Gromov-Witten invariants and the mod 2 spectral flow

Question

I'm reading Lee-Parker, “A structure theorem for the Gromov-Witten invariants of Kähler surfaces”, which studies Gromov-Witten invariants within symplectic geometry. Lee-Parker write (§9, p. 23) that

In Gromov-Witten theory, the GW invariant associated with a zero-dimensional space of stable maps is the signed count of the maps in that space with the sign of each map $f$ specified by the mod 2 spectral flow of the linearization $D_f$ (provided each $D_f$ is an isomorphism).

I would like to understand why this is, but they don't provide a reference, and I wasn't able to find anything discussing this relationship by searching online. I would guess the mod 2 spectral flow appears somehow in the construction of the virtual fundamental class, but I don't know enough about that construction to fill in the details.

Is there a reference that explains why this fact is true?

Answer 1

Spectral flow is the standard way a sign is associated to a point in a zero-dimensional moduli space of curves (I am not sure what you mean by VFC here, it's 0-dimensional). This involves orienting the deformation operators of the curves, i.e. coherently orienting the 0-dimensional moduli space.

I would hope every book on GW theory has to describe this: McDuff-Salamon's big book "J-holomorphic curves and symplectic topology" (2nd edition) definitely does, specifically Remark 3.2.5 and Proposition A.2.4 and the surrounding discussion in Appendix A (which is then used in chapter 7 on GW theory). Another reference is for the related Gromov invariants by Taubes, "Counting pseudo-holomorphic submanifolds in dimension 4" (specifically chapter 2).

It is then an informative exercise to show that index 0 J-holomorphic curves (in symplectic 4-manifolds) which are "automatically transverse/regular" are counted with sign +1.