Gromov-Witten Research Articles

Algebraic reduced genus one Gromov–Witten invariants for complete intersections in projective spaces, Part 2

In Lee and Oh (Preprint, arXiv:1809.10995, 2018), we provided an algebraic proof of Zinger's comparison formula (Geom. Topol. 12 (2008), no. 2, 1203–1241; J. Differential Geom. 83 (2009), no. 2, 407–460) between genus one Gromov–Witten invariants and reduced invariants when the target space is a complete intersection of dimension 2 or 3 in a projective space. In this paper, we extend the proof in Lee and Oh (Preprint, arXiv:1809.10995, 2018) to all dimensions and to descendant invariants.

Counts of (tropical) curves in $E \times \mathbb{P}^1$ and Feynman integrals

We study generating series of Gromov–Witten invariants of E \times \mathbb{P}^1 and their tropical counterparts. Using tropical degeneration and floor diagram techniques, we can express the generating series as sums of Feynman integrals, where each summand corresponds to a certain type of graph which we call a \textit{pearl chain} . The individual summands are – just as in the case of mirror symmetry of elliptic curves, where the generating series of Hurwitz numbers equals a sum of Feynman integrals – complex analytic path integrals involving a product of propagators (equal to the Weierstrass- \wp -function plus an Eisenstein series). We also use pearl chains to study generating functions of counts of tropical curves in E_\mathbb{T}\times\mathbb{P}^1_\mathbb{T} of so-called \textit{leaky degree}.

Categorical primitive forms of Calabi–Yau $$A_\infty $$-categories with semi-simple cohomology

We study categorical primitive forms for Calabi–Yau $$A_\infty $$ categories with semi-simple Hochschild cohomology. We classify these primitive forms in terms of certain grading operators on the Hochschild homology. We use this result to prove that, if the Fukaya category $${{\textsf {Fuk}}}(M)$$ of a symplectic manifold M has semi-simple Hochschild cohomology, then its genus zero Gromov–Witten invariants may be recovered from the $$A_\infty $$ -category $${{\textsf {Fuk}}}(M)$$ together with the closed-open map. An immediate corollary of this is that in the semi-simple case, homological mirror symmetry implies enumerative mirror symmetry.

THE LOCAL-ORBIFOLD CORRESPONDENCE FOR SIMPLE NORMAL CROSSING PAIRS

AbstractFor X a smooth projective variety and $D=D_1+\dotsb +D_n$ a simple normal crossing divisor, we establish a precise cycle-level correspondence between the genus $0$ local Gromov–Witten theory of the bundle $\oplus _{i=1}^n \mathcal {O}_X(-D_i)$ and the maximal contact Gromov–Witten theory of the multiroot stack $X_{D,\vec r}$ . The proof is an implementation of the rank-reduction strategy. We use this point of view to clarify the relationship between logarithmic and orbifold invariants.

Narrow quantum D-modules and quantum Serre duality

Given Y a non-compact manifold or orbifold, we define a natural subspace of the cohomology of Y called the narrow cohomology. We show that despite Y being non-compact, there is a well-defined and non-degenerate pairing on this subspace. The narrow cohomology proves useful for the study of genus zero Gromov-Witten theory. When Y is a smooth complex variety or Deligne-Mumford stack, one can define a quantum D-module on the narrow cohomology of Y. This yields a new formulation of quantum Serre duality.

Holomorphic anomalies, fourfolds and fluxes

We investigate holomorphic anomalies of partition functions underlying string compactifications on Calabi-Yau fourfolds with background fluxes. For elliptic fourfolds the partition functions have an alternative interpretation as elliptic genera of N = 1 supersymmetric string theories in four dimensions, or as generating functions for relative, genus zero Gromov-Witten invariants of fourfolds with fluxes. We derive the holomorphic anomaly equations by starting from the BCOV formalism of topological strings, and translating them into geometrical terms. The result can be recast into modular and elliptic anomaly equations. As a new feature, as compared to threefolds, we find an extra contribution which is given by a gravitational descendant invariant. This leads to linear terms in the anomaly equations, which support an algebra of derivatives mapping between partition functions of the various flux sectors. These geometric features are mirrored by certain properties of quasi-Jacobi forms. We also offer an interpretation of the physics from the viewpoint of the worldsheet theory, and comment on holomorphic anomalies at genus one.

