We construct a prequantum 2-Hilbert space for any line bundle gerbe whose Dixmier-Douady class is torsion. Analogously to usual prequantisation, this 2-Hilbert space has the category of sections of the line bundle gerbe as its underlying 2-vector space. These sections are obtained as certain morphism categories in Waldorf's version of the 2-category of line bundle gerbes. We show that these morphism categories carry a monoidal structure under which they are semisimple and abelian. We introduce a dual functor on the sections, which yields a closed structure on the morphisms between bundle gerbes and turns the category of sections into a 2-Hilbert space. We discuss how these 2-Hilbert spaces fit various expectations from higher prequantisation. We then extend the transgression functor to the full 2-category of bundle gerbes and demonstrate its compatibility with the additional structures introduced. We discuss various aspects of Kostant-Souriau prequantisation in this setting, including its dimensional reduction to ordinary prequantisation.

We suggest a geometric approach to quantisation of the twisted Poisson structure underlying the dynamics of charged particles in fields of generic smooth distributions of magnetic charge, and dually of closed strings in locally non-geometric flux backgrounds, which naturally allows for representations of nonassociative magnetic translation operators. We show how one can use the 2-Hilbert space of sections of a bundle gerbe in a putative framework for canonical quantisation. We define a parallel transport on bundle gerbes on R d and show that it naturally furnishes weak projective 2-representations of the translation group on this 2-Hilbert space. We obtain a notion of covariant derivative on a bundle gerbe and a novel perspective on the fake curvature condition.

Simplicial presheaves on cartesian spaces provide a general notion of smooth spaces. We define a corresponding smooth version of the singular complex functor, which maps smooth spaces to simplicial sets. We exhibit this functor as one of several Quillen equivalences between the Kan-Quillen model category of simplicial sets and a motivic-style R-localisation of the (projective or injective) model category of smooth spaces. These Quillen equivalences and their interrelations are powerful tools: for instance, they allow us to give a purely homotopy-theoretic proof of a Whitehead Approximation Theorem for manifolds. Further, we provide a functorial fibrant replacement in the R-local model category of smooth spaces. This allows us to compute the homotopy types of mapping spaces in this model category in terms of smooth singular complexes. We explain the relation of our fibrant replacement functor to the concordance sheaves introduced recently by Berwick-Evans, Boavida de Brito, and Pavlov. Finally, we show how the R-local model category of smooth spaces formalises the homotopy theory on sheaves used by Galatius, Madsen, Tillmann, and Weiss in their seminal paper on the homotopy type of the cobordism category.

We investigate instantons on sine-cones over Sasaki-Einstein and 3-Sasakian manifolds. It is shown that these conical Einstein manifolds are Kähler with torsion (KT) manifolds admitting Hermitian connections with totally antisymmetric torsion. Furthermore, a deformation of the metric on the sine-cone over 3-Sasakian manifolds allows one to introduce a hyper-Kähler with torsion (HKT) structure. In the large-volume limit these KT and HKT spaces become Calabi-Yau and hyper-Kähler conifolds, respectively. We construct gauge connections on complex vector bundles over conical KT and HKT manifolds which solve the instanton equations for Yang-Mills fields in higher dimensions.