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IIB Matrix Model

H. Aoki, S. Iso, H. Kawai, Y. Kitazawa, T. Tada, A. Tsuchiya

TL;DR

This work presents the IIB matrix model as a covariant, nonperturbative framework aiming to define type IIB superstring theory from a zero-dimensional, large-$N$ matrix model. By deriving loop (Schwinger–Dyson) equations for Wilson loops and analyzing their continuum, light-cone limit, the authors show how the Green–Schwarz type IIB string emerges in a double-scaling regime with $\alpha'^2 \sim {g^2 I\over \epsilon}$ and $g_{\rm st} \sim {1\over I\epsilon}$, with ${\cal N}=2$ supersymmetry fixing the structure of the free and interacting sectors. The dynamics of eigenvalues generates space-time from matrix degrees of freedom, with one-loop effective actions producing a connected network that can realize a four-dimensional space-time; low-energy physics then exhibits local gauge invariance and diffeomorphism invariance arising from permutation symmetry and block structure. The framework connects matrix dynamics to string perturbation theory and suggests a pathway to emergent gravity via a permutation-invariant, dynamically generated random lattice, while discussing extensions to backgrounds like AdS$_5\times S^5$ and topological/commercial issues. Overall, the paper provides a concrete mechanism for emergent space-time, gauge symmetries, and gravity within a single matrix model, bridging nonperturbative definitions to perturbative string theory.

Abstract

We review our proposal for a constructive definition of superstring, type IIB matrix model. The IIB matrix model is a manifestly covariant model for space-time and matter which possesses N=2 supersymmetry in ten dimensions. We refine our arguments to reproduce string perturbation theory based on the loop equations. We emphasize that the space-time is dynamically determined from the eigenvalue distributions of the matrices. We also explain how matter, gauge fields and gravitation appear as fluctuations around dynamically determined space-time.

IIB Matrix Model

TL;DR

This work presents the IIB matrix model as a covariant, nonperturbative framework aiming to define type IIB superstring theory from a zero-dimensional, large- matrix model. By deriving loop (Schwinger–Dyson) equations for Wilson loops and analyzing their continuum, light-cone limit, the authors show how the Green–Schwarz type IIB string emerges in a double-scaling regime with and , with supersymmetry fixing the structure of the free and interacting sectors. The dynamics of eigenvalues generates space-time from matrix degrees of freedom, with one-loop effective actions producing a connected network that can realize a four-dimensional space-time; low-energy physics then exhibits local gauge invariance and diffeomorphism invariance arising from permutation symmetry and block structure. The framework connects matrix dynamics to string perturbation theory and suggests a pathway to emergent gravity via a permutation-invariant, dynamically generated random lattice, while discussing extensions to backgrounds like AdS and topological/commercial issues. Overall, the paper provides a concrete mechanism for emergent space-time, gauge symmetries, and gravity within a single matrix model, bridging nonperturbative definitions to perturbative string theory.

Abstract

We review our proposal for a constructive definition of superstring, type IIB matrix model. The IIB matrix model is a manifestly covariant model for space-time and matter which possesses N=2 supersymmetry in ten dimensions. We refine our arguments to reproduce string perturbation theory based on the loop equations. We emphasize that the space-time is dynamically determined from the eigenvalue distributions of the matrices. We also explain how matter, gauge fields and gravitation appear as fluctuations around dynamically determined space-time.

Paper Structure

This paper contains 15 sections, 124 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Loop equations and scaling limit
  3. Wilson loops and light-cone setting
  4. Loop equations and loop space
  5. Light-cone Hamiltonian
  6. General proof
  7. Power counting and parity symmetry
  8. $\mathcal{N}$=2 supersymmetry
  9. Free parts of supercharges and Hamiltonian
  10. Interaction parts of supercharges and Hamiltonian
  11. Dynamics of eigenvalues and space-time generation
  12. Symmetries in the low energy theory
  13. Local gauge invariance
  14. Diffeomorphism invariance
  15. Discussion

Figures (2)

  • Figure 1: The light-cone setting. We consider the configurations of the Wilson loops which possess the identical light-cone time $x^{+}$. We also locate a group of the Wilson loops at $x^{+}=-\infty$ which represent a particular initial state. We put $k^{+}(\sigma)=1$ for all the Wison loops by using the reparametrization invariance.
  • Figure 2: Result of the numerical simulation, where $N=800$ and the number of 8-fold bond is fixed to be 140. The horizontal axis represents the number of iterations, the vertical axis the length of the ten principal axes in ten dimensions. Two of them are relatively large, suggesting the existence of anisotropy in space-time.