Gromov–Witten theory of [ℂ2∕ℤn+1] × ℙ1

We compute the relative orbifold Gromov-Witten invariants of $[\mathbb{C}^2/\mathbb{Z}_{n+1}]\times \mathbb{P}^1$, with respect to vertical fibers. Via a vanishing property of the Hurwitz-Hodge bundle, 2-point rubber invariants are calculated explicitly using Pixton's formula for the double ramification cycle, and the orbifold quantum Riemann-Roch. As a result parallel to its crepant resolution counterpart for $\mathcal{A}_n$, the GW/DT/Hilb/Sym correspondence is established for $[\mathbb{C}^2/\mathbb{Z}_{n+1}]$. The computation also implies the crepant resolution conjecture for relative orbifold Gromov-Witten theory of $[\mathbb{C}^2/\mathbb{Z}_{n+1}]\times \mathbb{P}^1$.

Gromov-Witten invariants of [formula omitted] coupled to a KdV tau function

We consider the pull-back of a natural sequence of cohomology classes Θg,n∈H2(2g−2+n)(M‾g,n,Q) to the moduli space of stable maps M‾g,n(P1,d). These classes are related to the Brézin-Gross-Witten tau function of the KdV hierarchy via ZBGW(ħ,t0,t1,...)=exp⁡∑ħ2g−2n!∫M‾g,nΘg,n⋅∏j=1nψjkj∏tkj. Insertions of the pull-backs of the classes Θg,n into the integrals defining Gromov-Witten invariants define new invariants which we show in the case of target P1 are given by a random matrix integral and satisfy the Toda equation.

Bilinear equations in Darboux transformations by Boson–Fermion correspondence

Bilinear equation is an important property for integrable nonlinear evolution equation. Many famous research objects in mathematical physics, such as Gromov–Witten invariants, can be described in terms of bilinear equations to show their connections with the integrable systems. Here in this paper, we mainly discuss the bilinear equations of the transformed tau functions under the successive applications of the Darboux transformations for the KP hierarchy, the modified KP hierarchy (Kupershmidt–Kiso version) and the BKP hierarchy, by the method of the Boson–Fermion correspondence. The Darboux transformations are considered in the Fermionic picture, by multiplying the different Fermionic fields on the tau functions. Here the Fermionic fields are corresponding to the (adjoint) eigenfunctions, whose changes under the Darboux transformations are showed to be the ones of the squared eigenfunction potentials in the Bosonic picture, used in the spectral representations of the (adjoint) eigenfunctions. Then the successive applications of the Darboux transformations are given in the Fermionic picture. Based upon this, some new bilinear equations in the Darboux chain are derived, besides the ones of (l−l′) -th modified KP hierarchy. The corresponding examples of these new bilinear equations are given.

Tri-Hamiltonian structure of the Ablowitz–Ladik hierarchy

We construct a local tri-Hamiltonian structure of the Ablowitz–Ladik hierarchy, and compute the central invariants of the associated bihamiltonian structures. We show that the central invariants of one of the bihamiltonian structures are equal to 124, and the dispersionless limit of this bihamiltonian structure coincides with the one that is defined on the jet space of the Frobenius manifold associated with the Gromov–Witten invariants of local P1. This result provides support for the validity of Brini’s conjecture on the relation of these Gromov–Witten invariants with the Ablowitz–Ladik hierarchy.

A Geometric Depiction of Solomon–Tukachinsky’s Construction of Open Gromov–Witten Invariants

A Geometric Depiction of Solomon–Tukachinsky’s Construction of Open Gromov–Witten Invariants

Effective computations of the Atiyah-Bott formula

We present an implementation of the Atiyah-Bott residue formula for M‾0,m(Pn,d). We use this implementation to compute a large number of Gromov-Witten invariants of genus 0, including intersection numbers of rational curves on general complete intersections. We also compute some numbers of rational contact curves satisfying suitable Schubert conditions. Our computations confirm known predictions made by Mirror Symmetry. The code we developed for these problems is publicly available and can also be used for other types of computations.

