The canonical quantization of the chiral Wess-Zumino-Novikov-Witten (WZNW) monodromy matrices (both the diagonal and the general one) requires additional numerical factors that can be attributed to renormalization. We discuss, for G=SU(n), the field-theoretic and algebraic aspects of this phenomenon and show that these renormalization factors are compatible with the natural definitions of quantum determinants possessing the factorization property (i.e., the determinant of a product is equal to the product of determinants, which is a non-trivial fact for matrices with non-commuting entries).
We consider the quantum difference equation of the Hilbert scheme of points in $\mathbb{C}^2$. This equation is the K-theoretic generalization of the quantum differential equation discovered by A. Okounkov and R. Pandharipande. We obtain two explicit descriptions for the monodromy of these equations - representation-theoretic and algebro-geometric. In the representation-theoretic description, the monodromy acts via certain explicit elements in the quantum toroidal algebra $\frak{gl}_1$. In the algebro-geometric description, the monodromy features as transition matrices between the stable envel…
We classified finite orbits of monodromies of the Fuchsian system for five $2\times 2$ matrices. The explicit proof of this result is given. We have proposed a conjecture for a similar classification for $6$ or more $2\times 2$ matrices. Cases in which all monodromy matrices have a common eigenvector are excluded from the consideration. To classify the finite monodromies of the Fuchsian system we combined two methods developed in this paper. The first is an induction method: using finite orbits of smaller number of monodromy matrices the method allows the construction of such orbits for bigger…
We describe the monodromy of the equivariant quantum differential equation of the cotangent bundle of a Grassmannian in terms of the equivariant K-theory algebra of the cotangent bundle. This description is based on the hypergeometric integral representations for solutions of the equivariant quantum differential equation. We identify the space of solutions with the space of the equivariant K-theory algebra of the cotangent bundle. In particular, we show that for any element of the monodromy group, all entries of its matrix in the standard basis of the equivariant K-theory algebra of the cotang…
In this paper we study quantum del Pezzo surfaces belonging to a certain class. In particular we introduce the generalised Sklyanin-Painlevé algebra and characterise its PBW/PHS/Koszul properties. This algebra contains as limiting cases the generalised Sklyanin algebra, Etingof-Ginzburg and Etingof-Oblomkov-Rains quantum del Pezzo and the quantum monodromy manifolds of the Painlevé equations.
The Champagne bottle is one of the simplest and typical examples of Liouville integrable systems that exhibit a non-trivial classical monodromy. This geometrical invariant perturbs globally the existence of action-angle coordinates on the phrase space. However, our work shows a new way to detect the geometric modification on the phrase space by looking at the spectrum of a single operator, that is small non-selfadjoint perturbation of a selfadjoint semiclassical operator accepting the Hamiltonian of the Champagne bottle as its principal symbol.
We propose a procedure to derive quantum spectral curves of AdS/CFT type by requiring that a specially designed analytic continuation around the branch point results in an automorphism of the underlying algebraic structure. In this way we derive four new curves. Two are based on SU(2|2) symmetry, and we show that one of them, under the assumption of square root branch points, describes Hubbard model. Two more are based on SU(2|2) x SU(2|2). In the special subcase of zero central charge, they both reduce to the unique nontrivial curve which furthermore has analytic properties compatible with PS…
Construction of Monodromy Matrix in the F- basis and Scalar products in Spin Chains
本文简单地介绍了Maillet和Sanches de Santos最近提出的自旋为1/2的链的因子分解算子理论。我们得到了分解算子的矩阵元素关于Monodromy矩阵元素的显式表达式。我们利用这个结果导出了量子自旋链的一般标积的表达式。本文对前人关于贝特本征态与任意对偶态的标积的测定作了评述。我们还建立了F-基上的标积计算与通常基上的标积计算之间的直接对应。
