Two-form gauge theory dual to scalar-tensor theory

We generalize the electromagnetic duality between a massless free scalar field and a two-form gauge field in a four-dimensional spacetime to scalar-tensor theories. We derive the action of a two-form gauge field that is dual to two kinds of scalar-tensor theories: the shift symmetric K-essence theory and the shift symmetric Horndeski theory up to quadratic in a scalar field. The former case, the dual two-form, has a nonlinear kinetic term. The latter case, the dual two-form, has nontrivial interactions with gravity through the Einstein tensor. In both cases, the duality relation is modified from the usual case, that is, the dual two-form field is not simply given by the Hodge dual of the gradient of the scalar field.