On the log–local principle for the toric boundary

Let X $X$ be a smooth projective complex variety and let D = D 1 + ⋯ + D l $D=D_1+\cdots +D_l$ be a reduced normal crossing divisor on X $X$ with each component D j $D_j$ smooth, irreducible and numerically effective. The log–local principle put forward in van Garrel et al. (Adv. Math. 350 (2019) 860–876) conjectures that the genus 0 log Gromov–Witten theory of maximal tangency of ( X , D ) $(X,D)$ is equivalent to the genus 0 local Gromov–Witten theory of X $X$ twisted by ⨁ j = 1 l O ( − D j ) $\bigoplus _{j=1}^{l}\mathcal {O}(-D_j)$ . We prove that an extension of the log–local principle holds for X $X$ a (not necessarily smooth) Q $\mathbb {Q}$ -factorial projective toric variety, D $D$ the toric boundary, and descendant point insertions.

Tropical Jucys covers

AbstractWe study monotone and strictly monotone Hurwitz numbers from a bosonic Fock space perspective. This yields to an interpretation in terms of tropical geometry involving local multiplicities given by Gromov-Witten invariants. Furthermore, this enables us to prove that a main result of Cavalieri-Johnson-Markwig-Ranganathan is actually equivalent to the Gromov-Witten/Hurwitz correspondence by Okounkov-Pandharipande for the equivariant Riemann sphere.

The spaces of rational curves on del Pezzo threefolds of degree one

We prove the irreducibility of moduli spaces of rational curves on a general del Pezzo threefold of Picard rank 1 and degree 1. As corollaries, we confirm Geometric Manin’s conjecture and enumerativity of certain Gromov–Witten invariants for these threefolds.

Gromov–Witten theory with maximal contacts

AbstractWe propose an intersection-theoretic method to reduce questions in genus 0 logarithmic Gromov–Witten theory to questions in the Gromov–Witten theory of smooth pairs, in the presence of positivity. The method is applied to the enumerative geometry of rational curves with maximal contact orders along a simple normal crossings divisor and to recent questions about its relationship to local curve counting. Three results are established. We produce counterexamples to the local-logarithmic conjectures of van Garrel–Graber–Ruddat and Tseng–You. We prove that a weak form of the conjecture holds for product geometries. Finally, we explicitly determine the difference between local and logarithmic theories, in terms of relative invariants for which efficient algorithms are known. The polyhedral geometry of the tropical moduli of maps plays an essential and intricate role in the analysis.

The Abelian/Nonabelian Correspondence and Gromov–Witten Invariants of Blow-Ups

Abstract We prove the abelian/nonabelian correspondence with bundles for target spaces that are partial flag bundles, combining and generalising results by Ciocan-Fontanine–Kim–Sabbah, Brown, and Oh. From this, we deduce how genus-zero Gromov–Witten invariants change when a smooth projective variety X is blown up in a complete intersection defined by convex line bundles. In the case where the blow-up is Fano, our result gives closed-form expressions for certain genus-zero invariants of the blow-up in terms of invariants of X. We also give a reformulation of the abelian/nonabelian Correspondence in terms of Givental’s formalism, which may be of independent interest.

Gromov–Witten theory and Noether–Lefschetz theory for holomorphic-symplectic varieties

Abstract We use Noether–Lefschetz theory to study the reduced Gromov–Witten invariants of a holomorphic-symplectic variety of $K3^{[n]}$ -type. This yields strong evidence for a new conjectural formula that expresses Gromov–Witten invariants of this geometry for arbitrary classes in terms of primitive classes. The formula generalizes an earlier conjecture by Pandharipande and the author for K3 surfaces. Using Gromov–Witten techniques, we also determine the generating series of Noether–Lefschetz numbers of a general pencil of Debarre–Voisin varieties. This reproves and extends a result of Debarre, Han, O’Grady and Voisin on Hassett–Looijenga–Shah (HLS) divisors on the moduli space of Debarre–Voisin fourfolds.

