Measuring the Number of Degrees of Freedom in 3-d CFT

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Measuring the Number of Degrees of Freedom in
3-d CFT

• Igor Klebanov
• Institute for Advanced Study and
• Princeton University
• Talk at Rutgers University
• September 27, 2011

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• The talk is based mainly on the papers
• C. Herzog, I.K., S. Pufu, T. Tesileanu,
  Multi-Matrix Models and Tri-Sasaki Einstein
  Spaces, 1011.5487.
• D. Jafferis, I.K., S. Pufu, B. Safdi, Towards the
  F-Theorem N2 Field Theories on the
  Three-Sphere, 1103.1181.
• I.K., S. Pufu, B. Safdi, F-Theorem without
  Supersymmetry, 1105.4598.

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• A deep problem in QFT is how to define a good
  measure of the number of degrees of freedom which
  decreases along RG flows and is stationary at
  fixed points.
• In two dimensions this problem was beautifully
  solved by Alexander Zamolodchikov who, using
  two-point functions of the stress-energy tensor,
  found the c-function which satisfies these
  properties.

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• At RG fixed points the c-function coincides with
  the Virasoro central charge, which is also the
  Weyl anomaly. It also determines the thermal free
  energy.
• For dgt2 it also seems physically reasonable to
  use the coefficient cT of the thermal free energy
  as the measure of the number of degrees of
  freedom
• It can be extracted from the Euclidean path
  integral on

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No cT Theorem!

• However, there are counterexamples to the
  hypothetical cT theorem in dgt2.
• In d3 Sachdev calculated the thermal free energy
  of the O(N) vector model,
• In the critical model m0, and

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• A relevant pertubation of this fixed point with
  makes it flow to the Goldstone phase
  described in the IR by N-1 free scalar fields.
• Hence, in the IR
• For large enough N this exceeds the UV value.
  This means that cT does not always decrease along
  RG flow.
• Another idea for generalizing the c-theorem to
  higher dimensions was proposed by Cardy.

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The a-theorem

• In d4 there are two Weyl anomaly coefficients,
  and one of them, called a is proportional to the
  4-d Euler characteristic. It can be extracted
  from the Euclidean part integral on the 4-d
  sphere.
• Cardy has conjectured that the a-coefficient
  decreases along any RG flow.
• No working counterexamples. A proof was recently
  proposed. Komargodski, Schwimmer

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• In theories with N1 SUSY, the a-coefficient is
  determined by the R-charges
• a Trf 3 (3R3 R)/32
• Intriligator and Wecht proposed that the
  R-symmetry is determined by locally maximizing a.
  This a-maximization principle has passed many
  consistency checks.
• In large N theories dual to type IIB strings on
  the a-coefficient is inversely
  proportional to the volume of Y5. AdS and CFT
  definitions of a agree.

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• How do we extend these successes to odd
  dimensions where there are no anomalies?
• This is clearly interesting, especially in d3
  where there is an abundance of conformal field
  theories, some of them describing critical points
  in statistical mechanics and condensed matter
  physics.
• It has been proposed that the good measure of
  the number of DOF is the free energy on the
  3-sphere Jafferis Jafferis, IK, Pufu, Safdi

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• In field theories with extended supersymmetry,
  the localization approach reduces the Euclidean
  path integral on a sphere to a finite dimensional
  integral, a matrix model. Pestun Kapustin,
  Willett, Yaakov Jafferis
• In d3 theories with N2 SUSY the marginality of
  superpotential often leaves some freedom in
  R-symmetry. Jafferis proposed that this freedom
  is fixed by locally extremizing (in fact,
  maximizing) F.
• This is the 3-d analogue of a-maximization.

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AdS/CFT Matching of F

• In large N models which have duals it
  is possible to compare the CFT result with the
  corresponding gravity calculation. After
  subtracting cubic and linear divergences, it
  gives
• The N3/2 scaling is a common feature of many
  leading order results in AdS4. IK, Tseytlin

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• The field theory calculations of F via large N
  matrix models reproduce this gravity results in a
  variety of models.
• The first success was achieved for the ABJM
  theory which is the U(N)k x U(N)-k
• Chern-Simons gauge theory dual to
  AdS4 x S7/Zk.

