String coupling and interactions in type IIB matrix model arXiv:0812 String coupling and interactions in type IIB matrix model arXiv:0812.3460[hep-th] Phys. Rev. D79 (2009) 106002 Satoshi Nagaoka (KEK) with Yoshihisa Kitazawa (KEK & Sokendai) YITP workshop on Development of Quantum Field Theory and String Theory /July 10 2009

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]

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.

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

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

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]

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

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]

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]

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.

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.

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.

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]

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.

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:

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.