Extending the Coates–Givental–Tseng localization formula and Virasoro reduction from toric bundles to general holomorphic torus actions

The Virasoro conjecture predicts an infinite system of differential constraints on the total descendant potential of a target in Gromov–Witten theory. Coates, Givental, and Tseng proved that the conjecture holds for the total space of a toric bundle $E \to B$ if and only if it holds for the base $B$. The proof rests on a virtual-localization formula expressing the torus-equivariant ancestor potential of $E$ through twisted ancestor potentials of the base and a single quantized loop-group operator, followed by a nonequivariant limit based on Brown's mirror theorem for toric fibrations. The authors state (Remark 3.7) that the equivariant localization formulas — the genus-zero factorization and the all-genus ancestor formula — extend in outline to an arbitrary Kähler manifold with a holomorphic torus action, but that this requires foundational work they did not undertake: a systematic theory of leg moduli spaces — moduli of stable maps onto one-dimensional torus orbits, rigid in the toric-bundle case but parameter-dependent in general — with their virtual fundamental cycles and localization contributions. The nonequivariant-limit step, by contrast, is carried out in the source only for toric bundles via Brown's mirror theorem, and no extension of that step to general torus actions is claimed there; the composite coordinate change between equivariant and nonequivariant parameters (the Dijkgraaf–Witten maps composed with the mirror maps) is described in the source as complicated but as holding automatically within the argument, without an explicit description. This problem asks for the general-torus-action extension of the localization formulas and the resulting reduction of the Virasoro conjecture to the fixed locus. In the literature through 2026 the question remains open: the toric-bundle theorem is used as a black box, and the leg-moduli foundations remain undeveloped.

Background

Gromov–Witten (GW) invariants of a smooth projective variety $X$ are intersection numbers on moduli spaces of stable maps — spaces of holomorphic maps from nodal Riemann surfaces with marked points to $X$, taken up to isomorphism; they are encoded in generating functions. The total descendant potential $\mathcal{D}_X$ collects all-genus GW invariants with insertions of $\psi$-classes (Chern classes of the cotangent lines at the marked points). Ancestor potentials $\mathcal{A}(\tau;t)$ are the analogous all-genus generating functions in which the $\psi$-classes are replaced by classes pulled back from the moduli space of stable curves via the stabilization map; they depend on an auxiliary parameter $\tau \in H^\bullet(X)$ and are related to the descendant potential by the dilaton-shifted identification furnished by the topological recursion relations (identities expressing classes pulled back from the moduli of stable curves in terms of boundary divisors).

The Virasoro conjecture (Eguchi–Hori–Xiong, 1997; formulated mathematically by Getzler, 1999) asserts that $\mathcal{D}_X$ is annihilated by an explicit infinite family of differential operators $L_m$, $m \geq -1$, forming a Virasoro algebra; its genus-zero part is a theorem. Proven cases include nonsingular curves (Okounkov–Pandharipande), targets whose quantum cohomology is generically semisimple (Givental's reconstruction combined with Teleman's classification of semisimple cohomological field theories — algebraic systems of bilinear products on cohomology satisfying the axioms of GW theory, classified in the semisimple case by their genus-zero data), toric varieties, and flag manifolds.

Givental's formalism encodes genus-zero GW theory in an overruled Lagrangian cone $\mathcal{L}_X$ inside a symplectic loop space built from $H^\bullet(X)((z^{-1}))$; quantization of quadratic Hamiltonians on this loop space defines an action of a completion of the twisted loop group — the group of $z$-linear symplectomorphisms of the loop space — on the Fock space of potentials. A key property is loop-group covariance: the Virasoro constraints are stable under quantized loop-group operators that respect the grading. Consequently, if the descendant potential of $E$ is obtained from that of another space by a grading-respecting quantized loop-group operator, the Virasoro conjecture transfers between the two.

A toric bundle is the total space $E \to B$ of a fiber bundle whose fiber is a smooth toric variety $X$ and whose structure group is a torus acting on $X$; the fiberwise torus $T$-action has fixed locus $E^T \cong \bigsqcup_{\alpha \in F} B$, indexed by the fixed points $\alpha$ of $X$. Coates–Givental–Tseng analyze the $T$-equivariant GW theory of $E$ using Graber–Pandharipande virtual localization: the equivariant virtual class localizes onto fixed loci described by decorated graphs whose vertices carry stable maps to $E^T \cong B$ and whose edges are legs — components mapped onto one-dimensional $T$-orbit closures via covers $z \mapsto z^k$ of multiplicity $k$. For toric bundles the legs are rigid (the multiplicity is the only datum), and the leg contributions assemble into one explicit operator.

