Non-Semisimple Topological Quantum Field Theories for 3-Manifolds with Corners

d + 1-dimensional manifold, whose is a union of d-dimensional boundary disjoint v manifolds and d, a linear : -+ The manifold -Zod V(Md+l) V(Zod) V(Zld). ma- is with the orientation. The axiom in that z0g, Zod opposite gluing [Ati88] requires if we two such d + 1-manifolds a common d-subma- glue together along (closed) fold of in their the linear for the has to be the boundaries, composite compo- map tion of the linear of the individual d + 1-manifolds. maps the of and as in we can state categories functors, [Mac88], Using language axioms as follows: concisely Atiyah's very Definition 0.1.1 A in dimension d is a ([Ati88]). topological quantumfield theory between monoidal functor symmetric categories [Mac881 asfollows: V : --+ k-vect. Cobd+1 finite Here k-vect denotes the whose are dimensional v- category, objects for field tor over a field k, which we assume to be instance, a perfect, spaces The of of characteristic 0. set between two vector is morphisms, simply spaces the set of linear with the usual The has as composition. category Cobd+1 maps manifolds. such closed oriented d-dimensional A between two objects morphism. Zd d oriented d 1-- d-manifolds and is a + 1-cobordism, an + Zod meaning gMd+l = Zd is the d- mensional manifold, Md+l, whose Lj boundary _ZOd of the d-manifolds. consider union two we as joint (Strictly speaking morphisms cobordisms modulo relative Given another or homeomorphisms diffeomorphisms).

List of contents

and Summary of Results.- The Double Category of Framed, Relative 3-Cobordisms.- Tangle-Categories and Presentation of Cobordisms.- Isomorphism between Tangle and Cobordism Double Categories.- Monoidal categories and monoidal 2-categories.- Coends and construction of Hopf algebras.- Construction of TQFT-Double Functors.- Generalization of a modular functor.- From Quantum Field Theory to Axiomatics.- Double Categories and Double Functors.- Thick tangles.

Product details

• Authors: Thomas Kerler, Volodymyr V. Lyubashenko
• Publisher: Springer, Berlin
• Languages: English
• Product format: Paperback / Softback
• Released: 26.06.2009
• EAN: 9783540424161
• ISBN: 978-3-540-42416-1
• No. of pages: 383
• Dimensions: 157 mm x 235 mm x 22 mm
• Weight: 591 g
• Illustrations: VI, 383 p.
• Series: Lecture Notes in Mathematics; Lecture Notes in Mathematics
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Arithmetic, algebra; Algebra, B, Mathematics and Statistics, Theoretical, Mathematical and Computational Physics, Mathematical physics, Commutative algebra, Commutative rings, Commutative Rings and Algebras, Manifolds and Cell Complexes (incl. Diff.Topology), Manifolds (Mathematics), Analytic geometry, Manifolds and Cell Complexes, Complex manifolds, Analytic topology, Mathematical foundations, Category theory (Mathematics), Category Theory, Homological Algebra, Homological algebra