2012/04/02 by Oliver Pfante, Pfante, Oliver
Mathematics · Physics and Astronomy · #46L87 #57R56 #58B34 #81T75 #Advanced Operator Algebra Research #Black Holes and Theoretical Physics #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Noncommutative and Quantum Gravity Theories #Operator Algebras (math.OA)
paper · pdf · doi:10.48550/arxiv.1204.0418
openalex publication_date 2012/04/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Witten constructed a topological quantum field theory with the Chern-Simons\naction as Lagrangian. We define a Chern-Simons action for 3-dimensional\nspectral triples. We prove gauge invariance of the Chern-Simons action, and we\nprove that it concurs with the classical one in the case the spectral triple\ncomes from a 3-dimensional spin manifold. In contrast to the classical\nChern-Simons action, or a noncommutative generalization of it introduced by A.\nH. Chamseddine, A. Connes, and M. Marcolli by use of cyclic cohomology, the\nformula of our definition contains a linear term which shifts the critical\npoints of the action, i. e. the solutions of the corresponding variational\nproblem. Additionally, we investigate and compute the action for a particular\nexample: the quantum group SUq(2). Two different spectral triples were\nconstructed for SUq(2). We investigate the Chern-Simons action, defined in the\npresent paper, in both cases, and conclude the non-topological nature of the\naction. Using the Chern-Simons action as Lagrangian we define and compute the\npath integral, at least conceptually.\n