Systematic study of the SO(10) symmetry breaking vacua in the matrix model for type IIB superstrings

Abstract

We study the properties of the space-time that emerges dynamically from the matrix model for type IIB superstrings in ten dimensions. We calculate the free energy and the extent of space-time using the Gaussian expansion method up to the third order. Unlike previous works, we study the SO(d) symmetric vacua with all possible values of d within the range 2 ≤ d ≤ 7, and observe clear indication of plateaus in the parameter space of the Gaussian action, which is crucial for the results to be reliable. The obtained results indeed exhibit systematic dependence on d, which turns out to be surprisingly similar to what was observed recently in an analogous work on the six-dimensional version of the model. In particular, we find the following properties: i) the extent in the shrunken directions is given by a constant, which does not depend on d; ii) the ten-dimensional volume of the Euclidean space-time is given by a constant, which does not depend on d except for d = 2;iii) The free energy takes the minimum value at d = 3. Intuitive understanding of these results is given by using the low-energy effective theory and some Monte Carlo results.

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

“…Moreover, the extent of space-time in the extended d directions and that in the shrunken (10 − d) directions turn out to have a finite ratio even in the large-N limit. 4 While these results reveal interesting dynamical properties of the Euclidean model, the connection to our real space-time is not very clear. This motivated us to consider the Lorentzian model.…”

“…Therefore, it is possible that a four-dimensional space-time is generated dynamically in this model. This issue has been studied in the Euclidean version of the model, 3,4,[9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] which can be obtained from (1) by making a Wick rotation…”

“…3 was used to calculate the free energy of the SO(d) symmetric vacua (2 ≤ d ≤ 7) and it was found that d = 3 gives the minimum. 4 This implies that the SO(10) symmetry of the Euclidean model is spontaneously broken down to SO (3). Moreover, the extent of space-time in the extended d directions and that in the shrunken (10 − d) directions turn out to have a finite ratio even in the large-N limit.…”

Recent developments in the type IIB matrix model

“…First in the Euclidean model, it was found that SSB indeed occurs, but the SO(10) symmetry is broken down to SO(3) [9]. The extent of space-time in the extended direction and that in the shrunken direction are calculated, and their ratio is found to be around five in the large-N limit.…”

“…1 we show the most recent results [9] based on the Gaussian expansion method [18] in the large-N limit. On the left we plot the free energy for the SO(d) symmetric vacuum.…”