Gromov-Witten Theory Of Orbifolds Research Articles

Quantum Curve and Bilinear Fermionic Form for the Orbifold Gromov–Witten Theory of ℙ[r

Quantum Curve and Bilinear Fermionic Form for the Orbifold Gromov–Witten Theory of ℙ[r

Double Ramification Cycles with Orbifold Targets

In this paper, we consider double ramification cycles with orbifold targets. An explicit formula for double ramification cycles with orbifold targets, which is parallel to and generalizes the one known for the smooth case, is provided. Some applications for orbifold Gromov—Witten theory are also included.

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 $[\mathbb{C}^2/\mathbb{Z}_{n+1}]\times \mathbb{P}^1$.

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.

On the polynomiality of orbifold Gromov–Witten theory of root stacks

In [25], higher genus Gromov–Witten invariants of the stack of r-th roots of a smooth projective variety X along a smooth divisor D are shown to be polynomials in r. In this paper we study the degrees and coefficients of these polynomials.

Orbifold Gromov-Witten theory of weighted blowups

Consider a compact symplectic sub-orbifold groupoid $\sf S$ of a compact symplectic orbifold groupoid $(\mathsf X,\omega)$. Let $\mathsf X_{\mathfrak a}$ be the weight-$\mathfrak a$ blowup of $\sf X$ along $\sf S$, and $\mathsf D_{\mathfrak a}=\mathsf{PN}_{\mathfrak a}$ be the exceptional divisor, where $\sf N$ is the normal bundle of $\sf S$ in $\sf X$. In this paper we show that the absolute orbifold Gromov--Witten theory of $\mathsf X_{\mathfrak a}$ can be effectively and uniquely reconstructed from the absolute orbifold Gromov--Witten theories of $\sf X$, $\sf S$ and $\mathsf D_{\mathfrak a}$, the natural restriction homomorphism $H^*_{\text{CR}}({\sf X})\rightarrow H^*_{\text{CR}}({\sf S})$ and the first Chern class of the tautological line bundle over $\mathsf D_{\mathfrak a}$. To achieve this we first prove similar results for the relative orbifold Gromov--Witten theories of $(\mathsf X_{\mathfrak a}|\mathsf D_{\mathfrak a})$ and $(\mathsf N_{\mathfrak a}|\mathsf D_{\mathfrak a})$. As applications of these results, we prove an orbifold version of a conjecture of Maulik--Pandharipande on the Gromov--Witten theory of blowups along complete intersections, a conjecture on the Gromov--Witten theory of root constructions and a conjecture on Leray--Hirsch result for orbifold Gromov--Witten theory of Tseng--You.

Exterior powers of the adjoint representation and the Weyl ring of E8

I derive explicitly all polynomial relations in the character ring of E8 of the form χ∧ke8−pk(χ1,…,χ8)=0, where ∧ke8 is an arbitrary exterior power of the adjoint representation and χi is the ith fundamental character. This has simultaneous implications for the theory of relativistic integrable systems, Seiberg–Witten theory, quantum topology, orbifold Gromov–Witten theory, and the arithmetic of elliptic curves. The solution is obtained by reducing the problem to a (large, but finite) dimensional linear problem, which is amenable to an efficient solution via distributed computation.

Stacky GKM graphs and orbifold Gromov–Witten theory

A smooth GKM stack is a smooth Deligne-Mumford stack equipped with an action of an algebraic torus $T$, with only finitely many zero-dimensional and one-dimensional orbits.(i) We define the stacky GKM graph of a smooth GKM stack, under the mild assumption that any one-dimensional $T$-orbit closure contains at least one $T$ fixed point. The stacky GKM graph is a decorated graph which contains enough information to reconstruct the $T$-equivariant formal neighborhood of the $1$-skeleton (union of zero-dimensional and one-dimensional $T$-orbits) as a formal smooth DM stack equipped with a $T$-action.(ii) We axiomize the definition of a stacky GKM graph and introduce abstract stacky GKM graphs which are more general than stacky GKM graphs of honest smooth GKM stacks. From an abstract GKM graph we construct a formal smooth GKM stack.(iii) We define equivariant orbifold Gromov–Witten invariants of smooth GKM stacks, as well as formal equivariant orbifold Gromov–Witten invariants of formal smooth GKM stacks. These invariants can be computed by virtual localization and depend only the stacky GKM graph or the abstract stacky GKM graph. Formal equivariant orbifold Gromov–Witten invariants of the stacky GKM graph of a smooth GKM stack $\mathcal{X}$ are refinements of equivariant orbifold Gromov–Witten invariants of $\mathcal{X}$.

