The recent data on abnormal excess of dileptons in central heavy ion collisions are surveyed. We propose a novel explanation for this dilepton excess observed in dense/hot nuclear matter at invariant masses below 1 GeV. We argue that the presence of local parity breaking (LPB) due to a slowly varying time-dependent isosinglet and/or isotriplet pseudoscalar condensate may substantially modify the dispersion relation of photons and vector mesons propagating in such a medium, resulting in a large excess of l+l− with respect to the common theoretical predictions. Various experimental options to prove or falsify this effect are discussed.