Related papers: Simply-laced quantum connections generalising KZ
We will exhibit a group of symmetries of the simply-laced quantum connections, generalising the quantum/Howe duality relating KZ and the Casimir connection. These symmetries arise as a quantisation of the classical symmetries of the…
We construct a flat connection on the elliptic configuration space associated to any complex semisimple Lie algebra g. This elliptic Casimir connection has logarithmic singularities, and takes values in the deformed double current algebra…
We study Knizhnik-Zamolodchikov (KZ) connection in the presence of irregular singularities, that is, poles of higher order. We consider both the case of a universal connection and the case when it is associated with a specific simple Lie…
We introduce isomonodromy Knizhnik-Zamolodchikov (KZ) connections with respect to the quantum Stokes matrices, and prove that the classical limit of the KZ type connections gives rise to the Dubrovin connections of semisimple Frobenius…
We present the construction of a new family of coherent states for quantum theories of connections obtained following the polymer quantization. The realization of these coherent states is based on the notion of graph change, in particular…
For each connected complex reductive group G, we find a family of new examples of complex quasi-Hamiltonian G-spaces with G-valued moment maps. These spaces arise naturally as moduli spaces of (suitably framed) meromorphic connections on…
Generalising work of Calaque-Enriquez-Etingof, we construct a universal KZB connection D_R for any finite (reduced, crystallographic) root system R. D_R is a flat connection on the regular locus of the elliptic configuration space…
We introduce the concept of regular quantum graphs and construct connected quantum graphs with discrete symmetries. The method is based on a decomposition of the quantum propagator in terms of permutation matrices which control the way…
In this paper, we study and build the Hamiltonian system attached to any $\mathfrak{gl}_2(\mathbb{C})$ meromorphic connection with an arbitrary number of non-ramified poles of arbitrary degrees. In particular, we propose the Lax pairs and…
We construct a twisted version of the genus one universal Knizhnik-Zamolodchikov-Bernard (KZB) connection introduced by Calaque-Enriquez-Etingof, that we call the ellipsitomic KZB connection. This is a flat connection on a principal bundle…
In the physics literature, Bilal--Fock--Kogan \cite{BFK} introduced the idea of parabolic reduced flat connections on a surface to give a geometric origin to $W$-algebras. In this paper, we combine these ideas with higher complex…
Let g be a complex, semisimple Lie algebra, G the corresponding simply-connected Lie group and H a maximal torus in G. We construct a flat connection on H with logarithmic singularities on the root hypertori and values in the Yangian Y(g)…
We review the Kohno-Drinfeld theorem as well as a conjectural analogue relating quantum Weyl groups to the monodromy of a flat connection D on the Cartan subalgebra of a complex, semi-simple Lie algebra g with poles on the root hyperplanes…
Motivated by a geometric decomposition of the vector field associated with the Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) equation for finite-level open quantum systems, we propose a generalization of the recently introduced contact…
Quantization together with quantum dynamics can be simultaneously formulated as the problem of finding an appropriate flat connection on a Hilbert bundle over a contact manifold. Contact geometry treats time, generalized positions and…
In the context of (2+1)--dimensional quantum gravity with negative cosmological constant and topology R x T^2, constant matrix--valued connections generate a q--deformed representation of the fundamental group, and signed area phases relate…
The Hamiltonian approach to isomonodromic deformation systems is extended to include generic rational covariant derivative operators on the Riemann sphere with irregular singularities of arbitrary Poincar\'e rank. The space of rational…
Graphs with given k vertices generate an (acyclic) simplicial complex. We describe the homology of its quotient complex, formed by all connected graphs, and demonstrate its applications to the topology of braid groups, knot theory,…
B. Fedosov has given a simple and very natural construction of a deformation quantization for any symplectic manifold, using a flat connection on the bundle of formal Weyl algebras associated to the tangent bundle of a symplectic manifold.…
Generalized Fourier transformation between the position and the momentum representation of a quantum state is constructed in a coordinate independent way. The only ingredient of this construction is the symplectic (canonical) geometry of…