In his quantum field theoretic interpretation of the Jones polynomial, E. Witten outlined a construction of a (projective) local system on the moduli space of curves and used the monodromy representation of this local system to define 'quantum' invariants of 3-manifolds called mapping tori. Amongst all mapping tori, pseudo-Anosov mapping tori are the only ones which are hyperbolic and it is expected that their quantum invariant is related to their hyperbolic volume. We will show how one can use Teichmueller curves to compute quantum invariants of pseudo-Anosov mapping tori.
For concreteness, we will recall the construction of the local system, which comes from geometric quantization of the moduli space of semi stable vector bundles or a 2-dimensional conformal field theory, in the case of genus two curves where it can be explicitly written down. We will compute the monodromy of this local system restricted to a particular Teichmueller curve and talk about the pseudo-Anosov mapping tori and their quantum invariants coming from this example.