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Search: MSC category 14N35 ( Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45] )

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1. CJM 2010 (vol 62 pp. 1246)

Chaput, P. E.; Manivel, L.; Perrin, N.
Quantum Cohomology of Minuscule Homogeneous Spaces III. Semi-Simplicity and Consequences
We prove that the quantum cohomology ring of any minuscule or cominuscule homogeneous space, specialized at $q=1$, is semisimple. This implies that complex conjugation defines an algebra automorphism of the quantum cohomology ring localized at the quantum parameter. We check that this involution coincides with the strange duality defined in our previous article. We deduce Vafa--Intriligator type formulas for the Gromov--Witten invariants.

Keywords:quantum cohomology, minuscule homogeneous spaces, Schubert calculus, quantum Euler class
Categories:14M15, 14N35

2. CJM 2008 (vol 60 pp. 875)

Mare, Augustin-Liviu
A Characterization of the Quantum Cohomology Ring of $G/B$ and Applications
We observe that the small quantum product of the generalized flag manifold $G/B$ is a product operation $\star$ on $H^*(G/B)\otimes \bR[q_1,\dots, q_l]$ uniquely determined by the facts that: it is a deformation of the cup product on $H^*(G/B)$; it is commutative, associative, and graded with respect to $\deg(q_i)=4$; it satisfies a certain relation (of degree two); and the corresponding Dubrovin connection is flat. Previously, we proved that these properties alone imply the presentation of the ring $(H^*(G/B)\otimes \bR[q_1,\dots, q_l],\star)$ in terms of generators and relations. In this paper we use the above observations to give conceptually new proofs of other fundamental results of the quantum Schubert calculus for $G/B$: the quantum Chevalley formula of D. Peterson (see also Fulton and Woodward ) and the ``quantization by standard monomials" formula of Fomin, Gelfand, and Postnikov for $G=\SL(n,\bC)$. The main idea of the proofs is the same as in Amarzaya--Guest: from the quantum $\D$-module of $G/B$ one can decode all information about the quantum cohomology of this space.

Categories:14M15, 14N35

3. CJM 2007 (vol 59 pp. 981)

Jiang, Yunfeng
The Chen--Ruan Cohomology of Weighted Projective Spaces
In this paper we study the Chen--Ruan cohomology ring of weighted projective spaces. Given a weighted projective space ${\bf P}^{n}_{q_{0}, \dots, q_{n}}$, we determine all of its twisted sectors and the corresponding degree shifting numbers. The main result of this paper is that the obstruction bundle over any 3\nobreakdash-multi\-sector is a direct sum of line bundles which we use to compute the orbifold cup product. Finally we compute the Chen--Ruan cohomology ring of weighted projective space ${\bf P}^{5}_{1,2,2,3,3,3}$.

Keywords:Chen--Ruan cohomology, twisted sectors, toric varieties, weighted projective space, localization
Categories:14N35, 53D45

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