# Relation of Betti numbers to Veneziano's scattering amplitude?

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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?

(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.

edited Dec 12, 2016

The formula is only an approximation of the Veneziano's formula, as explained in the G. Arutyunov paper : (" Thus, the scattering amplitude can be essentially written as...."). One simply replaces the original formula by the sum of (pole multiply by residue at the pole)

@Trimok: when the sum of the right hand side converges, it is really equal to the Veneziano amplitude $A(s,t)$. If we fix $t$ with a real part negative enough, the sum converges and one can consider the two sides of the equality as meromorphic functions in $s$. As these two functions have the same poles and residues, they differ at most by an entire function but comparison of the asymptotics at infinity shows that this entire function vanishes at infinity and so is zero. For this reason, I don't understand the "essentially" in "the scattering amplitude can be essentially written as".

@40227 : Thanks for the precision, I was thinking of differences between, for instance, an original function like $\dfrac{e^{-s}}{s}$, and product of pole and residue, which is $\frac{1}{s}$. However, it is true, that for $s \to -\infty$, these functions are not asymptotically equals.

Use the binomial expansion of the factor in the integral rep of the beta function, i.e., the Veneziano amplitude to show the eqivalence of the sum to the ratio of gamma fcts.

With google keywords "fatgraphs open string ﬁeld matrix model", it seems that there is a link between fatgraphs, open string field Feynman diagrams, topological string theory, and (random) matrix models (for instance the book of Brezin or this presentation of  Rastelli)

I wonder if there is any kind of  relationship with the Penner model (and extensions), where the free energy is a generating function for the orbifold Euler characteristics of the moduli space of Riemann surfaces of genus g with s punctures (as defined in this ref, for instance).

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