1. BRST quantization of string theory in pp-wave background with flux
Yoichi Chizaki,　Shigeaki Yahikozawa
Rikkyo University
Thank you Mr. chairman.
My name is Yoichi Chizaki from Rikkyo University.
I will be talking about our recent work on BRST quantization of string theory in pp-wave background with flux.
This work is collaboration with Shigeaki Yahikozawa at Rikkyo University.
This work will be published in the progress of theoretical physics, coming soon
＠ Hawaii 2006
hep-th/
Prog. Theor. Phys. Vol. 116, No.5

2. Section 1 Introduction
There are three well known methods to quantize string theory in pp-wave.
Firstly, canonical quantization of light cone gauge fixing.
Secondly, Conformal Field Theory
Thirdly, Path-integral method.
Then we construct a new quantization of string theory in pp-wave background with flux.
The key words are covariant gauge fixing and canonical quantization and BRST operator formalism.
In other words, we have applied Kato-Ogawa’s BRST formalism in the case of pp-wave background with flux.
According to the new quantization, we can see two new structures.
first, free mode representations, this means all modes are free modes,
next, we can obtain space-time dimensions and ordering constant from the nilpotency of BRST charge.

3. Section 2 Background and Action.
We treat this pp-wave background with B field flux.
Under the background, fixing the covariant gauge, the action become the simple form.
Here D-sub-plus-minus is world sheet covariant derivatives and Z and Z-bar are complex coordinates.
D-sub-plus-minus look like covariant derivatives of QED.
The ghost action is the usual form.

4. Section 3 Equations of motion and General solutions
We can check the Euler-Lagrange equations correspond to the Heisenberg equations in this system.
Firstly, X^{+} and X^{k} are free fields. We can solve easily.
Secondly, Z and Z-bar interact with X^{+} through the covariant derivatives D_{+ -} as follows.
Thirdly, X^{-} interacts with all fields except for X^{k} as follow.

5. Firstly X^{+} and X^{k} are the same as the ordinary free fields.
Secondly The general solution of Z is as follow.
And Z-bar is the Hermitian conjugate of Z.
According to the periodicity of Z, f and g become twisted fields.
Substituting the general solutions of Z and Z-bar,
the equation of X^{-} becomes very simple.
And the general solution of X^{-} is as follow.
We have to take care of the next two points
Each term becomes a periodic function of sigma by itself.
X^{-}L+X^{-}R is not a free field.

6. Section 4. Construction of free-mode representations
Firstly we expand the general solutions by modes.
Secondly imposing canonical commutation relations on all fields, we quantize the general solutions.
Thirdly we determine the commutation relations between all the modes which become free modes.
We call the mode expansions, free-mode representations.
Free-mode representations of X^{+} and X^{k} are the same as the ordinary free fields.

7. Free mode representations of Z, Z-bar and X^{-} are constructed as follows.
We impose the canonical commutation relation.
Especially the commutation relation between X^{-} and Z, strongly restrict the fields as follow.
commutation relation between X^{-} and g is the same as commutation relation between X^{-} and f.
We can construct the free mode representations by using these conditions.

8. Free mode representations of Z and Z-bar become as follows.
Here, we use the momentum representation
For free mode representations, we have to divide modes by the factor of square root.
If we use the mode A-hat which is not divided by the factor,
the commutation relation between small x^{-} and A-hat is not zero.
small x^{-} is a mode of large X^{-}

10. The free-mode representation of X^{-} is as follows.
In the general solutions of X^{-},
X^{-}_{L}+X^{-}_{R} can be divided into X^{-}_{0} and X^{-}_{1}
Here X^{-}_{0} satisfies the canonical commutation relations with X^{+} and the momentum of X^{-}
Moreover X^{-}_{1} is commutative with X^{+} and the momentum of X^{-}, and contains the modes of f and g
X^{-}_{2} is defined as this.
Substituting the free-mode representations of f and g, X^{-}_{2} becomes as follow.
Omega is defined as this, and :: is the normal ordering symbol.

11. The commutation relation between modes are as follows.
In the case of free mode representation all modes are free modes.
But the fields are not all free fields as follows.

