论文标题
$ u_q \ mathfrak {sl} _2 $ -invariant非压缩边界条件
$U_q\mathfrak{sl}_2$-invariant non-compact boundary conditions for the XXZ spin chain
论文作者
论文摘要
我们介绍了开放xxz旋转链的新$ u_q \ mathfrak {sl} _2 $ -Invariant边界条件。对于$ q $的通用值，我们将散装的哈密顿量与自旋链的一个或两个边界上的无限维verma模块相结合，以及$ q = e^{\ frac {iπ} {iπ} {p}} {p}} $ a $ 2p $ - $ 2p $ - $ 2p $ - unity $ - $ p $ -dim-dimentimentimential $ -dim-dimemensional actaroge。两种情况均由连续的“自旋” $α\ in \ mathbb {c} $进行参数。 为了激励我们的构建，我们首先专注于$ q = i $，在那里我们获得了带有量子群对称性的修改后的XX Hamiltonian，其频谱和缩放极限是使用免费费米斯明确计算的。在连续体中，该模型在上半平面上使用$（η，ξ）$ ghost CFT识别，其连续的真实轴上是连续不变的边界条件。识别出不可还原的virasoro表示。 回到通用$ Q $，我们调查了基础晶格代数的代数属性。我们表明，如果$ q^α\ notin \ pm q^{\ mathbb {z}} $，新的边界耦合提供了Blob Algebra的忠实表示，该表示是Schur-weyl dual dual dual of $ u_q \ u_q \ mathfrak {slfrak {sl}} _2 $。然后，为了修改左侧和右侧的边界条件，我们获得了通用两势temberley-lieb代数的表示。这些表示形式的发电机和参数是按$ Q $和$α$明确计算的。最后，在这种情况下，我们猜想了Schur-Weyl二元性的一般形式。 本文是一个系列中的第一个，我们将研究参数的所有值，频谱及其连续性极限，相关晶格代数的表示内容以及这些新旋转链的融合特性。
We introduce new $U_q\mathfrak{sl}_2$-invariant boundary conditions for the open XXZ spin chain. For generic values of $q$ we couple the bulk Hamiltonian to an infinite-dimensional Verma module on one or both boundaries of the spin chain, and for $q=e^{\frac{iπ}{p}}$ a $2p$-th root of unity $ - $ to its $p$-dimensional analogue. Both cases are parametrised by a continuous "spin" $α\in\mathbb{C}$. To motivate our construction, we first specialise to $q=i$, where we obtain a modified XX Hamiltonian with unrolled quantum group symmetry, whose spectrum and scaling limit is computed explicitly using free fermions. In the continuum, this model is identified with the $(η,ξ)$ ghost CFT on the upper-half plane with a continuum of conformally invariant boundary conditions on the real axis. The different sectors of the Hamiltonian are identified with irreducible Virasoro representations. Going back to generic $q$ we investigate the algebraic properties of the underlying lattice algebras. We show that if $q^α\notin\pm q^{\mathbb{Z}}$, the new boundary coupling provides a faithful representation of the blob algebra which is Schur-Weyl dual to $U_q\mathfrak{sl}_2$. Then, modifying the boundary conditions on both the left and the right, we obtain a representation of the universal two-boundary Temperley-Lieb algebra. The generators and parameters of these representations are computed explicitly in terms of $q$ and $α$. Finally, we conjecture the general form of the Schur-Weyl duality in this case. This paper is the first in a series where we will study, at all values of the parameters, the spectrum and its continuum limit, the representation content of the relevant lattice algebras and the fusion properties of these new spin chains.
