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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

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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.

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