Abstract:We describe two constructions giving rise to curved A ∞ -algebras. The first consists of deforming A ∞ -algebras, while the second involves transferring curved dg structures that are deformations of (ordinary) dg structures along chain contractions. As an application of the second construction, given a vector bundle on a polyhedron X, we exhibit a curved A ∞ -structure on the complex of matrix-valued cochains of sufficiently fine triangulations of X. We use this structure as a motivation to develop a homotopy associative version of Chern-Weil theory.

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Abstract. To every Hermitian vector bundle with connection over a compact Riemannian manifold M one can associate a corresponding connection Laplacian acting on the sections of the bundle. We define analogous combinatorial, metric dependent Laplacians associated to triangulations of M and prove that their spectra converge, as the mesh of the triangulations approaches zero, to the spectrum of the connection Laplacian.

We present a general infinite volume limit construction of probability measures obeying the Glimm-Jaffe axioms of Euclidean quantum field theory in arbitrary space-time dimension. In particular, we obtain measures that may be interpreted as corresponding to scalar quantum fields with arbitrary bounded continuous self-interaction. It remains however an open problem whether this general construction contains non-Gaussian measures.