Gromov-Witten Research Articles

Enumerative geometry of surfaces and topological strings

This survey covers recent developments on the geometry and physics of Looijenga pairs, namely pairs [Formula: see text] with [Formula: see text] a complex algebraic surface and [Formula: see text] a singular anticanonical divisor in it. I will describe a surprising web of correspondences linking together several a priori distant classes of enumerative invariants associated to [Formula: see text], including the log Gromov–Witten invariants of the pair, the Gromov–Witten invariants of an associated higher dimensional Calabi–Yau variety, the open Gromov–Witten invariants of certain special Lagrangians in toric Calabi–Yau threefolds, the Donaldson–Thomas theory of a class of symmetric quivers, and certain open and closed BPS-type invariants. I will also discuss how these correspondences can be effectively used to provide a complete closed-form solution to the calculation of all these invariants.

Generalized ILW hierarchy: solutions and limit to extended lattice GD hierarchy

The intermediate long wave (ILW) hierarchy and its generalization, labelled by a positive integer N, can be formulated as reductions of the lattice KP hierarchy. The integrability of the lattice KP hierarchy is inherited by these reduced systems. In particular, all solutions can be captured by a factorization problem of difference operators. A special solution among them is obtained from Okounkov and Pandharipande’s dressing operators for the equivariant Gromov-Witten theory of . This indicates a hidden link with the equivariant Toda hierarchy. The generalized ILW hierarchy is also related to the lattice Gelfand-Dickey hierarchy and its extension by logarithmic flows. The logarithmic flows can be derived from the generalized ILW hierarchy by a scaling limit as a parameter of the system tends to 0. This explains an origin of the logarithmic flows. A similar scaling limit of the equivariant Toda hierarchy yields the extended 1D/bigraded Toda hierarchy.

Gromov–Witten theory of complete intersections via nodal invariants

AbstractWe provide an inductive algorithm computing Gromov–Witten invariants in all genera with arbitrary insertions of all smooth complete intersections in projective space. We also prove that all Gromov–Witten classes of all smooth complete intersections in projective space belong to the tautological ring of the moduli space of stable curves. The main idea is to show that invariants with insertions of primitive cohomology classes are controlled by their monodromy and by invariants defined without primitive insertions but with imposed nodes in the domain curve. To compute these nodal Gromov–Witten invariants, we introduce the new notion of nodal relative Gromov–Witten invariants. We then prove a nodal degeneration formula and a relative splitting formula. These results for nodal relative Gromov–Witten theory are stated in complete generality and are of independent interest.

A Gromov–Witten Theory for Simple Normal-Crossing Pairs Without Log Geometry

We define a new Gromov–Witten theory relative to simple normal crossing divisors as a limit of Gromov–Witten theory of multi-root stacks. Several structural properties are proved including relative quantum cohomology, Givental formalism, Virasoro constraints (genus zero) and a partial cohomological field theory. Furthermore, we use the degree zero part of the relative quantum cohomology to provide an alternative mirror construction of Gross and Siebert (Intrinsic mirror symmetry, arXiv:1909.07649) and to prove the Frobenius structure conjecture of Gross et al. (Publ Math Inst Hautes Études Sci 122:65–168, 2015) in our setting.

Geometrical proof of generalized mirror transformation of projective hypersurfaces

In this paper, we propose a geometrical proof of the generalized mirror transformation of genus [Formula: see text] Gromov–Witten invariants of degree [Formula: see text] hypersurface in [Formula: see text].

The Doran-Harder-Thompson conjecture for toric complete intersections

Given a Tyurin degeneration of a Calabi-Yau complete intersection in a toric variety, we prove gluing formulas relating the generalized functional invariants, periods, and I-functions of the mirror Calabi-Yau family and those of the two mirror Landau-Ginzburg models. Our proof makes explicit the “gluing/splitting” of fibrations in the Doran-Harder-Thompson mirror conjecture. Our gluing formula implies an identity, obtained by composition with their respective mirror maps, that relates the absolute Gromov-Witten invariants for the Calabi-Yaus and relative Gromov-Witten invariants for the quasi-Fanos.

Invariant probability measures from pseudoholomorphic curves Ⅱ: Pseudoholomorphic curve constructions

In the previous work, we introduced a method for constructing invariant probability measures of a large class of non-singular volume-preserving flows on closed, oriented odd-dimensional smooth manifolds with pseudoholomorphic curve techniques from symplectic geometry. The technique requires existence of certain pseudoholomorphic curves satisfying some weak assumptions. In this work, we appeal to Gromov–Witten theory and Seiberg–Witten theory to construct large classes of examples where these pseudoholomorphic curves exist. Our argument uses neck stretching along with new analytical tools from Fish–Hofer's work on feral pseudoholomorphic curves.

