# Low Dimensional Topology

## April 20, 2013

### The next big thing in quantum topology?

Filed under: 3-manifolds,Hyperbolic geometry,Quantum topology,Triangulations — dmoskovich @ 11:02 pm

The place to be in May for a quantum topologist is Vietnam. After some wonderful-sounding mini-courses in Hanoi, the party with move to Nha Trang (dream place to visit) for a quantum topology conference.

I’d like to tell you very briefly about some exciting developments which I expect will be at the centre of the Nha Trang conference, and which I expect may significantly effect the landscape in quantum topology. The preprint in question is $1$-Efficient triangulations and the index of a cusped hyperbolic $3$-manifold by Garoufalidis, Hodgson, Rubinstein, and Segerman (with a list of authors like that, you know it’s got to be good!).

As discussed previously on this blog here, there is an exciting new paradigm in M-theory, of a family of maximally symmetric conformal field theories is six dimensions which underlie lower dimensional supersymmetric field theories, inducing geometric descriptions for them. The archetype for such a concept is Witten’s TQFT for the Jones polynomial (a $2+1$ dimensional TQFT undelying the Jones polynomial), and Witten’s extension of it which gives a gauge theoretical description for Khovanov homology (see his`Fivebranes and knots’ paper).

The six dimensional theories are not understood mathematically, but they do lead to mathematical predictions for relationships between lower dimensional theories, that are mathematically better understood. In particular, Dimofte-Giaotto-Gukov show that a six dimensional theory implies a correspondence between an `index invariant’ of a triangulated cusped hyperbolic $3$–manifold and a gauge theoretic invariant. The gauge theory is independent of the triangulation, and so you would expect from Physics that the index invariant ought to be a topological invariant of the $3$-manifold. The reason that you should care is that the index invariant is an analytic continuation of the coloured Jones polynomial in some physics sense (which I don’t understand), yet it is intimately connected to the geometry of a cusped hyperbolic $3$–manifold, in particular to its normal surfaces. I imagine that the point is that the coloured Jones polynomial corresponds to a perturbative expansion around the trivial flat connection, whereas the index invariant is associated to the whole gauge theory (and is thus non-perturbative).

The Volume Conjecture relates the asymptotic behaviour of coloured Jones polynomials with the hyperbolic volume; the topic of the `GHRS’ preprint is to relate the index invariant with normal surface theory of hyperbolic $3$-manifolds, which in particular proves that the index invariant is a topological invariant which does not depend on the triangulation, within a class of triangulations depending only on the manifold (can this be strengthener further?). The idea would then be that it is a `more natural’ object than the coloured Jones polynomial, because it comes straight from gauge theory rather than from some perturbative expansion around some connection; and that its relationship with hyperbolic geometry would be easier to understand. I imagine that the volume conjecture might somehow follow from a parallel and easier conjecture about the index invariant, but I don’t know how.

Maybe the index invariant will be the next big thing?

What I like about this work is that it links quantum topology to normal surface theory, and to the combinatorics of triangulations of cusped hyperbolic $3$-manifolds, in a compelling and explicit way (other connections between these fields have been investigated by others, but perhaps this way is more physically justified because the invariant is somehow non-perturbative). And I expect that this will be better understood by all following the Nha Trang conference.