A conformally invariant differential operator on Weyl tensor densities

We derive a tensorial formula for a fourth-order conformally invariant differential operator on conformal 4-manifolds. This operator is applied to algebraic Weyl tensor densities of a certain conformal weight, and takes its values in algebraic Weyl tensor densities of another weight. For oriented manifolds, this operator reverses duality: For example in the Riemannian case, it takes self-dual to anti-self-dual tensors and vice versa. We also examine the place that this operator occupies in known results on the classification of conformally invariant operators, and we examine some related operators.Comment: 17 pages, LaTe

Compatibility, multi-brackets and integrability of systems of PDEs

We establish an efficient compatibility criterion for a system of generalized complete intersection type in terms of certain multi-brackets of differential operators. These multi-brackets generalize the higher Jacobi-Mayer brackets, important in the study of evolutionary equations and the integrability problem. We also calculate Spencer delta-cohomology of generalized complete intersections and evaluate the formal functional dimension of the solutions space. The results are applied to establish new integration methods and solve several differential-geometric problems.Comment: Some modifications in sections 6.1-2; new references're adde

The extended algebra of observables for Dirac fields and the trace anomaly of their stress-energy tensor

We discuss from scratch the classical structure of Dirac spinors on an arbitrary globally hyperbolic, Lorentzian spacetime, their formulation as a locally covariant quantum field theory, and the associated notion of a Hadamard state. Eventually, we develop the notion of Wick polynomials for spinor fields, and we employ the latter to construct a covariantly conserved stress-energy tensor suited for back-reaction computations. We shall explicitly calculate its trace anomaly in particular.Comment: 65 page

Singular propagators in deformation quantization and Shoikhet-Tsygan formality

This paper adds some details to the seminal approach to logarithmic formality \cite{AWRT} and interpolation formality \cite{WR} by Alekseev, Rossi, Torossian and Willwacher: We prove that the interpolation family of Kontsevich formality maps extends to Shoikhet-Tsygan formality and a complex interpolation parameter. We show some elementary relations satisfied by this polynomials. We also compute some Kontsevich integral weights and reason on the number theoretic meaning of the invariance of Kontsevich's propagator under real translations and scalings in the case of the Merkulov $n$-wheels.Comment: Some typos corrected. A vanishing lemma on p. 46 got corrected by restricting to special graph