M Theory Lesson 45

Tao mentions a paper by Speyer on a proof of the honeycomb theorem that uses no representation theory at all. It contains a theorem by Klyachko [1] which states that the additive problem is solvable for spectra $(\lambda ,\mu ,\nu )$ iff the multiplicative problem is solvable for $(e^{\lambda },e^{\mu },e^{\nu })$.

Kholodenko attacks the Gromov-Witten invariants via the multiplicative problem. The 3x3 relation

${\lambda }_1+{\lambda }_2+{\lambda }_3+{\mu }_1+{\mu }_2+{\mu }_3={\nu }_1+{\nu }_2+{\nu }_3$

is replaced by the generalised expression

${\lambda }_1+{\lambda }_2+{\lambda }_3+{\mu }_1+{\mu }_2+{\mu }_3={\nu }_1+{\nu }_2+{\nu }_3+N(d_1+d_2+d_3)$

where the $d_i$ are associated to punctures on a sphere. Let $d$ be the sum of the $d_i$. Fusion rules then belong to quantum cohomology

${\sigma }_a*{\sigma }_b=\sum _{d,c}{q}^d{C}_{ab}^c\left(d\right){\sigma }_c$

with a new kind of product for classes. These coefficients give the Gromov-Witten invariants in the genus zero, three point case. In terms of monodromy matrices, Kholodenko writes

$\prod _{i=1}^n\text{exp}\left(2\pi i\frac{A_i}{d_i}\right)=\text{exp}\left(2\pi iI\right)$

where the $A_i$ are diagonalisable matrices that produce an eigenvalue set.

[1] A. Klyachko, Lin. Alg. Appl. 319 (2000) 37-59