Gromov-Witten Research Articles

Vortices on cylinders and warped exponential networks

We study 3d N=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {N}=2$$\\end{document}U(1) Chern–Simons-Matter QFT on a cylinder C×R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C\ imes \\mathbb {R}$$\\end{document}. The topology of C gives rise to BPS sectors of low-energy solitons known as kinky vortices, which interpolate between (possibly) different vacua at the ends of the cylinder and at the same time carry magnetic flux. We compute the spectrum of BPS vortices on the cylinder in an isolated Higgs vacuum, through the framework of warped exponential networks, which we introduce. We then conjecture a relation between these and standard vortices on R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^2$$\\end{document}, which are related to genus-zero open Gromov–Witten invariants of toric branes. More specifically, we show that in the limit of large Fayet–Iliopoulos coupling, the spectrum of kinky vortices on C undergoes an infinite sequence of wall-crossing transitions and eventually stabilizes. We then propose an exact relation between a generating series of stabilized CFIV indices and the Gromov–Witten disk potential and discuss its consequences for the structure of moduli spaces of vortices.

Global Kuranishi charts and a product formula in symplectic Gromov–Witten theory

Global Kuranishi charts and a product formula in symplectic Gromov–Witten theory

Tropical Refined Curve Counting with Descendants

We prove a q-refined tropical correspondence theorem for higher genus descendant logarithmic Gromov–Witten invariants with a λg\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda _g$$\\end{document} class in toric surfaces. Specifically, a generating series of such logarithmic Gromov–Witten invariants agrees with a q-refined count of rational tropical curves satisfying higher valency conditions. As a corollary, we obtain a geometric proof of the deformation invariance of this tropical count. In particular, our results give an algebro-geometric meaning to the tropical count defined by Blechman and Shustin. Our strategy is to use the logarithmic degeneration formula, and the key new technique is to reduce to computing integrals against double ramification cycles and connect these integrals to the non-commutative KdV hierarchy.

Quivers and curves in higher dimension

We prove a correspondence between Donaldson–Thomas invariants of quivers with potential having trivial attractor invariants and genus zero punctured Gromov–Witten invariants of holomorphic symplectic cluster varieties. The proof relies on the comparison of the stability scattering diagram, describing the wall-crossing behavior of Donaldson–Thomas invariants, with a scattering diagram capturing punctured Gromov–Witten invariants via tropical geometry.

X-y duality in topological recursion for exponential variables via quantum dilogarithm

For a given spectral curve, the theory of topological recursion generates two different families \omega_{g,n}ωg,n and \omega_{g,n}^\veeωg,n∨ of multi-differentials, which are for algebraic spectral curves related via the universal x-yx−y duality formula. We propose a formalism to extend the validity of the x-yx−y duality formula of topological recursion from algebraic curves to spectral curves with exponential variables of the form e^x=F(e^y)ex=F(ey) or e^x=F(y)e^{a y}ex=F(y)eay with FF rational and aa some complex number, which was in principle already observed in [Commun. Number Theory Phys. 13, 763 (2019); J. Lond. Math. Soc. 109, e12946 (2024)]. From topological recursion perspective the family \omega_{g,n}^\veeωg,n∨ would be trivial for these curves. However, we propose changing the n=1n=1 sector of \omega_{g,n}^\veeωg,n∨ via a version of the Faddeev’s quantum dilogarithm which will lead to the correct two families \omega_{g,n}ωg,n and \omega_{g,n}^\veeωg,n∨ related by the same x-yx−y duality formula as for algebraic curves. As a consequence, the x-yx−y symplectic transformation formula extends further to important examples governed by topological recursion including, for instance, Gromov-Witten invariants of \mathbb{C}^3ℂ3 (or, equivalently, triple Hodge integrals), orbifold Hurwitz numbers, and stationary Gromov-Witten invariants of \mathbb{P}^1ℙ1. The proposed formalism is related to the issue topological recursion encounters for specific choices of framings for the topological vertex curve.

Tropical Mirror

We describe the tropical curves in toric varieties and define the tropical Gromov-Witten invariants. We introduce amplitudes for the higher topological quantum mechanics (HTQM) on special trees and show that the amplitudes are equal to the tropical Gromov-Witten invariants. We show that the sum over the amplitudes in $A$-model HTQM equals the total amplitude in B-model HTQM, defined as a deformation of the $A$-model HTQM by the mirror superpotential. We derived the mirror superpotentials for the toric varieties and showed that they coincide with the superpotentials in the mirror Landau-Ginzburg theory. We construct the mirror dual states to the evaluation observables in the tropical Gromov-Witten theory.

