On the boundary of Q-systems : introduction to the KNS conjecture

talks

• Séminaire d'Algèbre, IHP, 3/12/2012

statement of the KNS conjecture

Let $X$ be a Dynkin diagram of type $ADE$ of rank $r$ and $I$ be the index set of it. There exists a unique solution $\mathbf{z}=(z^{(a)}_{i})$ of the level $k$ restricted $Q$-system of type $X$ satisfying $z_{i}^{(a)}>0$ for $0\leq i \leq k$ and $a\in I$. Moreover, it satisfies the following additional properties:

• (symmetry) $z^{(a)}_{i}=z^{(a)}_{k-i}$ for $0\leq i \leq k$ and $a\in I$.
• (unimodality) $z^{(a)}_{i-1}<z^{(a)}_{i}$ for $1\le i \le m=\lfloor\frac{k}{2}\rfloor$ and $a\in I$ where $\lfloor x\rfloor$ is the floor function.
• $z^{(a)}_{k+1}=z^{(a)}_{k+2}=\cdots =z^{(a)}_{k+h-1}=0$ for $a\in I$.

example : level $k=8$ restricted Q-system of type $D_5$

$$ \begin{bmatrix}

1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\
6.027 & 19.16 & 42.36 & 8.524 & 8.524 \\
17.16 & 112.0 & 401.6 & 30.30 & 30.30 \\
30.30 & 294.6 & 1380. & 60.60 & 60.60 \\
36.33 & 401.6 & 2050. & 75.65 & 75.65 \\
30.30 & 294.6 & 1380. & 60.60 & 60.60 \\
17.16 & 112.0 & 401.6 & 30.30 & 30.30 \\
6.027 & 19.16 & 42.36 & 8.524 & 8.524 \\
1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 

\end{bmatrix} $$

references

• Kuniba, A. (1993). Thermodynamics of the Uq(Xr(1)) Bethe ansatz system with q a root of unity. Nuclear Physics B, 389(1), 209–244. doi:10.1016/0550-3213(93)90291-V
• Kuniba, A., Nakanishi, T., & Suzuki, J. (2011). T -systems and Y -systems in integrable systems. Journal of Physics A: Mathematical and Theoretical, 44(10), 103001. High Energy Physics - Theory; Mathematical Physics; Mathematical Physics; Quantum Algebra; Exactly Solvable and Integrable Systems. doi:10.1088/1751-8113/44/10/103001

pdf

• introduction to the KNS conjecture about quantum dimension solution of Q-systems
• contains explicit examples of level $k$ restricted Q-systems
• File:IntroKNS.pdf