Let X be a (possibly nodal) K-trivial threefold moving in a fixed ambient space P. Suppose X contains a continuous family of curves, all of whose members satisfy certain unobstructedness conditions in P. A formula is given for computing the corresponding virtual number of curves, that is, the number of curves on a generic deformation of X "contributed by" the continuous family on X.
We study the enumerative geometry of algebraic curves on abelian surfaces and threefolds. In the abelian surface case, the theory is parallel to the well-developed study of the reduced Gromov-Witten theory of K3 surfaces. We prove complete results in all genera for primitive classes. The generating series are quasimodular forms of pure weight. Conjectures for imprimitive classes are presented. In genus 2, the counts in all classes are proven. Special counts match the Euler characteristic calculations of the moduli spaces of stable pairs on abelian surfaces by Göttsche-Shende. A formula for hyp…
We study BPS spectra of D-branes on local Calabi-Yau threefolds $\mathcal{O}(-p)\oplus\mathcal{O}(p-2)\to \mathbb{P}^1$ with $p=0,1$, corresponding to $\mathbb{C}^3/\mathbb{Z}_{2}$ and the resolved conifold. Nonabelianization for exponential networks is applied to compute directly unframed BPS indices counting states with D2 and D0 brane charges. Known results on these BPS spectra are correctly reproduced by computing new types of BPS invariants of 3d-5d BPS states, encoded by nonabelianization, through their wall-crossing. We also develop the notion of exponential BPS graphs for the simplest …
We study BPS states of 5d $\mathcal{N}=1$ $SU(2)$ Yang-Mills theory on $S^1\times \mathbb{R}^4$. Geometric engineering relates these to enumerative invariants for the local Hirzebruch surface $\mathbb{F}_0$. We illustrate computations of Vafa-Witten invariants via exponential networks, verifying fiber-base symmetry of the spectrum at certain points in moduli space, and matching with mirror descriptions based on quivers and exceptional collections. Albeit infinite, parts of the spectrum organize in families described by simple algebraic equations. Varying the radius of the M-theory circle inter…
We prove the elliptic transformation law of Jacobi forms for the generating series of Pandharipande--Thomas invariants of an elliptic Calabi--Yau 3-fold over a reduced class in the base. This proves part of a conjecture by Huang, Katz, and Klemm. For the proof we construct an involution of the derived category and use wall-crossing methods. We express the generating series of PT invariants in terms of low genus Gromov--Witten invariants and universal Jacobi forms. As applications we prove new formulas and recover several known formulas for the PT invariants of $\mathrm{K3} \times E$, abelian 3…
An arithmetic method of proving the irrationality of smooth projective 3-folds is described, using reduction modulo $p$. It is illustrated by an application to a cubic threefold, for which the hypothesis that its intermediate Jacobian is isomorphic to the Jacobian of a curve is contradicted by reducing modulo 3 and counting points over appropriate extensions of $\mathbb F_3$. As a spin-off, it is shown that the 5-dimensional Prym varieties arising as intermediate Jacobians of certain cubic 3-folds have the maximal number of points over $\mathbb F_q$ which attains Perret's and Weil's upper boun…
Let $X\to \mathbb P^2$ be the elliptic Calabi-Yau threefold given by a general Weierstrass equation. We answer the enumerative question of how many discrete rational curves lie over lines in the base, proving part of a conjecture by Huang, Katz, and Klemm. The key inputs are a modularity theorem of Kudla and Millson for locally symmetric spaces of orthogonal type and the deformation theory of $A_n$ singularities.