Crepant resolution and the holomorphic anomaly equation for [C3/Z3

We study the orbifold Gromov–Witten theory of the quotient [ C 3 / Z 3 ] in all genera. Our first result is a proof of the holomorphic anomaly equations in the precise form predicted by B-model physics. Our second result is an exact crepant resolution correspondence relating the Gromov–Witten theories of [ C 3 / Z 3 ] and local P 2 . The proof of the correspondence requires an identity proven in the Appendix by T. Coates and H. Iritani.

HIGHER GENUS GROMOV–WITTEN THEORY OF AND ASSOCIATED TO LOCAL CURVES

We study the higher genus equivariant Gromov–Witten theory of the Hilbert scheme of $n$ points of $\mathbb{C}^{2}$ . Since the equivariant quantum cohomology, computed by Okounkov and Pandharipande [Invent. Math. 179 (2010), 523–557], is semisimple, the higher genus theory is determined by an $\mathsf{R}$ -matrix via the Givental–Teleman classification of Cohomological Field Theories (CohFTs). We uniquely specify the required $\mathsf{R}$ -matrix by explicit data in degree $0$ . As a consequence, we lift the basic triangle of equivalences relating the equivariant quantum cohomology of the Hilbert scheme $\mathsf{Hilb}^{n}(\mathbb{C}^{2})$ and the Gromov–Witten/Donaldson–Thomas correspondence for 3-fold theories of local curves to a triangle of equivalences in all higher genera. The proof uses the analytic continuation of the fundamental solution of the QDE of the Hilbert scheme of points determined by Okounkov and Pandharipande [Transform. Groups 15 (2010), 965–982]. The GW/DT edge of the triangle in higher genus concerns new CohFTs defined by varying the 3-fold local curve in the moduli space of stable curves. The equivariant orbifold Gromov–Witten theory of the symmetric product $\mathsf{Sym}^{n}(\mathbb{C}^{2})$ is also shown to be equivalent to the theories of the triangle in all genera. The result establishes a complete case of the crepant resolution conjecture [Bryan and Graber, Algebraic Geometry–Seattle 2005, Part 1, Proceedings of Symposia in Pure Mathematics, 80 (American Mathematical Society, Providence, RI, 2009), 23–42; Coates et al., Geom. Topol. 13 (2009), 2675–2744; Coates & Ruan, Ann. Inst. Fourier (Grenoble) 63 (2013), 431–478].

On orbifold Gromov–Witten theory in codimension one

We propose a conjectural determination of the Gromov–Witten theory of a root stack along a smooth divisor. We verify our conjecture under an additional assumption.

Mirror symmetry for orbifold Hurwitz numbers

We study mirror symmetry for orbifold Hurwitz numbers. We show that the Laplace transform of orbifold Hurwitz numbers satisfy a dierential recursion, which is then proved to be equivalent to the integral recursion of Eynard and Orantin with spectral curve given by the r-Lambert curve. We argue that the r-Lambert curve also arises in the innite framing limit of orbifold Gromov-Witten theory of (C 3 =(Z=rZ)). Finally, we prove that the mirror model to orbifold Hurwitz numbers admits a quantum curve.

The tropical vertex

Elements of the tropical vertex group, introduced by Kontsevich and Soibelman, are formal families of symplectomorphisms of the 2-dimensional algebraic torus. We prove ordered product factorizations in the tropical vertex group are equivalent to calculations of certain genus 0 relative Gromov-Witten invariants of toric surfaces. The relative invariants which arise have full tangency to a toric divisor at a single unspecified point. The method uses scattering diagrams, tropical curve counts, degeneration formulas, and exact multiple cover calculations in orbifold Gromov-Witten theory.

On Virasoro Constraints for Orbifold Gromov-Witten Theory

Virasoro constraints for orbifold Gromov-Witten theory are described. These constraints are applied to the degree zero, genus zero orbifold Gromov-Witten potentials of the weighted projective stacks , , and to obtain formulas of descendant cyclic Hurwitz-Hodge integrals.

On the mathematics and physics of high genus invariants of $[\mathbb{C}^3 / \mathbb{Z}_3

This paper wishes to foster communication between mathematicians and physicists working in mirror symmetry and orbifold Gromov-Witten theory. We provide a reader friendly review of the physics computation in [arXiv:hep-th/0607100] that predicts Gromov-Witten invariants of [C^3/Z_3] in arbitrary genus, and of the mathematical framework for expressing these invariants as Hodge integrals. Using geometric properties of the Hodge classes, we compute the unpointed invariants for g=2,3, thus providing the first high genus mathematical check of the physics predictions.

Generating functions for Hurwitz–Hodge integrals

We describe explicit generating functions for a large class of Hurwitz–Hodge integrals. These are integrals of tautological classes on moduli spaces of admissible covers, a (stackily) smooth compactification of the Hurwitz schemes. Admissible covers and their tautological classes are interesting mathematical objects on their own, but recently they have proved to be a useful tool for the study of the tautological ring of the moduli space of curves, and the orbifold Gromov–Witten theory of DM stacks.