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 3×3 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 Responses so far »

  1. 1

    CarlBrannen said,

    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.

  2. 2

    Kea said,

    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 some kind 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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