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This paper is a continuation of the work on the spectral problem of Harper operator using algebraic geometry. We continue to discuss the local monodromy of algebraic Fermi curves based on Picard-Lefschetz formula. The density of states over approximating components of Fermi curves satisfies a Picard-Fuchs equation. By the property of Landen transformation, the density of states has a Lambert series as the quarter period. A $q$-expansion of the energy level can be derived from a mirror map as in the B-model., v2, 13 pages, minor changes have been made

A discretization of Painlev\'e VI equation was obtained by Jimbo and Sakai in 1996. There are two ways to quantize it: 1) use the affine Weyl group symmetry (of $D_5^{(1)}$) (Hasegawa, 2011), 2) Lax formalism i.e. monodromy preserving point of view. It turns out that the second approach is also successful and gives the same quantization as in the first approach., Comment: 14 pages with 4 figures

A linear system of ordinary differential equations corresponding by the isomonodromy deformation method to the third Painleve equation is considered. The surjectivity of the monodromy map generated by this system is proven using the Riemann–Hilbert factorization method.

The moduli space of curves that admit elliptic subcovers is described in several possible ways: by algebra, group theory, monodromy, and topology. Some results on the number of components are obtained by combining group theory and topology. Results surveyed or announced serve to demonstrate the applications of the space under study to integrable systems.

We show a natural relation between the monodromy formula for focus-focus singularities of integrable Hamiltonian systems and a formula of Duistermaat–Heckman, and extend the main results of our previous note ( $$\mathbb{S}$$ 1-action, monodromy, and topological classification) to the case of degenerate focus-focus singularities. We also consider the non-Hamiltonian case, local normal forms, etc.

This article is based on a talk given in Dijon for the 2000 summer school ‘Topological and Geometrical Methods: Applications to Dynamical Systems’. The standard definitions of the monodromy invariant for completely integrable classical systems are reviewed, and the link to the quantum monodromy observed in the joint spectrum of commuting operators is explained. The mathematical treatment relies on a modern and attractive version of the Bohr–Sommerfeld quantisation rules.

We generalize the BRS method for the (finite-dimensional) quantum gauge theory involved in the zero modes of the monodromy extended SU(2) WZNW model. The generalization consists of a nilpotent operator Q such that Qh = 0 (h = k + 2 = 2, 3, ... being the height of the current algebra representation) acting on an extended state space. The physical subquotient is identified with the direct sum ⊕ h-1 n=1 Ker(Qn)/Im(Qh−n).

In the natural correspondence between the self-dual Bianchi type IX metrics and solutions of Painleve VI, the self-dual Ricci-flat metrics or the nontrivial self-dual Einstein–Weyl metrics correspond to the classical solutions of Painleve VI that determine isomonodromic deformations with reducible monodromy.

The zero modes of the monodromy extended SU(2) WZNW model give rise to a gauge theory with a finite-dimensional state space. A generalized BRS operator A such that \(A^h = 0\left( {h = k + 2 = 3,4,...} \right.\) being the height of the current algebra representation) acts in \(\left( {2h - 1} \right)\)-dimensional indefinite metric space\({\mathcal{H}}_I\) of quantum group invariant vectors. The generalized cohomologies Ker \(A^n /{Im}\;A^{h - n} \left( {n = 1,...,h - 1} \right)\) are 1-dimensional. Their direct sum spans the physical subquotient of \({\mathcal{H}}_I\).

We study the monodromy representationsℬ k,Iof the mapping class groupℬ 4 acting on 4-point blocks satisfying the Knizhnik-Zamolodchikov equation for the levelk su2 current algebra. We classify all irreducibleℬ k,Iwhich are realized by finite groups; we also display finite irreducible components for the reducible representations corresponding tok = 10.

A generalized Toda lattice based on gl(n) is considered. The Poisson brackets are expressed in terms of a Lax connection, L=∂σ−Φ(σ) and a classical r-matrix, {Φ1,Φ2}=[r,Φ1+Φ2}. The essential point is that the local lattice transfer matrix is taken to be the ordinary exponential, T=eΦ; this assures the intepretation of the local and the global transfer matrices in terms of monodromy, which is not true of the T-matrix used for the sl(n) Toda lattice. To relate this exponential transfer matrix to the more manageable and traditional factorized form, it is necessary to make specific assumptions about the equal time operator product expansions. The simplest possible assumptions lead to an equivalent, factorized expression for T, in terms of operators in (an extension of) the enveloping algebra of gl(n). Restricted to sl(n), and to multiplicity-free representations, these operators satisfy the commutation relations of slq(n), which provides a very simple injection of slq(n) into the enveloping algebra of sl(n). A deformed coproduct, similar in form to the familiar coproduct on slq(n), turns gl(n) into a deformed Hopf algebra glq(n). It contains slq(n) as a subalgebra, but not as a sub-Hopf algebra.

We define the transformation of linear differential equations with rational function coefficients that fix monodromy data and change local multiplicities by any sequence of integers. This transformation that gives rise to Pade approximations, at the same time defines the Backlund transformation of Schlesinger equations.

Vertex operators (primary fields) are constructed for the conformal field theory on P1 by means of A1(1) modules. The commutation relations of vertex operators induce monodromy representations of the braid group on the spaces of vacuum expectations of compositions of vertex operators.

The problem of describing all the monodromy matrices for R matrices of the XXX and XXZ models is discussed. It is shown that the L operator of the lattice nonlinear Schrodinger model generates all possible monodromy matrices for the XXX R matrix.