12. Section 5. Proof of canonical commutation relation.
Canonical momentum are defined as follow.
Especially using X^{-}_{0} and f and g, the momenta P_{+} and P_{Z} become simple.
For this reason, we can completely prove the canonical commutation relations.

13. Proof of the non-trivial relation between X^{-} and X^{-}
In the case of tau equal zero, X^{-} becomes easier form as follows.
Moreover using the method of Fourier series,
we can prove the non-trivial commutation relation.

14. Section 6. VIrasoro anomaly and Nilpotency of BRST charge
In the case of free-mode representations, the Virasoro anomalies become as follows.
And the square of BRST charge is as follow.
If BRST operator has the property of nilpotency,
we can determine the space-time dimension and the ordering constant.

15. section 7. summary
We have canonically quantized string theory in pp-wave with the flux of $B_mu-nu$ by using the covariant BRST formalism.
We have constructed the free-mode representations about all covariant string coordinates.
We have determined the dimension of space-time and the ordering constant.
We want to apply the case of superstring.
That’s all, thank you.

18. The BMN correspondence is a correspondence between string theory in pp-wave and gauge theory.
But, as usual, we quantize string theory in pp-wave background
by using light-cone gauge, so that manifest covariant property is not exist.
If we can quantize string theory in pp-wave background
by using covariant gauge, there exist excitations of the light-cone directions and the ghosts.
In the gauge theory side, how should we understand the excitations.
In the other motivation,
what does the covariant quantization of string theory in background fields give us.

19. Then we construct a new quantization of string theory in pp-wave background with flux.
The key words are covariant gauge fixing and canonical quantization and BRST operator formalism.
In other words, we have applied Kato-Ogawa ‘s BRST formalism in the case of pp-wave background with flux.
According to the new quantization, we can see two new structures.
first, free mode representations, this means all modes are free modes,
next, we can obtain space-time dimensions and ordering constant from the nilpotency of BRST charge.

23. If BRST operator has the property of nilpotency

24. summary
We canonically quantize string theory in pp-wave with the flux of $B_{\mu\nu}$ by using the covariant BRST formalism.
We construct the free-mode representations about all covariant string coordinates.
We determine the dimension of space-time and the ordering constant.
We want to apply the case of superstring.
Thank you.

25. Next let us determine the background fields.
we determine pp-wave metric and B-field as follows.
Here we use space-time light cone coordinates.
In the case of pp-wave background, beta function becomes zero at all orders.

26. Moreover we use the following coordinates.
These covariant derivatives look like the covariant derivative of QED.
Using the coordinates, the action becomes the simple form.

27. Let us solve the equations of motion
First we can easily solve the equations of X^{+} and X^{k}
These general solutions are the normal d’Alembert’s solutions as follows.
Secondly we solve the equations of Z and Z-bar.
Z-bar is the Hermitian conjugate of Z.
X-tilde^{+} is defined as follows
Multiplying e^{I mu X-tilde {+}} from left side, the equation becomes solvable form.
The general solution is as follows.
According to the periodicity of Z, f and g become twisted fields.

28. Thirdly we solve the equation of X^{-}.
Substituting the general solutions of Z and Z-bar equation of X^{-} becomes the simple form.
Taking off the derivatives from left side, the equation becomes the solvable form.
The general solution is as follows.
Each term becomes a periodic function of sigma by itself.
X^{-}_{L}+X^{-}_{R} is not a free field.

29. The general solution of Z is as follows.
The twisted boundary conditions of f and g are as follows.
The free-mode representations of Z and Z-bar are as follows.
For the free-mode representations, we have to divide the modes by this.
Here mu alpha p^{+} is not integer and we use the momentum representations.
In the case of mu alpha p^{+} is integer we can also construct the free-mode representations.

30. Commutation relations between modes becomes as follows.
For the sign function in the commutation relations of A and B, normal ordering is defined as follows.
All modes are completely free modes,
But all modes are not free.
Because small x^{-} in X^{-}_{0} is not commutative with f and g.
But derivative of X^{-} is commutative with all fields.

31. The free-mode representations of X^{+} and X^{k} are as follows.
They are the mode expansions of the ordinary free fields.

32. We can obtain the canonical momentum from the action.
Using X^{-}_{0}, f, g , canonical momentum become very simple.
Especially P_{+} is the most simplified.
For this reason we can prove the canonical commutation relations.