We prove a genus zero Givental-style mirror theorem for all complete intersections in toric Deligne-Mumford stacks, which provides an explicit slice called big I-function on Givental’s Lagrangian cone for such targets. In particular, we remove a technical assumption called convexity needed in the previous mirror theorem for such complete intersections. In the realm of quasimap theory, our mirror theorem can be viewed as solving the quasimap wall-crossing conjecture for big I-function [13] for these targets. In the proof, we discover a new recursive characterization of the slice on Givental’s Lagrangian cone, which may be of self-independent interests.

We give a new proof of an unpublished result of Dale Peterson, proved by Lam and Shimozono, which identifies explicitly the structure constants, with respect to the quantum Schubert basis, for the T-equivariant quantum cohomology QH^{\bullet }_T(G/P)$QH^{\bullet }_T(G/P)$ of any flag variety G/P$G/P$ with the structure constants, with respect to the affine Schubert basis, for the T-equivariant Pontryagin homology H^T_{\bullet }(\mathcal {G}r)$H^T_{\bullet }(\mathcal {G}r)$ of the affine Grassmannian \mathcal {G}r$\mathcal {G}r$ of G, where G is any simple simply-connected complex algebraic group.

Our approach is to construct an H_T^{\bullet }(pt)$H_T^{\bullet }(pt)$-algebra homomorphism by Gromov-Witten theory and show that it is equal to Peterson’s map. More precisely, the map is defined via Savelyev’s generalized Seidel representations, which can be interpreted as certain Gromov-Witten invariants with input H^T_{\bullet }(\mathcal {G}r)\otimes QH_T^{\bullet }(G/P)$H^T_{\bullet }(\mathcal {G}r)\otimes QH_T^{\bullet }(G/P)$. We determine these invariants completely, in a way similar to how Fulton and Woodward did in their proof of the quantum Chevalley formula.

We define a new class of enumerative invariants called k-leaky double Hurwitz descendants, generalizing both descendant integrals of double ramification cycles and the k-leaky double Hurwitz numbers introduced in [CMR25]. These numbers are defined as intersection numbers of the logarithmic DR cycle against \psi $\psi$-classes and logarithmic classes coming from piecewise polynomials encoding fixed branch point conditions. We give a tropical graph sum formula for these new invariants, allowing us to show their piecewise polynomiality in any genus. Investigating the piecewise polynomial structure further (and restricting to genus zero for this purpose), we also show a wall-crossing formula. We also prove that in genus zero the invariants are always nonnegative and give a complete classification of the cases where they vanish.

We compute the open Gromov-Witten disk invariants and the relative quantum cohomology of the Chiang Lagrangian L_\triangle \subset \mathbb {C}P^3$L_\triangle \subset \mathbb {C}P^3$. Since L_\triangle $L_\triangle$ is not fixed by any anti-symplectic involution, the invariants may augment straightforward J-holomorphic disk counts with correction terms arising from the formalism of Fukaya A_\infty $A_\infty$-algebras and bounding cochains. These correction terms are shown in fact to be nontrivial for many invariants. Moreover, examples of nonvanishing mixed disk and sphere invariants are obtained.

We characterize a class of open Gromov-Witten invariants, called basic, which coincide with straightforward counts of J-holomorphic disks. Basic invariants for the Chiang Lagrangian are computed using the theory of axial disks developed by Evans-Lekili and Smith in the context of Floer cohomology. The open WDVV equations give recursive relations which determine all invariants from the basic ones. The denominators of all invariants are observed to be powers of 2$2$ indicating a nontrivial arithmetic structure of the open WDVV equations. The magnitude of invariants is not monotonically increasing with degree. Periodic behavior is observed with periods 8$8$ and 16.$16.$

We develop a theory of quasimaps to a moduli space of sheaves M on a surface S. Under some assumptions, we prove that moduli spaces of quasimaps are proper and carry a perfect obstruction theory. Moreover, they are naturally isomorphic to moduli spaces of sheaves on threefolds S\times C$S\times C$, where C is a nodal curve. Using Zhou’s theory of entangled tails, we establish a wall-crossing formula which therefore relates the Gromov–Witten theory of M and the Donaldson–Thomas theory of S\times C$S\times C$ with relative insertions. We evaluate the wall-crossing formula for Hilbert schemes of points S^{[n]}$S^{[n]}$, if S is a del Pezzo surface.

