Geodesic
The WDVV symmetries in two-primary models
Teoretičeskaâ i matematičeskaâ fizika, Tome 161 (2009) no. 3, pp. 367-381 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

From the bi-Hamiltonian standpoint, we investigate symmetries of Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations proposed by Dubrovin. These symmetries can be viewed as canonical Miura transformations between genus-zero bi-Hamiltonian systems of hydrodynamic type. In particular, we show that the moduli space of two-primary models under symmetries of the WDVV equations can be parameterized by the polytropic exponent $h$. We discuss the transformation properties of the free energy at the genus-one level.
Keywords: Frobenius manifold, WDVV equation, bi-Hamiltonian structure, primary free energy, dToda hierarchy, Benney hierarchy, dDym hierarchy, polytropic gas dynamics. 
@article{TMF_2009_161_3_a5,
     author = {Yu-Tung Chen and Niann-Chern Lee and Ming-Hsien Tu},
     title = {The~WDVV symmetries in two-primary models},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {367--381},
     year = {2009},
     volume = {161},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2009_161_3_a5/}
}
TY  - JOUR
AU  - Yu-Tung Chen
AU  - Niann-Chern Lee
AU  - Ming-Hsien Tu
TI  - The WDVV symmetries in two-primary models
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2009
SP  - 367
EP  - 381
VL  - 161
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TMF_2009_161_3_a5/
LA  - ru
ID  - TMF_2009_161_3_a5
ER  - 
%0 Journal Article
%A Yu-Tung Chen
%A Niann-Chern Lee
%A Ming-Hsien Tu
%T The WDVV symmetries in two-primary models
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2009
%P 367-381
%V 161
%N 3
%U http://geodesic.mathdoc.fr/item/TMF_2009_161_3_a5/
%G ru
%F TMF_2009_161_3_a5
Yu-Tung Chen; Niann-Chern Lee; Ming-Hsien Tu. The WDVV symmetries in two-primary models. Teoretičeskaâ i matematičeskaâ fizika, Tome 161 (2009) no. 3, pp. 367-381. http://geodesic.mathdoc.fr/item/TMF_2009_161_3_a5/

[1] I. M. Krichever, Comm. Math. Phys., 143:2 (1992), 415–429 | DOI | MR | Zbl

[2] B. Dubrovin, “Geometry of 2D topological field theories”, Integrable Systems and Quantum Group, Lecture Notes in Math., 1620, eds. M. Francaviglia, S. Greco, Berlin, Springer, 1996, 120–348 | DOI | MR | Zbl

[3] Yu. I. Manin, Frobenius Manifolds, Quantum Cohomology, and Moduli Spaces, Amer. Math. Soc. Colloq. Publ., 47, AMS, Providence, RI, 1999 | MR | Zbl

[4] N. Hitchin, “Frobenius manifolds”, With notes by D. Calderbank, Gauge Theory and Symplectic Geometry, NATO Adv. Sci. Inst. Ser. C, 488, eds. J. Hurtubise, F. Lalonde, G. Sabidussi, Kluwer, Dordrecht, 1997, 69–112 | MR | Zbl

[5] C. Hertling, Frobenius Manifolds and Moduli Spaces for Singularities, Cambridge Tracts in Math., 151, Cambridge Univ. Press, Cambridge, 2002 | MR | Zbl

[6] K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil, E. Zaslow, Mirror Symmetry, Clay Math. Monogr., 1, Providence, RI, AMS, 2003 | MR | Zbl

[7] I. A. B. Strachan, J. Math. Phys., 40:10 (1999), 5058–5079 | DOI | MR | Zbl

[8] E. Witten, Nucl. Phys. B, 340:2–3 (1990), 281–332 | DOI | MR

[9] R. Dijkgraaf, E. Verlinde, H. Verlinde, Nucl. Phys. B, 352:1 (1991), 59–86 | DOI | MR

[10] B. Dubrovin, Nucl. Phys. B, 379:3 (1992), 627–689 | DOI | MR

[11] G. B. Whitham, Linear and Nonlinear Waves, Pure Appl. Math., Wiley, New York–London–Sydney, 1974 | MR | Zbl

[12] P. J. Olver, Y. Nutku, J. Math. Phys., 29:7 (1988), 1610–1619 | DOI | MR | Zbl

[13] Y.-T. Chen, M.-H. Tu, Lett. Math. Phys., 63:2 (2003), 125–139 | DOI | MR | Zbl

[14] J.-H. Chang, M.-H. Tu, J. Phys. A, 34:2 (2001), 251–272 | DOI | MR | Zbl

[15] T. Eguchi, S.-K. Yang, Modern Phys. Lett. A, 9:31 (1994), 2893–2902 | DOI | MR | Zbl

[16] B. Dubrovin, Y. Zhang, Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov–Witten invariants, arXiv: math.DG/0108160

[17] B. Dubrovin, “On almost duality for Frobenius manifolds”, Geometry, Topology, and Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, 212, eds. V. M. Buchstaber, I. M. Krichever, Providence, RI, AMS, 2004, 75–132 | MR | Zbl

[18] A. Riley, I. A. B. Strachan, J. Nonlinear Math. Phys., 14:1 (2007), 82–94 | DOI | MR | Zbl

[19] K. Takasaki, T. Takebe, Rev. Math. Phys., 7:5 (1995), 743–808 | DOI | MR | Zbl

[20] S. Aoyama, Y. Kodama, Comm. Math. Phys., 182:1 (1996), 185–219 | DOI | MR | Zbl

[21] L.-C. Li, Comm. Math. Phys., 203:3 (1999), 573–592 | DOI | MR | Zbl

[22] B. A. Dubrovin, S. P. Novikov, DAN SSSR, 279:2 (1984), 294–297 | MR | Zbl

[23] B. Dubrovin, Y. Zhang, Comm. Math. Phys., 198:2 (1998), 311–361 | DOI | MR | Zbl

[24] R. Dijkgraaf, “Intersection theory, integrable hierarchies and topological field theory”, New Symmetry Principles in Quantum Field Theory, NATO Adv. Sci. Inst. Ser. B, 295, eds. J. Fröhlich, G.'t Hooft, A. Jaffe, G. Mack, P. K. Mitter, R. Stora, Plenum, New York, 1992, 95–158 | MR | Zbl

[25] Y.-T. Chen, N.-C. Lee, M.-H. Tu, J. Math. Phys., 48:7 (2007), 072702 | DOI | MR | Zbl

[26] E. Getzler, J. Amer. Math. Soc., 10:4 (1997), 973–998 | DOI | MR | Zbl

[27] I. A. B. Strachan, Int. Math. Res. Not., 2003:19 (2003), 1035–1051 | DOI | MR | Zbl

[28] M. V. Pavlov, TMF, 150:2 (2007), 263–285 | DOI | MR | Zbl