Emergent geometry of membranes

Abstract

In work [1], a surface embedded in flat R 3 is associated to any three hermitian matrices. We study this emergent surface when the matrices are large, by constructing coherent states corresponding to points in the emergent geometry. We find the original matrices determine not only shape of the emergent surface, but also a unique Poisson structure. We prove that commutators of matrix operators correspond to Poisson brackets. Through our construction, we can realize arbitrary noncommutative membranes: for example, we examine a round sphere with a non-spherically symmetric Poisson structure. We also give a natural construction for a noncommutative torus embedded in R 3 . Finally, we make remarks about area and find matrix equations for minimal area surfaces.

Kähler structure in the commutative limit of matrix geometry

“…See [17][18][19] for applications of coherent states in describing matrix geometries. 4 See also [21] for an application to describing various configurations of membranes and see [22] for a higher dimensional extension.…”

Commutative geometry for non-commutative D-branes by tachyon condensation

“…In summary, the tachyon condensation just picks up the zeros of the eigenfunction and as a result a defect remains on a region M. This is technically the same as the coherent state method mentioned in the introduction. In fact, the tachyon profile T (2.1) is exactly the same as the Dirac-like operator in the literature [12,13,14,17] and T 2 corresponds to the Hamiltonian in [15,16]. Our claim in this paper is that the tachyon condensation gives a new physical interpretation of this prescription, based on the dynamics of the non-BPS D-branes.…”

“…The energy of the open string can be measured by using a Dirac operator on the open strings and thus the shape of NC branes is defined as loci of zeros of the Dirac operator. See also [13,14] for analysis of this method.…”

Information metric, Berry connection, and Berezin-Toeplitz quantization for matrix geometry

“…The spectrum of (4.6) is derived in the Appendix B (See also [19,20,22]). There are three types of the eigenstates, …”

“…Since in all of these formulations, coherent states play very important roles, we call these methods collectively the coherent state methods in this paper. See [19][20][21] for some analysis using the coherent state methods. See also [22,23] for a nice interpretation of the coherent state methods in the system of non-Bogomol'nyi-Prasad-Sommerfield (BPS) D-branes.…”