In string theory, when we write the Feynman diagrams, we denote amplitude for open and closed string, as Veneziano and Virasoro-Shapiro amplitudes (a complex beta function) respectively (we will only discuss the former). Veneziano amplitude is an Euler beta function that obeys the crossing symmetry and looks*
where s and t are Mandelstem variables defined;
and $ \alpha(s) $ is Regge trajectory.
because any beta function of the form
can be written as
Note * has only one pole rather than two and ** is written in t poles. What we can do is writing ** in s poles, which then
And that is the duality. We can study various aspects of it by keeping t
fixed or s fixed. This is done in the very first paper on this by Veneziano,
here. Also, in integral representation, as like a beta function, this amplitude
can be written as
it is valid for a complex large s plane unless one gets too close to the
positive real line. This indicates that quantum corrections would be received
by the imaginary part. In large s and fixed t, one can also write $A(s,t) \sim
s^{\alpha(t)}$ (for linear Regge trajectory), and since in general Regge
theory $A(s,t)\sim s^J$ where J is averaged (effective) angular momentum, we
see
$$\boxed{ J = \alpha(t)}$$
For a good understanding of this amplitude (or string theory), you can read
Superstring Theory by Green, Schwarz, and Witten. Or Polchinksi's
volumes.
Nice post, keep it on!
Continuing your discussion what are the chances of it being calculated using the other algebra (VIrasoro)