The local Donaldson–Thomas theory of curves

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.