The paper's Theorem 3.5 states that the $T$-equivariant ancestor potential of $E$ factors as

$$\mathcal{A}^{\mathrm{eq}}_E(\tau;t) \;=\; \hat{R}(\tau) \bigotimes_{\alpha \in F} \mathcal{A}^{\alpha,\mathrm{tw}}_B\bigl(u_\alpha(\tau);\, t_\alpha\bigr),$$

where $\mathcal{A}^{\alpha,\mathrm{tw}}_B$ is the ancestor potential of the base $B$ twisted — in the standard sense that the virtual class of each moduli space is multiplied by the equivariant Euler class of a natural obstruction-type bundle — by the normal data at $\alpha$, where $\hat{R}(\tau)$ is the quantization of a loop-group element determined by genus-zero invariants of the fiber, and where $\tau \mapsto \bigoplus_\alpha u_\alpha(\tau)$ is a nonlinear change of variables (the Dijkgraaf–Witten map). Combining this with the Quantum Riemann–Roch theorem — the theorem of Coates and Givental expressing twisted descendant potentials as quantized loop-group operators applied to untwisted ones — and passing to the nonequivariant limit of the equivariant parameters (the delicate step, carried out via Brown's mirror theorem for toric fibrations, which supplies explicit hypergeometric $I$-functions and mirror maps, and stationary-phase asymptotics of oscillating integrals mirror to the toric fiber) yields Theorem 1.4: the descendant potential $\mathcal{D}_E$ is obtained from $\mathcal{D}_B^{\otimes |F|}$ by a grading-respecting quantized loop-group operator. By loop-group covariance this implies the Virasoro conjecture for $E$ if and only if it holds for $B$ (Corollary 1.5), giving new cases such as bases that are curves, K3 surfaces, or have generically semisimple quantum cohomology.

Two aspects of the argument are deliberately left incomplete in the source. First, Remark 3.7 observes that for a Kähler manifold with an arbitrary holomorphic torus action the same two formulas should hold — at genus zero, the fundamental solution matrix of the equivariant quantum differential equation should factor as $S_E(\tau,z) = R(\tau,z)\, S^{\mathrm{tw}}_{E^T}(u(\tau),z)$, and in all genera the ancestor formula above should hold with the fixed locus $E^T$ in place of the base — but for general actions the one-dimensional orbits (legs) come in families, and the required systematic description of leg moduli spaces and their virtual fundamental cycles has not been carried out; the authors state that this is the only reason the proof is restricted to toric bundles. Second, the nonequivariant limit of the composed operator is established only for a range of the ancestor parameter $\tau$ (a single value suffices for the application, since descendant potentials are $\tau$-independent), and the composite coordinate change from $\tau \in H^\bullet_T(E)$ to $\bigoplus_\alpha \tau^*_\alpha \in H^\bullet_T(E^T)$ — the Dijkgraaf–Witten maps composed with the mirror maps — is asserted to exist with the needed formal properties, but its explicit form, domains of convergence, and singularities are not given.

Problem Statement

Let $E$ be a compact Kähler manifold (or smooth projective variety over $\mathbb{C}$) equipped with a holomorphic action of an algebraic torus $T$ with nonempty fixed locus $E^T$, under no toric-bundle assumption and no GKM-type assumption on the orbit structure (GKM, after Goresky–Kottwitz–MacPherson, denotes the case where the equivariant cohomology is determined by the fixed locus together with finitely many one-dimensional orbit closures). Establish the general-torus-action analogue of the Coates–Givental–Tseng localization theory for Gromov–Witten theory. The target is the general-torus-action form of the two equivariant localization formulas, the genus-zero factorization (2) and the all-genus ancestor formula (3), resting on the leg-moduli foundation (1), together with the resulting reduction of the Virasoro conjecture for $E$ to fixed-locus data (4). The four components below are jointly required:

(1) Foundational localization input: construct the leg moduli spaces — moduli spaces of stable maps with the one-dimensional $T$-orbit closures as targets, including their multiplicities and their parameter dependence (which is absent in the rigid toric-bundle case) — together with perfect obstruction theories, virtual fundamental cycles, and the resulting localization contributions.

(2) Genus-zero factorization: prove that the $T$-equivariant fundamental solution matrix $S_E(\tau,z)$ of the quantum differential equation (the linear system encoding genus-zero equivariant Gromov–Witten invariants) satisfies $S_E(\tau,z) = R(\tau,z)\, S^{\mathrm{tw}}_{E^T}(u(\tau),z)$, where $R$ is an operator-valued power series in $z$ and $\tau \mapsto u(\tau)$ is a nonlinear change of variables over the coefficient ring, both determined by genus-zero Gromov–Witten invariants.

