# What is the difference between topological theta term and Wess-Zumino-Witten term?

It seems that they both proportional to some thing like $$\vec{n}\cdot \partial_{\tau}\vec{n}\, \times\,\partial_{s}\vec{n}$$.

References: Fradkin, Quantum field theory: an integrated approach

It depends on the number of arguments that the $$\vec{n}$$ field starts out with (is this a 1d theory or 2d?). In both situations integrating over $$\vec{n}\cdot \partial_{\tau}\vec{n}\, \times\,\partial_{s}\vec{n}$$ gives an area on the sphere that is the target space of $$\vec{n}$$.
If $$\vec{n}$$ just depends on one argument then with the appropriate boundary conditions $$\vec{n}(\tau)$$ will form a closed loop on a sphere. Then you can modify your action by integrating over the 2D interior of the loop (extending your field arbitrarily to some $$n(\tau,s)$$), and this is the WZW term. The ambiguity about which side is interior and exterior is why the coefficient of the WZW term needs to be quantized.
If $$\vec{n}$$ is already a function of $$\tau,s$$, then it already forms a 2D surface not a loop when mapped to the target space sphere. The boundary condition that it goes to the same point at infinity means it can only wind around the sphere a discrete integer amount of times. And putting a term in your action proportional to this winding number is a theta term.