Using the structure of a KZ-monad we create a general categorical workspace in which diagrams can be formally constructed. In particular this abstract framework of category theory is shown to provide a precise semantics for constructing the speciications of complex systems from their component parts.

Virtual reality (VR) training has become valuable in sports to improve motor behavior and train specific situations under standardized conditions. However, studies comparing conventional with VR are rare, especially for advanced athletes. Furthermore, it remains unclear whether the performance improvement achieved through can be transferred real world (RW). Therefore, we present a study analyzi...

We obtain the operator algebra of each twisted sector of all WZW orbifolds, including the general twisted current algebra and the algebra of the twisted currents with the twisted affine primary fields. Surprisingly, the twisted right and left mover current algebras are not a priori copies of each other. Using the operator algebra we also derive world-sheet differential equations for the twisted...

1.1. Shan has proved that the categories Oc(Wn) for rational Cherednik algebras of type Wn = W (G(`, 1, n)) = Snn(μ`) with n varying, together with decompositions of the parabolic induction and restriction functors of Bezrukavnikov-Etingof, provide a categorification of an integrable s̃le Fock space representation F(m), [18]. The parameters m ∈ Z` and e ∈ N ∪ {∞} arise from the choice of paramet...

Path integral quantization of generic two-dimensional dilaton gravity nonminimally coupled to a Dirac fermion is performed. After integrating out geometry exactly, perturbation theory is employed in the matter sector to derive the lowest order gravitational vertices. Consistency with the case of scalar matter is found and issues of relevance for bosonisation are pointed out. PACS numbers: 04.60...

We consider the Knizhnik-Zamolodchikov system of linear differential equations. The coefficients of this system are rational functions generated by elements of the symmetric group S n. We assume that parameter ρ = ±1. In previous paper [5] we proved that the fundamental solution of the corresponding KZ-equation is rational. Now we construct this solution in the explicit form.

It is a well-known paradigm to consider Vassiliev invariants as polynomials on the set of knots. We prove the following characterization: a rational knot invariant is a Vassiliev invariant of degree ≤ m if and only if it is a polynomial of degree ≤ m on every geometric sequence of knots. Here a sequence Kz with z ∈ Z is called geometric if the knots Kz coincide outside a ball B, inside of which...