Quantum cohomology of flag manifolds and Toda lattices. (English) Zbl 0828.55004

The quantum cohomology \(QH^* (X)\) of a compact Kaehler manifold \(X\) is a certain deformation of the cup product multiplication in the ordinary cohomology of \(X\). The authors study the relation of the quantum cohomology with the Floer homology and introduce the equivariant quantum cohomology. Then they compute the quantum cohomology algebras of the flag manifolds showing that it does coincide with the algebra of regular functions on an invariant Lagrangian variety of a Toda lattice.
Reviewer: V.Oproiu (Iaşi)


55N35 Other homology theories in algebraic topology
57R57 Applications of global analysis to structures on manifolds
81T99 Quantum field theory; related classical field theories
37-XX Dynamical systems and ergodic theory
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[1] [A] Atiyah, M.: Convexity and commuting hamiltonians. Bull. Lond. Math. Soc.23, 1–15 (1982) · Zbl 0482.58013
[2] [AB] Atiyah, M., Bott, R.: The moment map and equivariant cohomology. Topology23, 1–28 (1984) · Zbl 0521.58025
[3] [CV] Cecotti, S., Vafa, C.: Exact results for supersymmetric sigma models. Preprint HUTP-91/A062 · Zbl 0969.81634
[4] [D] Dubrovin, B.: Integrable systems in topological field theory. Nucl. Phys.B379, 627–685 (1992)
[5] [FF] Feigin, B., Frenkel, E.: Integrals of motion and quantum groups. Preprint, 1993
[6] [F1] Floer, A.: Morse theory and lagrangian intersections. J. Diff. Geom.28, 513–547 (1988) · Zbl 0674.57027
[7] [F2] Floer, A.: Symplectic fixed points and holomorphic spheres. Commun. Math. Phys.120, 575–611 (1989) · Zbl 0755.58022
[8] [G] Ginsburg, V.A.: Equivariant cohomology and Kahler geometry. Funct. Anal. Appl.21:4, 271–283 (1987) · Zbl 0656.53062
[9] [G1] Givental, A.: Periodic mappings in symplectic topology. Funct. Anal. Appl.23:4, 287–300 (1989) · Zbl 0724.58031
[10] [G2] Givental, A.: A symplectic fixed point theorem for toric manifolds. To appear in: Progress in Math., v.93, Basel: Birkhauser
[11] [G3] Givental, A.: Homological geometry and mirror symmetry. ICM94, Zürich
[12] [GH] Griffiths, P., Harris, J.: Principles of algebraic geometry. N.Y.: Wiley, 1978 · Zbl 0408.14001
[13] [Gr] Gromov, M.: Pseudo-holomorphic curves in almost complex manifolds. Invent. Math.82:2, 307–347 (1985) · Zbl 0592.53025
[14] [HS] Hofer, H., Salamon, D.: Floer homology and Novikov rings. Preprint, 1992 · Zbl 0842.58029
[15] [K] Kontsevich, M.:A algebras in mirror symmetry. Preprint, 1993
[16] [O] Ono, K.: On the Arnold conjecture for weakly monotone symplectic manifolds. Preprint, 1993 · Zbl 0823.53025
[17] [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)
[18] [Ru] Ruan, Y.: Topological sigma model and Donaldson type invariants in Gromov theory. Preprint · Zbl 0864.53032
[19] [S] Sadov, V.: On equivalence of Floer’s and quantum cohomology. Preprint HUTP-93/A027 · Zbl 0837.53059
[20] [V] Vafa C.: Topological mirrors and quantum rings. In: Yau, S.-T. (ed.), Essays on mirror manifolds Hong Kong: International Press Co., 1992 · Zbl 0827.58073
[21] [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 · Zbl 0843.57031
[22] [W] Witten, E.: Two-dimensional gravity and intersection theory on moduli space. Surveys in Diff. Geom.1, 243–310 (1991) · Zbl 0757.53049
[23] [W2] Witten, E.: Supersymmetry and Morse theory. J. Diff. Geom.17, 661–692 (1982) · Zbl 0499.53056
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