The quantum orbifold cohomology of weighted projective spaces
Let \(w_0,w_1,\dots, w_n\) be a sequence of positive integers and let \(\mathbb P^w\) be the weighted projective space \(\mathbb P(w_0,w_1,\dots, w_n)\), i.e., the quotient \([(\mathbb{C}^{n+ 1}\setminus\{0\})/\mathbb{C}^\times]\), where \(\mathbb{C}^\times\) acts with weights \(-w_0,-w_1,\dots, -w_n\). The authors calculate the small quantum orbifold cohomology ring of the weighted projective space \(\mathbb P(w_0,w_1,\dots, w_n)\). Their approach is essentially due to \textit{A. B. Givental} [Sel. Math., New Ser. 1, No.~2, 325--345 (1995; Zbl 0920.14028) in: Proceedings of the international congress of mathematicians, ICM `94, August 3--11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 472--480 (1995; Zbl 0863.14021) and Int. Math. Res. Not. 1996, No.~13, 613--663 (1996; Zbl 0881.55006)]. They begin with a heuristic argument relating the quantum cohomology of \(\mathbb P^w\) to the \(S^1\)-equivariant Floer cohomology of the loop space \(L\mathbb P^w\), and from this they conjecture a formula for a certain generating function -- the small J-function for genus-zero Gromov-Witten invariants of \(\mathbb P^w\). The small J-function determines the small quantum orbifold cohomology of \(\mathbb P^w\). Next, the authors prove that their conjectural formula for the small J-function is correct by analyzing the relationship between two compactifications of the space of parameterized rational curves in \(\mathbb P^w\) -- a toric compactification and the space of genus-zero stable maps to \(\mathbb P^w\times\mathbb P(1, r)\) of degree \(1/r\) with respect to the second factor. These compactifications carry naturally a \(\mathbb{C}^\times\)-action, which one can think of as arising from the rotation of loops, and there is a map between them which is \(\mathbb{C}^\times\)-equivariant. The authors formula for the small J-function can be expressed in terms of integrals of \(\mathbb{C}^\times\)-equivariant cohomology classes on the toric compactification. Then, using results of \textit{A. Bertram} [Invent. Math. 142, No.~3, 487--512 (2000; Zbl 1031.14027)], and localization in equivariant cohomology, they transform these into integrals of classes on the stable map compactification. This establishes the authors formula for the small J-function, and so allows them to determine the small quantum orbifold cohomology ring of \(\mathbb P^w\).
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