Geodesic
Integrable Structure Behind the WDVV Equations
Teoretičeskaâ i matematičeskaâ fizika, Tome 134 (2003) no. 1, pp. 18-31 Cet article a éte moissonné depuis la source Math-Net.Ru

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An integrable structure behind the Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations is identified with the reduction of the Riemann–Hilbert problem for the homogeneous loop group $\widehat{GL}(N,\mathbb C)$. The reduction requires the dressing matrices to be fixed points of an order-two loop group automorphism resulting in a subhierarchy of the $\widehat{gl}(N,\mathbb C)$ hierarchy containing only odd-symmetry flows. The model has Virasoro symmetry; imposing Virasoro constraints ensures the homogeneity property of the Darboux–Egoroff structure. Dressing matrices of the reduced model provide solutions of the WDVV equations.
Keywords: WDVV equations, dressing, Darboux–Egoroff metrics, Kadomtsev–Petviashvili hierarchies, tau functions, Riemann–Hilbert factorization. 
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Kh. Aratin; Zh. van de Ler. Integrable Structure Behind the WDVV Equations. Teoretičeskaâ i matematičeskaâ fizika, Tome 134 (2003) no. 1, pp. 18-31. http://geodesic.mathdoc.fr/item/TMF_2003_134_1_a2/

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

[2] B. Dubrovin, “Integrable systems and classification of 2-dimensional topological field theories”, Integrable Systems, The Verdier Memorial Conference (Luminy, France, July 1–5, 1991), Prog. Math., 115, eds. O. Babelon, O. Cartier, Y. Kosmann-Schwarzbach, Birkhäuser, Boston, MA, 1993, 313–359 | MR | Zbl

[3] B. Dubrovin, “Painleve transcendents and two-dimensional topological field theory”, The Painleve Property: One Century Later, CRM Ser. Math. Phys., ed. R. Conte, Springer, New York, 1999, 287–412 ; E-print math.ag/9803107 | MR | Zbl

[4] G. Wilson, “The $\tau$-Function of the $g$AKNS equations”, Integrable Systems, The Verdier Memorial Conference (Luminy, France, July 1–5, 1991), Prog. Math., 115, eds. O. Babelon, O. Cartier, Y. Kosmann-Schwarzbach, Birkhäuser, Boston, MA, 1993, 131–145 | MR | Zbl

[5] H. Aratyn, J. van de Leur, Solutions of the WDVV equations and integrable hierarchies of KP type, E-print hep-th/0104092 | MR

[6] J. van de Leur, Int. Math. Res. Notices, 11 (2001), 551–573 ; E-print nlin.si/0004021 | DOI | Zbl

[7] J. W. van de Leur, R. Martini, Commun. Math. Phys., 205 (1999), 587–616 ; E-print solv-int/9808008 | DOI | MR | Zbl

[8] B. Dubrovin, Y. Zhang, Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov–Witten invariants, E-print math.dg/0108160