Matrix regularization of embedded 4-manifolds

TL;DR

This paper introduces a matrix regularization method for embedded 4-manifolds, approximating their volume-preserving diffeomorphism algebra with tensor products of SU(N) matrices, enabling finite-dimensional representations.

Contribution

It develops a novel matrix regularization framework for 4-manifolds, extending techniques used for lower-dimensional cases and constructing explicit matrix representations.

Findings

Successfully approximates 4-volume preserving algebra with SU(N) tensor products

Constructs matrix representations for 4-sphere and 3-sphere algebras

Enables finite-dimensional analysis of embedded 4-manifolds

Abstract

We consider products of two 2-manifolds such as S^2 x S^2, embedded in Euclidean space and show that the corresponding 4-volume preserving diffeomorphism algebra can be approximated by a tensor product SU(N)xSU(N) i.e. functions on a manifold are approximated by the Kronecker product of two SU(N) matrices. A regularization of the 4-sphere is also performed by constructing N^2 x N^2 matrix representations of the 4-algebra (and as a byproduct of the 3-algebra which makes the regularization of S^3 also possible).