A topological view of Gromov-Witten theory

By Davesh Maulik, R. Pandharipande. 2004

174 citations recorded by the source corpus.

Abstract We study relative Gromov–Witten theory via universal relations provided by the degeneration and localization formulas. We find relative Gromov–Witten theory is completely determined by absolute Gromov–Witten theory. The relationship between the relative and absolute theories is guided by a strong analogy to classical topology. As an outcome, we present a mathematical determination of the Gromov–Witten invariants (in all genera) of the Calabi–Yau quintic 3-fold in terms of known theories.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/79a92aa9f39d6f9dc798eb3a9276b5bba424b285

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Citations in the corpus

A desingularization of the main component of the moduli space of genus-one stable maps into ℙn (2006, methodology): “ut not necessarily of the same dimension. 4 The last expression above is computable by localization. Remark: Another approach to computing positive-genus Gromov-Witten invariants has been proposed in [MaP]. In contrast to the approach of [LZ], it applies to arbitrary-genus invariants, but can at pre”
Gromov-Witten/pairs correspondence for the quintic 3-fold (2012, methodology): “Once the GW/P correspondences for these special relative geometries are established, then the degeneration scheme of [22] can be used to prove the GW/P correspondence for any compact Calabi-Yau 3-fold which admits a good degeneration.”
Lectures on K-theoretic computations in enumerative geometry (2015, background): “For a general introduction to enumerative geometry, my favorite source are the lectures by Pandharipande in [40], even though those lectures discuss topics that are essentially disjoint from our goals here.”
Curves on K3 surfaces and modular forms (2010, methodology): “The relative invariants of (E × P(1))/E are algorithmically determined from the Gromov–Witten theory of E using localization (via the C∗-action on P(1)) and the absolute/relative correspondence of [35].”
The local Donaldson–Thomas theory of curves (2005, methodology): “The technique, following Gromov–Witten theory (see Maulik–Pandharipande [26] and Okounkov–Pandharipande [30]), involves rigidification and topological recursion.”
Standard versus reduced genus-one Gromov–Witten invariants (2007, methodology): “ut not necessarily of the same dimension. 4 The last expression above is computable by localization. Remark: Another approach to computing positive-genus Gromov-Witten invariants has been proposed in [MaP]. In contrast to the approach of [LZ], it applies to arbitrary-genus invariants, but can at pre”
Logarithmic Gromov–Witten theory with expansions (2019, background): “A basic test for these developments would be to show that the logarithmic theory of (X,D) is determined by the absolute theory of the strata, generalizing the calculation schemes of Gathmann and Maulik– Pandharipande [26, 49].”
Orbifold techniques in degeneration formulas (2011, methodology): “tionship of the logarithmic construction with Li’s obstruction theory. Comparing the resulting Gromov–Witten invariants can be done indirectly using deformation invariance, localization, and the work [MP] of Maulik and Pandharipande. We plan to give details for all these comparison issues in [AF]. 0”
On the genus-one Gromov-Witten invariants of complete intersections (2005, methodology): “k is the cone corresponding to (L,∇) and NgY (A) = GWYg, 0(A;1). Two completely diﬀerent approaches to computing positive-genus GW-invariants of complete intersections have been proposed in [Ga2] and [MaP]. Both approaches use degenerations and relative Gromov-Witten invariants. The ﬁrst approach ca”
Fukaya A∞-structures Associated to Lefschetz Fibrations. II (2014, methodology): “ty k, and which carry an additional marked point. These give rise to classes (3.15) ~z(k) 2H2(F;K); with ~z (1)= z . All these relative invariants can be reduced to ordinary Gromov-Witten theory, see [27] for a general discussion. We will only appeal to the simplest special case of such a reduction:”
Landau–Ginzburg/Calabi–Yau correspondence for quintic three-folds via symplectic transformations (2008, methodology): “on (it is generally believed that the LG singularity model is relatively easy to compute). Here, we should mention that in Gromov–Witten theory several general methods have been recently found [LR01] [MP06] and can in principle determine Gromov–Witten invariants of all known examples. These methods,”
Algebraic cobordism revisited (2006, methodology): “nce the virtual dimension of the rubber space In(P/S−∪ S +,0)˜is −1, W−= 1. A discussion of virtual localization in relative Donaldson-Thomas theory and rubber moduli spaces can be found in [23]. See [24] for a construction of Ψ +. A local neighborhood of S + ⊂ P is given by the total space P + = P\”