Lorentzian fuzzy spheres

Abstract

We show that fuzzy spheres are solutions of Lorentzian IKKT matrix models. The solutions serve as toy models of closed noncommutative cosmologies where big bang/crunch singularities appear only after taking the commutative limit. The commutative limit of these solutions corresponds to a sphere embedded in Minkowski space. This 'sphere' has several novel features. The induced metric does not agree with the standard metric on the sphere, and moreover, it does not have a fixed signature. The curvature computed from the induced metric is not constant, has singularities at fixed latitudes (not corresponding to the poles) and is negative. Perturbations are made about the solutions, and are shown to yield a scalar field theory on the sphere in the commutative limit. The scalar field can become tachyonic for a range of the parameters of the theory.

Cosmological space-times with resolved Big Bang in Yang-Mills matrix models

“…[12][13][14][15][16][17]. Even compact solutions were found in [18,19], where it was pointed out that the induced metric on the brane can change from Euclidean to Minkowski signature. However, this alone is not sufficient to obtain a Big Bang, and it does not imply a rapid expansion.…”

On the structure of the emergent 3D expanding space in the Lorentzian type IIB matrix model

“…Due to the expansion of space, it is expected that the dominant configurations can be well approximated at late times by some classical solution of the Lorentzian type IIB matrix model. Indeed several types of classical solutions representing expanding space-time have been constructed [14][15][16][17][18][19][20][21]. Also, matrix configurations with various structures in the extra dimensions are considered to realize chiral fermions in the (3+1)d space-time.…”

Quantum gravity as a multitrace matrix model

“…However, exhibiting explicitly non-commutative gauge theories and emergent gravity on these fuzzy spaces by expanding around the uniform order is expected to be quite difficult. Extension to Lorentzian signature seems also to be within reach [53].…”