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

2. The talk is based mainly on the papers 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. 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. 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. I.K., S. Pufu, B. Safdi, F-Theorem without Supersymmetry, 1105.4598.

3. 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. 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. 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.

4. 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. 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 c T of the thermal free energy as the measure of the number of degrees of freedom: For d>2 it also seems physically reasonable to use the coefficient c T 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 It can be extracted from the Euclidean path integral on

5. No c T Theorem! No c T Theorem! However, there are counterexamples to the hypothetical c T theorem in d>2. However, there are counterexamples to the hypothetical c T theorem in d>2. In d=3 Sachdev calculated the thermal free energy of the O(N) vector model, In d=3 Sachdev calculated the thermal free energy of the O(N) vector model, In the critical model m=0, and In the critical model m=0, and

6. 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. 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 Hence, in the IR For large enough N this exceeds the UV value. This means that c T does not always decrease along RG flow. For large enough N this exceeds the UV value. This means that c T does not always decrease along RG flow. Another idea for generalizing the c- theorem to higher dimensions was proposed by Cardy. Another idea for generalizing the c- theorem to higher dimensions was proposed by Cardy.

7. The a-theorem 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. 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. Cardy has conjectured that the a-coefficient decreases along any RG flow. No working counterexamples. A proof was recently proposed. Komargodski, Schwimmer No working counterexamples. A proof was recently proposed. Komargodski, Schwimmer

8. In theories with N =1 SUSY, the a- coefficient is determined by the R-charges In theories with N =1 SUSY, the a- coefficient is determined by the R-charges a = Tr f 3 (3R 3 – R)/32 a = Tr f 3 (3R 3 – 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. 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 Y 5. AdS and CFT definitions of a agree. In large N theories dual to type IIB strings on the a-coefficient is inversely proportional to the volume of Y 5. AdS and CFT definitions of a agree.

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

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

11. AdS/CFT Matching of F 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 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 N 3/2 scaling is a common feature of many leading order results in AdS 4. IK, Tseytlin The N 3/2 scaling is a common feature of many leading order results in AdS 4. IK, Tseytlin

12. The field theory calculations of F via large N matrix models reproduce this gravity results in a variety of models. 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 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 AdS 4 x S 7 /Z k. Chern-Simons gauge theory dual to AdS 4 x S 7 /Z k.

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

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

15. The matrix model free energy The matrix model free energy agrees with the AdS formula after we use vol (S 7 /Z k ) = p 4 /(3k) agrees with the AdS formula after we use vol (S 7 /Z k ) = p 4 /(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 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

16. N =2 SUSY N =2 SUSY Now the R-charges are not fixed by supersymmetry. This offers nice oportunities to test the F-maximization, F- theorem and AdS/CFT. 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 As a function of the trial R-charges the matrix model free energy is Jafferis

17. For example, for the ABJM model with more general R-charges For example, for the ABJM model with more general R-charges the free energy is the free energy is Maximizing this we obtain the standard R- charges ½ and Maximizing this we obtain the 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

18. Performing the F-maximization in the IR theory we find 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 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

19. No SUSY 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. 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 F- coefficients for free massless fields should be positive. For consistency with F-theorem, the F- coefficients for free massless fields should be positive.

20. Conformal Scalar on S d Conformal Scalar on S d In any dimension In any dimension The eigenvalues and degeneracies are The eigenvalues and degeneracies are Using zeta-function regularization in d=3, Using zeta-function regularization in d=3,

21. A massless Dirac fermion A massless Dirac fermion The eigenvalues and degeneracies are The eigenvalues and degeneracies are Using zeta-function regularization Using zeta-function regularization For a chiral multiplet (complex scalar+fermion) F= (log 2)/2 For a chiral multiplet (complex scalar+fermion) F= (log 2)/2

22. Slightly Relevant Operators Slightly Relevant Operators Perturb a CFT by a relevant operator of dimension Perturb a CFT by a relevant operator of 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.

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

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

25. For C>0 there exists a robust IR fixed point at For C>0 there exists a robust IR fixed point at The 3-sphere free energy decreases The 3-sphere free energy decreases A similar calculation for d=1 provided initial evidence for the G-theorem conjectured by Affleck and Ludwig. 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 For a general odd dimension, what decreases along RG flow is

26. Double-Trace Flows 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 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 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 F can be calculated using the Hubbard- Stratonovich trick

27. The change in F between IR and UV is of order 1 and is computable Gubser, IK; Diaz, Dorn The change in F between IR and UV is of order 1 and is computable Gubser, IK; Diaz, Dorn In all odd dimensions In all odd dimensions For d=3 For d=3

28. The change in free energy is negative, in support of the F-theorem 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 The particular case D =1 corresponds to the critical O(N) model

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

30. 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. 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 Yet, there is no contradiction with the F- theorem since

31. Comments 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. 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 R 2. Myers, Sinha It appears to be equivalent to F. Casini, Huerta, Myers 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 R 2. Myers, Sinha It appears to be equivalent to F. Casini, Huerta, Myers