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"Topological quantum field theory", Edward Witten, 1988

Reviewed December 26, 2023

Citation: Witten, Edward. "Topological quantum field theory." Communications in Mathematical Physics 117.3 (1988): 353-386.

Web: https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-117/issue-3/Topological-quantum-field-theory/cmp/1104161738.full

Tags: Physical, Foundational, TQFT

This is the paper which introduced topological quantum field theory. It gives multiple descriptions of topological quantum field theory: in terms of philosophical ideas, in terms of Lagrangians, and in terms of Hamiltonians. The paper uses primarily the language of Floer homology. One treatment of TQFTs which is notably missing, however, is an axiomatic treatment. This had to wait until Atiyah's paper,

> Atiyah, Michael F. "Topological quantum field theory." Publications Mathématiques de l'IHÉS 68 (1988): 175-186.

A key question raised in this paper is whether or not there exist any non-trivial TQFTs. In the words of Witten, "One of the real mysteries is how to exhibit a manifestly generally covariant theory whose BRST gauge fixing (at least in some approximation) gives the 'topological quantum field theory' we have considered". The first constructions of TQFTs were those of Reshetikhin-Turaev type, as given in

> Reshetikhin, Nicolai, and Vladimir G. Turaev. "Invariants of 3-manifolds via link polynomials and quantum groups." Inventiones mathematicae 103.1 (1991): 547-597.