Matrix regularization for Riemann surfaces with magnetic fluxes

Laplacians on Fuzzy Riemann Surfaces

“…In this section, we consider the Berezin-Toeplitz quantization of scalar fields in the presence of nontrivial background gauge fields [8,9,14,15,20] (See also [16]). After defining the quantization map, we derive the large-N asymptotic expansion for Toeplitz operators.…”

“…Now, let us consider the regularized Laplacian (3.6). A direct calculation (for example in [16,24]) shows that the embedding functions are mapped to…”

“…The N × N matrix T N (f ) for f ∈ C ∞ (M) is called the Toeplitz operator of f . The Berezin-Toeplitz quantization was further generalized in [14,15] and applied to U(1) charged scalar fields on M [16], towards understanding the fuzzy description of D-branes 2 . When M has a nontrivial magnetic flux, charged scalar fields cannot be globally defined.…”

“…In this paper, we further investigate the Berezin-Toeplitz quantization by extending the work [16]. We consider a more general setup than [16], such that the scalar fields to be regularized take values in a general representation of an arbitrary gauge group. We will show that the regularization for such fields can also be achieved by rectangular matrices.…”

Matrix regularization for tensor fields

“…The generalization discussed in this paper is based on the work [23,24] (see also [25,26]), where the quantization of (sections of) vector bundles are proposed. We consider the case that the vector bundle to be quantized is a tensor product of the tangent and the cotangent bundles.…”