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

Abstract

There is a difficulty in defining the positions of the D-branes when the scalar fields on them are non-abelian. We show that we can use tachyon condensation to determine the position or the shape of D0-branes uniquely as a commutative region in spacetime together with non-trivial gauge flux on it, even if the scalar fields are non-abelian. We use the idea of the so-called coherent state method developed in the field of matrix models in the context of the tachyon condensation. We investigate configurations of noncommutative D2-brane made out of D0-branes as examples. In particular, we examine a Moyal plane and a fuzzy sphere in detail, and show that whose shapes are commutative R 2 and S 2 , respectively, equipped with uniform magnetic flux on them. We study the physical meaning of this commutative geometry made out of matrices, and propose an interpretation in terms of K-homology.

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

“…We next introduce another method based on a matrix Dirac-type operator [18,22,23]. While the method using Hamiltonian is based on the analogy with the quantum mechanics, the method using Dirac operator is based on some observations in string theories.…”

“…The second interpretation is provided by flat non-BPS D-brane systems in superstring theory [22,23] (see also [24,25]). The theory on the non-BPS D-branes generally contains the tachyon field TðyÞ, and the potential term of TðyÞ in the low energy action is proportional to the exponential factor e −TðyÞ 2 .…”

“…In the setup with D-branes studied in [18,22,23], the Berry connection is understood as the gauge field on the D-branes, as first noted in [24,26]. For Kähler manifolds, the information metric is the compatible world volume metric on the D-branes.…”

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

“…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.…”