A Labelling Scheme for Higher Dimensional Simplex Equations

We present a succinct way of obtaining all possible higher dimensional generalization of Quantum Yang-Baxter Equation (QYBE). Using the scheme, we could generate the two popular three-simplex equations, namely: Zamolodchikov’s tetrahedron equation (ZTE) and Frenkel and Moore equation (FME). E-mail address: [email protected] E-mail address: [email protected] The Quantum Yang-Baxter Equation (QYBE) is a non-linear equation which appears in various forms in areas like integrable statistical models, topological field theories, the theory of braid groups,the theory of knots and links and conformal field theory. Several higher dimensional generalizations of QYBE have been proposed. By considering the scattering of straight strings in 2 + 1 dimensions, Zamolodchikov proposed a higher dimensional generalization of QYBE, commonly called the tetrahedron equation (ZTE) [1]: R123R145R246R356 = R356R246R145R123, (1) where R123 = R ⊗ 1I etc and R ∈ End(V ⊗ V ⊗ V ) for some vector space V . As the QYBE is often called a two-simplex equation, this generalization of QYBE is a three-simplex equation. In general, there are d equations with d variables, where d is the dimension of the vector space V . Despite its complexity, Zamolodchikov has ingeniously proposed a spectral dependent solution which is subsequently confirmed by Baxter [1, 2]. The tetrahedron equation is not the only possible higher dimensional generalization. Frenkel and Moore [3] has proposed another higher dimensional generalization (FME), namely R123R124R134R234 = R234R134R124R123, (2) They have also given an analytical solution for their three-simplex equation, namely: R = qq[1 + (q − q)(h⊗ e⊗ f + f ⊗ e⊗ h− e⊗ hq ⊗ f)], (3) where e, f , h are generators of sl(2) satisfying [h, e] = 2e, [h, f ] = 2f , [e, f ] = h. The threesimplex equation is not equivalent to Zamolodchikov’s equation as explained by Frenkel and Moore in their paper [3]. Furthermore, solutions of the three-simplex equation do not in general satisfy Zamolodchikov’s tetrahedron equation (1). In fact, the expression (3) does not satisfy Zamolodchikov’s tetrahedron equation unless the parameter q approaches unity, giving the identity matrix.