### 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

$\sigma_{a} * \sigma_{b} = \sum_{d,c} q^d C_{ab}^{c} (d) \sigma_{c}$

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

$\prod_{i = 1}^{n} \textrm{exp} (2 \pi i \frac{A_i}{d_i}) = \textrm{exp} (2 \pi i I)$

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

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

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

$\sigma_{a} * \sigma_{b} = \sum_{d,c} q^d C_{ab}^{c} (d) \sigma_{c}$

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

$\prod_{i = 1}^{n} \textrm{exp} (2 \pi i \frac{A_i}{d_i}) = \textrm{exp} (2 \pi i I)$

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

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

## 2 Comments:

I'm glad to see you doing this. As I've said before, I suspect that it is going to be useful before we are through.

Meanwhile, I've got another guess for that damned number, and I think that this one makes sense. I put the calculational details up in physics forums cause it needs LaTex.

Thanks, Carl. Hmmm. Yes, the kappa approach sounds interesting. In fact, I'm not expecting that damned number to be clarified until we sort out

somekind of E dependence.Now kneemo mentioned the idea of roots-of-unity (for that number) at one point, and this is why I keep talking about these q factors, which are roots-of-unity as a rule. So I like the idea of varying some function over all roots and finding a minimum.

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