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

CFTCFT

Igor KlebanovIgor Klebanov

Institute for Advanced Study andInstitute for Advanced Study andPrinceton UniversityPrinceton University

Talk at Rutgers UniversityTalk at Rutgers UniversitySeptember 27, 2011September 27, 2011

The talk is based mainly on the papersThe talk is based mainly on the papers

•C. Herzog, I.K., S. Pufu, T. Tesileanu, C. Herzog, I.K., S. Pufu, T. Tesileanu, Multi-Matrix Models and Tri-Sasaki Multi-Matrix Models and Tri-Sasaki Einstein Spaces, Einstein Spaces, 1011.5487.1011.5487.•D. Jafferis, I.K., S. Pufu, B. Safdi, D. Jafferis, I.K., S. Pufu, B. Safdi, Towards Towards the F-Theorem: the F-Theorem: NN=2 Field Theories on the =2 Field Theories on the Three-Sphere,Three-Sphere, 1103.1181. 1103.1181.•I.K., S. Pufu, B. Safdi, I.K., S. Pufu, B. Safdi, F-Theorem without F-Theorem without Supersymmetry,Supersymmetry, 1105.4598. 1105.4598.

• A deep problem in QFT is how to A deep problem in QFT is how to define a `good’ measure of the define a `good’ measure of the number of degrees of freedom which number of degrees of freedom which decreases along RG flows and is decreases along RG flows and is stationary at fixed points.stationary at fixed points.

• In two dimensions this problem was In two dimensions this problem was beautifully solved by Alexander beautifully solved by Alexander Zamolodchikov who, using two-point Zamolodchikov who, using two-point functions of the stress-energy tensor, functions of the stress-energy tensor, found the found the c-functionc-function which satisfies which satisfies these properties.these properties.

• At RG fixed points the c-function coincides At RG fixed points the c-function coincides with the Virasoro central charge, which is with the Virasoro central charge, which is also the Weyl anomaly. It also determines also the Weyl anomaly. It also determines the thermal free energy.the thermal free energy.

• For d>2 it also seems physically For d>2 it also seems physically reasonable to use the coefficient creasonable to use the coefficient cTT of the of the thermal free energy as the measure of the thermal free energy as the measure of the number of degrees of freedom:number of degrees of freedom:

• It can be extracted from the Euclidean It can be extracted from the Euclidean path integral on path integral on

No cNo cTT Theorem! Theorem!• However, there are counterexamples to However, there are counterexamples to

the hypothetical cthe hypothetical cTT theorem in d>2. theorem in d>2.• In d=3 Sachdev calculated the thermal In d=3 Sachdev calculated the thermal

free energy of the O(N) vector model,free energy of the O(N) vector model,

• In the critical model m=0, and In the critical model m=0, and

• A relevant pertubation of this fixed A relevant pertubation of this fixed point with makes it flow to the point with makes it flow to the Goldstone phase described in the IR by Goldstone phase described in the IR by N-1 free scalar fields. N-1 free scalar fields.

• Hence, in the IRHence, in the IR• For large enough N this exceeds the UV For large enough N this exceeds the UV

value. This means that cvalue. This means that cTT does not does not always decrease along RG flow.always decrease along RG flow.

• Another idea for generalizing the c-Another idea for generalizing the c-theorem to higher dimensions was theorem to higher dimensions was proposed by Cardy. proposed by Cardy.

The a-theoremThe a-theorem• In d=4 there are two Weyl anomaly In d=4 there are two Weyl anomaly

coefficients, and one of them, called coefficients, and one of them, called aa is is proportional to the 4-d Euler characteristic. proportional to the 4-d Euler characteristic. It can be extracted from the Euclidean part It can be extracted from the Euclidean part integral on the 4-d sphere.integral on the 4-d sphere.

• Cardy has conjectured that the Cardy has conjectured that the aa--coefficient decreases along any RG flow. coefficient decreases along any RG flow.

• No working counterexamples. A proof was No working counterexamples. A proof was recently proposed. recently proposed. Komargodski, SchwimmerKomargodski, Schwimmer

• In theories with In theories with NN=1 SUSY, the a-=1 SUSY, the a-coefficient is determined by the R-chargescoefficient is determined by the R-charges

aa = Tr = Trff 3 (3R 3 (3R33 – R)/32 – R)/32• Intriligator and Wecht proposed that the Intriligator and Wecht proposed that the

R-symmetry is determined by locally R-symmetry is determined by locally maximizing maximizing aa. This . This aa-maximization -maximization principle has passed many consistency principle has passed many consistency checks.checks.

• In large N theories dual to type IIB strings In large N theories dual to type IIB strings on the on the aa-coefficient is inversely -coefficient is inversely proportional to the volume of Yproportional to the volume of Y55. AdS and . AdS and CFT definitions of CFT definitions of aa agree. agree.

