[Paper Review] Deforming Symmetric Product Orbifolds: A tale of moduli and higher spin currents

Luis Apolo, Alexandre Belin+3 more|arXiv (Cornell University)|Apr 15, 2022

Black Holes and Theoretical Physics48 references34 citations

TL;DR

This paper investigates how higher spin currents in symmetric product orbifolds of N = 2 CFTs are lifted when deformed by twisted, exactly marginal operators. Using conformal perturbation theory, it shows that such currents are universally lifted to non-chiral operators, but the magnitude of the lift depends non-universally on the seed CFT's central charge and the specific marginal operator. The results clarify the holographic transition from stringy to supergravity spectra in AdS3/CFT2.

ABSTRACT

We analyze how deforming symmetric product orbifolds of two-dimensional $\mathcal{N}=2$ conformal field theories by an exactly marginal operator lifts higher spin currents present at the orbifold point. We find on the one hand that these currents are universally lifted regardless of the underlying CFT. On the other hand the details of the lifting are surprisingly non-universal, with dependence on the central charge of the underlying CFT and the specific marginal operator in use. In the context of the AdS/CFT correspondence, our results illustrate the mechanism by which the stringy spectrum turns into a supergravity spectrum when moving through the moduli space. They also provide further evidence that symmetric product orbifolds of $\mathcal{N}=2$ minimal models are holographic.

Motivation & Objective

• To understand how higher spin currents—arising from the symmetric product structure—behave under deformations of the orbifold CFT.
• To determine whether the lifting of these currents is universal across different seed CFTs or depends on specific details such as central charge and marginal operator type.
• To clarify the mechanism by which the stringy spectrum in AdS3/CFT2 transitions to a supergravity spectrum as one moves through the moduli space.
• To provide evidence for the holographic nature of symmetric product orbifolds of N = 2 minimal models by analyzing their deformation properties.

Proposed method

• Applies conformal perturbation theory to compute the deformation of symmetric product orbifold CFTs by twisted, exactly marginal operators.
• Uses the large-N limit to analyze the behavior of correlation functions involving multi-trace operators and twisted sectors.
• Constructs permutation-invariant currents in SymN(C) from N copies of the seed CFT's N = 2 superconformal currents.
• Evaluates symmetrized correlation functions using group-theoretic techniques and root-of-unity sums, reducing to residue calculations on complex contours.
• Derives the N-dependence of normalization factors and matrix elements to isolate leading-order contributions in the large-N limit.
• Applies Ward identities and regularization techniques to handle divergences in correlation functions involving higher spin currents.

Experimental results

Research questions

• RQ1 Are higher spin currents present at the symmetric product orbifold point universally lifted when the theory is deformed by a twisted, exactly marginal operator?
• RQ2 How does the magnitude of the lifting of higher spin currents depend on the central charge of the seed CFT?
• RQ3 Does the specific choice of marginal operator in the twisted sector affect the lifting mechanism, and if so, how?
• RQ4 Can the transition from a stringy spectrum to a supergravity spectrum in AdS3/CFT2 be understood through the deformation of symmetric product orbifolds?
• RQ5 What is the role of the head length and trace structure in determining the leading-order behavior of matrix elements in the large-N limit?

Key findings

• Higher spin currents are universally lifted under deformation by twisted, exactly marginal operators, regardless of the underlying seed CFT.
• The magnitude of the lifting is non-universal and depends on the central charge of the seed CFT and the specific marginal operator used in the deformation.
• For states with different head lengths, matrix elements are subleading in 1/N, with the leading contribution coming only from head terms of the same length.
• The leading-order contribution to correlation functions scales as O(N⁻¹) for same-head-length states, while unequal head-length states contribute at O(N⁻¹/²|L₁−L₂|−¹), which is subleading.
• The computation of symmetrized correlation functions reduces to evaluating at most two residues on the complex plane, enabling exact analytic control in the large-N limit.
• The results support the holographic interpretation of symmetric product orbifolds of N = 2 minimal models, as the deformation mechanism aligns with the expected transition from stringy to supergravity spectra in AdS3.