Dale Peterson has discovered a surprising result that the quantum cohomology ring of the flag variety \operatorname {\mathrm {GL}}_n({\mathbb {C}})/B$\operatorname {\mathrm {GL}}_n({\mathbb {C}})/B$ is isomorphic to the coordinate ring of the intersection of the Peterson variety \operatorname {\mathrm {Pet}}_n$\operatorname {\mathrm {Pet}}_n$ and the opposite Schubert cell associated with the identity element \Omega _e^\circ $\Omega _e^\circ$ in \operatorname {\mathrm {GL}}_n({\mathbb {C}})/B$\operatorname {\mathrm {GL}}_n({\mathbb {C}})/B$. This is an unpublished result, so papers of Kostant and Rietsch are referred for this result. An explicit presentation of the quantum cohomology ring of \operatorname {\mathrm {GL}}_n({\mathbb {C}})/B$\operatorname {\mathrm {GL}}_n({\mathbb {C}})/B$ is given by Ciocan–Fontanine and Givental–Kim. In this paper, we introduce further quantizations of their presentation so that they reflect the coordinate rings of the intersections of regular nilpotent Hessenberg varieties \operatorname {\mathrm {Hess}}(N,h)$\operatorname {\mathrm {Hess}}(N,h)$ and \Omega _e^\circ $\Omega _e^\circ$ in \operatorname {\mathrm {GL}}_n({\mathbb {C}})/B$\operatorname {\mathrm {GL}}_n({\mathbb {C}})/B$. In other words, we generalize the Peterson’s statement to regular nilpotent Hessenberg varieties via the presentation given by Ciocan–Fontanine and Givental–Kim. As an application of our theorem, we show that the singular locus of the intersection of some regular nilpotent Hessenberg variety \operatorname {\mathrm {Hess}}(N,h_m)$\operatorname {\mathrm {Hess}}(N,h_m)$ and \Omega _e^\circ $\Omega _e^\circ$ is the intersection of certain Schubert variety and \Omega _e^\circ $\Omega _e^\circ$, where h_m=(m,n,\ldots ,n)$h_m=(m,n,\ldots ,n)$ for 1<m<n$1<m<n$. We also see that \operatorname {\mathrm {Hess}}(N,h_2) \cap \Omega _e^\circ $\operatorname {\mathrm {Hess}}(N,h_2) \cap \Omega _e^\circ$ is related with the cyclic quotient singularity.

In our previous paper, we gave a presentation of the torus-equivariant quantum K-theory ring QK_{H}(Fl_{n+1})$QK_{H}(Fl_{n+1})$ of the (full) flag manifold Fl_{n+1}$Fl_{n+1}$ of type A_{n}$A_{n}$ as a quotient of a polynomial ring by an explicit ideal. In this paper, we prove that quantum double Grothendieck polynomials, introduced by Lenart-Maeno, represent the corresponding (opposite) Schubert classes in the quantum K-theory ring QK_{H}(Fl_{n+1})$QK_{H}(Fl_{n+1})$ under this presentation. The main ingredient in our proof is an explicit formula expressing the semi-infinite Schubert class associated to the longest element of the finite Weyl group, which is proved by making use of the general Chevalley formula for the torus-equivariant K-group of the semi-infinite flag manifold associated to SL_{n+1}(\mathbb {C})$SL_{n+1}(\mathbb {C})$.

We compute odd-degree genus 1 quasimap and Gromov–Witten invariants of moduli spaces of Higgs {\rm{S}}{{\rm{L}}_2}${\rm{S}}{{\rm{L}}_2}$-bundles on a curve of genus g \geqslant 2$g \geqslant 2$. We also compute certain invariants for all prime ranks. This proves some parts of the author’s conjectures on quasimap invariants of moduli spaces of Higgs bundles. More generally, our methods provide a computation scheme for genus 1 quasimap and Gromov–Witten invariants in the case when degrees of maps are coprime to the rank. This requires an analysis of the localisation formula for certain Quot schemes parametrising higher-rank quotients on an elliptic curve. Invariants for degrees that are not coprime to the rank exhibit a very different structure for a reason that we explain.

