Abstract
We consider the matrix regularization of fields on a Riemann surface which couple to gauge fields with a nonvanishing magnetic flux. We show that such fields are described as rectangular matrices in the matrix regularization. We construct the matrix regularization explicitly for the case of the sphere and torus based on the Berezin-Toeplitz quantization, and also discuss a possible generalization to cases with higher genera. We also discuss the matrix version of the Laplacian acting on the rectangular matrices.
Highlights
The matrix regularization plays important roles in the matrix-model formulations of M-theory or superstring theory [1,2]
The first quantized theory of a membrane or a string is mapped by the matrix regularization to the matrix model [3], which is conjectured to give a nonperturbative formulation of M-theory or superstring theory
This is very similar to the Bergman kernel on D2, which we review in Appendix L, except that the factor ð1 − zw Þ−ð1þNÞ is given as a Poincareseries
Summary
The matrix regularization plays important roles in the matrix-model formulations of M-theory or superstring theory [1,2]. For Riemann surfaces, the matrix regularization satisfying (1.1) can be constructed by using the Berezin-Toeplitz quantization [5,6,7,8], which we will review later. We will consider the matrix regularization of locally defined fields, which couple to the gauge field of the nontrivial magnetic flux This setup will be relevant for describing D-branes in terms of matrices, on which there can exist nontrivial gauge fluxes. In physical terminology, (1.2) is just the property that Uð1Þ charged fields with the same charge form a vector space and a product of a Uð1Þ charged field and a neutral field gives another charged field with the same charge In this case, the left and right multiplication gives the same operation, so the local sections give left and right modules of the algebra C∞ðMÞ.
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