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In this note we report on recent differential geometric constructions aimed at devising representations of braid groups in various contexts, together with some applications in different domains of mathematical physics. First, the classical Kohno construction for the 3- and 4-strand pure braid groups \(P_3\) and \(P_4\) is explicitly implemented by resorting to the Chen-Hain-Tavares nilpotent connections and to hyperlogarithmic calculus, yielding unipotent representations able to detect Brunnian and nested Brunnian phenomena. Physically motivated unitary representations of Riemann surface braid groups are then described, relying on Bellingeri’s presentation and on the geometry of Hermitian–Einstein holomorphic vector bundles on Jacobians, via representations of Weyl-Heisenberg groups.

We give a formula for the one-parameter strongly continuous semigroup $e^{-tL},\,t>0$, generated by the Hermite operator $L$ on the Heisenberg group $\H1$ in terms of Weyl transforms, and use it to obtain an $L^2$ estimate for the solution of the initial value problem for the heat equation governed by $L$ in terms of the $L^p$ norm of the initial data for $1\leq p\leq \infty.$