Universality and the dynamical space-time dimensionality in the Lorentzian type IIB matrix model

Abstract

Abstract:The type IIB matrix model is one of the most promising candidates for a nonperturbative formulation of superstring theory. In particular, its Lorentzian version was shown to exhibit an interesting real-time dynamics such as the spontaneous breaking of the 9-dimensional rotational symmetry to the 3-dimensional one. This result, however, was obtained after regularizing the original matrix integration by introducing "infrared" cutoffs on the quadratic moments of the Hermitian matrices. In this paper, we generalize the form of the cutoffs in such a way that it involves an arbitrary power (2p) of the matrices. By performing Monte Carlo simulation of a simplified model, we find that the results become independent of p and hence universal for p 1.3. For p as large as 2.0, however, we find that large-N scaling behaviors do not show up, and we cannot take a sensible large-N limit. Thus we find that there is a certain range of p in which a universal large-N limit can be taken. Within this range of p, the dynamical space-time dimensionality turns out to be (3 + 1), while for p = 2.0, where we cannot take a sensible large-N limit, we observe a (5+1)d structure.

Exponential and power-law expansion of the Universe from the type IIB matrix model

“…and obtained universal results for R(t) 2 in the cases of different p with p > 1 [6]. The results are qualitatively the same as the case of p = 1, but quantitatively different.…”

“…By performing large scale parallel computations, we reach around twenty times larger matrix size than the case of the original model. In the VDM model, we observe an exponential expansion [6], which is reminiscent of the inflation. On the other hand, in the bosonic model, we observe a power-law expansion [5], which is reminiscent of the expansion in the radiation dominant era of the Universe.…”

“…In Fig. 2(Right), we plot the extent of the space R 2 (t) normalized by R 2 (t c ) against x = (t − t c ) /R (t c ) for N = 64 ,128 and 256 [6]. We see that the data for different N scale and that the behavior of R 2 (t) for t > t c is fitted to an exponential function.…”

“…We interpretĀ i (t) as representing the structure of the space at time t. For instance, we define the extent of the space at time t as 6) and an analog of the moment of inertia tensor at time t as…”

“…2(Left), we plot the expectation values of the nine eigenvalues of T i j (t) for N = 128, κ = 8.0, n = 10 [6]. We observe the spontaneous symmetry breaking of SO (9) to SO(3) after the critical time t c .…”