Gromov-Witten Theory of Elliptic Curves

• Gromov-Witten theory of elliptic curves is a branch of enumerative geometry that computes intersection numbers on moduli spaces using modular forms and combinatorial sums.
• The framework employs configuration-space integrals and iterated Eisenstein-Kronecker techniques to reveal quasi-modular and integrable structures.
• Recent algorithmic and categorical advances provide explicit computation and validation of modularity, Virasoro constraints, and mirror symmetry in the theory.

Gromov-Witten theory of elliptic curves investigates the structure of Gromov-Witten (GW) invariants for a smooth complex elliptic curve $E$, connecting enumerative geometry, modular forms, integrable systems, and mirror symmetry. The theory exhibits remarkable explicitness: all GW invariants, including descendants and cycle-valued classes, are expressible in terms of modular data—Eisenstein series, theta functions, and configuration-space integrals. Central techniques include configuration integrals, categorical/derived approaches, explicit algorithmic calculations, and deep symmetries with integrable hierarchies.

1. Structure of Gromov-Witten Theory for Elliptic Curves

The GW invariants of a complex elliptic curve $E$ are defined as intersection numbers on the moduli space of stable maps %%%%2%%%%. The (full, disconnected) generating function is

$$F^E(t;q) = \sum_{g\geq 0} F^E_g(t), \quad F^E_g(t) = \sum_{n,d} \frac{q^d}{n!} \left\langle \prod_{i=1}^n \tau_{k_i}(\phi_{a_i})\right\rangle_{g,n,d}^E \prod_{i=1}^n t_{a_i, k_i}$$

where $t_{a,k}$ are descendant variables corresponding to cohomology classes $\phi_a$ ($H^0$, $H^{1,0}$, $H^{0,1}$, $H^2$) and cotangent line powers.

Stationary invariants—where all insertions are the point class—coincide with completed-cycle Hurwitz numbers. The dimension constraint (i.e., $2g - 2 + n = \sum_i (k_i + \deg \phi_{a_i} - 1)$) ensures finiteness in each genus.

A crucial feature of the elliptic case is that every GW invariant, including descendant and non-stationary ones, can be written as a finite bilinear combination of stationary invariants (Hurwitz numbers), via explicit combinatorial sum-over-partitions formulas. These stationary generating functions are quasimodular forms in $E_2$, $E_4$, $E_6$ (Buryak, 2022).

2. Configuration-Space and Iterated Integral Descriptions

Two complementary analytic frameworks describe the GW generating series:

A. Configuration-Space Integral Representation

The generating series admit realization as explicit integrals over configuration spaces $Conf_n(E)$: $$T_n(z_1, \ldots, z_n; q) = \langle \gamma_n^{GW}, \varphi_n^{GW} \rangle = \int_{Conf_n(E)} \omega_n$$ where $\omega_n$ are holomorphic $n$-forms constructed from sections of the Poincaré bundle, theta functions, and determinants of Szegö kernels (Zhou, 2023). The sum-over-partitions formula organizes the generating functions by weight and modularity properties.

B. Iterated Eisenstein-Kronecker Integral Approach

Alternatively, as shown by Zhou, GW generating functions are iterated configuration-space integrals of Eisenstein-Kronecker forms arising from the Kronecker theta kernel: $\widehat T_n(w_1, \dots, w_n; q) = \,_{{E}^n} \omega_n(z, w)$ This construction carries a pure-weight decomposition and reveals the algebraic origin of quasimodularity—each $\widehat{e}_m(z)$ is a quasi-Jacobi form, and the regularized integrals encode the full modular structure (Zhou, 2023).

These descriptions clarify why the generating series have pure weights, strict quasi-elliptic/quasi-modular properties, and admit clean combinatorial expressions via Bell polynomials and moduli space partitionings.

3. Modular and Quasi-Modular Structures

The generating functions of stationary and non-stationary GW invariants of $E$ are polynomials in Eisenstein series (modular forms) with concrete, homogeneous weight: $F_g(q) = Q_g(E_2(q^2), E_4(q^2), E_6(q^2))$ where $Q_g$ is a polynomial of weight $6g-6$ for $g>1$ (Aga et al., 2023). This modularity is preserved under various degenerations, including tropicalization and Feynman graph expansions.

For elliptic orbifold lines such as $\mathbb{P}^1_{3,3,3}$, $\mathbb{P}^1_{4,4,2}$, $\mathbb{P}^1_{6,3,2}$, the GW cycle-valued generating functions are vector-valued quasi-modular forms; their weight is $2g - 2 + n + \sum \ell_i$, with $\Gamma \subset SL_2(\mathbb{Z})$ determined by monodromy (Milanov et al., 2012). Anti-holomorphic (modular) completions arise, and the ring of quasi-modular forms is generated by Eisenstein-type series appropriate to the monodromy group.

