The quantum orbifold cohomology of weighted projective spaces
type: Item
instance of: scholarly article
title: The quantum orbifold cohomology of weighted projective spaces
zbMATH Open document ID: 1213.53106
author: Tom Coates
author: Alessio Corti
author: Y.-P. Lee
author: Hsian-Hua Tseng
published in: Acta Mathematica
publication date: 2011-01-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0608481
review text: 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 resu…
reviewed by: Andrzej Szczepański
Mathematics Subject Classification ID: 53D40
Mathematics Subject Classification ID: 57R58
Mathematics Subject Classification ID: 14N35
Mathematics Subject Classification ID: 53C08
Mathematics Subject Classification ID: 53D45
Mathematics Subject Classification ID: 14J81
zbMATH DE Number: 5836541
zbMATH Keywords: weighted projective space
zbMATH Keywords: quantum cohomology
zbMATH Keywords: rational curves
zbMATH Keywords: Gromov-Witten invariants
MaRDI profile type: Publication
OpenAlex ID: W2033036947
arXiv ID: math/0608481
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DOI: 10.1007/S11511-009-0035-X
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