Complex Langevin analysis of the spontaneous symmetry breaking in dimensionally reduced super Yang-Mills models

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

doi:10.1007/jhep02(2018)151

Abstract

In recent years the complex Langevin method (CLM) has proven a powerful method in studying statistical systems which suffer from the sign problem. Here we show that it can also be applied to an important problem concerning why we live in four-dimensional spacetime. Our target system is the type IIB matrix model, which is conjectured to be a nonperturbative definition of type IIB superstring theory in ten dimensions. The fermion determinant of the model becomes complex upon Euclideanization, which causes a severe sign problem in its Monte Carlo studies. It is speculated that the phase of the fermion determinant actually induces the spontaneous breaking of the SO(10) rotational symmetry, which has direct consequences on the aforementioned question. In this paper, we apply the CLM to the 6D version of the type IIB matrix model and show clear evidence that the SO(6) symmetry is broken down to SO(3). Our results are consistent with those obtained previously by the Gaussian expansion method.

Highlights

With the other is that it is computationally less costly, which enables its application to much larger system size
While the original expectation was that the SO(10) symmetry is spontaneously broken to SO(4) in order to account for the appearance of four-dimensional spacetime [25, 26], explicit calculations based on the Gaussian expansion method (GEM) suggested that it is broken down to SO(3) instead [27]
Combining this argument with our results, we conclude that the SO(6) rotational symmetry is spontaneously broken to SO(3) in the undeformed model corresponding to mf = 0, which agrees with the prediction from the GEM

Summary

Brief review of dimensionally reduced super Yang-Mills models

) in each direction has d large values and (6 − d) small values, which implies that the dynamically generated spacetime has d extended directions and (6 − d) shrunken directions This quantity λμ was calculated up to the fifth order in the GEM as well, and the result for d = 3 turned out to be. Let us mention that the free energy for the SO(2) vacuum turned out to be substantially higher than that for the vacua with higher dimensionality [33] This is consistent with the fact that d = 2 configurations are suppressed by the fermion determinant.

Applying the CLM to the 6D type IIB matrix model

Results

Summary and discussions

A Details of the complex Langevin simulation