We examine to what extent heterotic string worldsheets can describe arbitrary $E_8 × E_8$ gauge fields. The traditional construction of heterotic strings builds each $E_8$ via a $Spin(16)/Z2$ subgroup, typically realized as a current algebra by left-moving fermions, and as a result, only $E_8$ gauge fields reducible to $Spin(16)/Z_2$ gauge fields are directly realizable in standard constructions. However, there exist perturbatively consistent $E_8$ gauge fields which cannot be reduced to $Spin(16)/Z_2$ and so cannot be described within standard heterotic worldsheet constructions. A natural question to then ask is whether there exists any $(0,2)$ superconformal field theory (SCFT) that can describe such $E_8$ gauge fields. To answer this question, we first show how each 10-dimensional $E_8$ partition function can be built up using other subgroups than $Spin(16)/Z_2$, then construct “fibered WZW models” which allow us to explicitly couple current algebras for general groups and general levels to heterotic strings. This technology gives us a very general approach to handling heterotic compactifications with arbitrary principal bundles. It also gives us a physical realization of some elliptic genera constructed recently by Ando and Liu.

The reduction of the $E_8$ gauge theory to ten dimensions leads to a loop group, which in relation to twisted $K$-theory has a Dixmier–Douady class identified with the Neveu–Schwarz $H$-field. We give an interpretation of the degree two part of the eta form by comparing the adiabatic limit of the eta invariant with the one loop term in type IIA. More generally, starting with a $G$-bundle, the comparison for manifolds with String structure identifies $G$ with $E_8$ and the representation as the adjoint, due to an interesting appearance of the dual Coxeter number. This makes possible a description in terms of a generalized Wess-Zumino-Witten model at the critical level. We also discuss the relation to the index gerbe, the possibility of obtaining such bundles from loop space, and the symmetry breaking to finite-dimensional bundles. We discuss the implications of this and we give several proposals.

We study the connections between link invariants, the chromatic polynomial, geometric representations of models of statistical mechanics, and their common underlying algebraic structure. We establish a relation between several algebras and their associated combinatorial and topological quantities. In particular, we define the chromatic algebra, whose Markov trace is the chromatic polynomial $χQ$ of an associated graph, and we give applications of this new algebraic approach to the combinatorial properties of the chromatic polynomial. In statistical mechanics, this algebra occurs in the low-temperature expansion of the $Q$-state Potts model. We establish a relationship between the chromatic algebra and the $SO(3)$ Birman–Murakami–Wenzl algebra, which is an algebra-level analogue of the correspondence between the $SO(3)$ Kauffman polynomial and the chromatic polynomial.

We investigate the analog of holomorphic vector bundles in the context of Sasakian manifolds.

We define and study a gluing procedure for Bridgeland stability conditions in the situation when a triangulated category has a semiorthogonal decomposition. As an application, we construct stability conditions on the derived categories of $Z_2$-equivariant sheaves associated with ramified double coverings of P3. Also, we study the stability space for the derived category of $Z_2$-equivariant coherent sheaves on a smooth curve $X$, associated with a degree 2 map $X → Y$ , where $Y$ is another smooth curve. In the case when the genus of $Y is ≥ 1$ we give a complete description of the stability space.

We present a construction of self-dual Yang–Mills connections on the Taub-NUT space. We illustrate it by finding explicit expressions for all $SU(2)$ instantons of instanton number one and generic monodromy at infinity.

We analyze the quantum ground state structure of a specific model of itinerant, strongly interacting lattice fermions. The interactions are tuned to make the model supersymmetric. Due to this, quantum ground states are in one-to-one correspondence with cohomology classes of the so-called independence complex of the lattice. Our main result is a complete description of the cohomology, and thereby of the quantum ground states, for a two-dimensional square lattice with periodic boundary conditions. Our work builds on results by Jonsson, who determined the Euler characteristic (Witten index) via a correspondence with rhombus tilings of the plane. We prove a theorem, first conjectured by Fendley, which relates dimensions of the cohomology at grade $n$ to the number of rhombus tilings with $n$ rhombi.

Using the Alexandrov–Kontsevich–Schwarz–Zaboronsky (AKSZ) prescription we construct 2D and 3D topological field theories associated to generalized complex manifolds. These models can be thought of as 2D and 3D generalizations of A- and B-models. Within the BV framework we show that the 3D model on a two-manifold cross an interval can be reduced to the 2D model.