On 27th February I will give an invited seminar in the TQFT study group at the University of Warsaw, in which I will discuss a construction of a 3-dimensional topological quantum field theory (3D-TQFT) in terms of modular functors, from which we obtain a theory of quantum invariants of 3-manifolds with embedded banded links.

Title: TQFTs and Invariants of 3-Manifolds

Abstract: In this talk we study invariants of 3-manifolds with embedded banded knots.
There are three well-known approaches to these invariants: Chern-Simmons
field theory, 2-dimensional conformal field theory, and quantum groups. We
shall pursue the latter in this talk.


Part I: We review basic notions in monoidal categories with braiding, twists
and dualities, in particular ribbon categories. Distinguishing a dominating
set of objects yields the notion of a modular category. The homsets in modular
categories have the structure of modules for a fixed ground ring K. The
dominating objects generalise the decomposition of modules into direct sum
of irreducible modules.

We give an abstract axiomatic characterisation of modular functors from the
category of 3-manifolds and homeomorphisms into the category of projective
K-modules and K-isomorphisms. 3-dimensional TQFTs extend these modular
functors by maps which assign to each 3-cobordisms a K-homomorphism. We
define non-degeneracy and anomalies at this abstract level. Quantum invariants
arise as maps that assign to 3-cobordisms values in a ground ring K, which -
in this part - we assume only to be unital and commutative.


Part II: We give a concrete construction of a 3-dimensional TQFT for
3-cobordisms with embedded ribbon graphs. The construction uses the decoration
of boundaries of 3-cobordisms with ribbon graphs that are coloured in a
modular category V with ground ring K. An extension of surfaces and gluing
by decoration gives rise to a 3-dimensional TQFT for the ground ring K. The
constructed TQFT is non-degenerate, but has anomalies by definition. We
look into these anomalies in detail in Part III.


Part III: First we introduce anomalies abstractly in terms of certain cocycles
of gluing patterns in abelian groups G. Given such a cocycle, we can tweak the
constructed TQFT by extending the cobordisms and the associated notion of
gluing by weights in G. The so obtained TQFT is anomaly-free. The TQFT
constructed in Part II yields such a 2-cocycle, that we can compute in
terms of the Maslov indices of Lagrangian subspaces of the homologies on the
boundary of decorated 3-cobordisms. The definition will solely depend on the
topological structure of the 3-cobordisms and the decorated notion of gluing.


The talk will come full circle with the theorem that isomorphism clases of
non-degenerate anomaly-free 3-dimensional TQFTs are in bijection with quantum
invariants of closed 3-cobordisms.


The talk will be largely based on the book "Quantum Invariants of Knots and
3-Manifolds" of Turaev, the series of lectures on "Integral TQFT" by Masbaum,
and more remotely on "Algebraic Topology" by Hatcher for the construction
of homologies on surfaces.