By Faddeev L., Moerbeke P.V., Lambert F. (eds.)

ISBN-10: 1402035012

ISBN-13: 9781402035012

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For the integrable systems on quad-graphs we consider in this section the ˆ are attached to the vertices of the graph. They are subject to fields z : V (D) → C an equation Q(z 1 , z 2 , z 3 , z 4 ) = 0, relating four fields sitting on the four vertices 46 Alexander I. Bobenko Figure 2. A face of the labelled quad-graph of an arbitrary face from F(D). The Hirota equation z4 αz 3 − βz 1 = z2 βz 3 − αz 1 (1) is such an example. We observe that the equation carries parameters α, β which can be naturally associated to the edges, and the opposite edges of an elementary quadrilateral carry equal parameters (see Figure 2).

57–84. 6. Hollowood, T. J. and Mavrikis, E. (1997) Nucl. Phys. B484, pp. 631–652, hepth/9606116. 7. Mattsson, P. and Dorey, P. (2000) J. Phys. A33, pp. 9065–9094, hep-th/0008071. 8. Chim, L. (1996) Int. J. Mod. Phys. A11, pp. 4491–4512, hep-th/9510008. 9. Ahn, C. and Koo, W. M. (1996) Nucl. Phys. B482, p. 675, hep-th/9606003. 10. Nepomechie, R. I. Supersymmetry in the boundary tricritical Ising field theory, preprint UMTG-234, hep-th/0203123 GEOMETRY OF DISCRETE INTEGRABILITY. THE CONSISTENCY APPROACH Alexander I.

Xn ), Hi j f (N ) (x1 , . . , xn ) = h i j f (N ) (x1 , . . , xn ). For example, let n = 3 and N = 2. Then f (2) (x1 , x2 , x3 ) = (2, 0, 0)x12 + (0, 2, 0)x22 + (0, 0, 2)x32 + 2(1, 1, 0)x1 x2 + 2(0, 1, 1)x2 x3 + 2(1, 0, 1)x1 x3 and E 12 f (2) (x1 , x2 , x3 ) = 2(0, 2, 0)x1 x2 + 2(1, 1, 0)x12 + 2(0, 1, 1)x1 x3 . Comparison of coefficients of monomials yields actions e12 (2, 0, 0) = 2(1, 1, 0), e12 (1, 1, 0) = (0, 2, 0), e12 (1, 0, 1) = (0, 1, 1), the others vanishing. (10) (11) 20 Chris Athorne This construction is slightly round the houses.

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