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

## talks

• Séminaire d'Algèbre, IHP, Paris, 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 all classical types
• there exists an upgraded formulation using the fusion ring

## example : level $k=4$ restricted Q-system of type $D_5$

### numerical computation

$$\begin{bmatrix} 1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\ 3.732 & 8.464 & 14.93 & 4.732 & 4.732 \\ 5.464 & 15.93 & 33.32 & 7.464 & 7.464 \\ 3.732 & 8.464 & 14.93 & 4.732 & 4.732 \\ 1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0  \end{bmatrix}$$

### expression in terms of the affine weights of level 4 inside the alcove

$$\begin{bmatrix} \mathcal{D}_{4 \hat{\omega }_0} & \mathcal{D}_{4 \hat{\omega }_0} & \mathcal{D}_{4 \hat{\omega }_0} & \mathcal{D}_{4 \hat{\omega }_0} & \mathcal{D}_{4 \hat{\omega }_0} \\ \mathcal{D}_{3 \hat{\omega }_0+\hat{\omega }_1} & \mathcal{D}_{4 \hat{\omega }_0}+\mathcal{D}_{2 \hat{\omega }_0+\hat{\omega }_2} & \mathcal{D}_{3 \hat{\omega }_0+\hat{\omega }_1}+\mathcal{D}_{2 \hat{\omega }_0+\hat{\omega }_3} & \mathcal{D}_{3 \hat{\omega }_0+\hat{\omega }_4} & \mathcal{D}_{3 \hat{\omega }_0+\hat{\omega }_5} \\ \mathcal{D}_{2 \hat{\omega }_0+2 \hat{\omega }_1} & \mathcal{D}_{4 \hat{\omega }_0}+\mathcal{D}_{2 \hat{\omega }_2}+\mathcal{D}_{2 \hat{\omega }_0+\hat{\omega }_2} & \mathcal{D}_{2 \hat{\omega }_0+2 \hat{\omega }_1}+\mathcal{D}_{2 \hat{\omega }_3}+\mathcal{D}_{\hat{\omega }_0+\hat{\omega }_1+\hat{\omega }_3} & \mathcal{D}_{2 \hat{\omega }_0+2 \hat{\omega }_4} & \mathcal{D}_{2 \hat{\omega }_0+2 \hat{\omega }_5} \\ \mathcal{D}_{\hat{\omega }_0+3 \hat{\omega }_1} & \mathcal{D}_{4 \hat{\omega }_0}+\mathcal{D}_{2 \hat{\omega }_0+\hat{\omega }_2} & \mathcal{D}_{\hat{\omega }_0+3 \hat{\omega }_1}+\mathcal{D}_{2 \hat{\omega }_1+\hat{\omega }_3} & \mathcal{D}_{\hat{\omega }_0+3 \hat{\omega }_4} & \mathcal{D}_{\hat{\omega }_0+3 \hat{\omega }_5} \\ \mathcal{D}_{4 \hat{\omega }_1} & \mathcal{D}_{4 \hat{\omega }_0} & \mathcal{D}_{4 \hat{\omega }_1} & \mathcal{D}_{4 \hat{\omega }_4} & \mathcal{D}_{4 \hat{\omega }_5} \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0  \end{bmatrix}$$ where $\mathcal{D}_{\hat{\lambda}}$ denotes the quantum dimension of $\hat{\lambda}$.

## references

### origin of the problem

• Bazhanov, V. V., and N. Yu. Reshetikhin. 1989. “Critical RSOS Models and Conformal Field Theory.” International Journal of Modern Physics A. Particles and Fields. Gravitation. Cosmology. Nuclear Physics 4 (1): 115–142. doi:10.1142/S0217751X89000042.
• A. N. Kirillov (1989), Identities for the Rogers Dilogarithm Function Connected with Simple Lie Algebras. Journal of Soviet Mathematics 47
• Bazhanov, V. V., and N. Reshetikhin. 1990. “Restricted Solid-on-solid Models Connected with Simply Laced Algebras and Conformal Field Theory.” Journal of Physics A: Mathematical and General 23 (9) (May 7): 1477. doi:10.1088/0305-4470/23/9/012.: 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

### review paper

• 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
• see chapters 13 and 14