Abstract
The theory of genus zero Gromov-Witten invariants associates to a compact symplectic manifold X a Frobenius manifold H (also known as the small phase space of X) whose underlying flat manifold is the cohomology space H*(X, ℂ). Higher genus Gromov-Witten invariants give rise to a sequence of generating functions F X g , one for each genus g > 0; these are functions on the large phase space
The manifold H∞ has a rich geometric structure: it is the jet space of curves in the Frobenius manifold H. (This identification is implicit in Dubrovin [9].)
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References
L. Abrams, The quantum Euler class and the quantum cohomology of the Grassmannians. Israel J. Math. 117 (2000), 335–352.
A. Bayer and Yu. Manin, (Semi)simple exercises in quantum cohomology. <mml:math. AG/0103164>
K. Behrend, Gromov-Witten invariants in algebraic geometry, Invent. Math. 127 (1997), 601–617. <alg-geom/9601011>
K. Behrend and Yu. Manin, Stacks of stable maps and Gromov-Witten invariants, Duke Math. J. 85 (1996), 1–60. <alg-geom/9506023>
D. Cox and S. Katz, “Mirror symmetry and algebraic geometry.” Mathematical Surveys and Monographs, 68. American Mathematical Society, Providence, RI, 1999.
T. Coates and A. Givental, Quantum Riemann-Roch, Lefschetz and Serre, <mml:math.AG/0110142>
R. Dijkgraaf and E. Witten, Mean field theory, topological field theory, and multi-matrix models, Nucl. Phys. B342 (1990), 486–522.
B. Dubrovin, Differential geometry of the space of orbits of a Coxeter group. Surveys in differential geometry: integrable systems, 181–211, Surv. Differ. Geom., IV, Int. Press, Boston, MA, 1998.
B. Dubrovin, Geometry of 2D topological field theories, in “Integrable systems and quantum groups, Montecalini Terme, 1993,” eds. M. Prancaviglia and S. Greco, Lect. Notes Math., vol. 1620, Springer-Verlag, Berlin, 1996, pp. 120–348.
B. Dubrovin, Flat pencils of metrics and Frobenius manifolds. In “Integrable systems and algebraic geometry (Kobe/Kyoto, 1997),” World Sci. Publishing, River Edge, NJ, 1998, pp. 47–72. <mml:math.DG/9803106>
B. Dubrovin, Y. Zhang, Bihamiltonian hierarchies in 2D topological field theory at one-loop approximation, Commun. Math. Phys., 198 (1998), 311–361. <hep-th/9712232>
B. Dubrovin and Y. Zhang, Frobenius manifolds and Virasoro constraints Selecta Math. (N.S.) 5 (1999), 423–466. <mml:math.AG/9808048>
B. Dubrovin and Y. Zhang, Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants. <mml:math.DG/0108160>
T. Eguchi, E. Getzler and C.-S. Xiong, Topological gravity in genus 2 with two primary fields. Adv. Theor. Math. Phys. 4 (2000), 981–998. <hep-th/0007194>
T. Eguchi, K. Hori and C.-S. Xiong, Quantum cohomology and Virasoro algebra, Phys. Lett. B 402 (1997), 71–80. <hep-th/9703086>
T. Eguchi, Y. Yamada and S.-K. Yang, On the genus expansion in the topological string theory, Rev. Math. Phys. 7 (1995), 279–309. <hep-th/9405106>
T. Eguchi and C.-S. Xiong, Quantum cohomology at higher genus: topological recursion relations and Virasoro conditions, Adv. Theor. Math. Phys. 2 (1998), 219–228. <hep-th/9801010>
E. Getzler, Intersection theory on M̄1,4 and elliptic Gromov-Witten invariants, J. Amer. Math. Soc. 10 (1997), 973–998. alg-geom/9612004
E. Getzler, Topological recursion relations in genus 2. In “Integrable systems and algebraic geometry (Kobe/Kyoto, 1997),” World Sci. Publishing, River Edge, NJ, 1998, pp. 73–106. <mml:math.AG/9801003>
E. Getzler, The Virasoro conjecture for Gromov-Witten invariants, “Algebraic geometry: Hirzebruch 70 (Warsaw, 1998),” Contemp. Math. 241, Amer. Math. Soc, Providence, RI, 1999, pp. 147–176. <mml:math.AG/9812026>
A. Givental, Gromov-Witten invariants and quantization of quadratic Hamiltonians, Mose. Math. J. 1 (2001), 551–568.
C. Hertling, “Frobenius manifolds and moduli spaces for singularities.” Cambridge Tracts in Mathematics, 151. Cambridge University Press, Cambridge, 2002.
C. Hertling and Yu. Manin, Weak Frobenius manifolds, Internat. Math. Res. Notices (1999), 277–286. <mml:math.QA/9810132>
K. Hori, Constraints for topological strings in D ≥ 1, Nucl. Phys. B439 (1995) 395–420. hep-th/9411135.
M. Kontsevich and Yu. I. Manin, Relations between the correlators of the topological sigmamodel coupled to gravity, Comm. Math. Phys. 196 (1998), 385–398. <alg-geom/9708024>.
X. Liu, Elliptic Gromov-Witten invariants and Virasoro conjecture, Comm. Math. Phys. 216, 705–728. <mml:math.AG/9907113>
Yu. Manin, “Frobenius manifolds, quantum cohomology, and moduli spaces,” American Mathematical Society Colloquium Publications, 47. American Mathematical Society, Providence, RI, 1999.
K. Saito, On a linear structure of the quotient variety by a finite reflection group. Publ. Res. Inst. Math. Sci. 29 (1993), 535–579.
E. Witten, Two dimensional gravity and intersection theory on moduli space, Surveys in Differential Geom. 1 (1991), 243–310.
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Getzler, E. (2004). The jet-space of a Frobenius manifold and higher-genus Gromov-Witten invariants. In: Hertling, K., Marcolli, M. (eds) Frobenius Manifolds. Aspects of Mathematics, vol 36. Vieweg+Teubner Verlag. https://doi.org/10.1007/978-3-322-80236-1_3
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