Abstract
We discuss relations of Vafa's quantum cohomology with Floer's homology theory, introduce equivariant quantum cohomology, formulate some conjectures about its general properties and, on the basis of these conjectures, compute quantum cohomology algebras of the flag manifolds. The answer turns out to coincide with the algebra of regular functions on an invariant lagrangian variety of a Toda lattice.
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[A] Atiyah, M.: Convexity and commuting hamiltonians. Bull. Lond. Math. Soc.23, 1–15 (1982)
[AB] Atiyah, M., Bott, R.: The moment map and equivariant cohomology. Topology23, 1–28 (1984)
[CV] Cecotti, S., Vafa, C.: Exact results for supersymmetric sigma models. Preprint HUTP-91/A062
[D] Dubrovin, B.: Integrable systems in topological field theory. Nucl. Phys.B379, 627–685 (1992)
[FF] Feigin, B., Frenkel, E.: Integrals of motion and quantum groups. Preprint, 1993
[F1] Floer, A.: Morse theory and lagrangian intersections. J. Diff. Geom.28, 513–547 (1988)
[F2] Floer, A.: Symplectic fixed points and holomorphic spheres. Commun. Math. Phys.120, 575–611 (1989)
[G] Ginsburg, V.A.: Equivariant cohomology and Kahler geometry. Funct. Anal. Appl.21:4, 271–283 (1987)
[G1] Givental, A.: Periodic mappings in symplectic topology. Funct. Anal. Appl.23:4, 287–300 (1989)
[G2] Givental, A.: A symplectic fixed point theorem for toric manifolds. To appear in: Progress in Math., v.93, Basel: Birkhauser
[G3] Givental, A.: Homological geometry and mirror symmetry. ICM94, Zürich
[GH] Griffiths, P., Harris, J.: Principles of algebraic geometry. N.Y.: Wiley, 1978
[Gr] Gromov, M.: Pseudo-holomorphic curves in almost complex manifolds. Invent. Math.82:2, 307–347 (1985)
[HS] Hofer, H., Salamon, D.: Floer homology and Novikov rings. Preprint, 1992
[K] Kontsevich, M.:A ∞-algebras in mirror symmetry. Preprint, 1993
[O] Ono, K.: On the Arnold conjecture for weakly monotone symplectic manifolds. Preprint, 1993
[R] Reyman, A.: Hamiltonian systems related to graded Lie algebras. In: Diff. Geom., Lie groups and Mechanics, III Zapiski Nauchn. Sem. LOMI,95, Nauka, 1980 (in Russian)
[Ru] Ruan, Y.: Topological sigma model and Donaldson type invariants in Gromov theory. Preprint
[S] Sadov, V.: On equivalence of Floer's and quantum cohomology. Preprint HUTP-93/A027
[V] Vafa C.: Topological mirrors and quantum rings. In: Yau, S.-T. (ed.), Essays on mirror manifolds Hong Kong: International Press Co., 1992
[Vt] Viterbo, C.: The cup-product on the Thom-Smale-Witten complex, and Floer cohomology. To appear in: Progress in Math., v.93, Basel: Birkhauser
[W] Witten, E.: Two-dimensional gravity and intersection theory on moduli space. Surveys in Diff. Geom.1, 243–310 (1991)
[W2] Witten, E.: Supersymmetry and Morse theory. J. Diff. Geom.17, 661–692 (1982)
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Communicated by R.H. Dijkgraaf
Supported by Alfred P. Sloan Foundation
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Givental, A., Kim, B. Quantum cohomology of flag manifolds and Toda lattices. Commun.Math. Phys. 168, 609–641 (1995). https://doi.org/10.1007/BF02101846
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DOI: https://doi.org/10.1007/BF02101846