• How do we extend these successes to odd How do we extend these successes to odd dimensions where there are no dimensions where there are no anomalies? anomalies?

• This is clearly interesting, especially in This is clearly interesting, especially in d=3 where there is an abundance of d=3 where there is an abundance of conformal field theories, some of them conformal field theories, some of them describing critical points in statistical describing critical points in statistical mechanics and condensed matter physics. mechanics and condensed matter physics.

• It has been proposed that the `good’ It has been proposed that the `good’ measure of the number of DOF is the free measure of the number of DOF is the free energy on the 3-sphere energy on the 3-sphere Jafferis; Jafferis, IK, Pufu, SafdiJafferis; Jafferis, IK, Pufu, Safdi

• In field theories with extended In field theories with extended supersymmetry, the localization approach supersymmetry, the localization approach reduces the Euclidean path integral on a reduces the Euclidean path integral on a sphere to a finite dimensional integral, a sphere to a finite dimensional integral, a matrix model. matrix model. Pestun; Kapustin, Willett, Yaakov; Jafferis; …Pestun; Kapustin, Willett, Yaakov; Jafferis; …

• In d=3 theories with In d=3 theories with NN=2 SUSY the =2 SUSY the marginality of superpotential often leaves marginality of superpotential often leaves some freedom in R-symmetry. Jafferis some freedom in R-symmetry. Jafferis proposed that this freedom is fixed by proposed that this freedom is fixed by locally extremizing (in fact, maximizing) locally extremizing (in fact, maximizing) F. F.

• This is the 3-d analogue of a-This is the 3-d analogue of a-maximization. maximization.

AdS/CFT Matching of FAdS/CFT Matching of F• In large N models which have In large N models which have

duals it is possible to compare the CFT duals it is possible to compare the CFT result with the corresponding gravity result with the corresponding gravity calculation. After subtracting cubic and calculation. After subtracting cubic and linear divergences, it giveslinear divergences, it gives

• The NThe N3/23/2 scaling is a common feature of scaling is a common feature of many leading order results in AdSmany leading order results in AdS44. . IK, TseytlinIK, Tseytlin

• The field theory calculations of F via The field theory calculations of F via large N matrix models reproduce this large N matrix models reproduce this gravity results in a variety of models.gravity results in a variety of models.

• The first success was achieved for The first success was achieved for the ABJM theory which is the U(N)the ABJM theory which is the U(N)kk x x U(N)U(N)-k-k

Chern-Simons gauge theory dual to Chern-Simons gauge theory dual to AdS AdS44 x S x S77/Z/Zkk. .

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

• In the large N limit where k is kept fixed, In the large N limit where k is kept fixed, the correct ansatz isthe correct ansatz is

• Cancellation of long-range forces on Cancellation of long-range forces on eigenvalues enables us to write a local eigenvalues enables us to write a local functionalfunctional

• We find and a We find and a constantconstant eigenvalue eigenvalue density.density.

• The matrix model free energyThe matrix model free energy

agrees with the AdS formula after agrees with the AdS formula after we use vol (S we use vol (S77/Z/Zkk) = ) = 44/(3k) /(3k)

Drukker, Marino, Putrov; Herzog, IK, Pufu, TesileanuDrukker, Marino, Putrov; Herzog, IK, Pufu, Tesileanu

• Reducing supersymmetry to Reducing supersymmetry to NN=3, =3, there exists a nice set of CS gauge there exists a nice set of CS gauge theories with `necklace’ quivers for theories with `necklace’ quivers for which exact agreement has also which exact agreement has also been obtainedbeen obtained

NN=2 SUSY=2 SUSY• Now the R-charges are not fixed by Now the R-charges are not fixed by

supersymmetry. This offers nice supersymmetry. This offers nice oportunities to test the F-oportunities to test the F-maximization, F-theorem and maximization, F-theorem and AdS/CFT. AdS/CFT.

• As a function of the trial R-charges As a function of the trial R-charges the matrix model free energy is the matrix model free energy is JafferisJafferis

• For example, for the ABJM model with For example, for the ABJM model with more general R-charges more general R-charges

• the free energy is the free energy is

• Maximizing this we obtain the Maximizing this we obtain the standard R-charges ½ and standard R-charges ½ and

• If we add a relevant operator If we add a relevant operator then in the gauge with then in the gauge with

• Performing the F-maximization in the Performing the F-maximization in the IR theory we findIR theory we find

• Consistent with the F-theorem and Consistent with the F-theorem and with AdS/CFT. The conjectured gravity with AdS/CFT. The conjectured gravity dual of the IR theory is Warner’s SU(3) dual of the IR theory is Warner’s SU(3) symmetric extremum of the gauged symmetric extremum of the gauged SUGRA. SUGRA. Benna, IK, Klose, Smedback Benna, IK, Klose, Smedback

No SUSYNo SUSY• The simplest CFT’s involve free The simplest CFT’s involve free

conformal scalar and fermion fields. conformal scalar and fermion fields. Adding mass terms makes such a Adding mass terms makes such a theory flow to a theory with no theory flow to a theory with no massless degrees of freedom in the IR massless degrees of freedom in the IR where F=0. where F=0.

