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Holographic Renormalization Group with Gravitational Chern-Simons Term

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 ). Introduction. “C-theorem“ is one of the most interesting features of 2-dim QFT.

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Holographic Renormalization Group with Gravitational Chern-Simons Term

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  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.

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