We use the language of differential cohomology to give an analytic description of the moduli space of classical vacua for heterotic string theory in eight dimensions. The complex structure of this moduli space is then related to the character of the appropriate Kac–Moody algebra.

An explicit description of the spectral data of stable ${\rm U}(n)$ vector bundles on elliptically fibered Calabi–Yau three-folds is given, extending previous work of Friedman, Morgan and Witten. The characteristic classes are computed and it is shown that part of the bundle cohomology vanishes. The stability and the dimension of the moduli space of the ${\rm U}(n)$ bundles are discussed. As an application, it is shown that the ${\rm U}(n)$ bundles are capable to solve the basic topological constraints imposed by heterotic string theory. Various explicit solutions of the Donaldson– Uhlenbeck–Yau equation are given. The heterotic anomaly cancellation condition is analyzed; as a result, an integral change in the number of fiber wrapping 5-branes is found. This gives a definite prediction for the number of 3-branes in a dual F-theory model. The net-generation number is evaluated, showing more flexibility compared with the ${\rm SU}(n)$ case.

We construct multiloop superparticle amplitudes in eleven dimensions (11d) using the pure spinor formalism, which we compare with the multiloop pure spinor superstring amplitudes prescription given by Berkovits. We then argue that this construction points to some evidence for the existence of a topological M-theory based on a relation between the ghost number of the full-fledged supersymmetric critical models and the dimension of the spacetime for topological models. In particular, we show that the extensions at higher orders of the previous results for the tree and one-loop level expansion for the superparticle in 11 dimensions is related to a topological model in 7 dimensions.

We prove that a Kähler supermetric on a supermanifold with one complex fermionic dimension admits a super Ricci-flat supermetric, if and only if the bosonic metric has vanishing scalar curvature. As a corollary, it follows that Yau’s theorem does not hold for supermanifolds.

We construct a version of Dijkgraaf–Witten theory based on a compact abelian Lie group within the formalism of Turaev’s homotopy quantum field theory. As an application we show that the $2+1$-dimensional theory based on $\rm U(1)$ classifies lens spaces up to homotopy type.