Complex Langevin analysis of the spontaneous breaking of 10D rotational symmetry in the Euclidean IKKT matrix model

Toshiyuki Okubo,Stratos Kovalkov Papadoudis,Konstantinos N Anagnostopoulos,Jun Nishimura,Yuta Ito,Takehiro Azuma

doi:10.1007/jhep06(2020)069

Abstract

The IKKT matrix model is a promising candidate for a nonperturbative formulation of superstring theory. In this model, spacetime is conjectured to emerge dynamically from the microscopic matrix degrees of freedom in the large-N limit. Indeed in the Lorentzian version, Monte Carlo studies suggested the emergence of (3+1)-dimensional expanding spacetime. Here we study the Euclidean version instead, and investigate an alternative scenario for dynamical compactification of extra dimensions via the spontaneous symmetry breaking (SSB) of 10D rotational symmetry. We perform numerical simulations based on the complex Langevin method (CLM) in order to avoid a severe sign problem. Furthermore, in order to avoid the singular-drift problem in the CLM, we deform the model and determine the SSB pattern as we vary the deformation parameter. From these results, we conclude that the original model has an SO(3) symmetric vacuum, which is consistent with previous results obtained by the Gaussian expansion method (GEM). We also apply the GEM to the deformed matrix model and find consistency with the results obtained by the CLM.

Highlights

Introduction[42, 43] for early work.) Recently, the conditions for the correct convergence were clarified [44,45,46,47,48,49] and various new techniques have been proposed to meet these conditions for a large space of parameters [50,51,52,53,54,55]
The deformation (3.11) breaks the SO(10) symmetry down to SO(7) × SO(3), and we examine if the SO(7) symmetry is broken down to a smaller group for mf = 3.0, 1.4, 1.0, 0.9, 0.7 to consider what happens for the undeformed IKKT model (3.1) corresponding to mf = 0
This does not cause any harm, as far as we find that λ4 and λ5 do not agree in the N → ∞ and → 0 limits, which implies that the SO(7) symmetry is broken to SO(4) or lower symmetries

Summary

Introduction

[42, 43] for early work.) Recently, the conditions for the correct convergence were clarified [44,45,46,47,48,49] and various new techniques have been proposed to meet these conditions for a large space of parameters [50,51,52,53,54,55] Thanks to these developments, the CLM has been applied successfully to many systems in lattice quantum field theory [49, 56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71] and matrix models [11, 21, 49, 53, 72,73,74,75,76] with complex actions. The CLM has been applied successfully to many systems in lattice quantum field theory [49, 56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71] and matrix models [11, 21, 49, 53, 72,73,74,75,76] with complex actions. (For a recent review on the CLM and related methods, see ref. [77].)

Objectives

Results

Conclusion