# Classical field theory with fields on different base spaces

Keeping things at a "basic level", a field is a function from a base manifold (of dimension D) to some other space. Usually the base manifold is the spacetime but may be something different (a lattice, etc etc). We all know how to construct the classical action of a complex scalar field over the usual 3+1 spacetime. Imagine now that we want to let this field interact with a classical membrane, or a string, or a point particle immersed in the spacetime. How to write the action for such a theory in which a "low dimensional field" (zero-dimensional in the case of the particle) is immersed into the spacetime? The situation is not conceptually different from when we write the action for a charged particle immersed into a background electromagnetic field. How to generalise the theory (the variational principle) in such a way to have the coupled equations of motion for the particle and the field?

OP's scenario is often the case. Perhaps an example is in order. The E&M gauge fields $$A_0,A_1,A_2,A_3:[t_i,t_f]\times \mathbb{R}^3\to \mathbb{R}$$ (whose base manifold is spacetime) and $$N$$ (non-relativistic) point charges $$q_1,\ldots, q_N$$ with positions $${\bf r}_1, \ldots, {\bf r}_N: [t_i,t_f]\to \mathbb{R}^3$$ (whose base manifold is a worldline) are described by the action $$S[A_0,A_1,A_2,A_3,{\bf r}_1, \ldots, {\bf r}_N]~=~\int \! dt ~L,$$ with Lagrangian $$L~=~\sum_{i=1}^N\left( \frac{m_i}{2}{\bf v}^2_i + q_i\{A_0({\bf r}_i) + {\bf v}_i\cdot {\bf A}({\bf r}_i)\} \right) -\frac{1}{4} \int d^3 {\bf x}\sum_{\mu,\nu=0}^3F_{\mu\nu}(t,{\bf x})F^{\mu\nu}(t,{\bf x})$$ [in $$(-,+,+,+)$$ Minkowski sign convention].