# How build a topological charge from of the mapping between physical and inner space?

How build a topological charge from the mapping between physical and inner space? When we make a mapping between two coordinates system, we normally relate both systems by coordinate transformation as, for example,$$x^\mu=f^\mu\left(\xi^\nu\right).$$ The Jacobian is easily obtained as being given by $$J=\textrm{det}\left(\frac{\partial{x^\mu}}{\partial\xi^\nu}\right).$$ However, it seems not to be trivial to build a mapping that relates to the physical and inner space. For example, consider a complex field $$\phi\left(\boldsymbol{x}\right)$$. I understand that $$\phi\left(\boldsymbol{x}\right)$$ is a relation that associate the vector $$\boldsymbol{x}=\left(x,y,z\right)$$ in the physical space with one number $$\phi$$ also in the physical space. On the other hand, I also can think about the $$\phi$$ as being the relation $$\phi=\phi_1+i\phi_2$$ such that is now $$\phi=\phi\left(\phi_1,\phi_2\right)$$ in the inner space. The question is now: how I can build a Jacobean that allow mapping the physical space in the inner space? It is known that the topological current can be built from this Jacobean.

• It's not clear what you asking. You say that you know how the topological current works in some cases. The general idea is that one a map $f$ from the physical space to the inner space, and a "topological" object in the inner space i.e a closed-but-not-exact p-form $\omega$, and then you pull this back to the physical space to get $f^*\omega$ which is topological object there. The jacobean is automatic since we are using $p$-forms. – mike stone Sep 22 '19 at 12:54
• The point is, I know how it works qualitatively, but I don't know how to build the topological charge quantitatively. From my point of view, I need an expression that maps between one space and another just as a coordinate transformation maps one coordinate system to another. – lucenalex Sep 22 '19 at 13:35
• This is too vague to be a helpful. It's just a pullback. – mike stone Sep 22 '19 at 23:34