Large N instantons from topological strings

The 1/N1/N expansion of matrix models is asymptotic, and it requires non-perturbative corrections due to large NN instantons. Explicit expressions for large NN instanton amplitudes are known in the case of Hermitian matrix models with one cut, but not in the multi-cut case. We show that the recent exact results on topological string instanton amplitudes provide the non-perturbative contributions of large NN instantons in generic multi-cut, Hermitian matrix models. We present a detailed test in the case of the cubic matrix model by considering the asymptotics of its 1/N1/N expansion, which we obtain at relatively high genus for a generic two-cut background. These results can be extended to certain non-conventional matrix models which admit a topological string theory description. As an application, we determine the large NN instanton corrections for the free energy of ABJM theory on the three-sphere, which correspond to D-brane instanton corrections in superstring theory. We also illustrate the applications of topological string instantons in a more mathematical setting by considering orbifold Gromov-Witten invariants. By focusing on the example of {\mathbb C}^3/{\mathbb Z}_3ℂ3/ℤ3, we show that they grow doubly-factorially with the genus and we obtain and test explicit asymptotic formulae for them.

Tropological sigma models

With the use of mathematical techniques of tropical geometry, it was shown by Mikhalkin some twenty years ago that certain Gromov-Witten invariants associated with topological quantum field theories of pseudoholomorphic maps can be computed by going to the tropical limit of the geometries in question. Here we examine this phenomenon from the physics perspective of topological quantum field theory in the path integral representation, beginning with the case of the topological sigma model before coupling it to topological gravity. We identify the tropicalization of the localization equations, investigate its geometry and symmetries, and study the theory and its observables using the standard cohomological BRST methods. We find that the worldsheet theory exhibits a nonrelativistic structure, similar to theories of the Lifshitz type. Its path-integral formulation does not require a worldsheet complex structure; instead, it is based on a worldsheet foliation structure.

A note on quantum K-theory of root constructions

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

Quantum K-theory of incidence varieties

We prove a conjecture of Buch and Mihalcea in the case of the incidence variety X=Fl(1,n-1;n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X=\ extrm{Fl}\\hspace{0.55542pt}(1,n-1;n)$$\\end{document} and determine the structure of its (T-equivariant) quantum K-theory ring. Our results are an interplay between geometry and combinatorics. The geometric side concerns Gromov–Witten varieties of 3-pointed genus 0 stable maps to X with markings sent to Schubert varieties, while on the combinatorial side are formulas for the (equivariant) quantum K-theory ring of X. We prove that the Gromov–Witten variety is rationally connected when one of the defining Schubert varieties is a divisor and another is a point. This implies that the (equivariant) K-theoretic Gromov–Witten invariants defined by two Schubert classes and a Schubert divisor class can be computed in the ordinary (equivariant) K-theory ring of X. We derive a positive Chevalley formula for the equivariant quantum K-theory ring of X and a positive closed formula for Littlewood–Richardson coefficients in the non-equivariant quantum K-theory ring of X. The Littlewood–Richardson rule in turn implies that non-empty Gromov–Witten varieties given by Schubert varieties in general position have arithmetic genus 0.

The Proper Landau–Ginzburg Potential, Intrinsic Mirror Symmetry and the Relative Mirror Map

Given a smooth log Calabi–Yau pair (X, D), we use the intrinsic mirror symmetry construction to define the mirror proper Landau–Ginzburg potential and show that it is a generating function of two-point relative Gromov–Witten invariants of (X, D). We compute certain relative invariants with several negative contact orders, and then apply the relative mirror theorem of Fan et al. (Sel Math (NS) 25(4): Art. 54, 25, 2019. https://doi.org/10.1007/s00029-019-0501-z) to compute two-point relative invariants. When D is nef, we compute the proper Landau–Ginzburg potential and show that it is the inverse of the relative mirror map. Specializing to the case of a toric variety X, this implies the conjecture of m Gräfnitz et al. (2022) that the proper Landau–Ginzburg potential is the open mirror map. When X is a Fano variety, the proper potential is related to the anti-derivative of the regularized quantum period.

Virasoro constraints for moduli of sheaves and vertex algebras

In enumerative geometry, Virasoro constraints were first conjectured in Gromov-Witten theory with many new recent developments in the sheaf theoretic context. In this paper, we rephrase the sheaf theoretic Virasoro constraints in terms of primary states coming from a natural conformal vector in Joyce’s vertex algebra. This shows that Virasoro constraints are preserved under wall-crossing. As an application, we prove the conjectural Virasoro constraints for moduli spaces of torsion-free sheaves on any curve and on surfaces with only (p,p)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(p,p)$\\end{document} cohomology classes by reducing the statements to the rank 1 case.

