On the CP1 topological sigma model and the Toda lattice hierarchy

Journal of Geometry and Physics

Volumn 40, Issue 3-4, 2002, Pages 215-232

  Zhang, Youjin ^a  

^a TSINGHUA UNIVERSITY   (China)

Author keywords

2D topological field theory; Bihamiltonian structure; Integrable system

Indexed keywords

EID: 0346532050     PISSN: 03930440     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0393-0440(01)00036-5     Document Type: Article

References (28)

1. [1] P. Belorosski, R. Pandharipande, A descendent relation in genus 2, math/9803072.
2. [2] Dijkgraaf, R., Verlinde, E., Verlinde, H., Topological strings in d<1. Nucl. Phys. B, 352, 1991, 59.
3. [3] Dijkgraaf, R., Witten, E., Mean field theory, topological field theory, and multi-matrix models. Nucl. Phys. B 342 (1990), 486–522.
4. [4] B. Dubrovin, Integrable systems and classification of 2-dimensional topological field theories, in: O. Babelon, O. Cartier, Y. Kosmann-Schwarbach (Eds.), Integrable Systems, Proceedings of Luminy 1991 Conference Dedicated to the Memory of J.-L. Verdier, Birkhäuser, 1993.
5. [5] Dubrovin, B., Integrable systems in topological field theory. Nucl. Phys. B 379 (1992), 627–689.
6. [6] B. Dubrovin, Geometry of 2D topological field theories, in: M. Francaviglia, S. Greco (Eds.), Integrable Systems and Quantum Groups, Montecalini, Terme, 1993, Springer Lecture Notes in Mathematics, Vol. 1620, Springer, Berlin, 1996, pp. 120–348.
7. [7] Dubrovin, B., Zhang, Y., Bihamiltonian hierarchies in 2D topological field theory at one loop approximation. Commun. Math. Phys. 198 (1998), 311–361.
8. [8] Dubrovin, B., Zhang, Y., Frobenius manifolds and Virasoro constraints. Selecta Math. New Ser. 5 (1999), 423–466.
9. [9] B. Dubrovin, Y. Zhang, Normal forms of integrable hierarchies and Gromov–Witten invariants, in press.
10. [10] Eguchi, T., Kanno, H., Toda lattice hierarchy and the topological description of the c=1 string theory. Phys. Lett. B 331 (1994), 330–334.
11. 1 model and the large-N matrix integral. Mod. Phys. Lett. A 9 (1994), 2893–2902.
12. [12] Eguchi, T., Yamada, Y., Yang, S.-K., On the genus expansion in the topological string theory. Rev. Math. Phys., 7, 1995, 279.
13. [13] Eguchi, T., Hori, K., Xiong, C.S., Gravitational quantum cohomology. Int. J. Mod. Phys. A 12 (1997), 1743–1782.
14. [14] Eguchi, T., Hori, K., Xiong, C.S., Quantum cohomology and Virasoro algebra. Phys. Lett. B 402 (1997), 71–80.
15. [15] Eguchi, T., Xiong, C.S., Quantum cohomology at higher genus: topological recursion relations and Virasoro conditions. Adv. Theor. Math. Phys. 2 (1998), 219–229.
16. [16] T. Eguchi, E. Getzler, C.S. Xiong, Topological gravity in genus 2 with two primary fields, hep-th/0007194.
17. [17] Getzler, E., Intersection theory on [Figure presented] and elliptic Gromov–Witten invariants, alg-geom/9612004. J. Am. Math. Soc. 10 (1997), 973–998.
18. [18] E. Getzler, Topological recursion relations in genus 2, math/9801003.
19. [19] E. Getzler, A Darboux theorem for Hamiltonian operators in the formal calculus of variations, math.DG/0002164.
20. [20] Kontsevich, M., Intersection theory on the moduli space of curves and the matrix Airy function. Commun. Math. Phys. 147 (1992), 1–23.
21. [21] B.A. Kupershmidt, Discrete Lax equations and differential difference calculus, Astérisque, 1985.
22. [22] Liu, X., Tian, G., Virasoro constraints for quantum cohomology. J. Diff. Geom. 50 (1998), 537–590.
23. [23] X. Liu, Elliptic Gromov–Witten invariants and Virasoro conjecture, math.AG/9907113.
24. [24] Magri, F., A simple model of the integrable Hamiltonian systems. J. Math. Phys. 19 (1978), 1156–1162.
25. [25] Toda, M., Wave propagation in anharmonic lattices. J. Phys. Soc. Jpn. 23 (1967), 501–506.
26. [26] Witten, E., On the structure of the topological phase of two-dimensional gravity. Nucl. Phys. B 340 (1990), 281–332.
27. [27] Witten, E., Two-dimensional gravity and intersection theory on moduli space. Surv. Diff. Geom. 1 (1991), 243–310.
28. [28] E. Witten, On the Kontsevich model and other models of two-dimensional gravity, Preprint IASSNS-HEP-91/24.