  Using the framework of twisted cohomology, we study twisted Riemann bilinear
relations (TRBRs) satisfied by multi-loop Feynman integrals and their cuts in
dimensional regularisation. After showing how to associate to a given family of
Feynman integrals a period matrix whose entries are cuts, we investigate the
TRBRs satisfied by this period matrix, its dual and the intersection matrices
for twisted cycles and co-cycles. For maximal cuts, the non-relative framework
is applicable, and the period matrix and its dual are related in a simple
manner. We then find that the TRBRs give rise to quadratic relations that
generalise quadratic relations that have previously appeared in the literature.
However, we find that the TRBRs do not allow us to obtain quadratic relations
for non-maximal cuts or completely uncut Feynman integrals. This can be traced
back to the fact that the TRBRs are not quadratic in the period matrix, but
separately linear in the period matrix and its dual, and the two are not simply
related in the case of a relative cohomology theory, which is required for
non-maximal cuts.
