New localization mechanism and Hodge duality for $q-$form field

In this paper, we investigate the problem of localization and the Hodge duality for a $q-$form field on a $p-$brane with codimension one. By a general Kaluza-Klein (KK) decomposition without gauge fixing, we obtain two Schr\"{o}dinger-like equations for two types of KK modes of the bulk $q-$form field, which determine the localization and mass spectra of these KK modes. It is found that there are two types of zero modes (the $0-$level modes): a $q-$form zero mode and a $(q-1)-$form one, which cannot be localized on the brane at the same time. For the $n-$level KK modes, there are two interacting KK modes, a massive $q-$form KK mode and a massless $(q-1)-$form one. By analyzing gauge invariance of the effective action and choosing a gauge condition, the $n-$level massive $q-$form KK mode decouples from the $n-$level massless $(q-1)-$form one. It is also found that the Hodge duality in the bulk naturally becomes two dualities on the brane. The first one is the Hodge duality between a $q-$form zero mode and a $(p-q-1)-$form one, or between a $(q-1)-$form zero mode and a $(p-q)-$form one. The second duality is between two group KK modes: one is an $n-$level massive $q-$form KK mode with mass $m_n$ and an $n-$level massless $(q-1)-$form mode; another is an $n-$level $(p-q)-$form one with the same mass $m_n$ and an $n-$level massless $(p-q-1)-$form mode. Because of the dualities, the effective field theories on the brane for the KK modes of the two dual bulk form fields are physically equivalent.