I heard from a recent talk that the "number of $n$-degree curves" in a Calabi-Yau manifold is an invariant of the space. But what does that mean? (Specifically I would like to ask the following.)
Does the sequence of "number of $n$-degree curves in a CY-manifold has a name? (i.e. the sequence for which Physics wins over Mathematics in calculating $n_3$.)
So does it make sense to ask, for example, 'What's the number of quadratic rational curves on a torus?'
Is this sequence related to something more familiar, like homology groups, Betti number, torsion, Euler characteristic (not its weighted version), etc.?
Thank you very much for your attention.
The theory of curve counting in CY manifolds is well established in dimension $3$, and it goes under the name of Donaldson-Thomas theory.
Example. The general quintic threefold (which is a CY) is expected to contain a finite number $N_d$ of degree $d$ rational curves, for any $d$. The history of this sequence $\{N_d\}$ is fascinating. The number of lines, $2875$, was known in the $19$th century. The number of conics, $609250$, was found by Katz, while the number of twisted cubics, $317206375$, was found by Ellingsrud and Strømme (and did agree with a conjectured formula for the $N_d$'s coming from the world of String Theory; such a formula turned out to be correct later in the $1990$'s). I do not know, however, whether the above sequence has a name.
Remark. The number $2875$ (say) is not an invariant of the quintic threefold $Q\subset \mathbb P^4$. Indeed, any quintic with the "correct" number of lines deforms to the Fermat quintic $$x_0^5+x_1^5+x_2^5+x_3^5+x_4^5=0,$$ which contains $50$ one-dimensional families of lines (this was proved Albano-Katz). There are other examples of quintic threefolds with infinitely many lines.
The idea of DT theory is to count curves "virtually", so that the count will always be a finite number, even when an actual count in a special situation would give the answer $\infty$.
For a projective CY$3$ $X$, one can look at one-dimensional subschemes $Z\subset X$ such that $\chi(\mathscr O_Z)=n$ (view this as "fixing the genus") and the homology class of the pure $Z$ (i.e. the class of $Z$ with embedded and isolated points removed) is some $\beta\in H_2(X,\mathbb Z)$. These $(n,\beta)$ are the discrete invariants one has to fix in order to get a "moduli space of degree $\beta$ curves in $X$". A standard notation for such moduli space is $I_n(X,\beta)$.
The virtual dimension of this moduli space is $-K_X\cdot\beta=0$, thanks to the CY condition, and the virtual count of its points, which is the Donaldson-Thomas invariant of $I_n(X,\beta)$, is an integer $$N_{n,\beta}=DT(I_n(X,\beta))\in\mathbb Z\,\,\,\,\,\,\,\,\,\,\,\,\,(\star)$$ which is invariant under deformations of the CY$3$. This is the main reason why such integer deserves to be called an "invariant".
Again, I do not know whether $\{N_{n,\beta}\}$ has a name, but I guess an equivalent datum, namely the generating series $$Z_{DT}(X;q,u)=\sum_{n,\beta}N_{n,\beta}q^nu^\beta$$ over all $n,\beta$ does have a name: it is called the Donaldson-Thomas partition function of $X$.
As for question $(2)$, an Abelian variety $A$ contains no rational curve, as every map $\mathbb P^1\to A$ is constant.
The only thing that comes to my mind for your question $(3)$ is the one you want to avoid, namely weighted Euler characteristics, which compute the numbers $(\star)$.