We elucidate the vector space (twisted relative cohomology) that is Poincar\'e dual to the vector space of Feynman integrals (twisted cohomology) in general dimension. These spaces are paired via an inner product called the intersection number -- an invariant which can be computed algebraically. In this language, reduction of an integrand modulo integration-by-part identities is simply its projection onto a chosen basis. The dual-forms turn out to be far simpler than their Feynman counterparts; they are uplifts of lower-dimensional forms supported on the maximal cuts of various sub-topologies (boundaries). The intersection numbers are then computed by taking residues around cuts where various subsets of propagators become on-shell, giving a systematic approach to generalized unitarity. Moreover, the dual integrals satisfy the transposed differential equation to their Feynman counterparts. Since the dual-forms are localized to cut the transposed differential equation is a much simpler to construct.

Generalized Unitarity and the Poincaré Duals of Feynman Integrals

<i>**Session Chair: **  Thomas Creutzig</i>

W3-2 Mathematical and Theoretical Physics (DTP) / Physique mathématique et physique théorique (DPT)

technical paper

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Generalized Unitarity and the Poincaré Duals of Feynman Integrals

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