Topological field theories induced by twisted R-Poisson structure in any dimension

We construct a class of topological field theories with Wess-Zumino term in spacetime dimensions ≥ 2 whose target space has a geometrical structure that suitably generalizes Poisson or twisted Poisson manifolds. Assuming a field content comprising a set of scalar fields accompanied by gauge fields of degree (1, p − 1, p) we determine a generic Wess-Zumino topological field theory in p + 1 dimensions with background data consisting of a Poisson 2-vector, a (p + 1)-vector R and a (p + 2)-form H satisfying a specific geometrical condition that defines a H-twisted R-Poisson structure of order p + 1. For this class of theories we demonstrate how a target space covariant formulation can be found by means of an auxiliary connection without torsion. Furthermore, we study admissible deformations of the generic class in special spacetime dimensions and find that they exist in dimensions 2, 3 and 4. The two-dimensional deformed field theory includes the twisted Poisson sigma model, whereas in three dimensions we find a more general structure that we call bi-twisted R-Poisson. This extends the twisted R-Poisson structure of order 3 by a non-closed 3-form and gives rise to a topological field theory whose covariant formulation requires a connection with torsion and includes a twisted Poisson sigma model in three dimensions as a special case. The relation of the corresponding structures to differential graded Q-manifolds based on the degree shifted cotangent bundle T*[p]T*[1]M is discussed, as well as the obstruction to them being QP-manifolds due to the Wess-Zumino term.

Superintegrable Systems on Moduli Spaces of Flat Connections

The main result of this paper is the construction of a family of superintegrable Hamiltonian systems on moduli spaces of flat connections on a principal G-bundle on a surface. The moduli space is a Poisson variety with Atiyah–Bott Poisson structure. Among particular cases of such systems are spin generalizations of Ruijsenaars–Schneider models.

Bi-Hamiltonian Structure of Spin Sutherland Models: The Holomorphic Case

We construct a bi-Hamiltonian structure for the holomorphic spin Sutherland hierarchy based on collective spin variables. The construction relies on Poisson reduction of a bi-Hamiltonian structure on the holomorphic cotangent bundle of $$\mathrm{GL}(n,\mathbb {C})$$ GL ( n , C ) , which itself arises from the canonical symplectic structure and the Poisson structure of the Heisenberg double of the standard $$\mathrm{GL}(n,\mathbb {C})$$ GL ( n , C ) Poisson–Lie group. The previously obtained bi-Hamiltonian structures of the hyperbolic and trigonometric real forms are recovered on real slices of the holomorphic spin Sutherland model.

Exploring the gravity sector of emergent higher-spin gravity: effective action and a solution

We elaborate the description of the semi-classical gravity sector of Yang-Mills matrix models on a covariant quantum FLRW background. The basic geometric structure is a frame, which arises from the Poisson structure on an underlying S2 bundle over space-time. The equations of motion for the associated Weitzenböck torsion obtained in [1] are rewritten in the form of Yang-Mills-type equations for the frame. An effective action is found which reproduces these equations of motion, which contains an Einstein-Hilbert term coupled to a dilaton, an axion and a Maxwell-type term for the dynamical frame. An explicit rotationally invariant solution is found, which describes a gravitational field coupled to the dilaton.

Quantum Orbit Method in the Presence of Symmetries

We review some of the main achievements of the orbit method, when applied to Poisson–Lie groups and Poisson homogeneous spaces or spaces with an invariant Poisson structure. We consider C∗-algebra quantization obtained through groupoid techniques, and we try to put the results obtained in algebraic or representation theoretical contexts in relation with groupoid quantization.

Poisson Principal Bundles

We semiclassicalise the theory of quantum group principal bundles to the level of Poisson geometry. The total space X is a Poisson manifold with Poisson-compatible contravariant connection, the fibre is a Poisson-Lie group in the sense of Drinfeld with bicovariant Poisson-compatible contravariant connection, and the base has an inherited Poisson structure and Poisson-compatible contravariant connection. The latter are known to be the semiclassical data for a quantum differential calculus. The theory is illustrated by the Poisson level of the q-Hopf fibration on the standard q-sphere. We also construct the Poisson level of the spin connection on a principal bundle.

L∞ - derivations and the argument shift method for deformation quantization algebras

The argument shift method is a well-known method for generating commutative families of functions in Poisson algebras from central elements and a vector field, verifying a special condition with respect to the Poisson bracket. In this notice we give an analogous construction, which gives one a way to create commutative subalgebras of a deformed algebra from its center (which is as it is well known describable in the terms of the center of the Poisson algebra) and an L∞-differentiation of the algebra of Hochschild cochains, verifying some additional conditions with respect to the Poisson structure.

Trigonometric Real Form of the Spin RS Model of Krichever and Zabrodin

We investigate the trigonometric real form of the spin Ruijsenaars–Schneider system introduced, at the level of equations of motion, by Krichever and Zabrodin in 1995. This pioneering work and all earlier studies of the Hamiltonian interpretation of the system were performed in complex holomorphic settings; understanding the real forms is a non-trivial problem. We explain that the trigonometric real form emerges from Hamiltonian reduction of an obviously integrable ‘free’ system carried by a spin extension of the Heisenberg double of the $${\mathrm{U}}(n)$$ U ( n ) Poisson–Lie group. The Poisson structure on the unreduced real phase space $${\mathrm{GL}}(n,{\mathbb {C}})\times {\mathbb {C}}^{nd}$$ GL ( n , C ) × C nd is the direct product of that of the Heisenberg double and $$d\ge 2$$ d ≥ 2 copies of a $${\mathrm{U}}(n)$$ U ( n ) covariant Poisson structure on $${\mathbb {C}}^n \simeq {\mathbb {R}}^{2n}$$ C n ≃ R 2 n found by Zakrzewski, also in 1995. We reduce by fixing a group valued moment map to a multiple of the identity and analyze the resulting reduced system in detail. In particular, we derive on the reduced phase space the Hamiltonian structure of the trigonometric spin Ruijsenaars–Schneider system and we prove its degenerate integrability.