The local Gromov-Witten theory of curves

By Jim Bryan, R. Pandharipande. 2004

202 citations recorded by the source corpus.

The local Gromov-Witten theory of curves is solved by localization and degeneration methods. Localization is used for the exact evalu- ation of basic integrals in the local Gromov-Witten theory of P 1 . A TQFT formalism is defined via degeneration to capture higher genus curves. Together, the results provide a compete and effective solution. The local Gromov-Witten theory of curves is equivalent to the lo- cal Donaldson-Thomas theory of curves, the quantum cohomology of the Hilbert scheme points of C 2 , and the orbifold quantum cohomol- ogy the symmetric product of C 2 . The results of the paper provide the local Gromov-Witten calculations required for the proofs of these equivalences.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4500a2fd9964137e36f18f99c4ff11d7cbc875f5

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Algebraic cobordism revisited (2006, background): “ Y c3(TY ⊗KY ). Finally, since Z(Y,0) = 1 and M(0) = 1, Z(Y,q) = M(−q) R Y c3(TY ⊗KY ), completing the proof. 14.2. Conjecture 1′. Next, we consider an equivariant version of Conjecture 1 proposed in [4]. Let X be a smooth quasi-projective 3-fold over C equipped with an action of an algebraic torus ”
The local Donaldson–Thomas theory of curves (2005, methodology): “For the Gromov–Witten side, we follow the definitions and notations of Bryan–Pandharipande [4].”
Black Holes, q-Deformed 2d Yang-Mills, and Non-perturbative Topological Strings (2004, methodology): “In a recent work [ 4 ] topological A-model string theory amplitudes”
Local Gromov-Witten invariants are log invariants (2017, background): “3],2 specializing the equivariant parameter t2 in [7] to 1, the left hand side of (2.”
Gromov-Witten/Donaldson-Thomas correspondence for toric 3-folds (2008, methodology): “The evaluation of (41) is a central result of [4].”
Quivers, curves, and the tropical vertex (2009, methodology): “The virtual aspects are handled by the multiple cover formulas of [3, 4] and the enumerative aspects by the tropical/holomorphic curve counts.”
Supersymmetric Gauge Theories, Intersecting Branes and Free Fermions (2007, methodology): “T2 curve, embedded as the zero section y = 0 in the geometry B deﬁned as the quotient of C∗ × Cby (x,y) ∼ (x+ t− ry,y). (4.29) Now the A-model topological string partition function can be computed as [65, 66] Ztop(t,λ) = e−t 3 /6r2λ2Q−1/24 X R Q |R qrκR 2, where Q= e−t with tthe Ka¨hler parameter of”
Double ramification cycles on the moduli spaces of curves (2016, methodology): “Local theory of curves The local Gromov-Witten theory of curves as developed in [2] starts with a fundamental pairing with a double ramification cycle (the resulting matrix then also appears in the quantum cohomology of the Hilbert scheme of points of C2 [30]).”
The Crepant Resolution Conjecture (2006, methodology): “The image of this map is called Hn ⊂ Mg,2 in [13] and the pairing of [Hn] against λgλg−1 is explicitly evaluated in [13] to yield the series above.”
Nonperturbative effects and nonperturbative definitions in matrix models and topological strings (2008, background): “It can be realized as a special limit of topological string theory on certain toric Calabi–Yau manifolds, see for example [12, 15] for detailed derivations.”
Lectures on non‐perturbative effects in large N gauge theories, matrix models and strings (2012, background): “It can be realized as a special limit of topological string theory on certain toric Calabi–Yau manifolds, see for example [30, 34] for detailed derivations.”
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”
Open string amplitudes and large order behavior in topological string theory (2006, methodology): “As explained in [ 17 ], the A model has to be defined equivariantly.”
K3 surfaces with Picard number one and infinitely many rational points (2005, methodology): “(This result differs from the analogous one in [Bryan and Pandharipande 2004] by a factor of −i , reflecting a normalization in their generating function conventions that we have not adopted.”
Curve counting via stable pairs in the derived category (2007, background): “of stable pairs in Section 2 is valid in the equivariant setting, [Pn(X,β)]vir ∈ AG cβ (Pn(X,β),Z). We may then deﬁne an equivariant theory of stable pairs and an equivariant correspondence following [8]. We leave the details to the reader. Another standard direction is the relative theory. While re”
Gromov-Witten/pairs correspondence for the quintic 3-fold (2012, methodology): “Factoring the quintic equation yields a degeneration, (76) T5 T4 ∪S4 P(3)[4, 5], where S4 ⊂ T4 is a linear section and S4 ⊂ P(3)[4, 5] is the strict transform of a quartic containing C4,5 ⊂ P(3).”