
  with respect to the U1 isometry group. The one form, omega, the
  scalar, V, and the metric, gamma, can be regarded as fields on B, the
  space of orbits of the isometry group.
  If B has homology in dimension two, the Kaluza Klein field strength,
  F, can have non-zero integrals over two cycles. This means that the
  one form, omega, will have Dirac strings in B. In turn, this will mean
  that the foliation of the spacetime, M, by surfaces of constant tau,
  will break down on surfaces of co-dimension two, called Misner
  strings.
  In order to do a Hamiltonian treatment using surfaces of constant tau,
  one has to cut out small neighbourhoods of the fixed point sets, and
  the Misner strings. This modifies the treatment, in two ways. First,
  the surfaces of constant tau now have boundaries at the fixed-point
  sets, and Misner strings, as well as the boundary at infinity. This
  means there can be additional surface terms in the Hamiltonian.
  In fact, the surface terms at the fixed-point sets are zero, because
  the shift and lapse vanish there. On the other hand, at a Misner
  string, the lapse vanishes, but the shift is non zero. The Hamiltonian
  can therefore have a surface term on the Misner string, which is the
  shift, times a component of the second fundamental form, of the
  constant tau surfaces. The total Hamiltonian, will be the sum of this
  Misner string Hamiltonian, and the Hamiltonian surface term at
  infinity.
  Consequences of non-foliation
  As before, the action will be beta times the Hamiltonian. However,
  this will be the action of the space-time, with the neighborhoods of
  the fixed-point sets and Misner strings removed. To get the action of
  the full space-time, one has to put back the neighborhoods. When one
  does so, the surface term associated with the Einstein Hilbert action,
  will give a contribution to the action, of minus area over 4G, for the
  bolts and Misner strings. Here G is Newton's constant in the dimension
  one is considering. The surface terms around lower dimensional
  fixed-point sets make no contribution to the action.
  The action of the space-time, will be the lowest order contribution to
  minus log Z, where Z is the partition function. But log Z is equal to
  the entropy, minus beta times the Hamiltonian at infinity. So the
  entropy is a quarter the area of the bolts and Misner strings, minus
  beta times the Hamiltonian on the Misner strings. In other words, the
  entropy is the amount by which the action is less than the value, beta
  times the Hamiltonian at infinity, that it would have if the surfaces
  of constant tau, foliated the space-time.
  This formula for the entropy applies in any dimension and for any
  class of boundary condition at infinity. Thus one can use it for
  rotating black holes, in anti de Sitter space. In this case, the
  reference background is just Euclidean anti de Sitter space,
  identified with imaginary time period, beta, and appropriate rotation.
  Four-dimensional Kerr-AdS
  The four-dimensional Kerr anti de Sitter solution, was found by
  Carter, and is shown on the slide. The parameter, a, determines the
  rate of rotation. When a-l approaches 1, the co-rotation velocity
  approaches the speed of light at infinity. It is therefore interesting
  to examine the behavior of the black hole action, and the conformal
  field theory partition function, in this limit.
  To calculate the action of the black hole is quite delicate, because
  one has to match it to rotating anti de Sitter space, and subtract one
  infinite quantity, from another.
  Euclidean action.
  Nevertheless, this can be done in a well-defined way, and the result
  is shown on the slide. As you might expect, it diverges at the
  critical angular velocity, at which the co-rotating velocity,
  approaches the speed of light.
  The boundary of rotating anti de Sitter, is a rotating Einstein
  universe, of one dimension lower. Thus it is straightforward in
  principle, to calculate the partition function for a free conformal
  field on the boundary. Someone like Dowker might have calculated the
  result exactly. However, as we are only human, we looked only at the
  divergence in the partition function, as one approaches the critical
  angular velocity.
  This divergence arises because in the mode sum for the partition
  function, one has Bose-Einstein factors with a correction because of
  the rotation. As one approaches the critical angular velocity, this
  causes a Bose-Einstein condensation in modes with the maximum axial
  quantum number, m.
  Conformal field theory
  The divergence in the conformal field theory partition function has
  the same divergence as the black hole action, at the critical angular
  velocity. I haven't compared the residues. This is difficult, because
  it is not clear what three-dimensional conformal field theory one
  should use on the boundary of four dimensional anti de Sitter.
