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Quantum \(D\)-modules and equivariant Floer theory for free loop spaces. (English) Zbl 1121.53062
Summary: The objective of this paper is to clarify the relationships between the quantum \(D\)-module and equivariant Floer theory. Equivariant Floer theory was introduced by A. Givental in his paper “Homological Geometry” [Sel. Math., New Ser. 1, No. 2, 325–345 (1995; Zbl 0920.14028)]. He conjectured that the quantum \(D\)-module of a symplectic manifold is isomorphic to the equivariant Floer cohomology for the universal cover of the free loop space. First, motivated by the work of Guest, we formulate the notion of abstract quantum \(D\)-module which generalizes the \(D\)-module defined by the small quantum cohomology algebra. Second, we define the equivariant Floer cohomology of toric complete intersections rigorously as a \(D\)-module, using Givental’s model. This is shown to satisfy the axioms of abstract quantum \(D\)-module. By Givental’s mirror theorem A. Givental [A mirror theorem for toric complete intersections. Prog. Math. 160, 141–175 (1998; Zbl 0936.14031)], it follows that the equivariant Floer cohomology defined here is isomorphic to the quantum cohomology \(D\)-module.

MSC:
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
53D40 Symplectic aspects of Floer homology and cohomology
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References:
[1] Atiyah, M. Bott,R.: The Moment Map and Equivariant Cohomology. Topology 23, 1–28 (1984) · Zbl 0521.58025
[2] Audin, M.: The Topology of Torus Actions on Symplectic Manifolds. Birkäuser. Progress in Mathematics 93. 1991 · Zbl 0726.57029
[3] Barannikov, S.: Quantum periods. I. Semi-infinite variations of Hodge structures. Internat. Math. Res. Notices (23), 1243–1264 (2001) · Zbl 1074.14510
[4] Brion, M.: Piecewise Polynomial Functions, Convex Polytopes and Enumerative Geometry. Parameter spaces (Warsaw, 1994), Banach Center Publ., 36, Polish Acad. Sci., Warsaw, 1996, pp. 25–44 · Zbl 0878.14035
[5] Coates, T., Givental, A.B.: Quantum Riemann-Roch, Lefschetz and Serre. math.AG/0110142 · Zbl 1189.14063
[6] Cohen, R.L. Jones, J.D.S. Segal, G.B. Floer’s infinite-dimensional Morse theory and homotopy theory. The Floer memorial volume, Progr. Math., 133, Birkäuser, Basel, 1995, pp. 297–325 · Zbl 0843.58019
[7] Cox, D.A., Katz, S.: Mirror Symmetry and Algebraic Geometry. Mathematical Surveys and Monographs, 68. American Mathematical Society, Providence, RI, 1999 · Zbl 0951.14026
[8] Givental, A.B. Homological Geometry. I. Projective Hypersurfaces. Selecta Math. (N.S.) 1, 325–345 (1995) · Zbl 0920.14028
[9] Givental, A.B.: Equivariant Gromov-Witten Invariants. Internat. Math. Res. Notices (13), 613–663 (1996) · Zbl 0881.55006
[10] Givental, A.B.: A Mirror Theorem for Toric Complete Intersections. Topological field theory, primitive forms and related topics (Kyoto, 1996), Progr. Math., 160, Birkhäuser Boston, Boston, MA, 1998, pp. 141–175 · Zbl 0936.14031
[11] Givental, A.B. Elliptic Gromov-Witten invariants and the generalized mirror conjecture. Integrable systems and algebraic geometry (Kobe/Kyoto, 1997), pp. 107–155, World Sci. Publishing, River Edge, NJ, 1998 · Zbl 0961.14036
[12] Guest, M.A. Introduction to Homological Geometry: part I. math.DG/0104274 · Zbl 1106.53058
[13] Guest, M.A. Introduction to Homological Geometry: part II. math.DG/0105032 · Zbl 1114.53073
[14] Guest, M.A.: Quantum Cohomology via D-modules. math.DG/0206212 · Zbl 1081.53077
[15] Iritani, H.: The S1 Equivariant Cohomology of the Universal Covering of Free Loop Spaces. master thesis, 2003
[16] Iritani, H.: Quantum D-modules and Generalized Mirror Transformations. (in preparation) · Zbl 1170.53071
[17] Kontsevich, M., Manin, Y.I.: Gromov-Witten Classes, Quantum Cohomology, and Enumerative Geometry. Comm. Math. Phys. 164, 525–562 (1994) · Zbl 0853.14020
[18] Lian, B.H., Liu, K., Yau, S.-T.: Mirror principle. I. Asian J. Math. 1, 729–763 (1997) · Zbl 0953.14026
[19] Lian, B.H., Liu, K., Yau, S.-T.: Mirror principle. II. Sir Michael Atiyah: a great mathematician of the twentieth century. Asian J. Math. 3, 109–146 (1999)
[20] Lian, B.H., Liu, K., Yau, S.-T.: Mirror principle. III. Asian J. Math. 3, 771–800 (1999) · Zbl 1009.14006
[21] Manin, Y.I: Frobenius Manifolds, Quantum Cohomology, and Moduli Spaces. American Mathematical Society Colloquium Publications, 47. American Mathematical Society, Providence, RI, 1999 · Zbl 0952.14032
[22] Pandharipande, R.: Rational curves on hypersurfaces (after A. Givental). Sminaire Bourbaki. Vol. 1997/98. Astérisque 252, Exp. No. 848, 5, 307–340 (1998) · Zbl 0932.14029
[23] Piunikhin, S. Salamon, D. Schwarz, M.: Symplectic Floer-Donaldson Theory and Quantum Cohomology. Contact and symplectic geometry (Cambridge, 1994), Publ. Newton Inst., 8, Cambridge Univ. Press, Cambridge, 1996, pp. 171–200 · Zbl 0874.53031
[24] x Ruan, Y., Tian, G.: Bott-type Symplectic Floer Cohomology and its Multiplication Structures. Math. Res. Lett. 2, 203–219 (1995) · Zbl 0844.57023
[25] Saito, K.: Period mapping associated to a primitive form. Publ. Res. Inst. Math. Sci. 19, 1231–1264 (1983) · Zbl 0539.58003
[26] Vlassopoulos, Y.: Quantum Cohomology and Morse Theory on the Loop Space of Toric Varieties. math.AG/0203083
[27] Witten, E.: Phases of N = 2 theories in two dimensions. Nuclear Phys. B 403, 159–222 (1993) · Zbl 0910.14020
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