Solutions to the Hull–Strominger System with Torus Symmetry

We construct new smooth solutions to the Hull–Strominger system, showing that the Fu–Yau solution on torus bundles over K3 surfaces can be generalized to torus bundles over K3 orbifolds. In particular, we prove that, for $$13 \le k \le 22$$ 13 ≤ k ≤ 22 and $$14\le r\le 22$$ 14 ≤ r ≤ 22 , the smooth manifolds $$S^1\times \sharp _k(S^2\times S^3)$$ S 1 × ♯ k ( S 2 × S 3 ) and $$\sharp _r (S^2 \times S^4) \sharp _{r+1} (S^3 \times S^3)$$ ♯ r ( S 2 × S 4 ) ♯ r + 1 ( S 3 × S 3 ) , have a complex structure with trivial canonical bundle and admit a solution to the Hull–Strominger system.

Torus bundles, automorphisms and T-duality

We reconsider some older constructions of T-duality, based on automorphisms of the worldsheet operator algebra, in a modern context. It has been long known that at special points in the moduli space of torus compactifications, the target space gauge symmetry may be enhanced. Away from such points the symmetry is broken and T-duality may be understood as a residual discrete gauge symmetry that survives this breaking. Drawing on work on connections over the space of string backgrounds, we discuss how to generalise this framework for T-duality to geometric and non-geometric backgrounds that are not full solutions of string theory, but may play an important role in exact backgrounds. Along the way we find an interesting algebraic structure and discuss its relationship with doubled geometry. We comment on non-isometric T-duality in this context.

T-dual solutions of the Hull–Strominger system on non-Kähler threefolds

We construct new examples of solutions of the Hull–Strominger system on non-Kähler torus bundles over K3 surfaces, with the property that the connection {\nabla} on the tangent bundle is Hermite–Yang–Mills. With this ansatz for the connection {\nabla}, we show that the existence of solutions reduces to known results about moduli spaces of slope-stable sheaves on a K3 surface, combined with elementary analytical methods. We apply our construction to find the first examples of T-dual solutions of the Hull–Strominger system on compact non-Kähler manifolds with different topology.

Braids, 3-Manifolds, Elementary Particles: Number Theory and Symmetry in Particle Physics

In this paper, we will describe a topological model for elementary particles based on 3-manifolds. Here, we will use Thurston’s geometrization theorem to get a simple picture: fermions as hyperbolic knot complements (a complement C ( K ) = S 3 \ ( K × D 2 ) of a knot K carrying a hyperbolic geometry) and bosons as torus bundles. In particular, hyperbolic 3-manifolds have a close connection to number theory (Bloch group, algebraic K-theory, quaternionic trace fields), which will be used in the description of fermions. Here, we choose the description of 3-manifolds by branched covers. Every 3-manifold can be described by a 3-fold branched cover of S 3 branched along a knot. In case of knot complements, one will obtain a 3-fold branched cover of the 3-disk D 3 branched along a 3-braid or 3-braids describing fermions. The whole approach will uncover new symmetries as induced by quantum and discrete groups. Using the Drinfeld–Turaev quantization, we will also construct a quantization so that quantum states correspond to knots. Particle properties like the electric charge must be expressed by topology, and we will obtain the right spectrum of possible values. Finally, we will get a connection to recent models of Furey, Stoica and Gresnigt using octonionic and quaternionic algebras with relations to 3-braids (Bilson–Thompson model).