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Long strings and chiral primaries in the hybrid formalism

Lorenz Eberhardt, Kevin Ferreira

TL;DR

The paper investigates AdS$_3$ string backgrounds with mixed NS-NS and RR flux, focusing on long strings and missing chiral primaries that appear on the pure NS-NS locus. Using the hybrid formalism and a detailed algebraic analysis of the PSU$(1,1|2)$ worldsheet CFT, it shows that continuous representations acquire non-unitary (imaginary) conformal weights away from the singular locus, removing long strings, while unitarity constraints loosen and previously missing chiral primaries reemerge when infinitesimally perturbing away from pure NS-NS flux. The results connect worldsheet representation theory to moduli-space physics (Higgs/C Coulomb tubes) and provide a no-ghost theorem-based perspective (R-sector) that clarifies the Maldacena–Ooguri bound. Together, these findings explain how the spectrum reorganizes across the singular locus, with potential implications for protected quantities and wall-crossing in the elliptic genus and related indices.

Abstract

We revisit two related phenomena in $\mathrm{AdS}_3$ string theory backgrounds. At pure NS-NS flux, the spectrum contains a continuum of long strings which can escape to the boundary of $\mathrm{AdS}_3$ at a finite cost of energy. Related to this are certain gaps in the BPS spectrum one computes from the RNS worldsheet description. One expects that both these effects disappear when perturbing slightly away from the pure NS-NS flux background. We employ the hybrid formalism for mixed flux backgrounds to demonstrate directly from the worldsheet that this is indeed the case.

Long strings and chiral primaries in the hybrid formalism

TL;DR

The paper investigates AdS string backgrounds with mixed NS-NS and RR flux, focusing on long strings and missing chiral primaries that appear on the pure NS-NS locus. Using the hybrid formalism and a detailed algebraic analysis of the PSU worldsheet CFT, it shows that continuous representations acquire non-unitary (imaginary) conformal weights away from the singular locus, removing long strings, while unitarity constraints loosen and previously missing chiral primaries reemerge when infinitesimally perturbing away from pure NS-NS flux. The results connect worldsheet representation theory to moduli-space physics (Higgs/C Coulomb tubes) and provide a no-ghost theorem-based perspective (R-sector) that clarifies the Maldacena–Ooguri bound. Together, these findings explain how the spectrum reorganizes across the singular locus, with potential implications for protected quantities and wall-crossing in the elliptic genus and related indices.

Abstract

We revisit two related phenomena in string theory backgrounds. At pure NS-NS flux, the spectrum contains a continuum of long strings which can escape to the boundary of at a finite cost of energy. Related to this are certain gaps in the BPS spectrum one computes from the RNS worldsheet description. One expects that both these effects disappear when perturbing slightly away from the pure NS-NS flux background. We employ the hybrid formalism for mixed flux backgrounds to demonstrate directly from the worldsheet that this is indeed the case.

Paper Structure

This paper contains 16 sections, 62 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The sigma-model description
  3. The D-brane setup
  4. The gauge theory description
  5. The worldsheet description
  6. The hybrid formalism
  7. The sigma-model on $\mathrm{PSU}(1,1|2)$
  8. The spectrum of the sigma-model
  9. The spectrum at the first level
  10. A unitarity bound
  11. Continuous representations
  12. Missing chiral primaries
  13. Discussion
  14. The (affine) Lie superalgebra $\boldsymbol{\mathfrak{psu}(1,1|2)}$
  15. The spectrum at the $\boldsymbol{n}$-th level
  16. ...and 1 more sections

Figures (2)

  • Figure 1: The structure of the moduli space on the singular locus and when slightly perturbed away from it. On the singular locus (left-hand picture), chiral primaries can escape to the Coulomb branch and are emitted as D1-branes from the system. The Higgs branch and the Coulomb branch are connected by an infinitely long tube, with the string coupling constant blowing up in the middle. Associated with the tube are long strings, which give rise to a continuum in the spectrum. When slightly perturbing the system away from the singular locus (right-hand picture), the moduli space approximation becomes good and the non-renormalization theorem makes the Higgs branch flat. The tube disappears and all chiral primaries are confined to the Higgs branch.
  • Figure 2: The two branches of the norm of $| \Psi \rangle$. At the WZW-point, the two branches intersect at $\ell=\tfrac{k-2}{2}$. For a slight perturbation away from the WZW-point, we have an 'avoided crossing' and the first branch has always positive norm.