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A uniqueness theorem for Frobenius manifolds and Gromov-Witten theory for orbifold projective lines. (English) Zbl 1323.53094

The article computes the orbifold Gromov-Witten invariants of an orbifold projective line \(\mathbb P^1_{(a_1,a_2,a_3)}\), i.e., the projective line with three orbifold singularities of order \(0<a_1 \leq a_2 \leq a_3\). Gromov-Witten theory in the orbifold setting was defined in [D. Abramovich et al., Am. J. Math. 130, No. 5, 1337–1398 (2008; Zbl 1193.14070)] and [W. Chen and Y. Ruan, Contemp. Math. 310, 25–85 (2002; Zbl 1091.53058)]. It is important to note that the Witten-Dijkgraaf-Verlinde-Verlinde equations (WDVV equations) are satisfied in the orbifold setting as well.
A major step of the computation consists of establishing a uniqueness statement for Frobenius algebras satisfying certain conditions. These conditions are subsequently shown to be satisfied by the Gromov-Witten invariants of the above orbifold projective lines, from which it then follows that they are uniquely determined. In particular, their Gromov-Witten invariants are determined by the WDVV equations together with certain initial conditions.
The computations of this article generalises previous computations in [P. Rossi, Math. Ann. 348, No. 2, 265–287 (2010; Zbl 1235.14053)].

MSC:

53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
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