We construct new topological theories related to sigma models whose target space is a seven-dimensional manifold of G₂ holonomy. We define a new type of topological twist and identify the BRST operator and the physical states. Unlike the more familiar six-dimensional case, our topological model is defined in terms of conformal blocks and not in terms of local operators of the original theory. We also present evidence that one can extend this definition to all genera and construct a seven-dimensional topological string theory. We compute genus zero correlation functions and relate these to Hitchin’s functional for three-forms in seven dimensions. Along the way we develop the analogue of special geometry for G₂ manifolds. When the seven-dimensional topological twist is applied to the product of a Calabi–Yau manifold and a circle, the result is an interesting combination of the six-dimensional A and B models.

We elaborate on the proposed general boundary formulation as an extension of standard quantum mechanics to arbitrary (or no) backgrounds. Temporal transition amplitudes are generalized to amplitudes for arbitrary space-time regions. State spaces are associated to general (not necessarily spacelike) hypersurfaces.

We give a detailed foundational exposition of this approach, including its probability interpretation and a list of core axioms. We explain how standard quantum mechanics arises as a special case. We include a discussion of probability conservation and unitarity, showing how these concepts are generalized in the present framework. We formulate vacuum axioms and incorporate space-time symmetries into the framework. We show how the Schrödinger–Feynman approach is a suitable starting point for casting quantum field theories into the general boundary form. We discuss the role of operators.