arXiv subjects
Publications and source records attributed to Yoshihiro Takeyama.
We give differential equations compatible with the rational qKZ equation with boundary reflection. The total system contains the trigonometric degeneration of the bispectral qKZ equation of type (C_{n}^{\vee}, C_{n}) which in the case of type GL_{n} was studied by van Meer and Stokman. We construct an integral formula for solutions to our compatible system in a special case.
We construct special solutions to the rational quantum Knizhnik-Zamolodchikov equation associated with the Lie algebra $gl_N$. The main ingredient is a special class of the shifted non-symmetric Jack polynomials. It may be regarded as a shifted version of the singular polynomials studied by Dunkl. We prove that our solutions contain those obtained as a scaling limit of matrix elements of the vertex operators of level one.
We prove some relations for the $q$-multiple zeta values ($q$MZV). They are $q$-analogues of the cyclic sum formula, the Ohno relation and the Ohno-Zagier relation for the multiple zeta values (MZV). We discuss the problem to determine the dimension of the space spanned by $q$MZV's over $\Q$, and present an application to MZV.
We obtain a new integral formula for solutions of the rational quantum Knizhnik-Zamolodchikov equation associated with Lie algebra sl_{N} at level zero. Our formula contains the integral representation of form factors of SU(N) invariant Thirring model constructed by F. Smirnov. We write down recurrence relations arising from the construction of the form factors. We check that the recurrence relations hold for the form factors of the energy momentum tensor.
We consider the q-hypergeometric equation with q^{N}=1 and $α, β, γ\in {\Bbb Z}$. We solve this equation on the space of functions given by a power series multiplied by a power of the logarithmic function. We prove that the subspace of solutions is two-dimensional over the field of quasi-constants. We get a basis for this space explicitly. In terms of this basis, we represent the q-hypergeometric function of the Barnes type constructed by Nishizawa and Ueno. Then we see that this function has logarithmic singularity at the origin. This is a difference between the q-hypergeometric functions with 0<|q|<1 and at |q|=1.
Integral formulae for form factors of a large family of charged local operators in SU(2) invariant Thirring model are given extending Smirnov's construction of form factors of chargeless local operators in the sine-Gordon model. New abelian symmetry acting on this family of local operators is found. It creates Lukyanov's operators which are not in the above family of local operators in general.
We construct the q-twisted cohomology associated with the q-multiplicative function of Jordan-Pochhammer type at |q|=1. In this framework, we prove the Heine's relations and a connection formula for the q-hypergeometric function of the Barnes type. We also prove an orthogonality relation of the q-little Jacobi polynomials at |q|=1.
We construct the hypergeometric solutions for the quantized KZ equation with values in a tensor product of vector representations of $U_q(sl_n)$ at $|q|=1$ and give an explicit formula for the corresponding determinant in terms of the double sine function.
We write the integral formula of Tarasov-Varchenko type for the solutions to the quantum Knizhnik-Zamolodchikov associated with a tensor product the of vector representations of sl_n. We consider the case where the deformation parameter q satisfies |q|=1. We use the bosonization of the type II vertex operators in order to find the hypergeometric pairing in this setting.
The fundamental matrix solution of the quantum Knizhnik-Zamolodchikov equation associated with quantum affine sl2 algebra is constructed for |q|=1. The formula for its determinant is given in terms of the double sine function.