The curvature of branes, currents and gravity in matrix models

Abstract

The curvature of brane solutions in Yang-Mills matrix models is expressed in terms of conserved currents associated with global symmetries of the model. This implies a relation between the Ricci tensor and the energy-momentum tensor due to the basic matrix model action, without invoking an Einstein-Hilbert term. The coupling is governed by the extrinsic curvature of the brane embedding, which arises naturally for compactified brane solutions. The effective gravity on the brane is thereby related to the compactification moduli, and protected from quantum corrections due to the relation with global symmetries.Comment: 28 pages. V2: typos fixed, published versio

2D fuzzy anti-de Sitter space from matrix models

“…Thus the normal component A ⊥ couples to the trace of the energymomentum tensor, while the tangential components couple to its derivative. This illustrates the observation [21,22] that a non-derivative coupling of the embedding perturbations to the energy-momentum tensor arises on branes with extrinsic curvature. Using the on-shell condition (4.6) for the background and…”

“…The crucial coupling to T µν arises due to the extrinsic curvature of the brane encoded in ∇ µ J a ν = K a µν , as pointed out in [21,22]; this is already seen in (5.19). Also, it is gratifying (and not evident) that the Newton constant turns out to be positive.…”

“…We study the perturbations around the AdS 2 solutions, and their dynamics in the presence of matter. This is interesting because the extrinsic curvature of the brane leads to a coupling of the linearized matrix perturbations to the energy-momentum tensor, 2 as pointed out in [21][22][23]. More precisely, the transversal perturbations of the brane couple to the trace of the energy-momentum tensor of matter, due to the extrinsic curvature.…”

“…In 4 and higher dimensions, the dynamics of the effective geometry is complicated due to a mixing between tangential and transversal brane perturbations [21,22], which prohibits a full understanding at present. A similar mixing is observed here, but we are able to disentangle the coupled wave equations, and thereby essentially solve the perturbative dynamics.…”

Kähler structure in the commutative limit of matrix geometry

“…There are also various interesting attempts to understand how the matrix models describe gravity [30][31][32][33][34]. It is important to consider whether and how our results can be applied in these contexts.…”