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
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: N=2 Field Theories on
the Three-Sphere, 1103.1181.
• I.K., S. Pufu, B. Safdi, F-Theorem without
Supersymmetry, 1105.4598.
• 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.
• 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 d>2 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
No cT Theorem!
• However, there are counterexamples to
the hypothetical cT theorem in d>2.
• In d=3 Sachdev calculated the thermal
free energy of the O(N) vector model,
• In the critical model m=0, and
• A relevant pertubation of this fixed point
with
makes it flow to the
Goldstone phase described in the IR by N1 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 ctheorem to higher dimensions was
proposed by Cardy.
The a-theorem
• In d=4 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
• In theories with N=1 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.
• How do we extend these successes to odd
dimensions where there are no anomalies?
• This is clearly interesting, especially in d=3
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
• 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 d=3 theories with N=2 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.
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
• 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.
• To gain some intuition, the eigenvalue
positions in the complex plane can be
studied numerically using the saddle point
equations
• 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
density.
and a constant eigenvalue
• 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 N=3,
there exists a nice set of CS gauge
theories with `necklace’ quivers for
which exact agreement has also
been obtained
N=2 SUSY
• Now the R-charges are not fixed by
supersymmetry. This offers nice
oportunities to test the F-maximization, Ftheorem and AdS/CFT.
• As a function of the trial R-charges the
matrix model free energy is Jafferis
• For example, for the ABJM model with
more general R-charges
• the free energy is
• Maximizing this we obtain the standard Rcharges ½ and
• If we add a relevant operator
then in the gauge with
• 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 Warner’s SU(3) symmetric
extremum of the gauged SUGRA. Benna, IK,
Klose, Smedback
No SUSY
• The simplest CFT’s 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 F=0.
• For consistency with F-theorem, the Fcoefficients for free massless fields should
be positive.
Conformal Scalar on Sd
• In any dimension
• The eigenvalues and degeneracies are
• Using zeta-function regularization in d=3,
A massless Dirac fermion
• The eigenvalues and degeneracies are
• Using zeta-function regularization
• For a chiral multiplet (complex
scalar+fermion) F= (log 2)/2
Slightly Relevant Operators
• Perturb a CFT by a relevant operator of
dimension
• The path integral on a sphere is
• The 1-pt function vanishes.
• The 2- and 3-pt function are determined
by conformal invariance in terms of the
chordal distance
• The change in the free energy is
• The beta function for the dimensionless
coupling
is
• Integrating the RG equation and setting
the scale to inverse sphere radius
• For C>0 there exists a robust IR fixed
point at
• The 3-sphere free energy decreases
• A similar calculation for d=1 provided
initial evidence for the G-theorem
conjectured by Affleck and Ludwig.
• For a general odd dimension, what
decreases along RG flow is
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 HubbardStratonovich trick
• The change in F between IR and UV is of
order 1 and is computable
• In all odd dimensions
• For d=3
Gubser, IK; Diaz, Dorn
• The change in free energy is negative, in
support of the F-theorem
• The particular case D=1 corresponds to
the critical O(N) model
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
• Recall that the flow from the critical to the
Goldstone phase provided a counterexample to the proposal that the thermal
free energy decreases along RG flow.
• Yet, there is no contradiction with the Ftheorem since
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