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I came across Veneziano's famous formula for the scattering amplitude for four tachyons written as

$$A(s,t)= \sum_{n \geq0} \frac{(-1)^n}{n-1 + \alpha^{'}s}\frac{P_n(\alpha^{'}t)}{n!},\:\:\:\:P_n(x)=\frac{(x-2)!}{(x-n-2)!}$$

(G. Arutyunov, Lectures on String Theory, pg. 55) and noted that the polynomial $P_n(x)$ is the generating function for the Betti numbers of smooth Riemann surfaces of genus $0$ with marked points, ${\mathcal{M}}_{0,n+2}$ (R. Murri, Fatgraph Algorithms and the Homology of the Kontsevich Complex, pg. 3). Is there any physical relation of the Betti numbers to the physics?

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