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

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.

New perspectives on the emergence of (3+1)D expanding space-time 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 [13] used successfully in studying the Euclidean version [16,18] is worth trying, though. We consider that the dominance of classical solutions at late times [35,36] 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.…”

“…In particular, it was applied successfully to the Euclidean version of the 6D type IIB matrix model [16], and the spontaneous breaking of the rotational SO(6) symmetry to SO(3) suggested by the Gaussian expansion method [17] was confirmed. Recently this work has been extended to the original Euclidean 10D type IIB matrix model [18]. Similarly to the 6D version, the rotational SO(10) symmetry was found to be spontaneously broken to SO(3) as suggested by the Gaussian expansion method [19].…”

“…18) and the PfM (A) in (2.10) should be replaced by PfM (e −ikπ/2 A 0 , A i ). The Lorentzian model is retrieved at (s, k) = (0, 0), whereas the Euclidean model corresponds to setting (s, k) = (1, 1).…”

Lefschetz thimbles and quantum phases in zero-dimensional bosonic models

“…The IKKT matrix model, a zerodimensional supersymmetric non-Abelian quantum field theory, serves as a promising candidate for a nonperturbative formulation of superstring theory. However, this model is shown to have a complex fermion operator [4][5][6] making the effective bosonic action complex. An investigation based on the thimble formalism would certainly turn out to be fruitful.…”

“…There are also quantum theories with complex actions that are invariant under PT symmetry [1][2][3]. In the context of string theory, the IKKT matrix model, a zero-dimensional supersymmetric quantum field theory that serves as a promising candidate for a nonperturbative formulation of superstring theory, is shown to have a complex fermion operator [4][5][6]. Investigating the nonperturbative structure of such theories using traditional path-integral Monte Carlo methods is unreliable due to the presence of the sign problem.…”