Holographic Renormalization Group with Gravitational Chern-Simons Term

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

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 • monotonicallydecreasing 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 • monotonicallydecreasing 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 • monotonicallydecreasing 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 • monotonicallydecreasing 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 • monotonicallydecreasing 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 c-functions : length scale At the fixed point, coincide with two central charges. Weyl anomaly

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

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

13. Holographic Renormalization Group

14. Holographic Renormalization Group

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

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 : auxiliary fields • Since the action contains the third derivative of , we treat • as independent dynamical variables.

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

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

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

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

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

25. Hamilton-Jacobi Equation contain and also 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 .

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 from which we can remove . Moreover, by using a classical action, we can also remove from Hamiltonian. where the classical solution is substituted into .

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

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: [Fukuma, Matsuura, Sakai] We regard as a quantum action of dual field theory, which might contain non-local terms.

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 determine weight-zero counterterms : where From terms in , we can obtain the RG equation in 2-dim: : constant

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

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

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

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

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.