• For consistency with F-theorem, the F-For consistency with F-theorem, the F-coefficients for free massless fields coefficients for free massless fields should be positive.should be positive.

Conformal Scalar on SConformal Scalar on Sdd

• In any dimensionIn any dimension

• The eigenvalues and degeneracies areThe eigenvalues and degeneracies are

• Using zeta-function regularization in Using zeta-function regularization in d=3,d=3,

A massless Dirac fermionA massless Dirac fermion

• The eigenvalues and degeneracies areThe eigenvalues and degeneracies are

• Using zeta-function regularizationUsing zeta-function regularization

• For a chiral multiplet (complex For a chiral multiplet (complex scalar+fermion) F= (log 2)/2scalar+fermion) F= (log 2)/2

Slightly Relevant Slightly Relevant OperatorsOperators• Perturb a CFT by a relevant operator Perturb a CFT by a relevant operator

of dimensionof dimension

• The path integral on a sphere is The path integral on a sphere is

• The 1-pt function vanishes.The 1-pt function vanishes.

• The 2- and 3-pt function are The 2- and 3-pt function are determined by conformal invariance determined by conformal invariance in terms of the chordal distancein terms of the chordal distance

• The change in the free energy isThe change in the free energy is

• The beta function for the The beta function for the dimensionless coupling isdimensionless coupling is

• Integrating the RG equation and Integrating the RG equation and setting the scale to inverse sphere setting the scale to inverse sphere radius radius

• For C>0 there exists a robust IR fixed point For C>0 there exists a robust IR fixed point atat

• The 3-sphere free energy decreases The 3-sphere free energy decreases

• A similar calculation for d=1 provided initial A similar calculation for d=1 provided initial evidence for the G-theorem conjectured by evidence for the G-theorem conjectured by Affleck and Ludwig.Affleck and Ludwig.

• For a general odd dimension, what For a general odd dimension, what decreases along RG flow is decreases along RG flow is

Double-Trace FlowsDouble-Trace Flows• If we perturb a large N CFT by a relevant If we perturb a large N CFT by a relevant

double-trace operator, it flows to another double-trace operator, it flows to another fixed point in the IR fixed point in the IR

• If in the UV the dimension of If in the UV the dimension of is is , in the IR , in the IR it is d- it is d-

• F can be calculated using the Hubbard-F can be calculated using the Hubbard-Stratonovich trickStratonovich trick

• The change in F between IR and UV The change in F between IR and UV is of order 1 and is computable is of order 1 and is computable Gubser, Gubser, IK; Diaz, DornIK; Diaz, Dorn

• In all odd dimensionsIn all odd dimensions

• For d=3For d=3

• The change in free energy is The change in free energy is negative, in support of the F-theoremnegative, in support of the F-theorem

• The particular case The particular case =1 corresponds =1 corresponds to the critical O(N) model to the critical O(N) model

O(N) Model ReduxO(N) Model Redux• The critical O(N) model is obtained via The critical O(N) model is obtained via

a double-trace perturbation of the a double-trace perturbation of the theory of N free real scalarstheory of N free real scalars

• Using our free and double-trace resultsUsing our free and double-trace results

• A further relevant perturbation takes it A further relevant perturbation takes it to the Goldstone phase whereto the Goldstone phase where

• Recall that the flow from the critical Recall that the flow from the critical to the Goldstone phase provided a to the Goldstone phase provided a counter-example to the proposal that counter-example to the proposal that the thermal free energy decreases the thermal free energy decreases along RG flow.along RG flow.

• Yet, there is no contradiction with the Yet, there is no contradiction with the F-theorem sinceF-theorem since

CommentsComments• The `F-theorem’ has passed some The `F-theorem’ has passed some

consistency checks both via field theory and consistency checks both via field theory and using gauge/gravity duality. More should be using gauge/gravity duality. More should be done to search for counterexamples, or done to search for counterexamples, or perhaps even prove it.perhaps even prove it.

• Another recent proposal for measuring the Another recent proposal for measuring the degrees of freedom, this time in Lorentzian degrees of freedom, this time in Lorentzian signature, is the entanglement entropy of a signature, is the entanglement entropy of a disk with its complement in Rdisk with its complement in R22. . Myers, SinhaMyers, Sinha It It appears to be equivalent to F. appears to be equivalent to F. Casini, Huerta, MyersCasini, Huerta, Myers