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Journal of the London Mathematical Society 1998 58(1):208-228; doi:10.1112/S0024610798006383
© 1998 by London Mathematical Society
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© The London Mathematical Society

S-Structures for k-Linear Categories and the Definition of a Modular Functor

Ulrike Tillmann

Mathematical Institute, University of Oxford 24–29 St Giles Street, Oxford OX1 3LB. E-mail: tillmann{at}maths.ox.ac.uk

Received 15 August 1995. Revision received 15 March 1996.

Ideas from string theory and quantum field theory have been the motivation for new invariants of knots and 3-dimensional manifolds which have been constructed from complex algebraic structures such as Hopf algebras [17, 22], monoidal categories with additional structure [24], and modular functors [14, 23]. These constructions are closely related. Here we take a unifying categorical approach based on a natural 2-dimensional generalisation of a topological field theory in the sense of Atiyah [1], and show that the axioms defining these complex algebraic structures are a consequence of the underlying geometry of surfaces.


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