The beta function

$B(a,b) = \frac{\Gamma (a) \Gamma (b)}{\Gamma (a + b)}$

has an integral representation, which can be found by using polar coordinates in the expression

$m! n! = \int_{0}^{2 \pi} \int_{0}^{\infty} e^{- r^2} | r \textrm{cos} \theta |^{2m + 1} | r \textrm{sin} \theta |^{2n + 1} r \textrm{d} r \textrm{d} \theta$

and pulling out the $r$ factors to obtain

$2 (m + n + 1)! \int_{0}^{\frac{\pi}{2}} \textrm{cos}^{2m+1} \theta \textrm{sin}^{2n+1} \theta \textrm{d} \theta$.

It could be fun to play around with higher dimensional analogues of such integrals using, for instance, Euler angles. In a series of papers, Kholodenko looks at multidimensional analogues, and he concludes that Veneziano-like amplitudes are capable of reproducing spectra for both open and closed strings. This involves a study of amplitudes using period integrals for Fermat hypersurfaces. On an historial note, he mentions the original paper of Chowla and Selberg on elliptic integrals. Recall that Chowla is the guy who introduced Montgomery to Dyson at tea.

The Veneziano amplitude takes the form

$V(s,t,u) = V(s,t) + V(s,u) + V(t,u)$

for $V(a,b) = B (- \alpha (a) , – \alpha (b))$. When the condition

$\alpha (s) + \alpha (t) + \alpha (u) = -2$

holds, the amplitude takes the form

$V(s,t,u) = \frac{\zeta (1 + 0.5 \alpha (s)) \zeta (1 + 0.5 \alpha (t)) \zeta (1 + 0.5 \alpha (u))}{\zeta (- 0.5 \alpha (s)) \zeta (- 0.5 \alpha (t)) \zeta (- 0.5 \alpha (u))}$.

Kholodenko discusses the generalised condition

$\alpha (s) m + \alpha (t) n + \alpha (u) l + k = 0$

where the Veneziano case is recovered when $k$ is associated to the degree $N$ of the Fermat surface, while the Shapiro-Virasoro condition arises for degree $2N$.

More recently, Kholodenko has been looking at honeycombs, which were used by Terence Tao and Allen Knutson to solve the long outstanding Horn conjecture on the spectra of $n \times n$ Hermitean matrices.

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## Anonymous said,

March 6, 2007 @ 4:38 pm

I would like to add that the Euler-type integrals used in Kholodenko’s works belong to the hypergeometric-type integrals considered in detail by Aomoto and Gelfand (with collaborators). In the honeycomb paper arguments are presented in favour of universality of such type of integrals. That is to say, all physical problems-from quantum mechanics to quantim field and string theory- should involve such type of integrals. This observation enables one to build a string theory model living in normal dimensions which also accounts for mirror symmetry. No compactification and/or dimensional reduction problems exist for such a model.

## Kea said,

March 6, 2007 @ 10:47 pm

This observation enables one to build a string theory model living in normal dimensions which also accounts for mirror symmetry.Why, thank you for your comment, anonymous! Yes, we are hoping to understand these things better from a more categorical viewpoint. Any further comments are appreciated.