Extremal Gromov-Witten invariants of the Hilbert scheme of Points

AbstractWe determine all the extremal Gromov-Witten invariants of the Hilbert scheme of$3$points on a smooth projective complex surface. Our result for the genus-$1$case verifies a conjecture that we propose for the genus-$1$extremal Gromov-Witten invariant of the Hilbert scheme ofnpoints withnbeing arbitrary. The main ideas in the proofs are to use geometric arguments involving the cosection localization theory of Kiem and J. Li [17, 23], algebraic manipulations related to the Heisenberg operators of Grojnowski [13] and Nakajima [34], and the virtual localization formulas of Gromov-Witten theory [12, 20, 30].

The higher-dimensional tropical vertex

We study log Calabi-Yau varieties obtained as a blow-up of a toric variety along hypersurfaces in its toric boundary. Mirrors to such varieties are constructed by Gross-Siebert from a canonical scattering diagram built by using punctured log Gromov-Witten invariants of Abramovich-Chen-Gross-Siebert. We show that there is a piecewise linear isomorphism between the canonical scattering diagram and a scattering diagram defined algortihmically, following a higher dimensional generalisation of the Kontsevich-Soibelman construction. We deduce that the punctured log Gromov-Witten invariants of the log Calabi-Yau variety can be captured from this algorithmic construction. As a particular example, we compute these invariants for a non-toric blow-up of the three dimensional projective space along two lines. This generalizes previous results of Gross-Pandharipande-Siebert on "The Tropical Vertex" to higher dimensions.

On equivariant Gromov–Witten invariants of resolved conifold with diagonal and anti-diagonal actions

We propose two conjectural relationships between the equivariant Gromov-Witten invariants of the resolved conifold under diagonal and anti-diagonal actions and the Gromov-Witten invariants of $\mathbb{P}^1$, and verify their validity in genus zero approximation. We also provide evidences to support the validity of these relationships in genus one and genus two.

Strong Positivity for the Skein Algebras of the 4-Punctured Sphere and of the 1-Punctured Torus

The Kauffman bracket skein algebra is a quantization of the algebra of regular functions on the $${\textit{SL}}_2$$ character variety of a topological surface. We realize the skein algebra of the 4-punctured sphere as the output of a mirror symmetry construction based on higher genus Gromov–Witten theory and applied to a complex cubic surface. Using this result, we prove the positivity of the structure constants of the bracelets basis for the skein algebras of the 4-punctured sphere and of the 1-punctured torus. This connection between topology of the 4-punctured sphere and enumerative geometry of curves in cubic surfaces is a mathematical manifestation of the existence of dual descriptions in string/M-theory for the $$\mathcal {N}=2$$ $$N_f=4$$ SU(2) gauge theory.

Genus Two Siegel Quasi-Modular Forms and Gromov–Witten Theory of Toric Calabi–Yau Threefolds

We first develop theories of differential rings of quasi-Siegel modular and quasi-Siegel Jacobi forms for genus two. Then we apply them to the Eynard-Orantin topological recursion of certain local Calabi-Yau threefolds equipped with branes, whose mirror curves are genus two hyperelliptic curves. By the proof of the Remodeling Conjecture, we prove that the corresponding open- and closed- Gromov-Witten potentials are essentially quasi-Siegel Jacobi and quasi-Siegel modular forms for genus two, respectively.

A mirror theorem for multi-root stacks and applications

Let X be a smooth projective variety with a simple normal crossing divisor D:=D_1+D_2+cdots +D_n, where D_isubset X are smooth, irreducible and nef. We prove a mirror theorem for multi-root stacks X_{D,{overrightarrow{r}}} by constructing an I-function lying in a slice of Givental’s Lagrangian cone for Gromov–Witten theory of multi-root stacks. We provide three applications: (1) We show that some genus zero invariants of X_{D,overrightarrow{r}} stabilize for sufficiently large overrightarrow{r}. (2) We state a generalized local-log-orbifold principle conjecture and prove a version of it. (3) We show that regularized quantum periods of Fano varieties coincide with classical periods of the mirror Landau–Ginzburg potentials using orbifold invariants of X_{D,overrightarrow{r}}.

Logarithmic Gromov–Witten theory with expansions

We construct a version of relative Gromov-Witten theory with expanded degenerations in the normal crossings setting and establish a degeneration formula for the resulting invariants. Given a simple normal crossings pair $(X,D)$, we construct virtually smooth and proper moduli spaces of curves in $X$ with prescribed boundary conditions along $D$. Each point in such a moduli space parameterizes maps from nodal curves to expanded degenerations of $X$ that are dimensionally transverse to the strata. We use the expanded formalism to reconstruct the virtual class attached to a tropical map in terms of spaces of maps to expansions attached to the vertices.

Disk counting and wall-crossing phenomenon via family Floer theory

We use the wall-crossing formula in the non-archimedean SYZ mirror construction (arXiv: 2003.06106) to compute the Landau-Ginzburg superpotential and the one-pointed open Gromov-Witten invariants for a Chekanov-type Lagrangian torus in any smooth toric Fano compactification of C^n. It agrees with the works of Auroux, Chekanov-Schlenk, and Pascaleff-Tonkonog.

