The local Donaldson–Thomas theory of curves

By A. Okounkov, R. Pandharipande. 2005

90 citations recorded by the source corpus.

Let X be a nonsingular projective variety of dimension 3 over C . Gromov–Witten theory is defined by integration over the moduli space of stable maps to X , and Donaldson–Thomas theory is defined by integration over the moduli space of ideal sheaves of X (see Donaldson–Thomas [5], Maulik et al [24; 25] and Thomas [34]). If X is quasi-projective, the Gromov–Witten and Donaldson–Thomas theories may not be well-defined. However, if X is the total space of a rank 2 bundle over a nonsingular projective curve, N ! C; local Gromov–Witten and Donaldson–Thomas theories are defined via equivariant residues (see Bryan–Pandharipande [4] and Maulik et al [24]). The Gromov–Witten and Donaldson–Thomas theories of X relative to a nonsingular surface S X are defined via moduli spaces of maps and sheaves with boundary conditions along S . See Eliashberg–Givental–Hofer [6], Ionel–Parker [12], Li–Ruan [16], Li [17; 18] and Maulik et al [25] for various treatments of the subject.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b9b788436016d2bc0f79c11e70368db3728f5daa

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Citations in the corpus

Gromov-Witten/Donaldson-Thomas correspondence for toric 3-folds (2008, methodology): “The Gromov-Witten and Donaldson-Thomas capped edges are matched in Section 3.3 by strengthening the local curves correspondence of [4, 30 ].”
The Crepant Resolution Conjecture (2006, background): “n associative Frobenius algebra [11]. Similarly, on obtains a Frobenius algebra from the partition function for the degree n equivariant Donaldson-Thomas invariants of P1 × C2 relative to {0,1,∞}× C2 [22]. Theorem 3.11 completes the following tetrahedron of equivalences. ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ Equivari”
Lectures on K-theoretic computations in enumerative geometry (2015, background): “Such a relation indeed exists and has been already profitably used in cohomology [57,58,71,72].”
Stability conditions and curve counting invariants on Calabi–Yau 3-folds (2011, methodology): “Also GW/DT correspondence for toric Calabi-Yau 3-folds and local curves are proved in [30] and [31] respectively, by using torus localization and degeneration formula.”
Lectures on black holes, topological strings and quantum attractors (2006, background): “This conjecture has been recently proven for any toric three-fold X [76].”
Gromov-Witten/pairs correspondence for the quintic 3-fold (2012, background): “The argument of [21] uses • the proof of the MNOP conjecture for local curves established in the papers [5, 28, 29], • the proof of the MNOP conjecture for 3-folds An × P(1), where An is the holomorphic symplectic resolution of the standard An-surface singularity, in the papers [16, 19, 20].”
Curve Counting on K3 × E, The Igusa Cusp Form χ10, and Descendent Integration (2016, background): “ii) for Xis straightforward to conjecture. Because reduced theories are considered, the correspondence here is not directly a special case of the standard GW/P correspondence for 3-folds [20, 33]. In [6, 29, 30], a triangle of parallel equivalences relating the GromovWitten and Donaldson-Thomas theo”
Curves on K3 surfaces and modular forms (2010, methodology): “ed vertex have already been proven. Only the parallel results for the local stable pairs theory of An-resolutions over curves are required. For the latter, we follow, step by step, the arguments from [33, 43]. Stable pairs have the same formal properties (with respect to localization and degeneratio”