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

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 fourdimensional 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.

On the structure of the emergent 3D expanding space in the Lorentzian type IIB matrix model

“…As a promising approach to overcome this problem, we may use the complex Langevin method [38,39], which has attracted much attention recently [40][41][42][43][44][45][46]. It was successful also in investigating the SSB of rotational symmetry in the 6d Euclidean type IIB matrix model [7]. Preliminary results [31] for the bosonic Lorentzian model suggest that by approaching θ = π/2, one obtains clear deviations from the Pauli-matrix structure without losing the (3+1)d expanding behavior.…”

“…The model, therefore, has the potential to clarify a possible nonperturbative mechanism for dynamical compactification in superstring theory. The Euclidean version of the model was investigated from this viewpoint, and the spontaneous breaking of SO(10) rotational symmetry was suggested by various approaches [2][3][4][5][6][7]. However, latest calculations based on the Gaussian expansion method suggested that the SO(10) symmetry is broken down to SO(3) instead of SO(4) [5].…”

Complex Langevin analysis of the space-time structure in the Lorentzian type IIB matrix model

“…Unfortunately, this is not straightforward since the complex Langevin method may suffer from the singular-drift problem due to the near-zero eigenvalues of the Dirac operator. The deformation technique [12] used successfully in studying the Euclidean version [13] is worth trying, though. We consider that the dominance of classical solutions at late times [26,27] supported by our results is important because it enables us to understand possible late-time behaviors of this model by solving classical equations of motion.…”

“…Recently the complex Langevin method (CLM) [4,5] has been attracting much attention as a promising approach to this problem [6][7][8][9][10][11][12]. In particular, it has been applied successfully to the Euclidean version of the 6D type IIB matrix model [13], and the spontaneous breaking of the rotational SO(6) symmetry to SO(3) suggested by the Gaussian expansion method [14] has been confirmed.…”

Complex Langevin simulations of zero-dimensional supersymmetric quantum field theories

“…Complex Langevin dynamics has been used successfully in various models in the recent past [13][14][15][16][17][18][19][20][21][22]. There have also been studies of supersymmetric matrix models based on complex Langevin dynamics [23][24][25]. In Ref.…”