How to compute Conley-Zehnder indices on prequantization spaces?

Question

My question is pretty much as in the title. On page 100 of his thesis, Bourgeois gives a computation of the CZ(or I suppose more correctly this is the Robbin-Salamon index) index of the Reeb orbits and their multiple covers on a prequantization space.

http://tel.archives-ouvertes.fr/docs/00/04/51/90/PDF/tel-00002421.pdf

His method involves constructing some capping of class $\beta_1$ and evaluating the first Chern class on that capping. The same exact computation appears in Eliashberg-Givental and Hofer's paper "Introduction to Symplectic Field Theory"

http://arxiv.org/pdf/math/0010059v1.pdf

in section 2.9.1. The only definition I know of these indices is given very clearly for example at the end of Oancea's paper http://arxiv.org/abs/math/0403377 and seems quite different a priori from what the above authors do. The definition in Oancea's work seems difficult to compute.

The only idea I can think of is that one wants to say that given a holomorphic sphere in the symplectic base $V$ and a choice of point through which the sphere passes, one can lift it to a unique holomorphic cap of the Reeb orbit over $V$, if one chooses a specific complex structure on the total space of the symplectization $\mathbb{R} \times Y$. Then one can get this formula by comparing expected dimensions of moduli spaces. However it is not clear to me whether the complex structure on the total space is sufficiently generic to really make this a rigorous computation.

Presumably, there is a very simple relation between the two definitions since neither author group comments much. How can one link the definition that I mention to the computation that is provided?

Answer 1

It seems to me that the computation in EGH or in Bourgeois's thesis only need the axioms that Oancea lists at the top of page 19. A pair of more recent references I like for a discussion of the Conley-Zehnder index are two papers by Jean Gutt:

• http://arxiv.org/abs/1201.3728
• http://arxiv.org/abs/1307.7239

Your discussion of transversality above is a red herring -- the CZ index of an orbit only depends on the trivialization of $\xi$ over the orbit. Furthermore, the index formula (expected dimension) only features Conley-Zehnder indices (and relative Chern numbers and Euler characteristic). This means that you can compute it using an arbitrary curve, not just a transverse one.

Here is a way of seeing the construction from Bourgeois or from EGH. Consider a point in $V$. The fibre above this point is then a closed Reeb orbit. There is a natural trivialization of the contact structure over the orbit, given by the fact that $\xi$ is a horizontal distribution. Indeed, if I trivialize $T_p V$, this induces a trivialization of $\xi$ at each point of the fibre over $p$. It is clear that this frame is invariant under the action of the Reeb flow, so the linearized flow is the constant identity map. You need to decide how to perturb this (this is the data coming from the Morse function, and is why the index of the critical point shows up). This is true for any multiplicity of the fibre.

Now, take a multiple of the fibre that is null-homolous, and cap it. We want to compare the above trivialization of $\xi$ over the orbit to the trivialization that extends. By the axioms of the CZ index, the difference between the two should be $\pm$ twice the Maslov index of the associated loop of symplectic matrices. (I say $\pm$ because you have to decide which way the matrices go, etc. so you should be careful here.)

If you look at the projection of this cap, this gives you a surface $S$ in $V$. A trivialization of $\xi$ over the cap then gives you a trivialization of $TV|_{S\setminus \{ p \} }$ away from $p$. The information about this Maslov index of symplectic matrices is really measuring the failure of this trivialization to extend across the point $p$. This is given by $c_1( TV)$ paired with $[S]$ (maybe there is a factor of 2, I never remember off the top of my head).

All of this is easy to understand in the case where $V$ is a Riemann surface (since it is just a discussion of winding numbers). For higher dimensional $V$, it is easiest (for me anyway) to think of winding numbers of sections of the determinant bundle of $TV$.