Gromov–Witten Invariants and Quantum Cohomology¶

Job: develop the moduli of stable maps with the virtual fundamental class, the Gromov–Witten invariants and the quantum cohomology / Frobenius manifold, and the mirror theorem (Givental / Lian–Liu–Yau) with the quintic genus-0 curve counts as the worked example.

Core¶

Moduli of stable maps¶

For a smooth projective variety $X$ and a class $\beta\in H_2(X,\mathbb{Z})$, the Deligne–Mumford–Knudsen moduli stack $\overline{M}_{g,n}(X,\beta)$ is the stack of stable maps $f:(C,p_1,\dots,p_n)\to X$ from an $n$-pointed genus-$g$ (nodal, projective, connected) curve $C$ to $X$ with $f_*[C]=\beta$, where stable means the automorphism group of $(C,p_1,\dots,p_n,f)$ is finite (equivalently every genus-0 component has $\ge3$ special points and every genus-1 component has $\ge1$). The stack is a proper Deligne–Mumford stack (properness = the stable reduction theorem; Deligne–Mumford from the unramified deformation theory).

Data	Description
$\overline{M}_{g,n}(X,\beta)$	stable maps $f:(C,p_1,\dots,p_n)\to X$, $f_*[C]=\beta$, stable
Stability	finite automorphism group; equivalently $\#$special points $\ge3$ on genus-0 components
Boundary strata	nodal curves: the gluing $M_{g,n}(X,\beta)\supset\bigcup \overline{M}_{g_1,n_1+1}(X,\beta_1)\times_{X}\overline{M}_{g_2,n_2+1}(X,\beta_2)$ ($\beta_1+\beta_2=\beta$); the splitting principle
Evaluation	$\mathrm{ev}_i:\overline{M}_{g,n}(X,\beta)\to X$, $\mathrm{ev}_i(f)=f(p_i)$
Psi-class	$\psi_i\in H^2(\overline{M}_{g,n}(X,\beta))$ the first Chern class of the cotangent line $\mathbb{L}_i$ at $p_i$ (the cotangent to $C$ at $p_i$)

The virtual fundamental class¶

The moduli $\overline{M}_{g,n}(X,\beta)$ has the wrong dimension in general — the deformation theory is obstructed (the tangent-obstruction theory $R^\bullet\pi_*f^*TX$ has $H^1(C,f^*TX)$ the obstructions). The virtual fundamental class corrects this.

Theorem (Li–Tian; Behrend–Fantechi). There is a canonical virtual fundamental class $$ [\overline{M}{g,n}(X,\beta)]^{\mathrm{vir}}\in H) $$ of (real) dimension }}(\overline{M}_{g,n}(X,\beta),\mathbb{Q$\mathrm{vdim}=(\dim_\mathbb{C}X-3)(1-g)+\int_\beta c_1(TX)+n$, constructed via the perfect obstruction theory $E^\bullet\to\mathbb{L}_{\overline{M}}$ (the cotangent complex) with $E^\bullet=(R\pi_*f^*TX)^\vee$, and the intrinsic normal cone (Behrend–Fantechi). The VFC reduces to the ordinary fundamental class when the moduli is smooth of the expected dimension.

The construction: the perfect obstruction theory gives a cone $C_{\overline{M}}\subset E_1$ (the intrinsic normal cone in the vector bundle $E_1=H^1(C,f^*TX)^\vee$), and the VFC is the refined Gysell pullback of $[C_{\overline{M}}]$ along the zero section $0:E_1\to\overline{M}$. The virtual dimension is $\mathrm{rk}(E^\vee)-\mathrm{rk}(E^\bullet)$.