Universal Equations for Higher Genus Gromov–Witten Invariants from Hodge Integrals

Universal Equations for Higher Genus Gromov–Witten Invariants from Hodge Integrals

Gopakumar–Vafa Type Invariants of Holomorphic Symplectic 4-Folds

Using reduced Gromov–Witten theory, we define new invariants which capture the enumerative geometry of curves on holomorphic symplectic 4-folds. The invariants are analogous to the BPS counts of Gopakumar and Vafa for Calabi–Yau 3-folds, Klemm and Pandharipande for Calabi–Yau 4-folds, and Pandharipande and Zinger for Calabi–Yau 5-folds. We conjecture that our invariants are integers and give a sheaf-theoretic interpretation in terms of reduced 4-dimensional Donaldson–Thomas invariants of one-dimensional stable sheaves. We check our conjectures for the product of two K3 surfaces and for the cotangent bundle of P2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {P}}^2$$\\end{document}. Modulo the conjectural holomorphic anomaly equation, we compute our invariants also for the Hilbert scheme of two points on a K3 surface. This yields a conjectural formula for the number of isolated genus 2 curves of minimal degree on a very general hyperkähler 4-fold of K3[2]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K3^{[2]}$$\\end{document}-type. The formula may be viewed as a 4-dimensional analogue of the classical Yau–Zaslow formula concerning counts of rational curves on K3 surfaces. In the course of our computations, we also derive a new closed formula for the Fujiki constants of the Chern classes of tangent bundles of both Hilbert schemes of points on K3 surfaces and generalized Kummer varieties.

Correction to: d-elliptic loci in genus 2 and 3

Abstract We correct an error in Proposition 5.1 of the named paper, which was brought to the author’s attention by Aaron Pixton. This affects various other formulas in the paper, including Theorem 1.4, which are also corrected.

Logarithmic Gromov–Witten theory and double ramification cycles

Abstract We examine the logarithmic Gromov–Witten cycles of a toric variety relative to its full toric boundary. The cycles are expressed as products of double ramification cycles and natural tautological classes in the logarithmic Chow ring of the moduli space of curves. We introduce a simple new technique that relates the Gromov–Witten cycles of rigid and rubber geometries; the technique is based on a study of maps to the logarithmic algebraic torus. By combining this with recent work on logarithmic double ramification cycles, we deduce that all logarithmic Gromov–Witten pushforwards, for maps to a toric variety relative to its full toric boundary, lie in the tautological ring of the moduli space of curves. A feature of the approach is that it avoids the as yet undeveloped logarithmic virtual localization formula, instead relying directly on piecewise polynomial functions to capture the structure that would be provided by such a formula. The results give a common generalization of work of Faber and Pandharipande, and more recent work of Holmes and Schwarz as well as Molcho and Ranganathan. The proof passes through general structure results on the space of stable maps to the logarithmic algebraic torus, which may be of independent interest.

Twisted Sectors for Lagrangian Floer Theory on Symplectic Orbifolds

The notion of twisted sectors play a crucial role in orbifold Gromov-Witten theory. We introduce the notion of dihedral twisted sectors in order to construct Lagrangian Floer theory on symplectic orbifolds and discuss related issues.

Marked relative invariants and GW/PT correspondences

We introduce marked relative Pandharipande-Thomas (PT) invariants for a pair (X,D) of a smooth projective threefold and a smooth divisor. These invariants are defined by integration over the moduli space of r-marked stable pairs on (X,D), and appear naturally when degenerating diagonal insertions via the Li-Wu degeneration formula. We propose a Gromov-Witten (GW) / PT correspondence for marked relative invariants. We show compatibility of the conjecture with the degeneration formula and a splitting formula for relative diagonals. The results provide new tools to prove GW/PT correspondences for varieties with vanishing cohomology.As an application we prove the GW/PT correspondence for:(i)all Fano complete intersections, and(ii)the reduced theories of (S×C,S×{z1,…,zN}) where S is a K3 surface and C is a curve, for all curve classes which have divisibility at most 2 over the K3 surface.In the appendix we introduce a notion of higher-descendent invariants which can be seen as an analogue of the nodal Gromov-Witten invariants defined by Argüz, Bousseau, Pandharipande and Zvonkine in [3]. We show that the higher-descendent invariants reduce to marked relative invariants with diagonal insertions.

Virasoro Constraints for Toric Bundles

Abstract We show that the Virasoro conjecture in Gromov–Witten theory holds for the the total space of a toric bundle $E \to B$ 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.

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