(3) All-genus ancestor formula: prove that the $T$-equivariant ancestor potential satisfies $\mathcal{A}^{\mathrm{eq}}_E(\tau;t) = \hat{R}(\tau)\, \mathcal{A}^{\mathrm{tw}}_{E^T}(u(\tau);t)$, expressing it as the action of a single quantized loop-group operator on the twisted ancestor potential of the fixed locus.

(4) Nonequivariant limit and Virasoro reduction: establish the nonequivariant limit of the resulting operators — as in the toric-bundle case, existence of the limit for a suitable range of the ancestor parameter $\tau$ suffices for the application, since descendant potentials are $\tau$-independent — and verify grading-respectivity, so that the Virasoro conjecture for $E$ reduces to the corresponding Virasoro-type constraints for the twisted Gromov–Witten theory of the fixed locus $E^T$.

References

1. Tom Coates, Alexander Givental, Hsian-Hua Tseng, Virasoro Constraints for Toric Bundles, Forum of Mathematics, Pi 12 (2024), Paper No. e4, DOI: 10.1017/fmp.2024.2, https://doi.org/10.1017/fmp.2024.2 (arXiv:1508.06282, https://arxiv.org/abs/1508.06282)
2. Jeff Brown, Gromov–Witten Invariants of Toric Fibrations, International Mathematics Research Notices 2014, no. 19, 5437–5482, DOI: 10.1093/imrn/rnt030, https://doi.org/10.1093/imrn/rnt030
3. Hsian-Hua Tseng, A note on Virasoro constraints for products, arXiv:2603.22486 (2026), https://arxiv.org/abs/2603.22486
4. Jiun-Cheng Chen, Hsian-Hua Tseng, Descendant and Fourier-Mukai equivalences for simple flops, arXiv:2604.09962 (2026), https://arxiv.org/abs/2604.09962
5. Yu Wang, Fenglong You, Gromov–Witten theory beyond maximal contacts, Journal of Algebraic Geometry 35 (2026), 741–782, DOI: 10.1090/jag/863, https://doi.org/10.1090/jag/863 (arXiv:2403.17200, https://arxiv.org/abs/2403.17200)
6. Yuki Koto, A mirror theorem for non-split toric bundles, Mathematische Annalen 393 (2025), 3337–3394, DOI: 10.1007/s00208-025-03302-7, https://doi.org/10.1007/s00208-025-03302-7 (arXiv:2310.09888, https://arxiv.org/abs/2310.09888)
7. Benjamin Zhou, Higher genus Gromov-Witten invariants from projective bundles on smooth log Calabi-Yau pairs, arXiv:2503.17713 (2025), https://arxiv.org/abs/2503.17713
8. Bohui Chen, Cheng-Yong Du, On the Tautological Property of Relative Gromov–Witten Classes and Applications, Results in Mathematics 80 (2025), DOI: 10.1007/s00025-025-02361-1, https://doi.org/10.1007/s00025-025-02361-1
9. Tohru Eguchi, Kentaro Hori, Chuan-Sheng Xiong, Quantum cohomology and Virasoro algebra, Physics Letters B 402 (1997), 71–80, DOI: 10.1016/S0370-2693(97)00401-2, https://doi.org/10.1016/S0370-2693(97)00401-2
10. Ezra Getzler, The Virasoro conjecture for Gromov-Witten invariants, Contemporary Mathematics 241 (1999), 147–176, DOI: 10.1090/conm/241/03634, https://doi.org/10.1090/conm/241/03634
11. Andrei Okounkov, Rahul Pandharipande, Virasoro constraints for target curves, Inventiones Mathematicae 163 (2006), 47–108, DOI: 10.1007/s00222-005-0455-y, https://doi.org/10.1007/s00222-005-0455-y
12. Constantin Teleman, The structure of 2D semi-simple field theories, Inventiones Mathematicae 188 (2012), 525–588, DOI: 10.1007/s00222-011-0352-5, https://doi.org/10.1007/s00222-011-0352-5
13. Tom Graber, Rahul Pandharipande, Localization of virtual classes, Inventiones Mathematicae 135 (1999), 487–518, DOI: 10.1007/s002220050293, https://doi.org/10.1007/s002220050293
14. Alexander Givental, Gromov-Witten Invariants and Quantization of Quadratic Hamiltonians, Moscow Mathematical Journal 1 (2001), 551–568, DOI: 10.17323/1609-4514-2001-1-4-551-568, https://doi.org/10.17323/1609-4514-2001-1-4-551-568