Gromov–Witten invariants¶

The (descendant) Gromov–Witten invariants are $$ \langle\tau_{a_1}(\gamma_1)\cdots\tau_{a_n}(\gamma_n)\rangle_{g,\beta}=\int_{[\overline{M}{g,n}(X,\beta)]^{\mathrm{vir}}}\prod_i^}^n\mathrm{ev(\gamma_i)\,\psi_i^{a_i},\qquad \gamma_i\in H^(X),\ a_i\ge0. $$ The primary invariants are $a_i=0$. The genus-0 three-point invariants $\langle\gamma_i,\gamma_j,\gamma_k\rangle_{0,\beta}=\int_{[\overline{M}_{0,3}(X,\beta)]^{\mathrm{vir}}}\mathrm{ev}_1^*\gamma_i\,\mathrm{ev}_2^*\gamma_j\,\mathrm{ev}_3^*\gamma_k$ define the quantum product. The invariants are rational numbers in general (the VFC is in $\mathbb{Q}$-homology); they count (with virtual multiplicity) the curves of class $\beta$ passing through cycles $\gamma_i$ with the descendant conditions.

Quantum cohomology¶

The small quantum cohomology is $$ QH^(X)=H^(X,\mathbb{C})\otimes_\mathbb{C}\mathbb{C}[q_1,\dots,q_r] $$ where $q_i$ are the Novikov variables for a basis $\{\beta_i\}$ of the effective cone of $H_2(X)$ (the monomials $q^\beta=\prod q_i^{\beta_i}$ track the curve classes), with the quantum product $$ \gamma_i\gamma_j=\sum_{\beta}\sum_{l}\langle\gamma_i,\gamma_j,\gamma_k\rangle_{0,\beta}\,g^{kl}\,\gamma_l\,q^\beta, $$ the structure constants being the genus-0 three-point GW invariants ($g^{kl}$ the Poincaré dual metric). The quantum product is *associative** (a non-trivial consequence of the geometry of $\overline{M}_{0,n}(X,\beta)$).

Theorem (WDVV / associativity). The quantum product is associative: $(\gamma_i*\gamma_j)*\gamma_k=\gamma_i*(\gamma_j*\gamma_k)$. This is the WDVV equation, the boundary-stratum relation on $\overline{M}_{0,4}(X,\beta)$: the two ways of degenerating a 4-pointed genus-0 stable map to a nodal curve (splitting $\beta=\beta_1+\beta_2$) give the same associativity identity.

The big quantum cohomology is the Frobenius manifold structure on $H^*(X)$ from the genus-0 potential $$ F_0(t,q)=\sum_{\beta,n}\frac{1}{n!}\langle t,\dots,t\rangle_{0,\beta}\,q^\beta,\qquad \nabla_i\nabla_j\partial_{t^k}F_0=C_{ij}^{\;\;l}g_{lk}, $$ the Dubrovin–Givental Frobenius manifold (a unital, commutative, associative product $\circ_t$ on each tangent space, the unit $e$, the Euler / grading vector field, the flat metric). The associativity is the WDVV equation; the potential $F_0$ is the prepotential.

Equation	Statement	Geometric origin
WDVV (associativity)	$\partial_i\partial_j\partial_m F_0\cdot g^{mn}\partial_n\partial_k\partial_l F_0=$ cyclic	boundary of $\overline{M}_{0,4}$
String equation	$\langle 1,\gamma_1,\dots,\gamma_n\rangle=\sum_i\langle\gamma_1,\dots,\gamma_{i-1},\gamma_{i+1},\dots\rangle$ with shifted degrees	the unit in $\overline{M}_{0,n+1}$
Dilaton equation	$\langle\psi_1,\gamma_1,\dots,\gamma_n\rangle=(n-2)\langle\gamma_1,\dots,\gamma_n\rangle$	the cotangent at the dilaton point
Topological recursion relation (TRR)	$\langle\psi_1^{k+1}\cdots\rangle_g=\sum\langle\psi_1^k\cdots\rangle_{g-1}\langle\cdots\rangle_0+\dots$	the boundary of $\overline{M}_{g,n}$

The mirror theorem (Givental / Lian–Liu–Yau)¶

The $J$-function (the A-model side) is the one-point descendant generating series $$ J(q,z)=1+\sum_{\beta\ne0}q^\beta\,\Big\langle\frac{\phi}{z-\psi}\Big\rangle_{0,\beta}^{\mathrm{red}}\in H^(X)\otimes\mathbb{C}[z^{-1}][[q]], $$ the *reduced** one-point invariant with the tangent insertion (the "frame"). The $I$-function (the B-model / mirror side) is the hypergeometric modification: for a toric variety $X$ (or a CY hypersurface in a toric variety) with the Batyrev mirror, the $I$-function is the hypergeometric series $$ I(q,z)=z\,\frac{\prod_{\alpha}\Gamma(1+\alpha\cdot\beta+D/z)}{\prod_{\alpha}\Gamma(1+\tilde\alpha\cdot\beta+D/z)}\,q^\beta, $$ summed over $\beta$, where $\alpha$ ranges over the Mori cone generators (the $I$-function is a hypergeometric series in $q$ with coefficients in $H^*(X)$).

