Gromov-Witten, Gopakumar-Vafa, and Donaldson-Thomas invariants of Calabi-Yau threefolds

@article{Katz2004GromovWittenGA,
  title={Gromov-Witten, Gopakumar-Vafa, and Donaldson-Thomas invariants of Calabi-Yau threefolds},
  author={Sheldon Katz},
  journal={arXiv: Algebraic Geometry},
  year={2004}
}
  • S. Katz
  • Published 19 August 2004
  • Mathematics
  • arXiv: Algebraic Geometry
Gromov-Witten, Gopakumar-Vafa, and Donaldson-Thomas invariants of Calabi-Yau threefolds are compared. In certain situations, the Donaldson-Thomas invariants are very easy to handle, sometimes easier than the other invariants. This point is illustrated in several ways, especially by revisiting computations of Gopakumar-Vafa invariants by Katz, Klemm, and Vafa in a rigorous mathematical framework. This note is based on my talk at the 2004 Snowbird Conference on String Geometry. 

Super-rigid Donaldson-Thomas Invariants

We solve the part of the Donaldson-Thomas theory of Calabi-Yau threefolds which comes from super-rigid rational curves. As an application, we prove a version of the conjectural

THE GROMOV–WITTEN AND DONALDSON–THOMAS CORRESPONDENCE FOR TRIVIAL ELLIPTIC FIBRATIONS

We study the Gromov–Witten and Donaldson–Thomas correspondence conjectured in [16, 17] for trivial elliptic fibrations. In particular, we verify the Gromov–Witten and Donaldson–Thomas correspondence

Stability conditions and curve counting invariants on Calabi–Yau 3-folds

The purpose of this paper is twofold: first we give a survey on the recent developments of curve counting invariants on Calabi-Yau 3-folds, e.g. Gromov-Witten theory, Donaldson-Thomas theory and

Khovanov-Rozansky Homology and Topological Strings

We conjecture a relation between the sl(N) knot homology, recently introduced by Khovanov and Rozansky, and the spectrum of BPS states captured by open topological strings. This conjecture leads to

Elliptic Calabi-Yau threefolds over a del Pezzo surface

We consider certain elliptic threefolds over the projective plane (more generally over certain rational surfaces) with a section in Weierstrass normal form. In particular, over a del Pezzo surface of

Towards refining the topological strings on compact Calabi-Yau 3-folds

We make a proposal for calculating refined Gopakumar-Vafa numbers (GVN) on elliptically fibered Calabi-Yau 3-folds based on refined holomorphic anomaly equations. The key examples are smooth elliptic

Topological string amplitudes, complete intersection Calabi-Yau spaces and threshold corrections

We present the most complete list of mirror pairs of Calabi-Yau complete intersections in toric ambient varieties and develop the methods to solve the topological string and to calculate higher genus

On certain moduli spaces of ideal sheaves and Donaldson-Thomas invariants

We determine the structure of certain moduli spaces of ideal sheaves by generalizing an earlier result of the first author. As applications, we compute the (virtual) Hodge polynomials of these moduli

Stable pairs on local $K3$ surfaces

We prove a formula which relates Euler characteristic of moduli spaces of stable pairs on local K3 surfaces to counting invariants of semistable sheaves on them. Our formula generalizes

References

SHOWING 1-10 OF 16 REFERENCES

Gromov–Witten theory and Donaldson–Thomas theory, II

We discuss the Gromov–Witten/Donaldson–Thomas correspondence for 3-folds in both the absolute and relative cases. Descendents in Gromov–Witten theory are conjectured to be equivalent to Chern

Gromov–Witten theory and Donaldson–Thomas theory, I

We conjecture an equivalence between the Gromov–Witten theory of 3-folds and the holomorphic Chern–Simons theory of Donaldson and Thomas. For Calabi–Yau 3-folds, the equivalence is defined by the

Hodge integrals and Gromov-Witten theory

Integrals of the Chern classes of the Hodge bundle in Gromov-Witten theory are studied. We find a universal system of differential equations which determines the generating function of these

Quantum Calabi-Yau and Classical Crystals

We propose a new duality involving topological strings in the limit of the large string coupling constant. The dual is described in terms of a classical statistical mechanical model of crystal

The Topological Vertex

We construct a cubic field theory which provides all genus amplitudes of the topological A-model for all non-compact toric Calabi-Yau threefolds. The topology of a given Feynman diagram encodes the

M theory, topological strings and spinning black holes

We consider M-theory compactification on Calabi-Yau threefolds. The recently discovered connection between the BPS states of wrapped M2 branes and the topological string amplitudes on the threefold

Three Questions in Gromov-Witten Theory

Three conjectural directions in Gromov-Witten theory are discussed: Goren­ stein properties, BPS states, and Virasoro constraints. Each points to basic structures in the subject which are not yet

Recovering the good component of the Hilbert scheme

We give an explicit construction, for a at map X ! S of algebraic spaces, of an ideal in the n’th symmetric product of X over S. Blowing up this ideal is then shown to be isomorphic to the schematic

Hodge Integrals and Degenerate Contributions

Abstract:Hodge integral techniques are used to compute the degree 1 degenerate contributions of curves of arbitrary genus in the Gromov–Witten theory of 3-folds. In the Calabi–Yau case, the

M theory and topological strings. 2.

The $R^2 F^{2g-2}$ terms of Type IIA strings on Calabi-Yau 3-folds, which are given by the corresponding topological string amplitudes (a worldsheet instanton sum for all genera), are shown to have a