I'm reading [Lee-Parker, “A structure theorem for the Gromov-Witten invariants of Kähler
surfaces”](https://arxiv.org/abs/math/0610570), 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?



Gromov-Witten invariants and the mod 2 spectral flow
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.


