One of the foundational papers in Quantum Topology, and one of the main reasons that the subject is called *Quantum Topology*, is Edward Witten’s landmark paper Quantum field theory and the Jones polynomial. One of the things Witten did in that paper was to define a –manifold invariant as a partition function with action functional proportional to the Chern-Simons –form. A partition function is a path integral, so Witten’s invariant is a physical construction rather than a mathematical one. Quantum topology of –manifolds is, to a large extent, the field whose goal is to mathematically reconstruct, and to understand, Witten’s invariant. Meanwhile, for –manifolds with a metric, Witten defined a –manifold invariant as a partition function in another landmark paper Topological quantum field theory.

I should warn you that I don’t know any physics so some (all?) of what I say below might be rubbish. Still, pressing boldly ahead…

Up until recently, mathematicians only understood tiny corners of Witten’s invariants, or, more broadly, of invariants (topological or otherwise) of manifolds (with or without extra structure) which come from quantum field theory partition functions. But I’ve recently glanced through two papers which seem to finally be going further, seeing more. The tiny corners we have seen already give rise to mathematical invariants of preternatural power (surely that’s the best word to describe it!), such as Ohtsuki series of rational homology –spheres (), Donaldson invariants, and Seiberg–Witten invariants.

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