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• To gain some intuition, the eigenvalue positions
  in the complex plane can be studied numerically
  using the saddle point equations

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• In the large N limit where k is kept fixed, the
  correct ansatz is
• Cancellation of long-range forces on eigenvalues
  enables us to write a local functional
• We find and a constant eigenvalue
  density.

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• The matrix model free energy
• agrees with the AdS formula after we
  use vol (S7/Zk) p4/(3k)
• Drukker, Marino, Putrov Herzog, IK, Pufu,
  Tesileanu
• Reducing supersymmetry to N3, there exists a
  nice set of CS gauge theories with necklace
  quivers for which exact agreement has also
  been obtained

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N2 SUSY

• Now the R-charges are not fixed by supersymmetry.
  This offers nice oportunities to test the
  F-maximization, F-theorem and AdS/CFT.
• As a function of the trial R-charges the matrix
  model free energy is Jafferis

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• For example, for the ABJM model with more general
  R-charges
• the free energy is
• Maximizing this we obtain the standard R-charges
  ½ and
• If we add a relevant operator
• then in the gauge with

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• Performing the F-maximization in the IR theory we
  find
• Consistent with the F-theorem and with AdS/CFT.
  The conjectured gravity dual of the IR theory is
  Warners SU(3) symmetric extremum of the gauged
  SUGRA. Benna, IK, Klose, Smedback

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No SUSY

• The simplest CFTs involve free conformal scalar
  and fermion fields. Adding mass terms makes such
  a theory flow to a theory with no massless
  degrees of freedom in the IR where F0.
• For consistency with F-theorem, the
  F-coefficients for free massless fields should be
  positive.

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Conformal Scalar on Sd

• In any dimension
• The eigenvalues and degeneracies are
• Using zeta-function regularization in d3,

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A massless Dirac fermion

• The eigenvalues and degeneracies are
• Using zeta-function regularization
• For a chiral multiplet (complex scalarfermion)
  F (log 2)/2

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Slightly Relevant Operators

• Perturb a CFT by a relevant operator of dimension
• The path integral on a sphere is
• The 1-pt function vanishes.

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• The 2- and 3-pt function are determined by
  conformal invariance in terms of the chordal
  distance
• The change in the free energy is

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• The beta function for the dimensionless coupling
  is
• Integrating the RG equation and setting the scale
  to inverse sphere radius

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• For Cgt0 there exists a robust IR fixed point at
• The 3-sphere free energy decreases
• A similar calculation for d1 provided initial
  evidence for the G-theorem conjectured by Affleck
  and Ludwig.
• For a general odd dimension, what decreases along
  RG flow is

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Double-Trace Flows

• If we perturb a large N CFT by a relevant
  double-trace operator, it flows to another fixed
  point in the IR
• If in the UV the dimension of F is D, in the IR
  it is d- D
• F can be calculated using the Hubbard-Stratonovich
  trick

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• The change in F between IR and UV is of order 1
  and is computable Gubser, IK Diaz, Dorn
• In all odd dimensions
• For d3

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• The change in free energy is negative, in support
  of the F-theorem
• The particular case D1 corresponds to the
  critical O(N) model

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O(N) Model Redux

• The critical O(N) model is obtained via a
  double-trace perturbation of the theory of N free
  real scalars
• Using our free and double-trace results
• A further relevant perturbation takes it to the
  Goldstone phase where

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• Recall that the flow from the critical to the
  Goldstone phase provided a counter-example to the
  proposal that the thermal free energy decreases
  along RG flow.
• Yet, there is no contradiction with the F-theorem
  since

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Comments

• The F-theorem has passed some consistency
  checks both via field theory and using
  gauge/gravity duality. More should be done to
  search for counterexamples, or perhaps even prove
  it.
• Another recent proposal for measuring the degrees
  of freedom, this time in Lorentzian signature, is
  the entanglement entropy of a disk with its
  complement in R2. Myers, Sinha It appears to
  be equivalent to F. Casini, Huerta, Myers