Feynman Integrals and Intersection Theory

We introduce the tools of intersection theory to the study of Feynman integrals, which allows for a new way of projecting integrals onto a basis. In order to illustrate this technique, we consider the Baikov representation of maximal cuts in arbitrary space-time dimension. We introduce a minimal basis of differential forms with logarithmic singularities on the boundaries of the corresponding integration cycles. We give an algorithm for computing a basis decomposition of an arbitrary maximal cut using so-called intersection numbers and describe two alternative ways of computing them. Furthermore, we show how to obtain Pfaffian systems of differential equations for the basis integrals using the same technique. All the steps are illustrated on the example of a two-loop non-planar triangle diagram with a massive loop.Comment: 13 pages, published versio

Intersection theory and the Alesker product

Alesker has introduced the space $\mathcal V^\infty(M)$ of {\it smooth valuations} on a smooth manifold $M$, and shown that it admits a natural commutative multiplication. Although Alesker's original construction is highly technical, from a moral perspective this product is simply an artifact of the operation of intersection of two sets. Subsequently Alesker and Bernig gave an expression for the product in terms of differential forms. We show how the Alesker-Bernig formula arises naturally from the intersection interpretation, and apply this insight to give a new formula for the product of a general valuation with a valuation that is expressed in terms of intersections with a sufficiently rich family of smooth polyhedra.Comment: further revisons, now 23 page