Quasi-modularity manifests in holomorphic anomaly equations: $$\frac{\partial G}{\partial G_2} = \text{(lower weight modular forms)}$$ This structure is intrinsic to the recursive determination and the partition formula organization of GW invariants.

4. Categorical and Derived Descriptions

The B-model Gromov-Witten theory can also be formulated categorically. Costello's construction begins with a cyclic $A_\infty$-model of the derived category $D(E)$, as given by Polishchuk, and constructs a categorical GW invariant using Hochschild chains, Connes' operator, and the Weyl algebra formalism:

• Hochschild chains $CC_\bullet(A)[d]$ with a circle action from the Connes $B$-operator;
• Periodic cyclic complexes and higher residue pairings via modular forms and theta series;
• Fock modules and ribbon graph actions encode modularity, with the quantum master equation satisfied by string vertices.

The explicit computation of the genus-one, one-point B-model categorical GW invariant gives $F_{1,1}^B(\tau) = -\frac{1}{24} E_2(\tau)$, perfectly matching the classical (A-model) invariant computed by Dijkgraaf: $F_{1,1}^A(q) = -\frac{1}{24} E_2(q)$ after the mirror map $q = e^{2\pi i \tau}$. This agreement provides strong support for homological mirror symmetry at higher genus (Caldararu et al., 2017).

The process illustrates the necessity of choosing a monodromy-invariant (Gauss–Manin flat) splitting of the Hodge filtration—mirroring the analytic modular approach.

5. Integrable Hierarchies and Quantum Structures

Elliptic curve GW theory produces a quantum double ramification (qDR) hierarchy:

• The state space $H^*(E)$ splits into even (bosonic) and odd (fermionic) generators, with explicit Poincaré pairing;
• Hamiltonians are written in terms of intersection numbers of double ramification cycles, GW classes, and Hodge classes (notably only $\lambda_{g-1}$ contributes);
• The DR potential

$$\mathcal{G} = \int \left[ \frac{1}{2}(u^1)^2 u^4 + u^1 u^2 u^3 + i\hbar( \mathcal{S}_\epsilon(u^1_x)\mathcal{S}_\epsilon(u^4_x)+\mathcal{S}_\epsilon(u^2_x)\mathcal{S}_\epsilon(u^3_x) )\exp( \mathcal{S}_\epsilon(u^4) D_q ) \right] \odot \sum_{g\geq 0} \epsilon^{2g} G_{2g+2}(q) \, dx$$

encodes all commuting Hamiltonians $G_{\alpha, d}$. Two of the four fields are genuinely fermionic. All flows mutually commute under the quantum bracket; the principal (dispersionless) hierarchy is recovered as $\epsilon \to 0$ (Rossi et al., 4 Dec 2025).

This constitutes the first explicit, modular-integrable quantum hierarchy from a cohomological field theory containing fermionic fields, with all structures expressible in terms of Eisenstein series and modular operations.

6. Explicit Computation and Algorithmic Advances

Algorithmic computation of GW invariants and Hurwitz numbers—the stationary sector—has seen major advances. New algorithms reframe the classical Feynman path-integral expansion (mirror symmetry side) using multivariate coefficient extractions, flip-signature reduction, and combinatorial orbit grouping:

• The expansion is organized in terms of trivalent graphs, with each graph $\Gamma$ contributing via a Feynman path integral expressed in terms of the propagator $P(z, q)$ and its expansion in modular data.
• The implementation in OSCAR and Singular achieves speed-ups by factors of $10^2$–$10^4$ at genus $g=2$–$4$, allowing computation to record degrees and genuses unattainable before (Aga et al., 2023).

This computational efficiency enables detailed verification of modularity, detection of congruences, and systematic paper of functional relations in the $q$-expansions of GW series.

7. Virasoro Constraints and Complete Solubility

The entire system of GW invariants for elliptic curves is determined by Virasoro-type constraints:

• For $E$, all descendant GW invariants are uniquely reconstructed from the stationary sector via explicit differential operators $L_m, D_m, D'_m$ acting on the partition function $Z^E$ (Buryak, 2022).
• The final closed formula for the full GW potential is a sum over partitions of $2g-2$,

$$F^E_g(t) = \sum_{\lambda \vdash (2g-2)} \frac{1}{\prod_j m_j(\lambda)!} C_\lambda(qe^{t_{1,0}}) \prod_{i=1}^{\ell(\lambda)}\left( \sum_{a, b} \eta^{ab} t_{b, \lambda_i} \right)$$

where $C_\lambda$ are the stationary generating series (Hurwitz numbers) and the combinatorics encodes all descendant information.

Thus, the elliptic GW theory is fully solved in closed analytic form, with all enumerative, analytic, and modular features accessible in terms of explicit algebraic quantities.

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References:

(Caldararu et al., 2017, Zhou, 2023, Zhou, 2023, Milanov et al., 2012, Rossi et al., 4 Dec 2025, Aga et al., 2023, Buryak, 2022)