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?

(Edit in response to Trimok's google results.) The web of relations presented in MathOverflow questions Q-181284 and Q-145555 motivated me to pose this question. Related papers are "Constructive motives and scattering" by Marni Sheppeard and "Motivic Amplitudes and Cluster Coordinates" by Golden et al., both with extensive references.

This post imported from StackExchange Physics at 2014-09-24 13:21 (UTC), posted by SE-user Tom Copeland

(Edit 9/2015) For some more notes and links on the Euler beta function integral rep of the Veneziano amplitude, see MO-Q Connections to physics, geometry, geometric probability theory of Euler's beta function. For more on $P_n(x)$, OEIS-049444 and OEIS-A074060. )

(Edit 12/2016) Related to answer to MO-Q Why does bosonic string theory require 26 spacetime dimensions?

$ P_n (x)$ is also simply related to A094638 which in turn is related to the diff op reps of the Witt, or centerless Virasoro, algebra, the braid group, and special Jack symmetric polynomials.