A Spin Chain for the Symmetric Product CFT_2

TL;DR

The paper develops a position-space spin-chain framework for gauge-invariant single-cycle operators in Sym^N T^4, outlining how twist fields and color permutations define spin-chain sites and impurities. It reports nontrivial tree-level mixing among impurity configurations and formulates a one-loop deformation preserving N=(4,4) SUSY, using a covering-surface map and a t_L(t) function to compute correlators. A key result is the non-renormalization of the chiral vacuum at one loop, combined with indications that one-loop impurity interactions exhibit a nearest-neighbor-like structure on the covering surface, suggesting a controlled, potentially integrable structure in this CFT2. The work provides a concrete computational framework for analyzing operator dimensions and mixing in AdS3/CFT2 and highlights important open questions about orthogonal impurity bases and higher-loop generalizations.

Abstract

We consider "gauge invariant" operators in Sym^N T^4, the symmetric product orbifold of N copies of the 2d supersymmetric sigma model with T^4 target. We discuss a spin chain representation for single-cycle operators and study their two point functions at large N. We perform systematic calculations at the orbifold point ("tree level"), where non-trivial mixing is already present, and some sample calculations to first order in the blow-up mode of the orbifold ("one loop").

Figures (10)

• Figure 1: (1) Young tableau representation of a state in the untwisted sector, ${\cal O}=\sum_{I\neq J\neq K\neq L}\,e^{i\left(\chi_I+\chi_J+\chi_K+\chi_L\right)}$. Here ${\cal O}$ has only a ${\mathcal{G}}$ part. (2) Representation of a state in a twisted sector, ${\cal O}=\sum_{I\neq J\neq K\neq L}\,e^{i\left(\chi_I+\chi_J+\chi_K+\chi_L\right)}\sigma_{(I\, J\, K\, L)}$. Here there is only an ${\mathcal{F}}$ part. (3) Representation of the state ${\cal O}=\sum_{I\neq J\neq K}\,e^{i\left(\chi_I+\chi_J+\chi_K\right)}\sigma_{(I\, J)}$. The ${\mathcal{G}}$ dressing is ${\mathcal{G}}=e^{i\chi_K}$, and the ${\mathcal{F}}$ dressing is ${\mathcal{F}}=e^{i\left(\chi_I+\chi_J\right)}$. (4) Representation of the state ${\cal O}=\sum_{I\neq J\neq K\neq L}\,e^{i\left(\chi_I+\chi_J+\chi_K+\chi_L\right)}\sigma_{(I\, J)}\sigma_{( K\, L)}$. There is only ${\mathcal{F}}$ dressing, ${\mathcal{F}} = e^{i\left(\chi_I+\chi_J+\chi_K+\chi_L\right)}$.
• Figure 2: A vertex corresponding to a seven-cycle operator ${\cal O}_7$. The red circles are color loops. We can think of the red dots as sites of a spin chain.
• Figure 3: A qualitative picture of the two point diagram. The two solid lines correspond to dressed twist field. We have taken the length of the cycles to be large and have drawn the circles corresponding to twist fields as infinite lines. The diagrams of Pakman:2009zz are obtained by identifying the vertices according to the dashed lines.
• Figure 4: The diagrams of map $b$ in the vicinity of $x=0$. The bottom diagram is the generic one, there are $n-2$ of these.
• Figure 5: The diagrams of map $a$ in the vicinity of $x=0$. The bottom diagram is the generic one, there are $n-2$ of these.