A Chern-Simons action for noncommutative spaces in general with the example SU_q(2)
Abstract
Witten constructed a topological quantum field theory with the Chern-Simons action as Lagrangian. We define a Chern-Simons action for 3-dimensional spectral triples. We prove gauge invariance of the Chern-Simons action, and we prove that it concurs with the classical one in the case the spectral triple comes from a 3-dimensional spin manifold. In contrast to the classical Chern-Simons action, or a noncommutative generalization of it introduced by A. H. Chamseddine, A. Connes, and M. Marcolli by use of cyclic cohomology, the formula of our definition contains a linear term which shifts the critical points of the action, i. e. the solutions of the corresponding variational problem. Additionally, we investigate and compute the action for a particular example: the quantum group SU_q(2). Two different spectral triples were constructed for SU_q(2). We investigate the Chern-Simons action, defined in the present paper, in both cases, and conclude the non-topological nature of the action. Using the Chern-Simons action as Lagrangian we define and compute the path integral, at least conceptually.
Cite
@article{arxiv.1204.0418,
title = {A Chern-Simons action for noncommutative spaces in general with the example SU_q(2)},
author = {Oliver Pfante},
journal= {arXiv preprint arXiv:1204.0418},
year = {2012}
}
Comments
35 pages. Due to publication this paper was replaced by a version which merges the previous one with another paper of me: arXiv:1204.0408; "A Chern-Simons action for noncommutative spaces". arXiv admin note: text overlap with arXiv:math/0209142, arXiv:1204.0411