  Five-dimensional Kerr-AdS
  The case of rotating black holes in anti de Sitter five, is broadly
  similar, but with some differences. One of these is that, because the
  spatial rotation group, O4, is of rank 2, there are two rotation
  parameters, a & b. Each of these must have absolute value less than l
  to the minus one, for the co-rotation velocity to be less than the
  speed of light, all the way out to infinity. If just one of a & b,
  approaches the limiting value, the action of the black hole, and the
  partition function of the conformal field theory, both diverge in a
  manner similar to the four dimensional case.
  Action of five-dimensional Kerr-AdS
  But if a = b, and they approach the limit together, the action and the
  partition function, both have the same stronger divergence. Again, I
  haven't compared residues, but this might be worth doing. It may be
  that in the critical angular velocity limit, the interactions between
  the particles of super Yang Mills theory, become unimportant. If this
  is the case, one would expect the action and partition function to
  agree, rather than differ by a factor of four thirds, as in the non
  rotating case.
  Asymptotically locally flat
  I now turn the case of Nut charge. For asymptotically locally flat
  metrics in four dimensions, the reference background is the self-dual
  Taub Nut solution. The Taub bolt solution, has the same asymptotic
  behavior, but with the zero-dimensional nut fixed point, replaced by a
  two-dimensional bolt. The area of the bolt is 12 pi N squared, where N
  is the Nut charge. The area of the Misner string is minus 6 pi N
  squared. That is to say, the area of the Misner string in Taub bolt,
  is infinite, but it is less than the area of the Misner string in Taub
  nut, in a well-defined sense. The Hamiltonian on the Misner string, is
  N over 8. Again the Misner string Hamiltonian is infinite, but the
  difference from Taub nut, is finite. And the period, beta, is 8pi N.
  Thus the entropy, is pi N squared. Note that this is less than a
  quarter the area of the bolt, which would give 3 pi N squared. It is
  the effect of the Misner string that reduces the entropy.
  Taub Nut Anti de Sitter
  We would like to confirm the effect of Misner strings on entropy, by
  seeing what effect they have on conformal field theories, on the
  boundary of anti de Sitter space. For this purpose we constructed
  versions of Taub nut and Taub bolt, with a negative cosmological
  constant. The Taub nut anti de Sitter metric is shown on the
  transparancy. The parameter E, is the squashing of the three-sphere at
  infinity. If E=1, the three spheres are round, and the metric is
  Euclidean anti de Sitter space. However, if E is not equal to one, the
  metric can not be matched to anti de Sitter space at large distance.
  Each value of E, therefore, defines a different sector of ALADS
  metrics. This is an important point, which did not seem to have been
  realized by Chamblin et al. Taub Bolt Anti de Sitter One can also find
  a family of Taub bolt anti de Sitter metrics, with the same asymptotic
  behavior. These are characterized by an integer, k, and a positive
  quantity, s. These determine the asymptotic squashing parameter, E,
  and the area of the bolt, A-. K is the self-intersection number of the
  bolt. Thus the spaces do not have spin structure if k is odd. At
  infinity, the squashed three sphere has k points identified around the
  U1 fiber. This means that the reference background, is Taub nut anti
  de Sitter, with k points identified. If k is greater than one, the
  reference background will have an orbifold singularity at the nut.
  However, as I said earlier, I shall take it that such singularities
  are harmless.
  Action
  To calculate the action, one matches the Taub bolt solution on a
  squashed three sphere, to a Taub nut solution. To do this, one has to
  re-scale the squashing parameter, E, as a function of radius. The
  surface term in the action, is the same asymptotically for Taub nut
  and Taub bolt. Thus the action comes entirely from the difference in
  volumes.
  Action for k = 1
  For k greater than one, the action is always negative, while for k=1,
  it is positive for small areas of the bolt, and negative for large
  areas. This behavior is similar to that for Schwarzschild anti de
  Sitter space, and might indicate a phase transition in the
  corresponding conformal field theory. However, as I will argue later,
  there are problems with the ADS, CFT duality, if k=1. On the other
  hand, our results seem to indicate that there will be no phase
  transition, if more than one point is identified around the fiber. It
  will be interesting to see if this is indeed the case, for a conformal
  field theory on an identified squashed three-sphere.
