On the massless modes of the AdS3/CFT2 integrable systems

Abstract

We make a proposal for incorporating massless modes into the spin-chain of the AdS3/CFT2 integrable system. We do this by considering the alpha to 0 limit of the alternating d(2,1;alpha)^2 spin-chain constructed in arXiv:1106.2558. In the process we encounter integrable spin-chains with non-irreducible representations at some of their sites. We investigate their properties and construct their R-matrices in terms of Yangians.

Figures (13)

• Figure 1: Pictorial representations of the $P$, $S$ and $R$ modules. The grayed out and dashed lines indicates the action of generators that vanish on a specific state, rendering the corresponding module reducible.
• Figure 2: The choice of (distinguished) Dynkin diagram of $\mathfrak{sl}(2|1)$ used in the text.
• Figure 3: An example of a state in a reducible homogeneous spin-chain. The squares indicate sites in a non-trivial irrep, such as ${\bf 2}$ in section \ref{['sec:homogenous-toy-model']} or the $R$ module in section \ref{['sec:homogenous-R-spin-chain']}. The dots indicate sites in a trivial (singlet) representation. In the notation introduced in equations (\ref{['partition']}) and (\ref{['ntildedef']}) this state has ${\bf N}=\{1,2,3,4,5,6,7,8,9,10\}$, ${\bf n}=\{3,4,5,9\}$ and ${\bf {\tilde{n}}}=\{1,2,7,8,10\}$ as well as $N=10$, $n=4$ and ${\tilde{n}}=6$. The states are drawn according to the ordering given by the site number.
• Figure 4: The spin-chain state from figure \ref{['fig:redchain']} drawn in a way that reflects the notion of locality as dictated by the integrable structure. The nearest-neighbors are lined up along the horizontal direction. For example, the nearest neighbors of the site 2 are sites 1 and 6, and the nearest neighbors of the site 7 are sites 6 and 8. The Hamiltonian of the spin-chain is local with respect to this notion of locality. The monodromy matrix $\tau(u)$ acts non-trivially only on the sites on which the non-singlet representations sit, which, in this case are $1,2,6,7,8,10$.
• Figure 5: An example of a state in a reducible alternating spin-chain of the type discussed in sections \ref{['sec:alternating-toy-model']} and \ref{['sec:alternating-R-spin-chain']}. For such spin-chains the odd sites are in the ${\bf r}$ or $R$ representations respectively (denoted in the figure by a square). The even sites are either singlets (denoted by a dot) or in the ${\bf r}$ (respectively $R$) representation. In the notation introduced in equations (\ref{['partition']}) and (\ref{['ntildedef']}) this state has ${\bf N}=\{1,2,3,4,5\}$, ${\bf n}=\{2,3,5\}$ and ${\bf {\tilde{n}}}=\{1,4\}$ as well as $N=5$, $n=3$ and ${\tilde{n}}=1$. In the notation introduced in equations (\ref{['eqMdef']}) and (\ref{['eqmdef']}) this state has ${\bf M}=\{1,2,3,4,5,6,7,8,9,10\}$, ${\bf n}=\{4,6,10\}$ and ${\bf {\tilde{n}}}=\{1,2,3,5,7,8,9\}$ as well as $M=10$, $m=3$ and ${\tilde{m}}=7$. The states are drawn according to the ordering given by the site number.