Book Review
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Book Information:
Author:
Yuri I. Manin
Title:
Frobenius manifolds, quantum cohomology, and moduli spaces
Additional book information:
Amer. Math. Soc.,
Providence, RI,
1999,
xiii + 303 pp.,
ISBN 0-8218-1917-8,
$55.00$
1. K. Behrend, Algebraic Gromov-Witten invariants. ``New trends in algebraic geometry (Warwick, 1996),'' London Math. Soc. Lecture Note Ser. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 19-70. CMP 2000:02
2. D. Cox and S. Katz, ``Mirror symmetry and algebraic geometry.'' Math. Surveys Monogr. 68, Amer. Math. Soc., Providence, RI, 1999. MR 1677117
3. B. Dubrovin, Geometry of 
D topological field theories. ``Integrable systems and quantum groups (Montecatini Terme, 1993),'' Lecture Notes in Math. 1620, Springer, Berlin, 1996, pp. 120-348. MR 1397274
4. B. Dubrovin, Painlevé transcendents in two-dimensional topological field theory. ``The Painlevé property,'' CRM Ser. Math. Phys., Springer, New York, 1999, 287-412. CMP 2000:02
5. B. Dubrovin, Geometry and analytic theory of Frobenius manifolds. ``Proceedings of the International Congress of Mathematicians, (Berlin, 1998),'' Doc. Math. 1998, Extra Vol. II, pp. 315-326. MR 1648082
6. B. Dubrovin and Y. Zhang, Bihamiltonian hierarchies in 2D topological field theory at one-loop approximation, Commun. Math. Phys. 198 (1998), 311-361. MR 1672512
7. T. Eguchi, K. Hori and C. S. Xiong, Quantum cohomology and Virasoro algebra, Phys. Lett. B402 (1997), 71-80. MR 1454328
8. W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology. ``Algebraic geometry--Santa Cruz 1995,'' Proc. Sympos. Pure Math. 62, Part 2, Amer. Math. Soc., Providence, RI, 1997, pp. 45-96. MR 1492534
9. A. B. Givental, On the WDVV-equation in quantum K-theory, arXiv.org/abs/math/0003158.
10. M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307-347. MR 0809718
11. J. Harris and I. Morrison. ``Moduli of curves.'' Graduate Texts in Mathematics 187, Springer, New York, 1998. MR 1631825
12. N. Hitchin, Frobenius manifolds. ``Gauge theory and symplectic geometry (Montreal, 1995),'' NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 488, Kluwer, Dordrecht, 1997, 69-112. MR 1461570
13. Y. Ruan, Topological sigma model and Donaldson-type invariants in Gromov theory, Duke Math. J. 83 (1996), no. 2, 461-500. MR 1390655
14. K. Saito, Period mapping associated to a primitive form, Publ. Res. Inst. Math. Sci. 19 (1983), 1231-1264. MR 0723468
15. E. Witten, Two dimensional gravity and intersection theory on moduli space, Surveys in Differential Geom. 1 (1991), 243-310. MR 1144529
- 1.
- K. Behrend, Algebraic Gromov-Witten invariants. ``New trends in algebraic geometry (Warwick, 1996),'' London Math. Soc. Lecture Note Ser. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 19-70. CMP 2000:02
- 2.
- D. Cox and S. Katz, ``Mirror symmetry and algebraic geometry.'' Math. Surveys Monogr. 68, Amer. Math. Soc., Providence, RI, 1999. MR 1677117
- 3.
- B. Dubrovin, Geometry of D topological field theories. ``Integrable systems and quantum groups (Montecatini Terme, 1993),'' Lecture Notes in Math. 1620, Springer, Berlin, 1996, pp. 120-348. MR 1397274
- 4.
- B. Dubrovin, Painlevé transcendents in two-dimensional topological field theory. ``The Painlevé property,'' CRM Ser. Math. Phys., Springer, New York, 1999, 287-412. CMP 2000:02
- 5.
- B. Dubrovin, Geometry and analytic theory of Frobenius manifolds. ``Proceedings of the International Congress of Mathematicians, (Berlin, 1998),'' Doc. Math. 1998, Extra Vol. II, pp. 315-326. MR 1648082
- 6.
- B. Dubrovin and Y. Zhang, Bihamiltonian hierarchies in 2D topological field theory at one-loop approximation, Commun. Math. Phys. 198 (1998), 311-361. MR 1672512
- 7.
- T. Eguchi, K. Hori and C. S. Xiong, Quantum cohomology and Virasoro algebra, Phys. Lett. B402 (1997), 71-80. MR 1454328
- 8.
- W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology. ``Algebraic geometry--Santa Cruz 1995,'' Proc. Sympos. Pure Math. 62, Part 2, Amer. Math. Soc., Providence, RI, 1997, pp. 45-96. MR 1492534
- 9.
- A. B. Givental, On the WDVV-equation in quantum K-theory, arXiv.org/abs/math/0003158.
- 10.
- M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307-347. MR 0809718
- 11.
- J. Harris and I. Morrison. ``Moduli of curves.'' Graduate Texts in Mathematics 187, Springer, New York, 1998. MR 1631825
- 12.
- N. Hitchin, Frobenius manifolds. ``Gauge theory and symplectic geometry (Montreal, 1995),'' NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 488, Kluwer, Dordrecht, 1997, 69-112. MR 1461570
- 13.
- Y. Ruan, Topological sigma model and Donaldson-type invariants in Gromov theory, Duke Math. J. 83 (1996), no. 2, 461-500. MR 1390655
- 14.
- K. Saito, Period mapping associated to a primitive form, Publ. Res. Inst. Math. Sci. 19 (1983), 1231-1264. MR 0723468
- 15.
- E. Witten, Two dimensional gravity and intersection theory on moduli space, Surveys in Differential Geom. 1 (1991), 243-310. MR 1144529
Review Information:
Reviewer:
Ezra Getzler
Affiliation:
Northwestern University
Email:
getzler@math.northwestern.edu
Journal:
Bull. Amer. Math. Soc.
38 (2001), 101-108
Published electronically:
October 2, 2000
Review copyright:
© Copyright 2000
American Mathematical Society
