18

The answer to your first question: "What is the relationship between $G_2$ and $\Delta$?" is $$q\frac{d}{dq} \log \Delta = -24G_2 $$ where $$\Delta = q\prod_{m=1}^\infty (1-q^m)^{24}$$ and $$G_2 = -\frac{1}{24} +\sum_{d=1}^\infty \sum_{k|d}k q^d$$ (note that I've included the constant -1/24 which is in usual definition of $G_2$ which you admitted). ...


18

Here is a purely mathematical reason why we prefer to put $u^{2g-2}$ in our generating function instead of, say, $u^g$. The generating function you write, $$\sum_{g,\beta} GW_{g,\beta}(X) u^{2g-2}Q^{\beta},$$ is the generating function for the connected Gromov-Witten invariants of $X$ (I've thrown in another variable that tracks the class $\beta$). ...


10

Here is an attempt at an overview of tropical curve counts by someone who has been involved in the story for a while but certainly hasn't followed everything that has happened. I look forward to being corrected by others. Toric surfaces are done: That's Mikhalkin's work and has since been redone in from many perspectives. Genus zero curves in any toric ...


10

The history of this is as follows. In the paper by Candelas, Lynker and Schimmrigk there are two weighted hypersurfaces whose cohomology is mirror to that of the quintic. These therefore are two potential candidates for the mirror quintic. The question then was how to decide whether they provide mirror partners to the quintic or not. This was addressed in a ...


8

The account Brian Greene gives in The Elegant Universe (a surprisingly useful reference for the early history of mirror symmetry) is that he and Plesser were studying techniques to create new Calabi-Yau manifolds using orbifolds. The mirror of the Fermat quintic arises naturally in this context. Nearly simultaneously, Candelas, Lynker, and Schimmrigk ...


6

Here are two ways how physicists think about the string coupling constant: 1) In usual quantum field theory defined by quantization of a classical field theory, the partition function is defined by a path integral of the form $Z= \int D\phi e^{\frac{S(\phi)}{\hbar}}$ where $S$ is the classical action. In particular, in the classical limit $\hbar \rightarrow ...


6

I believe that the paper http://arxiv.org/abs/alg-geom/9611012 by Pandharipande and Gottsche addresses exactly this question. The short answer is yes, the genus 0 invariants are enumerative up to k=8. For k>8, one can still conclude that the genus 0 GW invariants are weakly enumerative, meaning that for $\delta_\beta$ generic points, there are a finite ...


6

The point is that if the moduli space is non-singular, then the tangent sheaf $h^0(\mathcal{E}^\vee)$ is locally free and so the map $E^{-1}\to E^0$ must be of constant rank. This implies that the cokernel is also locally free. The difference between the dimensions of fibers of the tangent sheaf and the obstruction sheaf is always constant --- it is the ...


6

Here are a couple papers on oriented orbifold cobordism. The first gives rational invariants and generators and also shows that rationally odd dimensional orbifolds bound. The second paper develops machinery for handling the torsion (all dimensions) and applies that to show that every oriented three-orbifold bounds. K.S. Druschel. Oriented Orbifold ...


6

I think of the word "inertia" in "inertia stack" as representing the same idea as the "inertia" in "inertia group" (which presumably came first). This latter group typically comes up when one has a ramified Galois cover $X\rightarrow Y$ (say, of algebraic varieties over an algebraically closed field $k$ of characteristic 0). If the cover has Galois group $G$,...


5

The history is described here: The quintic mirror was constructed by Greene and Plesser as one of a few hundred mirror manifold pairs. Candelas et al. acknowledge in their article that they got the quintic mirror from Greene and Plesser. There is an interesting quote by Brian Greene why he did not himself pursue the enumeration problem solved by Candelas et ...


5

1) Huang, Klemm and Quackenbush computed the BPS invariants of the quintic 3-fold for low genera via the BCOV technique in http://arxiv.org/abs/hep-th/0612125. We can easily convert their data to get the GW invariants. 2) I think the bound is not a theorem, but an observation. We often assume such a vanishing condition to effectively solve the BCOV ...


5

It is certainly possible to have $n_d$ be negative or zero. For example if $X=K_{\mathbb{P}^2}$, the total space of the canonical bundle over $\mathbb{P}^2$, then $n_2=-6$. Your $n_d$ are genus 0 Gopakumar-Vafa invariants (or BPS numbers) and they have a description in terms of sheaves on $X$ which is (conjecturally) equivalent to your formula in terms of GW ...


4

You are right that it's equivalent to compute the degree of the forgetful map. The degree is $3!$, since it's the quotient by $S_3$ permuting the last three markings. If you divide by $S_4$ you get the moduli stack of genus one curves with a distinguished degree $2$ map to $\mathbf P^1$, which is not the same as $\overline M_{1,1}$. In fact there is a ...


