# On TQFT

### abstract

It is appropriate to recall some objectives from statistical mechanics and
quantum field theory.
It is convenient to cast these in a categorical framework.
Computation in these models is hard, so one is motivated to consider
simplifications. TQFT is a massive simplification... but not quite trivial.
And relatively tractable; and mathematically interesting.
Plan:

1. Summary
1.1. axioms (from two persectives, math and physical);
1.2. applications

2. notations and context for axioms
2.1. categories
2.2. cobordism
2.2.1. manifolds
2.3 examples

### Some TQFT references

Some reading:

Kock's notes.
Collateral reading:

Turaev Viro paper (Topology, 1992)
here

arXiv:1106.6033
The full equivalence proof between string nets and the Turaev Viro model by Kirillov Jr.

arXiv:cond-mat/0506438, Appendix E, often with physical motivations for some of the axioms.

arXiv:hep-th/0401076
Freidel and Louapre
,
Ponzano-Regge model of 3D quantum gravity

arXiv:gr-qc/0410141
The same model as above. Amplitudes are proved to be given by the Reshetikhin-Turaev evaluation of a coloured chain mail links based on the D(SU(2)) quantum group.

Roberts (Topology, 1995) The proof that (square of absolute values of) amplitudes of SU(2) Chern-Simons theory are given by Turave-Viro invariants (based on SU_q(2).

Karowski and Schrader (Comm. Math. Phys. 1992)
http://link.springer.com/article/10.1007%2FBF02096773?LI=true#page-1
Here observables are also in the game.

Ambjorn, Durhuus, Jonsoon: Quantum geometry: a statistical field theory approach

Karowski et al ;

and see ...

BACK