1. String coupling and interactions in type IIB matrix model arXiv:0812
String coupling and interactions in type IIB matrix model arXiv: [hep-th] Phys. Rev. D79 (2009)
Satoshi Nagaoka (KEK)
with Yoshihisa Kitazawa (KEK & Sokendai)
YITP workshop on
Development of Quantum Field Theory and String Theory
/July

2. Introduction String coupling constant in IIB matrix model
[Ishibashi-Kawai-Kitazawa-Tsuchiya ’96]
Two dimensional noncommutative solutions are obtained in IIB matrix model.
Strings appear in these solutions in the IR limit.
We read the coupling constant from the interactions between strings.
cf) twist field in matrix string theory [Dijkgraaf-Verlinde-Verlinde ’97]

3. Introduction Gauge/string duality
p<3 cases [Itzhaki-Maldacena-Sonnenschein-Yankielowicz `98]
Supergravity solution
The dilaton depends on U, which represents the running of the effective coupling constant.
The solution has large curvature for large U.

4. 2 dim super Yang-Mills[Itzhaki-Maldacena-Sonnenschein-Yankielowicz ’98]
D1 brane solution:
S-duality
F1 solution:

5. Plan of Talk 1. Introduction
2. The action of multiple strings in IIB matrix model
3. String coupling and the recombination
4. Summary

6. The action of superstrings in IIB matrix model
Expanding this action around 2-dim background: 2D N=8 U(n) noncommutative Yang-Mills theory [Aoki-Ishibashi-Iso-Kawai-Kitazawa-Tada ’99]

7. The action of multiple strings
By taking the IR limit, the NC Yang-Mills becomes commutative Yang-Mills
Mapping the coordinate system from R2 into R1×S1:
We identify
w : winding numbers along σ direction
(~ light-cone momenta in a T-dual formul.)
Multiple strings are obtained in general
n=∑a wa

8. The action of multiple strings
By the field redefinition:
and the rescaling:
we obtain the following action from the commutative super Yang-Mills.
cf) Diagonal part reduces to the Green-Schwarz light-cone superstring action by the identification [Kitazawa-SN ’07]

9. Scaling behavior and the effective action
The bosonic part of the action is
On the analogy of matrix string theory, gs will behave
We interpret this relation as representing the equivalence between the IR limit and the weak coupling limit.
cf) Matrix string theory [Dijkgraaf-Verlinde-Verlinde ’97]

10. Scaling behavior and the effective action
Supergravity solution of fundamental strings :
We can read the scaling behavior of x as
In the IR limit , string coupling vanishes.
For simplicity, we fix the scale z as |z|=zr=1. Thus, we map the coordinates by the analytic continuation
The action becomes
where we consider SU(2) since recombination is a local process which involves two strings.

11. String recombination The solution is locally described as
φ： a relative angle between two intersecting strings
For a generic value of zr ,
By taking the suitable rescaling,
we can absorb this factor.

12. Fluctuation analysis around the solution
We turn on the off-diagonal part of the fluctuations
Eigenfunctions of lowest modes are described as gaussian. [Hashimoto-SN ’03]
C(t) satisfies the equation
By diagonalizing the scalar field,
we can read the location of the strings and confirm that recombination happens.

13. Probability of the recombination
Substituting into the effective action, we obtain the action
This action can be regarded as a quantum mechanics of a particle moving in the inverse harmonic oscillator.
[Hanany-Hashimoto ’05]

14. Probability of the recombination
In the inverse harmonic potential, the parameter can be interpreted as
Mass :
Frequency :
Schrödinger equation is given by
For large t, the wave function behaves [Guth-Pi ’85]
where φ labels the initial condition.

15. Probability of the recombination
The recombination probability at a time t is estimated as
The probability per unit time is
In the small q limit, it is proportional to
This should be proportional to gs2 from perturbative string.
Thus we identify
At a large time , in the small q limit,
higher order corrections:

16. Summary
In the IR limit, 2-dim noncommutative solution in IIB matrix model reduces to Green-Schwarz superstring action.
The action to describe the recombination is obtained as SU(2) gauge theory.
From the fluctuation analysis, we have estimated the probability of the recombination and by comparing that obtained in the perturbative string theory, we have obtained the result q ∝ gs2 .
The scaling behavior of the worldsheet plays a crucial role.