We explore two-dimensional sigma models with (0, 2) supersymmetry through their chiral algebras. Perturbatively, the chiral algebras of (0, 2) models have a rich infinite-dimensional structure described by the cohomology of a sheaf of chiral differential operators. Nonperturbatively, instantons can deform this structure drastically. We show that under some conditions they even annihilate the whole algebra, thereby triggering the spontaneous breaking of supersymmetry. For a certain class of Kähler manifolds, this suggests that there are no harmonic spinors on their loop spaces and gives a physical proof of the Höhn–Stolz conjecture.

Let $C \to M$ be the bundle of connections of a principal bundle on $M$. The solutions to Hamilton–Cartan equations for a gauge-invariant Lagrangian density $\Lambda$ on $C$ satisfying a weak condition of regularity, are shown to admit an affine fibre-bundle structure over the set of solutions to Euler–Lagrange equations for $\Lambda$. This structure is also studied for the Jacobi fields and for the moduli space of extremals.

We obtain a new solution of the star–triangle relation with positive Boltzmann weights, which contains as special cases all continuous and discrete spin solutions of this relation, that were previously known. This new master solution defines an exactly solvable two lattice model of statistical mechanics, which involves continuous spin variables, living on a circle, and contains two temperature-like parameters. If one of the these parameters approaches a root of unity (corresponds to zero temperature), the spin variables freezes into discrete positions, equidistantly spaced on the circle. An absolute orientation of these positions on the circle slowly changes between lattice sites by overall rotations. Allowed configurations of these rotations are described by classical discrete integrable equations, closely related to the famous $Q_4$-equations by Adler, Bobenko and Suris. Fluctuations between degenerate ground states in the vicinity of zero temperature are described by a rather general integrable lattice model with discrete spin variables. In some simple special cases, the latter reduces to the Kashiwara–Miwa and chiral Potts models.

We study the recently discovered isomorphisms between hyperbolic Weyl groups and modular groups over integer domains in normed division algebras. We show how to realize the group action via fractional linear transformations on generalized upper half-planes over the division algebras, focusing on the cases involving quaternions and octonions. For these we construct automorphic forms, whose explicit expressions depend crucially on the underlying arithmetic properties of the integer domains. Another main new result is the explicit octavian realization of $W+(E_{10})$, which contains as a special case a new realization of $W+(E_8)$ in terms of unit octavians and their automorphism group.

What are called secondary characteristic classes in Chern–Weil theory are a refinement of ordinary characteristic classes of principal bundles from cohomology to differential cohomology. We consider the problem of refining the construction of secondary characteristic classes from cohomology sets to cocycle spaces; and from Lie groups to higher connected covers of Lie groups by smooth $\infty$-groups, i.e., by smooth groupal $A_\infty$- spaces. Namely, we realize differential characteristic classes as morphisms from $\infty$-groupoids of smooth principal $\infty$-bundles with connections to $\infty$-groupoids of higher $U(1)$-gerbes with connections. This allows us to study the homotopy fibres of the differential characteristic maps thus obtained and to show how these describe differential obstruction problems. This applies in particular to the higher twisted differential spin structures called twisted differential string structures and twisted differential fivebrane structures.

Using mirror symmetry, we resolve an old puzzle in the linear sigma model description of the spacetime Higgs mechanism in a heterotic string compactification with (2,2) worldsheet supersymmetry. The resolution has a nice spacetime interpretation via the normalization of physical fields and suggests that with a little care deformations of the linear sigma model can describe heterotic Higgs branches.

Based on a weak action of a finite group $J$ on a finite group $G$, we present a geometric construction of $J$-equivariant Dijkgraaf–Witten theory as an extended topological field theory. The construction yields an explicitly accessible class of equivariant modular tensor categories. For the action of a group $J$ on a group $G$, the category is described as the representation category of a $J$-ribbon algebra that generalizes the Drinfel’d double of the finite group $G$.