Double dimensional reduction of M5-brane action in Sen formalism

In this paper, we present double dimensional reduction of the complete M5-brane action in the Sen formalism of self-dual form. Although in this formalism the gravity couples to the independent pseudo-$2$-form and pseudo-$3$-form fields in a non-standard and very complicated way, the double dimensional reduction on the complete action can be carried out. This is by dualising some components of the pseudo-$2$-form field and integrating out the pseudo-$3$-form field. We show explicitly that the double dimensional reduction on a circle gives rise to the complete D4-brane or the complete dual D4-brane action depending on what components of the pseudo-$2$-form field are dualised. The duality between the D4-brane and dual D4-brane actions are realised in the viewpoint of the Sen formalism as the swapping of the roles of the components of the pseudo-$2$-form field between those which are dualised and those which remain. We also make a brief outline on how to generalise this to the cases of dimensional reduction on some other spaces as well as how to understand the duality of reduced action from the viewpoint of the Sen formalism. An explicit example for dimensional reduction of the quadratic six-dimensional action on a torus is given and the realisation of its S-duality is also discussed.