Virasoro constraints for stable pairs on toric threefolds

Abstract Using new explicit formulas for the stationary Gromov–Witten/Pandharipande–Thomas ( $\mathrm {GW}/{\mathrm {PT}}$ ) descendent correspondence for nonsingular projective toric threefolds, we show that the correspondence intertwines the Virasoro constraints in Gromov–Witten theory for stable maps with the Virasoro constraints for stable pairs proposed in [18]. Since the Virasoro constraints in Gromov–Witten theory are known to hold in the toric case, we establish the stationary Virasoro constraints for the theory of stable pairs on toric threefolds. As a consequence, new Virasoro constraints for tautological integrals over Hilbert schemes of points on surfaces are also obtained.

Open-closed Gromov-Witten invariants of 3-dimensional Calabi-Yau smooth toric DM stacks

Abstract We study open-closed orbifold Gromov-Witten invariants of 3-dimensional Calabi-Yau smooth toric Deligne-Mumford stacks (with possibly nontrivial generic stabilisers K and semi-projective coarse moduli spaces) relative to Lagrangian branes of Aganagic-Vafa type. An Aganagic-Vafa brane in this paper is a possibly ineffective $C^\infty $ orbifold that admits a presentation $[(S^1\times \mathbb {R} ^2)/G_\tau ]$ , where $G_\tau $ is a finite abelian group containing K and $G_\tau /K \cong \boldsymbol {\mu }_{\mathfrak {m}}$ is cyclic of some order $\mathfrak {m}\in \mathbb {Z} _{>0}$ . 1. We present foundational materials of enumerative geometry of stable holomorphic maps from bordered orbifold Riemann surfaces to a 3-dimensional Calabi-Yau smooth toric DM stack $\mathcal {X}$ with boundaries mapped into an Aganagic-Vafa brane $\mathcal {L}$ . All genus open-closed Gromov-Witten invariants of $\mathcal {X}$ relative to $\mathcal {L}$ are defined by torus localisation and depend on the choice of a framing $f\in \mathbb {Z} $ of $\mathcal {L}$ . 2. We provide another definition of all genus open-closed Gromov-Witten invariants in (1) based on algebraic relative orbifold Gromov-Witten theory, which agrees with the definition in (1) up to a sign depending on the choice of orientation on moduli of maps in (1). This generalises the definition in [57] for smooth toric Calabi-Yau 3-folds and specifies an orientation on moduli of maps in (1) compatible with the canonical orientation on moduli of relative stable maps determined by the complex structure. 3. When $\mathcal {X}$ is a toric Calabi-Yau 3-orbifold (i.e., when the generic stabiliser K is trivial), so that $G_\tau =\boldsymbol {\mu }_{\mathfrak {m}}$ , we define generating functions $F_{g,h}^{\mathcal {X},(\mathcal {L},f)}$ of open-closed Gromov-Witten invariants of arbitrary genus g and number h of boundary circles; it takes values in $H^*_{ {\mathrm {CR}} }(\mathcal {B} \boldsymbol {\mu }_{\mathfrak {m}}; \mathbb {C} )^{\otimes h}$ , where $H^*_{ {\mathrm {CR}} }(\mathcal {B} \boldsymbol {\mu }_{\mathfrak {m}}; \mathbb {C} )\cong \mathbb {C} ^{\mathfrak {m}}$ is the Chen-Ruan orbifold cohomology of the classifying space $\mathcal {B} \boldsymbol {\mu }_{\mathfrak {m}}$ of $\boldsymbol {\mu }_{\mathfrak {m}}$ . 4. We prove an open mirror theorem that relates the generating function $F_{0,1}^{\mathcal {X},(\mathcal {L},f)}$ of orbifold disk invariants to Abel-Jacobi maps of the mirror curve of $\mathcal {X}$ . This generalises a conjecture by Aganagic-Vafa [6] and Aganagic-Klemm-Vafa [5] (proved in full generality by the first and the second authors in [33]) on the disk potential of a smooth semi-projective toric Calabi-Yau 3-fold.