Andrzej Pokraka (Ï㽶ÊÓƵ)
Title:Â The duals of Feynman integrals.
´¡²ú²õ³Ù°ù²¹³¦³Ù:ÌýWe elucidate the vector space (twisted relative cohomology) that is Poincaré 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. As an application of our formalism, we derive compact differential equations satisfied by arbitrary one-loop integrals in non-integer spacetime dimension and show how to use the intersection number to express a scattering amplitude in terms of a minimal basis of integrals. We also examine the 4-dimensional limit of our formalism and provide prescriptions for extracting rational terms.