Emergent metric space-time from matrix theory

Abstract

The IKKT matrix model yields an emergent space-time. We further develop these ideas and give a proposal for an emergent metric. Based on previous numerical studies of this model, we provide evidence that the emergent space-time is continuous and infinite in extent, both in space and in time, and that the metric is spatially flat. The time evolution describes the transition from a string-theoretic emergent phase to a phase in which the SO(9) symmetry of the model is spontaneously broken to SO(6) × SO(3), with three dimensions of space expanding, becoming classical and at later times evolving like in a radiation-dominated universe, and the remaining six dimensions of space stabilized at the string scale. We speculate on how this analysis can be extended to yield an early universe cosmology which, in addition to the above-mentioned properties, also leads to a roughly scale-invariant spectrum of cosmological fluctuations and gravitational waves.

Emergent gravity from the IIB matrix model and cancellation of a cosmological constant

“…The simultaneous appearance of locality and (restricted) diffeomorphism invariance needs, of course, to be studied further. The same holds for other approaches to metric-field extraction from the IIB matrix model [35][36][37].…”

“…For completeness, we mention that there have also been other approaches on getting an effective metric field from the IIB matrix model (see, e.g. [35][36][37] and references therein).…”

Towards an “AdS1/CFT0” correspondence from the D(−1)/D7 system?

“…Interestingly the IKKT matrix model [31], which has been suggested as a non-perturbative definition of IIB string theory, seems to allow for dynamical solutions in which 3 large JHEP09(2022)249 classical dimensions emerge [32]. This approach to early-universe physics captures actual non-trivial string theory effects and vastly differs from the usual 4d EFT-like approaches used to inflation [33,34]. In this regard it is fascinating that our proposal for an "AdS 1 " vacuum leads to a matrix model that is an extension of the IKKT matrix model with extra interactions that make it conformal.…”

Superstring cosmology — a complementary review

“…The approach of [10] (see [187,188] for short reviews) is to start with the BFSS model in a high temperature state and use the n = 0 modes of the matrices to extract an emergent space-time. There is a prescription to obtain an induced spatial metric [189]. In the N → ∞ limit, the induced time is continuous and runs from −∞ to +∞, the induced space infinite and continuous in the same limit, and the induced spatial metric is flat.…”

“…Thus, in the N → ∞ limit, continous time which is both past and future infinite emerges. Next, we turn to emergent space [189]. Continuing to work in the basis in which A 0 is diagonal, we make use of the fact that the off-diagonal elements of the spatial matrices decay sufficiently far from the diagonal The proposal of [191][192][193] is to define spatial submatrices Āi (t, n i ) by taking the matrix A i , running down the diagonal to a position which corresponds to the time t of the A 0 matrix, and to take a n i × n i matrix centered at this point.…”