Tangent curves to degenerating hypersurfaces

Abstract We study the behaviour of rational curves tangent to a hypersurface under degenerations of the hypersurface. Working within the framework of logarithmic Gromov–Witten theory, we extend the degeneration formula to the logarithmically singular setting, producing a virtual class on the space of maps to the degenerate fibre. We then employ logarithmic deformation theory to express this class as an obstruction bundle integral over the moduli space of ordinary stable maps. This produces new refinements of the logarithmic Gromov–Witten invariants, encoding the degeneration behaviour of tangent curves. In the example of a smooth plane cubic degenerating to the toric boundary we employ localisation and tropical techniques to compute these refinements. Finally, we leverage these calculations to describe how embedded curves tangent to a smooth cubic degenerate as the cubic does; the results obtained are of a classical nature, but the proofs make essential use of logarithmic Gromov–Witten theory.

Difference equation for the Gromov-Witten potential of the resolved conifold

A difference equation is proved for the Gromov-Witten potential of the resolved conifold. Using the Gopakumar-Vafa resummation of the Gromov-Witten invariants of any Calabi-Yau threefold, it is further shown that similar difference equations are satisfied by the part of the resummed potential containing the contribution of the genus zero GV invariants.

Giant gravitons, Harish-Chandra integrals, and BPS states in symplectic and orthogonal mathcal{N} = 4 SYM

We find generating functions for half BPS correlators in mathcal{N} = 4 SYM theories with gauge groups Sp(2N), SO(2N + 1), and SO(2N) by computing the norms of a class of BPS coherent states. These coherent states are built from operators involving Harish-Chandra integrals. Such operators have an interpretation as localized giant gravitons in the bulk of anti-de-Sitter space. This extends the analysis of [1] to Sp(2N), SO(2N + 1), and SO(2N) gauge theories. We show that we may use ordinary Schur functions as a basis for the sector of states with no cross-caps in these theories. This is consistent with the construction of these theories as orientifold projections of an SU(2N) theory. We make note of some relations between the symmetric functions that appear in the expansion of these coherent states and symplectic Schur functions. We also comment on some connections to Schubert calculus and Gromov-Witten invariants, which suggest that the Harish-Chandra integral may be extended to such problems.

The Moduli Space of Stable Coherent Sheaves via Non-archimedean Geometry

We provide a construction of the moduli space of stable coherent sheaves in the world of non-archimedean geometry, where we use the notion of Berkovich non-archimedean analytic spaces. The motivation for our construction is Tony Yue Yu’s non-archimedean enumerative geometry in Gromov—Witten theory. The construction of the moduli space of stable sheaves using Berkovich analytic spaces will give rise to the non-archimedean version of Donaldson—Thomas invariants. In this paper we give the moduli construction over a non-archimedean field $${\mathbb{K}}$$ . We use the machinery of formal schemes, that is, we define and construct the formal moduli stack of (semi)-stable coherent sheaves over a discrete valuation ring R, and taking generic fiber we get the non-archimedean analytic moduli space of semistable coherent sheaves over the fractional non-archimedean field $${\mathbb{K}}$$ . We generalize Joyce’s d-critical scheme structure in [37] or Kiem—Li’s virtual critical manifolds in [38] to the world of formal schemes, and Berkovich non-archimedean analytic spaces. As an application, we provide a proof for the motivic localization formula for a d-critical non-archimedean $${\mathbb{K}}$$ -analytic space using global motive of vanishing cycles and motivic integration on oriented formal d-critical schemes. This generalizes Maulik’s motivic localization formula for the motivic Donaldson—Thomas invariants.

Open/closed correspondence via relative/local correspondence

We establish a correspondence between the disk invariants of a smooth toric Calabi-Yau 3-fold X with boundary condition specified by a framed Aganagic-Vafa outer brane (L,f) and the genus-zero closed Gromov-Witten invariants of a smooth toric Calabi-Yau 4-fold X˜, proving the open/closed correspondence proposed by Mayr and developed by Lerche-Mayr. Our correspondence is the composition of two intermediate steps:•First, a correspondence between the disk invariants of (X,L,f) and the genus-zero maximally-tangent relative Gromov-Witten invariants of a relative Calabi-Yau 3-fold (Y,D), where Y is a toric partial compactification of X by adding a smooth toric divisor D. This correspondence can be obtained as a consequence of the topological vertex (Li-Liu-Liu-Zhou) and Fang-Liu where the all-genus open Gromov-Witten invariants of (X,L,f) are identified with the formal relative Gromov-Witten invariants of the formal completion of (Y,D) along the toric 1-skeleton. Here, we present a proof without resorting to formal geometry.•Second, a correspondence in genus zero between the maximally-tangent relative Gromov-Witten invariants of (Y,D) and the closed Gromov-Witten invariants of the toric Calabi-Yau 4-fold X˜=OY(−D). This can be viewed as an instantiation of the log-local principle of van Garrel-Graber-Ruddat in the non-compact setting.