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1. CJM 2010 (vol 62 pp. 1246)
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)
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)
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 |