Difference between revisions of "On the boundary of Q-systems : introduction to the KNS conjecture"
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− | == | + | ==level $k$ restricted $Q$-system== |
− | + | ===notations=== | |
− | * (symmetry) $z^{(a)}_{ | + | * $X$ : a simply-laced Dynkin diagram (i.e. type $ADE$) of rank $r$ |
− | * (unimodality) $z^{(a)}_{ | + | * $I$ : 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 type $A_r$ and $D_r$ | ||
+ | * almost done for type $B_r$ and $C_r$ (work in progress) | ||
==example : level $k=8$ restricted Q-system of type $D_5$== | ==example : level $k=8$ restricted Q-system of type $D_5$== | ||
− | |||
$$ | $$ | ||
\begin{bmatrix} | \begin{bmatrix} | ||
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* 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. (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 | * 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, [http://arxiv.org/abs/1210.1669 arXiv:1210.1669] | ||
Revision as of 06:14, 16 January 2013
talks
- Séminaire d'Algèbre, IHP, 3/12/2012
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 type $A_r$ and $D_r$
- almost done for type $B_r$ and $C_r$ (work in progress)
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
- A Proof of the KNS conjecture : $D_r$ case, arXiv:1210.1669
- introduction to the KNS conjecture about quantum dimension solution of Q-systems
- contains explicit examples of level $k$ restricted Q-systems
- File:IntroKNS.pdf