Matrix regularization for tensor fields

Abstract

We propose a novel matrix regularization for tensor fields. In this regularization, tensor fields are described as rectangular matrices and both 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

“…In this paper, we investigate the Berezin-Toeplitz quantization of vector bundles over a general closed Kähler manifold. We show that the asymptotic properties of the Toeplitz operator given in [13,14] also exist in higher dimensional manifolds. We derive a large-p asymptotic expansion of the product T p (ϕ)T p (χ) for arbitrary sections of vector bundles (general fields) ϕ, χ, up to the second order in 1/p.…”

“…In this paper, we studied the Berezin-Toeplitz quantization of vector bundles over a general closed connected Kähler manifold, which is a continuation of our previous studies of two-dimensional cases [13,14]. In our formalism, we treated a vector bundle as a homomorphism bundle and treat its sections as some linear operator between suitable twisted spinor fields.…”

“…The Toeplitz operators for more general fields than functions were proposed in [8][9][10][11][12]. In more recent studies [13,14], it is shown that the Toeplitz operator of general fields on a closed Riemann surface enjoys beautiful properties, which are a natural generalization of (1.2).…”