We develop a theory of Gopakumar-Vafa (GV) invariants for a Calabi-Yau threefold (CY3) $X$ which is equipped with an involution $\imath$ preserving the holomorphic volume form. We define integers $n_{g,h}(β) $ which give a virtual count of the number of genus $g$ curves $C$ on $X$, in the class $β\in H_{2}(X)$, which are invariant under $\imath$, and whose quotient $C/\imath$ has genus $h$. We give two definitions of $n_{g,h}(β) $ which we conjecture to be equivalent: one in terms of a version of Pandharipande-Thomas theory and one in terms of a version of Maulik-Toda theory. We compute our in…
We use Joyce's theory of motivic Hall algebras to prove that reduced Donaldson-Thomas curve-counting invariants on Calabi-Yau threefolds coincide with stable pair invariants, and that the generating functions for these invariants are Laurent expansions of rational functions.
A CHL model is the quotient of $\mathrm{K3} \times E$ by an order $N$ automorphism which acts symplectically on the K3 surface and acts by shifting by an $N$-torsion point on the elliptic curve $E$. We conjecture that the primitive Donaldson-Thomas partition function of elliptic CHL models is a Siegel modular form, namely the Borcherds lift of the corresponding twisted-twined elliptic genera which appear in Mathieu moonshine. The conjecture matches predictions of string theory by David, Jatkar and Sen. We use the topological vertex to prove several base cases of the conjecture. Via a degenerat…
Given a quasi-projective scheme M over complex numbers equipped with a perfect obstruction theory and a morphism to a nonsingular quasi-projective variety B, we show it is possible to find an affine bundle M'/ M that admits a perfect obstruction theory relative to B. We study the resulting virtual cycles on the fibers of M'/B and relate them to the image of the virtual cycle [M]^vir under refined Gysin homomorphisms. Our main application is when M is a moduli space of stable codimension 1 sheaves with a fixed determinant L on a nonsingular projective surface or Fano threefold, B is the linear …
In this paper, we continue the study of boundedness questions for (simply connected) smooth Calabi-Yau threefolds commenced in arXiv:1706.01268. The diffeomorphism class of such a threefold is known to be determined up to finitely many possibilities by the integral middle cohomology and two integral forms on the integral second cohomology, namely the cubic cup-product form and the linear form given by cup-product with the second chern class. The question addressed in both papers is whether knowledge of these cubic and linear forms determines the threefold up to finitely many families, that is …
Conjectural results for cohomological invariants of wild character varieties are obtained by counting curves in degenerate Calabi-Yau threefolds. A conjectural formula for E-polynomials is derived from the Gromov-Witten theory of local Calabi-Yau threefolds with normal crossing singularities. A refinement is also conjectured, generalizing existing results of Hausel, Mereb and Wong as well as recent joint work of Donagi, Pantev and the author for weighted Poincaré polynomials of wild character varieties.
We show a version of the DT/PT correspondence relating local curve counting invariants, encoding the contribution of a fixed smooth curve in a Calabi-Yau threefold. We exploit a local study of the Hilbert-Chow morphism about the cycle of a smooth curve. We determine, via Quot schemes, the global Donaldson-Thomas theory of a general Abel-Jacobi curve of genus $3$.
We prove a formula for the Hodge numbers of square-free divisors of Calabi-Yau threefold hypersurfaces in toric varieties. Euclidean branes wrapping divisors affect the vacuum structure of Calabi-Yau compactifications of type IIB string theory, M-theory, and F-theory. Determining the nonperturbative couplings due to Euclidean branes on a divisor $D$ requires counting fermion zero modes, which depend on the Hodge numbers $h^i({\cal{O}}_D)$. Suppose that $X$ is a smooth Calabi-Yau threefold hypersurface in a toric variety $V$, and let $D$ be the restriction to $X$ of a square-free divisor of $V$…
We derive an approximate analytic relation between the number of consistent heterotic Calabi-Yau compactifications of string theory with the exact charged matter content of the standard model of particle physics and the topological data of the internal manifold: the former scaling exponentially with the number of Kahler parameters. This is done by an estimate of the number of solutions to a set of Diophantine equations representing constraints satisfied by any consistent heterotic string vacuum with three chiral massless families, and has been computationally checked to hold for complete inter…
