Spontaneous Breaking of the Rotational Symmetry in Dimensionally Reduced Super Yang-Mills Models

Abstract

We investigate the spontaneous breaking of the SO(D) symmetry in matrix models, which can be obtained by the zero-volume limit of pure SU(N ) super Yang-Mills theory in D = 6, 10 dimensions. The D = 10 case corresponds to the IIB matrix model, which was proposed as a non-perturbative formulation of type IIB superstring theory, and the spontaneous breaking corresponds to the dynamical compactification of space-time suggested in that model. First we study the D = 6 case by the Gaussian expansion method, which turns out to yield clearer results than the previous results for the D = 10 case for certain technical reasons. By comparing the free energy of the SO(d) symmetric vacua for d = 2, 3, 4, 5, we conclude that the breaking SO(6) → SO(3) actually occurs. We find that the extent of space-time in the shrunken directions is almost independent of d. In units of this universal scale, the extended directions seem to have large but still finite extents depending on d. We show that these results for the extent of space-time can be explained quantitatively by an argument based on the low-energy effective theory. With these new insights, we reconsider the previous results for the IIB matrix model, and find that they are also consistent with our argument based on the low-energy effective theory. Thus we arrive at comprehensive understanding and some quantitative predictions concerning the nature of the spontaneous symmetry breaking taking place in these models. The space-time picture that emerges from the IIB matrix model and its implication on possible interpretations of the model are also discussed.Subject Index: 125 §1. Introduction It has long been considered that matrix models provide a non-perturbative formulation of string theories analogous to lattice gauge theory for QCD. After the revolution triggered by the discovery of D-branes, there appeared concrete proposals for critical superstring theories and M theory, 1)-3) which take the form of large-N reduced models. For instance, the IIB matrix model, 2) which is proposed as a non-perturbative formulation of type IIB superstring theory, can be obtained by the zero-volume limit of pure SU (N ) super Yang-Mills theory in 10 dimensions.As a possible non-perturbative phenomenon analogous to quark confinement in * )

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

“…In this paper, we extend this work to the 6D version of the type IIB matrix model, which can be obtained by dimensionally reducing 6D super Yang-Mills theory to a point in the same way as one obtains the type IIB matrix model by dimensionally reducing 10D super Yang-Mills theory. The 6D version also suffers from the sign problem due to a complex determinant, which appears after integrating out the fermionic matrices, and its phase is speculated to cause the SSB of SO(6) rotational symmetry to SO(3) according to the GEM [34]. We show that the CLM indeed enables us to address this issue from first principles with the aid of the deformation technique, and our results turn out to be consistent with those of the GEM.…”

Recent developments in the type IIB matrix model

“…From the intersecting point, we can obtain the extent of spacetime in the shrunken directions. We find that it is given by x ∼ 0.35, which should be compared with the value obtained by the Gaussian expansion method [19] …”

“…The Gaussian expansion method has been applied also to a 6d version of the type IIB matrix model, which can be obtained by dimensionally reducing 6d super Yang-Mills (SYM) theory to a point [19], and it was found that the SO(6) symmetry is spontaneously broken down to SO(3). The mechanism of SSB is demonstrated by Monte Carlo studies in this case [20].…”

(3+1)-Dimensional Expanding Universe From a Lorentzian Matrix Model for Superstring Theory in (9+1)-Dimensions

“…25 There the value of the free energy tends to decrease slightly as one goes from order 3 to order 5. Considering such artifacts due to truncation, we speculate that the Krauth-Nicolai-Staudacher conjecture 24 actually refers to the partition function for the SO(10) symmetric vacuum (see Fig.…”