Theorem (mirror theorem, Givental; Lian–Liu–Yau mirror principle). The $J$-function equals the $I$-function up to the mirror map: $$ J(\tau(q),z)=I(q,z),\qquad \tau(q)=\frac{I_1(q)}{I_0(q)} $$ (the mirror map $\tau$ is the quotient of the two hypergeometric solutions $I_0$ (the constant / log solution) and $I_1$ (the non-constant solution); it identifies the A-model Kähler parameter with the B-model complex-structure parameter). The genus-0 GW invariants are extracted from the B-model period integrals.

Mirror theorem status	Setting
Genus-0, toric CY (Givental, LLY)	proven; the $J$-/$I$-function equality + the mirror map
Quintic threefold (CdOGP, Givental)	the genus-0 mirror theorem; $N_d$ from periods
Higher-genus BCOV (Bershadsky–Cecotti–Ooguri–Vazirani)	partly conjectural; the B-model recursion / holomorphic anomaly equations
Non-toric CY	open / partial

See it¶

Quintic threefold genus-0 mirror theorem. Let $X\subset\mathbb{P}^4$ be the quintic Calabi–Yau threefold ($h^{1,1}=1$, the only Kähler parameter $q$ tracking the degree). The $I$-function is the hypergeometric series (the Picard–Fuchs solution on the mirror) $$ I(q,z)=z\,q^D\sum_{d\ge0}q^d\,\frac{\prod_{k=1}^{5d}(5z+ kz)}{\prod_{k=1}^d (z+kz)^5}\cdot\frac{1}{z} = z\sum_{d\ge0}q^d\,\frac{(5d)!}{(d!)^5} $$ (up to the standard normalization, $I_0(q)=\sum_d q^d\frac{(5d)!}{(d!)^5}$). The mirror map is $$ \tau(q)=\frac{I_1(q)}{I_0(q)}=\frac{\log q}{5}+\sum_{d\ge1}q^d\,\frac{(5d)!}{(d!)^5}\cdot\frac{1}{I_0(q)}\cdot(\text{constant}) $$ (the quotient of the log / constant solutions of the Picard–Fuchs equation). The genus-0 three-point invariant (the Yukawa coupling on the B-side, identified with the A-model quantum product) is $$ \partial_\tau^3 F_0 = \frac{5}{\psi_0^3}\Bigl(1+\sum_{d\ge1}N_d\,\frac{d^3 q^d}{1-q^d}\Bigr), $$ the CdOGP formula for the number $N_d$ of rational curves of degree $d$ on the quintic (the multiplier $d^3/(1-q^d)$ the multiple-cover formula, confirmed by mirror symmetry). The first values, extracted from the mirror B-model period computation: $$ N_1=2875,\qquad N_2=609250,\qquad N_3=317206375,\qquad N_4=242467530000,\dots $$ The key point: the number of rational curves on the quintic is computed from the period integrals on the mirror — the operational content of mirror symmetry at genus 0. The $N_1=2875$ (lines on the quintic, classical) and $N_2=609250$ (conics, predicted by mirror symmetry before the direct enumeration) were the historical confirmation.

Check¶

Sketch why the WDVV equation is the boundary-stratum relation on $\overline{M}_{0,4}(X,\beta)$.

