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On Quantum Cohomology
, 1996
"... Abstract We discuss a general quantum theoretical example of quantum cohomology and show that various mathematical aspects of quantum cohomology have quantum mechanical and also observable significance. 1 The quantum cohomology is one of the most fundamental and intressting mathematicalphysical fie ..."
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Abstract We discuss a general quantum theoretical example of quantum cohomology and show that various mathematical aspects of quantum cohomology have quantum mechanical and also observable significance. 1 The quantum cohomology is one of the most fundamental and intressting mathematical
GromovWitten classes, quantum cohomology, and enumerative geometry
 Commun. Math. Phys
, 1994
"... The paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry. In particular, it contains an axiomatic treatment of Gromov–Witten classes, and a discussion of their properties for Fano varieties. Cohomological ..."
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Cited by 484 (3 self)
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The paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry. In particular, it contains an axiomatic treatment of Gromov–Witten classes, and a discussion of their properties for Fano varieties. Cohomological
Quantum cohomology of complete intersections
, 1995
"... The quantum cohomology algebra of a projective manifold X is the cohomology of X endowed with a different algebra structure, which takes into account the geometry of rational curves in X. This structure has been first defined heuristically ..."
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Cited by 41 (0 self)
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The quantum cohomology algebra of a projective manifold X is the cohomology of X endowed with a different algebra structure, which takes into account the geometry of rational curves in X. This structure has been first defined heuristically
Quantum cohomology rings of toric manifolds
, 1993
"... We compute the quantum cohomology ring H ∗ ϕ(PΣ,C) of an arbitrary ddimensional smooth projective toric manifold PΣ associated with a fan Σ. The multiplicative structure of H ∗ ϕ (PΣ,C) depends on the choice of an element ϕ in the ordinary cohomology group H 2 (PΣ,C). There are many properties of q ..."
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Cited by 95 (2 self)
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We compute the quantum cohomology ring H ∗ ϕ(PΣ,C) of an arbitrary ddimensional smooth projective toric manifold PΣ associated with a fan Σ. The multiplicative structure of H ∗ ϕ (PΣ,C) depends on the choice of an element ϕ in the ordinary cohomology group H 2 (PΣ,C). There are many properties
Quantum cohomology of Grassmannians
 Compositio Math
"... The purpose of this paper is to give simple proofs of the main theorems about quantum cohomology of Grassmannians. This first of all includes Bertram’s quantum versions of the Pieri and Giambelli formulas [1]. Bertram’s proofs of these theorems required the use of quot schemes. Our proof of the quan ..."
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Cited by 36 (8 self)
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The purpose of this paper is to give simple proofs of the main theorems about quantum cohomology of Grassmannians. This first of all includes Bertram’s quantum versions of the Pieri and Giambelli formulas [1]. Bertram’s proofs of these theorems required the use of quot schemes. Our proof
Quantum Cohomology Of Flag Varieties
 Internat. Math. Res. Notices
, 1995
"... Introduction The quantum cohomology ring of a Kahler manifold X is a deformation of the usual cohomology ring which appears naturally in theoretical physics in the study of the supersymmetric nonlinear sigma models with target X. In [W], Witten introduces the quantum multiplication of cohomology cl ..."
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Cited by 41 (2 self)
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Introduction The quantum cohomology ring of a Kahler manifold X is a deformation of the usual cohomology ring which appears naturally in theoretical physics in the study of the supersymmetric nonlinear sigma models with target X. In [W], Witten introduces the quantum multiplication of cohomology
On the quantum cohomology of adjoint varieties
 Proc. of the London Math. Soc., arχiv:0904:4824v1
"... We study the quantum cohomology of quasiminuscule and quasicominuscule homogeneous spaces. The product of any two Schubert cells does not involve powers of the quantum parameter higher than 2. With the help of the quantum to classical principle we give presentations of the quantum cohomology algeb ..."
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Cited by 10 (4 self)
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We study the quantum cohomology of quasiminuscule and quasicominuscule homogeneous spaces. The product of any two Schubert cells does not involve powers of the quantum parameter higher than 2. With the help of the quantum to classical principle we give presentations of the quantum cohomology
Notes On Stable Maps And Quantum Cohomology
, 1996
"... Contents 0. Introduction 1 1. Stable maps and their moduli spaces 10 2. Boundedness and a quotient approach 12 5. The construction of M g;n (X; fi) 25 6. The boundary of M 0;n (X; fi) 29 7. GromovWitten invariants 31 8. Quantum cohomology 34 9. Applications to enumerative geometry 38 10. Varia ..."
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Cited by 141 (14 self)
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Contents 0. Introduction 1 1. Stable maps and their moduli spaces 10 2. Boundedness and a quotient approach 12 5. The construction of M g;n (X; fi) 25 6. The boundary of M 0;n (X; fi) 29 7. GromovWitten invariants 31 8. Quantum cohomology 34 9. Applications to enumerative geometry 38 10
The quantum cohomology of homogeneous varieties
 J. Algebraic Geom
, 1997
"... The notion of quantum cohomology was first proposed by Witten [Va, Wi], based on topological field theory. Its mathematical theory was only established recently by Y. Ruan and the second named author [RT, Ru], where they proved the existence of the quantum rings on semipositive symplectic manifolds ..."
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Cited by 20 (1 self)
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The notion of quantum cohomology was first proposed by Witten [Va, Wi], based on topological field theory. Its mathematical theory was only established recently by Y. Ruan and the second named author [RT, Ru], where they proved the existence of the quantum rings on semipositive symplectic
Quantum cohomology of flag manifolds
"... In this paper, we study the (small) quantum cohomology ring of the partial flag manifold. We give proofs of the presentation of the ring and of the quantum Giambelli formula for Schubert varieties. These are known results, but our proofs are more natural and direct than the previous ones. ..."
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Cited by 8 (2 self)
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In this paper, we study the (small) quantum cohomology ring of the partial flag manifold. We give proofs of the presentation of the ring and of the quantum Giambelli formula for Schubert varieties. These are known results, but our proofs are more natural and direct than the previous ones.
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