1. Graviton Propagators in Supergravity and Noncommutative Gauge Theory
hep-th/
Phys. Rev. D75 (2007)
Satoshi Nagaoka (KEK)
with Yoshihisa Kitazawa (KEK, Sokendai)
KEK theory workshop 2007/March 14th, 2007

2. Introduction Type IIB matrix model
[Ishibashi-Kawai-Kitazawa-Tsuchiya ’96]
Nonperturbative formulation of superstring theory
Noncommutative (NC) gauge theory emerges by considering classical background in type IIB matrix model.

3. graviton propagator in supersymmetric NC gauge theory.
Introduction
Classical background in matrix model = (Higher dimensional) D-brane
Our universe may be constructed on D-brane.
Stability of the solution
How to describe graviton (propagator) in 4-dimension? …
We focus on
graviton propagator in supersymmetric NC gauge theory.

4. Introduction D=4 maximally supersymmetric NC gauge theory
Nonperturbative effect
Holographic dual description
[Hashimoto-Itzhaki ’99, Maldacena-Russo ’99, Ishibashi-Iso-Kawai-Kitazawa ’00]
Correlation functions in gauge theory
Boundary to boundary propagators in gravity theory
[Gubser-Klebanov-Polyakov ’98, Witten ’98]
Where is the screen of the holography in SUSY NC gauge theory?

5. 1. Introduction Plan of Talk
2. Supergravity description of NC gauge theory
3. Wilson line correlators in NC gauge theory
4. Boundary condition and graviton propagator
5. Summary

6. Supergravity description of NC gauge theory
Supergravity solution which is dual to 4D euclidean NC Yang-Mills:
[Hashimoto-Itzhaki ’99, Maldacena-Russo ’99]
Ordinary AdS space is obtained by taking the commutative limit

7. Supergravity description of NC gauge theory
Equation of motion for gravity mode (scalar mode) :
By changing the valuables,
we obtain Mathieu’s differential equation:

8. Supergravity description of NC gauge theory
Two independent solutions : [Gubser-Hashimoto ’98]
: regular in the region 0<r<R
: regular in the region R<r
: massless graviton
: KK modes
where
We focus on the S wave on S5 . ( mode )

9. Supergravity description of NC gauge theory
The coordinate r in AdS space corresponds to a length scale in the dual gauge theory.
Since we consider NC gauge theory, the minimum length is NC scale.
We do not need to impose the boundary condition at
Rather, a natural idea is that we impose the b. c. at
[Ishibashi-Iso-Kawai-Kitazawa ’00]

10. Supergravity description of NC gauge theory
We introduce the coordinate system which is conformally flat in the five dimensional subspace where has the maximum at

11. Supergravity description of NC gauge theory
We define Green function in the region 0<r<R.
We look for the prescription which
i) is smoothly connected with that in the ordinary AdS/CFT correspondence with respect to IR contribution.
ii) reproduces UV behavior which is seen in NC gauge theory due to UV/IR mixing effect.
Next, we will see the dual boundary NC gauge theory.

12. Wilson line correlators in NC gauge theory [Kitazawa-SN hep-th/0512204]
Leading nonanalytic behavior (UV contribution) of graviton propagator:
: a spherical harmonics on G/H homogeneous coset space
G/H Regularization of the correlators
Fuzzy G/H spaces approach flat space in large N limit.

13. Wilson line correlators in NC gauge theory
Power counting of the operator dimension does not show this type of behavior.
Indeed, the IR contribution to the correlators behaves
Operators with dimension behave

14. Graviton propagators on fuzzy G/H
leading contribution
where we define
where
We simplify

15. Graviton propagators on fuzzy G/H

16. Completeness condition
While Y depends on a particular G/H, the following relation is universal

17. for
When the momenta are equally shared as
On CP(2),
We have obtained the 1/k2 dependence.

18. Neumann boundary condition
For the Green function,
we impose the boundary condition as
Neumann b. c. at r=R
Small momentum expansion for mode gives

19. Commutative limit In the momentum region, We find
The dominant part of Green function which is well-defined in the bulk is
The nonanalytic behavior of this Green function becomes k4 log k, which is consistent with that in CFT.

20. Kaluza-Klein modes For :
In the dual NC gauge theory, the Kaluza-Klein modes receive the infrared contribution as
where is a dimension of the operator.
The operators correspond to the
KK modes.

21. Summary
We have investigated the Green function of the graviton mode in the type IIB supergravity background which is dual to NC gauge theory.
We have shown that the leading nonanalytic term which comes from UV contribution behaves as 1/k2 by imposing Neumann boundary condition on the Green function.
It may suggest that brane is located at r=R, which comes from the minimum length scale in NC gauge theory.
The nonanalytic behaviors of other Kaluza-Klein modes are not affected by the Neumann boundary condition.