Perturbed symmetric-product orbifold: first-order mixing and puzzles for integrability

TL;DR

The paper analyzes the marginal RR deformation of the symmetric-product orbifold Sym$_N(T^4)$ and computes first-order anomalous dimensions for low-lying states via conformal perturbation theory and covering-map techniques. It finds that a substantial but finite subset of states experiences $O(\lambda)$ mixing, signaling wrapping effects and challenging naive expectations from integrability. The authors perform a detailed construction of the mixing matrix, including twist-field, bosonic, and fermionic contributions, and reveal a sizable pattern of state mixing organized into $SU(2)_{\bullet}\times SU(2)_{\circ}$ multiplets, with half-BPS states protected. They also scrutinize the large-$w$ limit linked to integrability, identifying a potential flaw in prior work on off-shell states and arguing for a refined off-shell framework to properly derive the integrable S-matrix and dressing factors for the perturbed orbifold.

Abstract

We study the marginal deformation of the symmetric-product orbifold theory Sym$_N(T^4)$ which corresponds to introducing a small amount of Ramond-Ramond flux into the dual $AdS_3\times S^3\times T^4$ background. Already at first order in perturbation theory, the dimension of certain single-cycle operators is corrected, indicating that wrapping corrections from integrability must come into play earlier than expected. Our results provide a test for integrability computations from the mirror Thermodynamic Bethe Ansatz or Quantum Spectral Curve, akin to the computation of the Konishi anomalous dimension in $\mathcal{N}=4$ supersymmetric Yang--Mills theory. We also discuss a flaw in the original derivation of the integrable structure of the perturbed orbifold. Together, these observations suggest that more needs to be done to correctly identify and exploit the integrable structure of the perturbed orbifold CFT.

Figures (4)

• Figure 1: Free conformal dimensions $\Delta=h+\bar{h}$ and spins $s=h-\bar{h}$ of light ($\Delta\leq3$) states in the symmetric-product orbifold Sym$_N(T^4)$. We marked sectors with non-trivial mixing under first-order RR-deformation as orange circles. We can observe the relative scarcity in comparison with the states that are corrected only at higher orders in perturbation theory which are denoted by the blue triangles.
• Figure 2: Density of non-vanishing eigenvalues of the mixing matrix, for each of the five groups of states listed in \ref{['eq:mixingsecs']}.
• Figure 3: The asymptotic function $\mathcal{V}_\infty (p)$ as well as the $w=10$ real (circles) and imaginary parts (triangles) of appropriately rescaled $\mathcal{V}_{w}(n)$, respectively. We notice a good approximation already at relatively small twist $w$. The zeros at negative integer values are needed to cancel the poles in the propagator.
• Figure 4: Left: Structure of half-BPS operators and their definitions starting from $\Sigma^{--}_w$. Right: Dimensions $(h,\bar{h})$ (or equivalently R-charges $(j,\bar{\jmath})$) of the corresponding operators in the left figure. The indices $\dot{A}$ and $\dot{B}$ are $SU(2)_{\circ}$-indices.