Abstract:We study the fuzzy hyperboloids AdS 2 and dS 2 as brane solutions in matrix models. The unitary representations of SO(2, 1) required for quantum field theory are identified, and explicit formulae for their realization in terms of fuzzy wavefunctions are given. In a second part, we study the (A)dS 2 brane geometry and its dynamics, as governed by a suitable matrix model. In particular, we show that trace of the energy-momentum tensor of matter induces transversal perturbations of the brane and of the Ricci scalar. This leads to a linearized form of Henneaux-Teitelboim-type gravity, illustrating the mechanism of emergent gravity in matrix models.

The Einstein-Hilbert action in three dimensions and the transformation rules for the dreibein and spin connection can be naturally described in terms of gauge theory. In this spirit, we use covariant coordinates in noncommutative gauge theory in order to describe 3D gravity in the framework of noncommutative geometry. We consider 3D noncommutative spaces based on SU(2) and SU(1,1), as foliations of fuzzy 2-spheres and fuzzy 2-hyperboloids respectively. Then we construct a U(2) × U(2) and a GL(2,C) gauge theory on them, identifying the corresponding noncommutative vielbein and spin connection. We determine the transformations of the fields and an action in terms of a matrix model and discuss its relation to 3D gravity.1 This is also true in any dimension for what concerns the transformation rules of the fields. 2 We employ the standard convention that antisymmetrizations are taken with weight 1.

We analyse a specific, duality-based generalization of the hermitean matrix model. The existence of two collective fields enables us to describe specific excitations of the hermitean matrix model. By using these two fields, we construct topologically nontrivial solutions (BPS solitons) of the model. We find the low-energy spectrum of quantum fluctuations around the uniform solution. Furthermore, we construct the wave functional of the ground state and obtain the corresponding Green function.

We present a method for solving BPS equations obtained in the collective-field approach to matrix models. The method enables us to find BPS solutions and quantum excitations around these solutions in the one-matrix model, and in general for the Calogero model. These semiclassical solutions correspond to giant gravitons described by matrix models obtained in the framework of AdS/CFT correspondence. The two-field model, associated with two types of giant gravitons, is investigated. In this duality-based matrix model we find the finite form of the n-soliton solution. The singular limit of this solution is examined and a realization of open-closed string duality is proposed.

We discuss a dynamical matrix model by which probability distribution is associated with Gaussian ensembles from random matrix theory. We interpret the matrix M as a Hamiltonian representing interaction of a bosonic system with a single fermion. We show that a system of secondquantized fermions influences the ground state of the whole system by producing a gap between the highest occupied eigenvalue and the lowest unoccupied eigenvalue.