Related papers: Seiberg-Witten curves and double-elliptic integrab…
We consider three 'classical doubles' of any semisimple, connected and simply connected compact Lie group $G$: the cotangent bundle, the Heisenberg double and the internally fused quasi-Poisson double. On each double we identify a pair of…
We investigate an interacting Pais-Uhlenbeck oscillator with a Landau-Ginzburg type interaction term and analyse its classical dynamics from a geometric and numerical point of view. We show that the resulting fourth-order equation of motion…
We summarize recent results on the resolution of two intimately related problems, one physical, the other mathematical. The first deals with the resolution of the non-perturbative low energy dynamics of certain N=2 supersymmetric Yang-Mills…
We find the commutant of a pure contractive semigroup on a Hilbert space. We demonstrate that any tuple of doubly commuting pure contractive semigroups can be dilated to a tuple of doubly commuting pure isometric semigroups. En route, we…
The possibility of noncommutative topological gravity arising in the same manner as Yang-Mills theory is explored. We use the Seiberg-Witten map to construct such a theory based on a SL(2,C) complex connection, from which the Euler…
The SW map problem is formulated and solved in the BRST cohomological approach. The well known ambiguities of the SW map are shown to be associated to distinct cohomological classes. This analysis is applied to the noncommutative…
The Lie-Hamilton approach for $t$-dependent Hamiltonians is extended to cover the so-called nonlinear Lie-Hamilton systems, which are no longer related to a linear $t$-dependent combination of a basis of a finite-dimensional Lie algebra of…
Schroedinger equation on a Hilbert space ${\cal H}$, represents a linear Hamiltonian dynamical system on the space of quantum pure states, the projective Hilbert space $P {\cal H}$. Separable states of a bipartite quantum system form a…
Via a non degenerate symmetric bilinear form we identify the coadjoint representation with a new representation and so we induce on the orbits a simplectic form. By considering Hamiltonian systems on the orbits we study some features of…
We show that quadratic Hamiltonians in involution coming from a St\"ackel system are quantizable, in the sense that one can construct commutative self-adjoint operators whose symbols are the quadratic Hamiltonians. Moreover, they allow…
We consider the Lie-algebraic notion of commutant in the setting of Poisson algebra. This provides a framework for deforming Hamiltonian differential equations. By taking a subalgebra of the algebra of integrals, and considering the set of…
We study the relation between class S theories on punctured tori and isomonodromic deformations of flat SL(N) connections on the two dimensional torus with punctures. Turning on the self dual $\Omega$-background corresponds to a…
The mother functions for the eigenfunctions of the Koroteev-Shakirov version of quantum double-elliptic (Dell) Hamiltonians can be presented as infinite series in Miwa variables, very similar to the recent conjecture due to J. Shiraishi.…
We address two linked problems at the interface of quantum topology and number theory: deriving asymptotic expansions of the Witten--Reshetikhin--Turaev invariants for 3-manifolds and establishing quantum modularity of false theta…
We prove the integrability and superintegrability of a family of natural Hamiltonians which includes and generalises those studied in some literature, originally defined on the 2D Minkowski space. Some of the new Hamiltonians are a perfect…
The standard generators of tridiagonal algebras, recently introduced by Terwilliger, are shown to generate a new (in)finite family of mutually commuting operators which extends the Dolan-Grady construction. The involution property relies on…
The topology of complex classical paths is investigated to discuss quantum tunnelling splittings in one-dimensional systems. Here the Hamiltonian is assumed to be given as polynomial functions, so the fundamental group for the Riemann…
Infinite dimensional Hamiltonian systems appear naturally in the rich algebraic structure of Symplectic Field Theory. Carefully defining a generalization of gravitational descendants and adding them to the picture, one can produce an…
We review new aspects of integrable systems discovered recently in N=2 supersymmetric gauge theories and their topologically twisted versions. The main topics are (i) an explicit construction of Whitham deformations of the Seiberg-Witten…
In this paper, we present a general scheme to construct integrable systems based on realization in the coboundary dynamical Poisson groupoids of Etingof and Varchenko. We also present a factorization method for solving the Hamiltonian…