Dynamical covariant quantum spacetime with fuzzy extra dimensions in the IKKT model

A bstract We consider general k = −1 FLRW covariant quantum spacetimes $$ {\mathcal{M}}^{3,1}\times \mathcal{K} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>M</mml:mi> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>×</mml:mo> <mml:mi>K</mml:mi> </mml:math> with fuzzy extra dimensions $$ \mathcal{K} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> as classical solutions of the IKKT matrix model. The coupled equations of motion are recast in terms of conservation laws, which allow to determine the evolution of spacetime in a transparent way. We show that $$ \mathcal{K} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> is stabilized as a classical solution in the presence of a large R charge, corresponding to internal angular momentum. This provides a mechanism to maintain a large hierarchy between UV and IR scales. We also argue that the evolution of spacetime is determined by a balance between classical and quantum effects, leading to a cosmic scale factor a ( t ) ~ t and constant dilaton at late times. On such a background, the undeformed IKKT model leads to a higher-spin gauge theory including gravity.