Given any smooth germ of a 3-fold flopping contraction, we first give a combinatorial characterisation of which Gopakumar–Vafa (GV) invariants are non-zero, by prescribing multiplicities to the walls in the movable cone. On the Gromov–Witten (GW) side, this allows us to describe, and even draw, the critical locus of the associated quantum potential. We prove that the critical locus is the infinite hyperplane arrangement of Iyama and the second author and, moreover, that the quantum potential can be reconstructed from a finite fundamental domain. We then iterate, obtaining a combinatorial description of the matrix that controls the transformation of the non-zero GV invariants under a flop. There are three main ingredients and applications: (1) a construction of flops from simultaneous resolution via cosets, which describes how the dual graph changes; (2) a closed formula, which describes the change in dimension of the contraction algebra under flop; and (3) a direct and explicit isomorphism between quantum cohomologies of different crepant resolutions, giving a Coxeter-style, visual proof of the Crepant Transformation Conjecture for isolated cDV singularities.

We study the enumerativity of Gromov–Witten invariants where the domain curve is fixed in moduli and required to pass through the maximum possible number of points. We say a Fano manifold satisfies asymptotic enumerativity if such invariants are enumerative whenever the degree of the curve is sufficiently large. Lian and Pandharipande speculate that every Fano manifold satisfies asymptotic enumerativity. We give the first counterexamples, as well as some new examples where asymptotic enumerativity holds. The negative examples include special hypersurfaces of low Fano index and certain projective bundles, and the new positive examples include many Fano threefolds and all smooth hypersurfaces of degree d \leq (n+3)/3$d \leq (n+3)/3$ in {\mathbb P}^n${\mathbb P}^n$.

We obtain a new interpretation of the cohomological Hall algebra \mathcal {H}_Q$\mathcal {H}_Q$ of a symmetric quiver Q in the context of the theory of vertex algebras. Namely, we show that the graded dual of \mathcal {H}_Q$\mathcal {H}_Q$ is naturally identified with the underlying vector space of the principal free vertex algebra associated to the Euler form of Q. Properties of that vertex algebra are shown to account for the key results about \mathcal {H}_Q$\mathcal {H}_Q$. In particular, it has a natural structure of a vertex bialgebra, leading to a new interpretation of the product of \mathcal {H}_Q$\mathcal {H}_Q$. Moreover, it is isomorphic to the universal enveloping vertex algebra of a certain vertex Lie algebra, which leads to a new interpretation of Donaldson–Thomas invariants of Q (and, in particular, re-proves their positivity). Finally, it is possible to use that vertex algebra to give a new interpretation of CoHA modules made of cohomologies of non-commutative Hilbert schemes.

We study a quiver description of the nested Hilbert scheme of points on the affine plane and its higher rank generalization – that is, the moduli space of flags of framed torsion-free sheaves on the projective plane. We show that stable representations of the quiver provide an ADHM-like construction for such moduli spaces. We introduce a natural torus action and use equivariant localization to compute some of their (virtual) topological invariants, including the case of compact toric surfaces. We conjecture that the generating function of holomorphic Euler characteristics for rank one is given in terms of polynomials in the equivariant weights, which, for specific numerical types, coincide with (modified) Macdonald polynomials. From the physics viewpoint, the quivers we study describe a class of surface defects in four-dimensional supersymmetric gauge theories in terms of nested instantons.

This paper is the first part in a series of three papers devoted to the study of enumerative invariants of abelian surfaces through the tropical approach. In this paper, we consider the enumeration of genus g curves of fixed degree passing through g points. We compute the tropical multiplicity provided by a correspondence theorem due to T. Nishinou and show that it is possible to refine this multiplicity in the style of the Block–Göttsche refined multiplicity to get tropical refined invariants.

