Contributions à l'Etude du Vertex Topologique en Théorie ... - Toubkal

344 L.B. Drissi et al. / Nuclear Physics B 801 [FS] (2008) 316–345where p j are given byp j =(p − 1)!, j = 0,...,p− 1.j!(p − j − 1)!(A.37)(ii) Using the decomposition for Γ (p)− (z),Γ (p)− (z) = O 0(x 0 )O 1 (x 1 )O 2 (x 2 ) ···O p−1 (x p−1 )(A.38)we getO j+1 (x j+1 ) =∏p ji=1Γ (j+1) (− q j z ) , j = 0,...,p− 1.(A.39)Appendix B. Combinatorial Eq. (5.17)Here we want to derive the identity (5.17) namely,s∑k=1C p−2k+p−3 = Cp−1 s+p−2, p 2.This is a standard combinatorial identity; its proof follows from basic property [37],C k n+1 = Ck−1 n + C k n .(B.1)(B.2)Applying this identity to Cn k and putting it back into the above relation, we get,(B.3)Cn+1 k = Ck−1 n + C k−1n−1 + Ck n−1 .By induction, it results,C k n+1 =n∑j=k−1C k−1j.Setting k = p − 1 and n = s + p − 3, we recover the identity (B.1).(B.4)References[1] H.N. Temperley, Statistical mechanics and the partition of numbers I: The transition to liquid helium, Proc. R. Soc.London A 199 (1949) 361;H.N. Temperley, Statistical mechanics and the partition of numbers II: The form of crystal surfaces, Proc. CambridgePhilos. Soc. 48 (1952) 683.[2] E.J. van Rensburg, The Statistical Mechanics of Interacting Walks, Polygons, Animals and Vesicles, Oxford Univ.Press, Oxford, 2000.[3] C. Weiss, M. Holthaus, From number theory to statistical mechanics: Bose–Einstein condensation in isolated traps,Chaos Solitons Fractals 10 (1999) 795.[4] V. Elser, Solution of the dimer problem on a hexagonal lattice with boundary, J. Phys. A 17 (1984) 1509.[5] A. Okounkov, N. Reshetikhin, C. Vafa, Quantum Calabi–Yau and classical crystals, hep-th/0309208.[6] R. Kenyon, An introduction to the dimer model, math.CO/0310326.[7] R. Kenyon, A. Okounkov, S. Sheffield, Dimers and amoebae, math-ph/0311005.[8] D. Ghoshal, C. Vafa, c = 1 string as the topological theory of the conifold, Nucl. Phys. B 453 (1995) 121, hepth/9506122.[9] E. Witten, Ground ring of two-dimensional string theory, Nucl. Phys. B 373 (1992) 187, hep-th/9108004.