  In these Taub bolt anti de Sitter metrics, one can calculate the area
  of the Misner string, and the Hamiltonian surface term. Both will be
  infinite, but if one matches to Taub nut anti de Sitter on a large
  squashed three-sphere, the differences will tend to finite limits. As
  in the asymptotically locally flat case, the entropy is less than a
  quarter the area of the bolt, because of the effect of the Misner
  string.
  Entropy
  One can also calculate the entropy from the partition function, by the
  usual thermodynamic relations. The mass will be given by taking the
  derivative of the action with respect to beta. This is equal to the
  Hamiltonian surface term at infinity. The mass or energy, is the only
  charge that is constrained in the ensemble. The nut charge is fixed by
  the boundary conditions, and so doesn't need a La-grange multiplier.
  Thus the entropy is beta M, minus I. This agrees with the entropy
  calculated from the bolts and Misner strings, showing our definition,
  is consistent.
  Formally at least, one can regard Euclidean conformal field theory on
  the squashed three sphere, as a twisted 2+1 theory, at a temperature,
  beta to the minus one. Thus one would expect the entropy to be
  proportional to beta to the minus two, at least for small beta. This
  has been confirmed by calculations by Dowker, of the determinants of
  scalar and fermion operators on the squashed three sphere, for k=1.
  Dowker has not so far calculated the higher k cases, but one would
  expect that these would have similar leading terms, but with beta
  replaced by beta over k. The next leading order terms in the
  determinant, are beta to the minus one, log beta. No terms like this
  appear in the bulk theory, so if there really is an ADS, CFT duality
  in this situation, the log beta terms have to cancel between the
  different spins.
  In fact, the scalar and fermion log beta terms will cancel each other,
  if there are twice as many scalars as fermions. This would be implied
  by super symmetry, suggesting that super symmetry is indeed necessary
  for the ADS, CFT duality.
  The Misner string contributions to the entropy are of order beta
  squared. Thus Dowker's calculations will have to be extended to this
  order, to k greater than one, to fermion fields with anti periodic
  boundary conditions, and to spin one fields. All this is quite
  possible, but it will probably require Dowker to do it.
  One might ask, how can a conformal field theory on the Euclidean
  squashed three sphere, correspond to a theory in a spacetime of
  Lorentzian signature. The answer is that, unlike the Schwarzschild
  anti de Sitter case, one has to continue the period, beta, to
  imaginary values. This makes the spacetime periodic in real time,
  rather than imaginary time. One gets a 2+1 rotating spacetime, rather
  like the Goedel universe, with closed time like curves. Although field
  theory in such a spacetime, may seem pathological, it can be obtained
  by analytical continuation, and is well defined despite the lack of
  causality. It is interesting that the analytically continued entropy,
  is negative, suggesting that causality violating spacetimes, are
  quantum suppressed. However, it is probably a mistake, to attach
  physical significance, to the Lorentzian conformal field theory.
  To sum up, I discussed the ADS, CFT duality in two new contexts. That
  of rotating black holes and that of solutions with nut charge. I
  showed how gravitational entropy can be defined in general. The
  partition function for a thermodynamic ensemble can be defined by a
  path integral over periodic metrics. The lowest order contributions to
  the partition function will come from metrics with a U1 isometry, and
  given behavior at infinity. The entropy of such metrics will receive
  contributions from horizons or bolts, and from Misner strings, which
  are the Dirac strings of the U1 isometry, under Kaluza Klein
  reduction. One would like to relate this gravitational entropy, to the
  entropy of a conformal field theory on the boundary. For this reason,
  we considered a new class of asymptotically locally anti de Sitter
  spaces. Other people have investigated the Maldacena conjecture, by
  deforming the compact part of the metric, but this is the first time
  deformed anti de Sitter boundary conditions, have been considered.
  We studied Taub bolt anti de Sitter solutions, with Taub nut anti de
  Sitter, as the reference background. The entropy we obtained obeyed
  the right thermodynamic relations, and had the right temperature
  dependence, to be the entropy of a conformal field theory, on the
  squashed three sphere. Because the Taub bolt solutions for odd k, do
  not have spin structures, this may indicate that the anti de Sitter,
  conformal field theory correspondence, does not depend on super
  symmetry.
  I will end by saying that gravitational entropy, is alive and well, 34
  years on. But there's more to entropy, than just horizon area. We need
  to look at the nuts and bolts.
  Title silde

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