# Corners in the worldsheet and Ricci scalar term in the Polyakov action

I have a question about a passage in Polchinski's textbook [1], regarding the topological term in the Polyakov action.

In the Polyakov action for a closed manifold, we can add a term proportional to the Euler characteristic, which we can write as a local functional using the Gauss-Bonnet theorem: $$\Delta S_{\text{closed}} = \lambda\chi = \frac{\lambda}{4\pi}\int_\Sigma |g|^{1/2}R\text d^2 \sigma.$$

This is invariant under diffeomorphisms and Weyl transformations. For a manifold with boundary, we have to include a boundary term in order to preserve Weyl invariance. This is also the formula for the Euler characteristic given by the Gauss-Bonnet theorem: $$\Delta S_{\text{open}} = \lambda \chi = \frac{\lambda}{4\pi}\int_\Sigma |g|^{1/2}R\text d^2 \sigma + \frac{\lambda}{2\pi}\oint_{\partial M}k\text ds.$$

Now, if there are corners on the boundary, with inner angles $$\alpha_i$$ (Polchinski considers only $$\alpha_i = \pi/2$$), the Gauss-Bonnet theorem becomes: $$\chi = \underset{\tilde \chi}{\underbrace{\frac{1}{4\pi}\int_\Sigma |g|^{1/2}R\text d^2 \sigma + \frac{1}{2\pi}\oint_{\partial M}k\text ds }}+ \frac{1}{2\pi}\sum_i (\pi -\alpha_i).$$

Now, it is not clear (to me) what should be the term in the action : the topological term $$\lambda\chi$$ or only the contribution from the integrals $$\lambda\tilde\chi$$, disregarding the contributions from the corners, as both are Weyl-invariant.

In Polchinski's textbook [1,p.84], the author claims that the latter is correct. He says:

This follows from unitarity: the $$\lambda$$-dependence must be unchanged if we cut through a world-sheet, leaving fewer internal lines and more external sources. This will be the case for the weight $$\exp(-\lambda \tilde \chi)$$ because the surface integrals cancels on the cut edges.

Why does unitarity imply that the "$$\lambda$$-dependence" be unchanged under this cutting procedure?

[1] String Theory: Volume 1, An introduction to the bosonic string. J. Polchinski; Cambridge Monographs on Mathematical Physics.