4

Let $X$ be a smooth projective variety. Actually, as long as the quantum cohomology of $X$ is semisimple, the partition function of the (descendent) GW-invariants of $X$ is always identified with a tau-function of the Dubrovin--Zhang integrable hierarchy associated to $X$. So for the case of $\mathbb{P}^N$, any $N$, there is a nice integrable hierarchy whose ...


4

I'm not sure whether what I'm going to describe is good enough for your purpose. But the following will be the first thing that I would think about if I'm asked to compute your quantum rings. I apologize if you have been already aware of this stuff. The keyword is Quantum Lefschetz hyperplane theorem (for concave bundle). Although computing the quantum ...


4

First of all, I don't understand why you say that there are no holomorphic curves - a typical example of such a space is $T^*{\mathbb C}{\mathbb P}^1$ and it perfectly has non-trivial holomoprhic curves. What is true though, is that you can choose a different complex structure (still compatible with the same real symplectic form) so that all holomorphic ...


4

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: ...


3

I am not sure what "primary" means. However, I believe the answer to your first question is "no". For a sufficiently general quintic hypersurface $X$ in $\mathbb{C}P^4$, for sufficiently small curve classes $A$, all genus $0$ curves in $X$ of class $A$ are pairwise disjoint and smooth with normal bundle $\mathcal{O}_{\mathbb{P}^1}(-1)\oplus \mathcal{O}_{\...


3

Without loss of generality $L$ is very ample. This gives an embedding $X \to \mathbb P^n$. The genus $g\leq M$ degree $d\leq M$ curves in $X$ live in finitely many finite-dimensional moduli spaces (it's a subspace of finitely many components of the Hilbert scheme of $\mathbb P^n$). For each curve $C$, the codimension of the space of hypersurfaces of degree $...


3

One simple example (although it is a genus 0 example) is the following. Consider a global quotient $\mathscr{X} = [X/(\mathbb{Z}/2)]$. Then if we look at the genus 0 GW theory of $\mathscr{X}$ where the source curve has $2g + 2$ stacky $\mathbb{Z}/2$ points whose evaluations lie in the twisted sector of $\mathscr{X}$, then this will (at least in principle) ...


3

Asking for a deformation over $\mathbb{A}^1$ is quite restrictive. Even asking for formal deformations / deformations over an étale cover of $\mathbb{A}^1$ is nontrivial. The "standard" obstruction group for deforming a stable map is the hyper-Ext group $\mathbf{R}Hom^2_{\mathcal{O}_{C_0}}(L^\bullet_{f_0},\mathcal{O}_{C_0})$, where $L^\bullet_{f_0}$ ...


3

As Dan says, you can use the divisor equation: $$\langle e_{\alpha_1}, \ldots, e_{\alpha_n}, \ell \rangle_{g,d} = d\ \langle e_{\alpha_1}, \ldots, e_{\alpha_n} \rangle_{g,d} $$ where the $e_{\alpha_i}$ are any cohomology classes of $\mathbb{CP}^2$, to reduce your invariant $\langle p,p,\ell \rangle_{0,1}$ to $\langle p,p\rangle_{0,1}$, the number of lines ...


3

I would say that basically everything you wrote is correct, and in particular the equation $F=\lim_{\epsilon\to 0} \epsilon^2 \log \tau|_{t^{\alpha,p>0}=0,t^{\alpha,0}=t^\alpha}$. It is true that, more in general, $\epsilon^2 \log \tau$ gives you the total descendant potential $\mathcal F(t^{*,*};\epsilon)$ at all genera. Notice that you don't even need ...


3

You can compute the GW invariants associated to maps with fixed domain curve $C$ of genus $g$ in terms of genus 0 invariants. The conceptual idea is that the invariants are independent of the choice of the fixed curve $C$, and so one can choose $C$ to be a rational curve with $g$ nodes, and then use the usual gluing axioms to rewrite the invariant as a genus ...


3

A reference would be Pandharipande's ICM address Cohomological field theory calculations, section 1.3. The $R$-matrix is $$ R(z)=\exp\bigg(-\sum_{k=1}^\infty \frac{B_{2k}}{2k(2k-1)}z^{2k-1}\bigg). $$


3

This answer might just be a list of references, but I hope it helps. The most explicit computations of which I am aware exploit a torus action on the Calabi-Yau 3-fold in question, where the calculus can be reduced to a combinatorial problem: for example, the famous paper Gromov–Witten theory and Donaldson–Thomas theory. I (MSN) by Maulik, Nekrasov, Okounkov,...


2

The parameter $m$ appears if you study everything equivariantly with respect to the ${\mathbb C}^*$-action which acts in the standard way on the fibers of the cotangent bundle.


2

For $D$ a surface, there is work of Bogomolov. The results are weaker for higher dimensional varieties, but they do rule out genus $0$ and genus $1$ curves, cf. the work of G. Xu, Lawrence Ein, Claire Voisin and Gianluca Pacienza.


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