Geometry from matrices via D-branes

Matsumoto

et al. 2018

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.

Section: Introductionsupporting

confidence: 53%

“…In fact, = exp(−iθe −iϕJ 3 J 2 e iϕJ 3 ) = e −iθ(cos ϕJ 2 −sin ϕJ 1 ) = e i(−θ)(− sin ϕJ 1 +cos ϕJ 2 ) , (A. 23) which says that R generates (a). By using J ± = J 1 ± iJ 2 , R is also written as R = e − 1 2 θ(e −iϕ J + −e iϕ J − ) .…”

Section: Conclusion and Discussionmentioning

confidence: 99%

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

Asakawa

Matsumoto

et al. 2018

Matrix regularization for tensor fields

Adachi

Kanno

et al. 2022

We propose a novel matrix regularization for tensor fields. In this regularization, tensor fields are described as rectangular matrices and both of area-preserving diffeomorphisms and local rotations of the orthonormal frame are realized as unitary similarity transformations of matrices in a unified way. We also show that the matrix commutator corresponds to the covariantized Poisson bracket for tensor fields in the large-N limit.

Vector bundles on fuzzy Kähler manifolds

Adachi

Kanno

2023

We propose a matrix regularization of vector bundles over a general closed Kähler manifold. This matrix regularization is given as a natural generalization of the Berezin-Toeplitz quantization and gives a map from sections of a vector bundle to matrices. We examine the asymptotic behaviors of the map in the large-N limit. For vector bundles with algebraic structure, we derive a beautiful correspondence of the algebra of sections and the algebra of corresponding matrices in the large-N limit. We give two explicit examples for monopole bundles over a complex projective space CPn and a torus T2n.