Gromov-Witten invariants can be viewed as a generalization of various enumerative problems in algebraic geometry, motivated by ideas from gauge theory. They form a mathematical testing ground for ideas from string theory and they have been at the center of many of the most recent developments in mathematics. Quantum cohomology and the WDVV equations have been very effective in genus zero calculations. Similarly localization has been very effective in the presence of a strong torus action. In this paper, the authors consider computations of Gromov-Witten invariants in higher genus with no strong torus action.
The tool that they use in this case is the Mayer-Vietoris principle from A. M. Li and Y. Ruan [Invent. Math. 145, No. 1, 151–218 (2001; Zbl 1062.53073)], E.-N. Ionel and T. H. Parker [Ann. Math. (2) 157, No. 1, 45–96 (2003; Zbl 1039.53101)] and J. Li [J. Differential Geom. 60, No. 2, 199–293 (2002; Zbl 1063.14069)], D. Maulik and R. Pandharipande; [Topology 45, No. 5, 887-918 (2006; Zbl 1112.14065)] expressing the non-vanishing Gromov-Witten invariants of a space \(X\) in terms of the invariants of \(X_1\), \(X_2\) and \(Y\) when there is a good degeneration from \(X\) to \(X_1\cup_YX_2\).
They apply this idea to three basic examples. The first is double branched covers of \(\mathbb{P}^2\) over a curve of degree \(2n\) where the degeneration comes from a smooth curve of degree \(2n\) degenerating to the square of a curve of degree \(n\). The second example is the Enriques surface. They study this via a degeneration to a union of the rational elliptic surface and an elliptic fibration with two double fibers and no singular fibers. The third example is the Enriques Calabi-Yau. This degenerates into two copies of a quotient of the product of a \(K3\) with a projective line. After these examples, Maulik and Pandharipande consider applications of Seiberg-Witten theory to the Gromov-Witten invariants of surfaces. In particular they conjecture formula for the local theory of curves in surfaces in degrees \(1\) and \(2\). They finally consider the \(g\leq 2\) fiber class invariants of the Enriques Calabi-Yau, and discuss the holomorphic anomaly.