HEP Theory Journal Club

Duals of Feynman Integrals, I: Differential Equations

Andrzej Pokraka

McGill

We elucidate the vector space (twisted relative cohomology) that is Poincare' dual to the vector space of Feynman integrals (twisted cohomology) in general spacetime dimension. The pairing between these spaces - an algebraic invariant called the intersection number - extracts integral coefficients for a minimal basis, bypassing the generation of integration-by-parts identities. Dual forms turn out to be much simpler than their Feynman counterparts: they are supported on maximal cuts of various sub-topologies (boundaries). Thus, they provide a systematic approach to generalized unitarity: the reconstruction of amplitudes from on-shell data. In this talk, we will introduce the idea of dual forms and study their mathematical structures. As an application, we derive compact differential equations satisfied by arbitrary one-loop integrals in non-integer spacetime dimension. Our methods are expected to generalize to higher loop order and we provide a simple 2-loop example as evidence.