We define p$p$-adic \mathrm {BPS}$\mathrm {BPS}$ or p\mathrm {BPS}$p\mathrm {BPS}$ invariants for moduli spaces \operatorname {M}_{\beta,\chi }$\operatorname {M}_{\beta,\chi }$ of one-dimensional sheaves on del Pezzo and K3 surfaces by means of integration over a non-archimedean local field F$F$. Our definition relies on a canonical measure \mu _{\rm can}$\mu _{\rm can}$ on the F$F$-analytic manifold associated to \operatorname {M}_{\beta,\chi }$\operatorname {M}_{\beta,\chi }$ and the p\mathrm {BPS}$p\mathrm {BPS}$ invariants are integrals of natural {\mathbb {G}}_m${\mathbb {G}}_m$ gerbes with respect to \mu _{\rm can}$\mu _{\rm can}$. A similar construction can be done for meromorphic and usual Higgs bundles on a curve. Our main theorem is a \chi$\chi$-independence result for these p\mathrm {BPS}$p\mathrm {BPS}$ invariants. For one-dimensional sheaves on del Pezzo surfaces and meromorphic Higgs bundles, we obtain as a corollary the agreement of p\mathrm {BPS}$p\mathrm {BPS}$ with usual \mathrm {BPS}$\mathrm {BPS}$ invariants through a result of Maulik and Shen [Cohomological \chi$\chi$-independence for moduli of one-dimensional sheaves and moduli of Higgs bundles, Geom. Topol. 27 (2023), 1539–1586].

In this article, we study quasimaps to moduli spaces of sheaves on a K3$K3$ surface S. We construct a surjective cosection of the obstruction theory of moduli spaces of \epsilon $\epsilon$-stable quasimaps. We then establish reduced wall-crossing formulas which relate the reduced Gromov–Witten theory of moduli spaces of sheaves on S and the reduced Donaldson–Thomas theory of S\times C$S\times C$, where C is a nodal curve. As applications, we prove the Hilbert-schemes part of the Igusa cusp form conjecture; higher-rank/rank-one Donaldson–Thomas correspondence with relative insertions on S\times C$S\times C$, if g(C)\leq 1$g(C)\leq 1$; Donaldson–Thomas/Pandharipande–Thomas correspondence with relative insertions on S\times \mathbb {P}^1$S\times \mathbb {P}^1$.

Let X be a smooth and projective threefold with a simple normal crossings divisor D. We construct the Donaldson–Thomas theory of the pair (X|D)$(X|D)$ enumerating ideal sheaves on X relative to D. These moduli spaces are compactified by studying subschemes in expansions of the target geometry, and the moduli space carries a virtual fundamental class leading to numerical invariants with expected properties. We formulate punctual evaluation, rationality and wall-crossing conjectures, in parallel with the standard theory. Our formalism specializes to the Li–Wu theory of relative ideal sheaves when the divisor is smooth and is parallel to recent work on logarithmic Gromov–Witten theory with expansions.

We consider K-theoretic Gromov-Witten theory of root constructions. We calculate some genus 0$0$ K-theoretic Gromov-Witten invariants of a root gerbe. We also obtain a K-theoretic relative/orbifold correspondence in genus 0$0$.

We show that the Virasoro conjecture in Gromov–Witten theory holds for the the total space of a toric bundle E \to B$E \to B$ if and only if it holds for the base B. The main steps are: (i) We establish a localization formula that expresses Gromov–Witten invariants of E, equivariant with respect to the fiberwise torus action in terms of genus-zero invariants of the toric fiber and all-genus invariants of B, and (ii) we pass to the nonequivariant limit in this formula, using Brown’s mirror theorem for toric bundles.

Quasi-BPS categories appear as summands in semiorthogonal decompositions of DT categories for Hilbert schemes of points in the three-dimensional affine space and in the categorical Hall algebra of the two-dimensional affine space. In this paper, we prove several properties of quasi-BPS categories analogous to BPS sheaves in cohomological DT theory.

We first prove a categorical analogue of Davison’s support lemma, namely that complexes in the quasi-BPS categories for coprime length and weight are supported over the small diagonal in the symmetric product of the three-dimensional affine space. The categorical support lemma is used to determine the torsion-free generator of the torus equivariant K-theory of the quasi-BPS category of coprime length and weight.

We next construct a bialgebra structure on the torsion free equivariant K-theory of quasi-BPS categories for a fixed ratio of length and weight. We define the K-theoretic BPS space as the space of primitive elements with respect to the coproduct. We show that all localized equivariant K-theoretic BPS spaces are one-dimensional, which is a K-theoretic analogue of the computation of (numerical) BPS invariants of the three-dimensional affine space.

In this paper, we provide a new approach to prove some weighted-blowup formulae for genus zero orbifold Gromov–Witten invariants. As a consequence, we show the invariance of symplectically rational connectedness with respect to weighted-blowup along positive centers. Furthermore, we use this method to give a new proof to the genus zero relative-orbifold correspondence of Gromov–Witten invariants.