1. Holographic Renormalization Group with Gravitational Chern-Simons Term Takahiro Nishinaka ( Osaka U.) (Collaborators: K. Hotta, Y. Hyakutake, T. Kubota and H. Tanida ) ( arXiv: 0906.1255 [hep-th] )

2. Introduction  “C-theorem“ is one of the most interesting features of 2-dim QFT. c- function : # degrees of freedom

3. Introduction  “C-theorem“ is one of the most interesting features of 2-dim QFT. c- function : # degrees of freedom monotonically decreasing along the renormalization group flow

4. Introduction  “C-theorem“ is one of the most interesting features of 2-dim QFT. c- function : # degrees of freedom monotonically decreasing along the renormalization group flow  By virtue of holography, we can analyze this from 3-dim gravity. pure gravity + scalar

5. Introduction  “C-theorem“ is one of the most interesting features of 2-dim QFT. c- function : # degrees of freedom monotonically decreasing along the renormalization group flow  By virtue of holography, we can analyze this from 3-dim gravity. pure gravity + scalar Weyl anomaly calculation from gravity

6. Introduction  “C-theorem“ is one of the most interesting features of 2-dim QFT. c- function : # degrees of freedom monotonically decreasing along the renormalization group flow  By virtue of holography, we can analyze this from 3-dim gravity. pure gravity + scalar Weyl anomaly calculation from gravity  C-theorem is, however, known to be satisfied even when. Now is constant along the renormalization group.

7. Introduction  “C-theorem“ is one of the most interesting features of 2-dim QFT. c- function : # degrees of freedom monotonically decreasing along the renormalization group flow  By virtue of holography, we can analyze this from 3-dim gravity. pure gravity + scalar Weyl anomaly calculation from gravity  C-theorem is, however, known to be satisfied even when. Now is constant along the renormalization group. As a dual gravity set-up, we consider Topologically Massive Gravity (TMG) + scalar

8. Parity-Violating 2-dim QFT c-functions : length scale At the fixed point, coincide with two central charges.

9. Parity-Violating 2-dim QFT c-functions : length scale At the fixed point, coincide with two central charges.

10. Parity-Violating 2-dim QFT Weyl anomaly c-functions : length scale At the fixed point, coincide with two central charges.

11. Parity-Violating 2-dim QFT Weyl anomaly Gravitational anomaly c-functions : length scale At the fixed point, coincide with two central charges.

12. Parity-Violating 2-dim QFT Weyl anomaly Gravitational anomaly Bardeen-Zumino polynomial (making energy-momentum tensor covariant) c-functions : length scale At the fixed point, coincide with two central charges.

13. Holographic Renormalization Group

15. This is a dual description of the RG-flow of 2-dimensional QFT. UV IR Holographic Renormalization Group

16. TMG + Scalar scalar gravitational Chern-Simons term

17. TMG + Scalar scalar gravitational Chern-Simons term ADM decomposition We here decompose metric into the radial direction and 2-dim spacetime.

18. TMG + Scalar : auxiliary fields

19. TMG + Scalar  Since the action contains the third derivative of, we treat as independent dynamical variables. : auxiliary fields

20. TMG + Scalar  Since the action contains the third derivative of, we treat as independent dynamical variables. : auxiliary fields

21. TMG + Scalar : auxiliary fields  Since the action contains the third derivative of, we treat as independent dynamical variables.

22. TMG + Scalar  Since the action contains the third derivative of, we treat as independent dynamical variables. Momenta conjugate to them are : auxiliary fields

23. Hamilton-Jacobi Equation Hamiltonian is given by constraints: contain and also

24. Hamilton-Jacobi Equation Hamiltonian is given by constraints: Constraints from path integration over auxiliary fields are contain and also

25. Hamilton-Jacobi Equation Hamiltonian is given by constraints: Constraints from path integration over auxiliary fields are In order to see the physical meanings of these constraints, we have to express only in terms of the boundary conditions. contain and also

26. Hamilton-Jacobi Equation First, path integration over leads to from which we can remove.

27. Hamilton-Jacobi Equation First, path integration over leads to Moreover, by using a classical action, we can also remove from Hamiltonian. where the classical solution is substituted into. from which we can remove.

28. Hamilton-Jacobi Equation First, path integration over leads to Moreover, by using a classical action, we can also remove from Hamiltonian. where the classical solution is substituted into. Then are from which we can remove.

29. Holographic Renormalization The bulk action is a functional of boundary conditions.

30. Holographic Renormalization The bulk action is a functional of boundary conditions. We divide according to weight. includes only terms with weight.

31. Holographic Renormalization The bulk action is a functional of boundary conditions. We divide according to weight. includes only terms with weight. The weight is assigned as follows: [Fukuma, Matsuura, Sakai]

32. Holographic Renormalization The bulk action is a functional of boundary conditions. We divide according to weight. includes only terms with weight. The weight is assigned as follows: We regard as a quantum action of dual field theory, which might contain non-local terms. [Fukuma, Matsuura, Sakai]

33. We now study the physical meanings of, or by comparing weights of both sides.

34. Hamiltonian Constraint and Weyl Anomaly From terms in, we can determine weight-zero counterterms : where

35. Hamiltonian Constraint and Weyl Anomaly From terms in, we can obtain the RG equation in 2-dim: : constant From terms in, we can determine weight-zero counterterms : where

36. Hamiltonian Constraint and Weyl Anomaly From terms in, we can obtain the RG equation in 2-dim: And we can also read off the Weyl anomaly in the 2-dim QFT: : constant From terms in, we can determine weight-zero counterterms : where

37. Hamiltonian Constraint and Weyl Anomaly From terms in, we can obtain the RG equation in 2-dim: And we can also read off the Weyl anomaly in the 2-dim QFT: : constant cf.) In 2-dim, From terms in, we can determine weight-zero counterterms : where

38. Momentum Constraint and Gravitational Anomaly From weight three terms of the second constraint, we can read off the gravitational anomaly in the 2-dim QFT.

39. Momentum Constraint and Gravitational Anomaly From weight three terms of the second constraint, we can read off the gravitational anomaly in the 2-dim QFT. In pure gravity case, the RHS is zero which means energy-momentum conservation.

40. Momentum Constraint and Gravitational Anomaly From weight three terms of the second constraint, we can read off the gravitational anomaly in the 2-dim QFT. cf.) In 2-dim, In pure gravity case, the RHS is zero which means energy-momentum conservation.

41. Momentum Constraint and Gravitational Anomaly From weight three terms of the second constraint, we can read off the gravitational anomaly in the 2-dim QFT. cf.) In 2-dim, Bardeen-Zumino term: non-covariant terms which make energy-momentum tensor general covariant. In pure gravity case, the RHS is zero which means energy-momentum conservation.

42. Holographic c-functions We can define left-right asymmetric c-functions as follows: where depends on the radial coordinate and is constant along the renormalization group flow !!

43. Summary  We study Topologically Massive Gravity (TMG) + scalar system in 3 dimensions as a dual description of the RG-flow of 2-dimensional QFT.  Due to the gravitational Chern-Simons coupling, We can obtain left-right asymmetric c-functions holographically.  is constant along the renormalization group flow, which is consistent with the property of 2-dim QFT.  The Bardeen-Zumino polynomial is also seen in gravity side.

44. That‘s all for my presentation. Thank you very much.