Difference between revisions of "On the boundary of Q-systems : introduction to the KNS conjecture"
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===status=== | ===status=== | ||
* proved for all classical types | * proved for all classical types | ||
− | + | * there exists an upgraded formulation using the fusion ring | |
+ | * you can find talk slides in a newer setting at [[Kirillov-Reshetikhin modules and fusion rings from CFT]] | ||
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* contains explicit examples of level $k$ restricted Q-systems | * contains explicit examples of level $k$ restricted Q-systems | ||
* there exists an upgraded formulation using the fusion ring | * there exists an upgraded formulation using the fusion ring | ||
− | * you can find talk | + | * you can find talk slides in a newer setting at [[Kirillov-Reshetikhin modules and fusion rings from CFT]] |
* [[File:IntroKNS.pdf]] | * [[File:IntroKNS.pdf]] |
Revision as of 07:48, 9 July 2013
talks
- Séminaire d'Algèbre, IHP, Paris
level $k$ restricted $Q$-system
notations
- $X$ : a simply-laced Dynkin diagram (i.e. type $ADE$) of rank $r$
- $I$ : the index set of $X$
- $\mathcal{I}(X)$ : the adjacency matrix of $X$
definition
We define the level $k$ restricted $Q$-system of type $X$ to be the system of equations $$ \left\{ \begin{array}{lll} Q^{(a)}_{0} =1 & a\in I \\ \left(Q^{(a)}_{m}\right)^2=\prod _{b\in I} \left(Q^{(b)}_{m}\right)^{\mathcal{I}(X)_{ab}}+Q^{(a)}_{m-1}Q^{(a)}_{m+1} & 1\le m \le k-1, a\in I\\ Q^{(a)}_{k} =1 & a\in I \end{array} \right. $$ in variables $\{Q^{(a)}_{m}| a\in I , 0\leq m \leq k\}$.
- there is a more general version including all multiply-laced dynkin diagrams.
the KNS conjecture
known results
There exists a unique solution $\mathbf{z}=(z^{(a)}_{m})$ of the level $k$ restricted $Q$-system of type $X$ satisfying $z_{m}^{(a)}>0$ for $0\leq m \leq k$ and $a\in I$. Moreover, it satisfies the following additional properties :
- (symmetry) $z^{(a)}_{m}=z^{(a)}_{k-m}$ for $0\leq m \leq k$ and $a\in I$.
- (unimodality) $z^{(a)}_{m-1}<z^{(a)}_{m}$ for $1\le m \le \lfloor\frac{k}{2}\rfloor$ and $a\in I$ where $\lfloor x\rfloor$ is the floor function.
the statement of the KNS conjecture
The unique positive solution $\mathbf{z}=(z^{(a)}_{m})$ is given by the quantum dimension $\mathcal{D}^{(a)}_{m}$ of the KR module $W^{(a)}_{m}$. For each $a\in I$, $\mathcal{D}^{(a)}_{k+1}=\mathcal{D}^{(a)}_{k+2}=\cdots =\mathcal{D}^{(a)}_{k+h^{\vee}-1}=0$.
status
- proved for all classical types
- there exists an upgraded formulation using the fusion ring
- you can find talk slides in a newer setting at Kirillov-Reshetikhin modules and fusion rings from CFT
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
- A. N. Kirillov (1989), Identities for the Rogers Dilogarithm Function Connected with Simple Lie Algebras. Journal of Soviet Mathematics 47 : 2450–2459. doi:10.1007/BF01840426.
- 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
- A proof of the KNS conjecture : $D_r$ case, J. Phys. A: Math. Theor. 46 165201 doi:10.1088/1751-8113/46/16/165201, arXiv:1210.1669
- introduction to the KNS conjecture about quantum dimension solution of Q-systems
- contains explicit examples of level $k$ restricted Q-systems
- there exists an upgraded formulation using the fusion ring
- you can find talk slides in a newer setting at Kirillov-Reshetikhin modules and fusion rings from CFT
- File:IntroKNS.pdf