$\overline{M}_{0,4}(X,\beta)$ has boundary divisors where the 4-pointed genus-0 curve splits as a nodal union $C_1\cup C_2$ with $p_i,p_j$ on $C_1$ and $p_k,p_l$ on $C_2$, the class splitting $\beta=\beta_1+\beta_2$. There are three such pairings $(ij|kl)$, $(ik|jl)$, $(il|jk)$. Integrating $\mathrm{ev}_1^*\gamma_a\mathrm{ev}_2^*\gamma_b\mathrm{ev}_3^*\gamma_c\mathrm{ev}_4^*\gamma_d$ over $[\overline{M}_{0,4}(X,\beta)]^{\mathrm{vir}}$ and reducing to the boundary (where the VFC restricts to the fiber product over $X$) gives, on each boundary divisor, a product of two three-point invariants. The three pairings give three expressions for the same integral; equating them is the WDVV / associativity relation $\sum_{\beta_1+\beta_2}\langle\gamma_a,\gamma_b,\gamma_m\rangle_{\beta_1}g^{mn}\langle\gamma_n,\gamma_c,\gamma_d\rangle_{\beta_2}=$ cyclic.

Why is the $I$-function a hypergeometric series, and what is the role of the mirror map?

The $I$-function is hypergeometric because the mirror family (the Batyrev / Greene–Plesser mirror of the toric CY) has Picard–Fuchs equations that are (generalized) hypergeometric — the periods are $\sum_d q^d\frac{\prod\Gamma(\cdots)}{\prod\Gamma(\cdots)}$ from the GKZ / toric hypergeometric structure. The mirror map $\tau=I_1/I_0$ is the change of variables identifying the A-model Kähler parameter (the coordinate $q$ on $\mathcal{M}_K$) with the B-model complex-structure parameter (the period coordinate $\tau$ on $\mathcal{M}_{\mathrm{cx}}$); without it, the $I$-function is a B-model period but not the A-model $J$-function — the mirror map is the identification of the two moduli.

Pitfalls¶

The virtual fundamental class requires the perfect obstruction theory (Behrend–Fantechi); for the moduli without it (e.g. non-projective $X$, or unstable maps), the VFC is not defined. The VFC is in $\mathbb{Q}$-homology, not $\mathbb{Z}$ — the GW invariants are rational, not integer (the multiple-cover phenomenon).
The genus-0 mirror theorem is for toric / CY (Givental, LLY); the non-toric case is open / partial (some Fano / complete-intersection cases, but no general theorem).
The multiple-cover formula $N_d q^d/(1-q^d)$ (the Faber–Pandharipande / Aspinwall–Morrison formula) is the genus-0 multiple-cover contribution; it is confirmed by mirror symmetry but the "bare" curve count (without the multiple-cover factor) is not the GW invariant directly.
The descendant invariants ($\psi_i$ insertions) are not just bookkeeping — they enter the WDVV / TRR relations and the mirror theorem (the $J$-function is a descendant series); do not reduce all invariants to primary ones.
The higher-genus BCOV is partly conjectural — the holomorphic anomaly equations + the boundary conditions (the gap condition) reconstruct the higher-genus amplitudes but the full mathematical proof (the BCOV reconstruction) is partial.
Novikov variables $q_i$ are formal (the formal neighborhood of the large-radius limit); convergence of the GW series is not guaranteed (and often divergent), so $QH^*$ is a formal deformation, not a convergent one.

Refs¶

Kontsevich–Manin, "Gromov–Witten classes, quantum cohomology, and enumerative geometry" (CMP 1994)
Li–Tian, "Virtual moduli cycles and Gromov–Witten invariants" (JAMS 1998); Behrend–Fantechi, "The intrinsic normal cone" (Invent. 1997)
Givental, "Equivariant Gromov–Witten invariants" (IMRN 1996); Lian–Liu–Yau, "Mirror principle I–IV" (Asian J. Math. 1997–)
Candelas–de la Ossa–Green–Parkes, "A pair of Calabi–Yau manifolds as an exactly soluble superconformal theory" (Nucl. Phys. B 1991)
Cox–Katz, Mirror Symmetry and Algebraic Geometry (AMS 1999); Hori et al., Mirror Symmetry (Clay 2003)

Next¶

MS index · Foundations · Fukaya Category & HMS · SYZ & Gross–Siebert · Hitchin & Geometric Langlands · TG / moduli & enumerative — the tropical / log GW as the tropical limit / the Mikhalkin / Siebert–Nishinou correspondence · Symplectic / Floer · Kähler geometry · Algebraic geometry