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$\mathcal{SW}$-algebras and strings with torsion

Xenia de la Ossa, Mateo Galdeano, Enrico Marchetto

TL;DR

The paper establishes a concrete link between worldsheet SW-algebras and target-space $G$-structures with torsion in string backgrounds, focusing on NS flux $H$ and $ abla^{+}$-compatible geometries. By comparing classical Noether currents from a $(1,0)$ sigma-model with the quantum SW-algebra data, it shows how scalar torsion classes deform specific OPE coefficients at leading order in the string length $\ell_s$, thereby connecting geometric torsion to quantum symmetry data. It systematically analyzes $O(d-n)$, Spin(7), $G_2$, SU(2), and SU(3) structures, recovering known special-holonomy algebras in the torsion-free limit and uncovering one-parameter or infinitesimal torsion deformations that encode scalar torsion information. The approach provides a geometric interpretation of SW-algebra couplings and identifies null fields as essential tools to close deformations, offering a framework to study holographic and non-Kähler backgrounds and suggesting directions for higher-order corrections and gauge-bundle interactions. Overall, the work clarifies how intrinsic torsion classes, especially scalar ones, govern the worldsheet symmetry algebras of flux backgrounds and lays groundwork for further exploration of nontrivial geometries in string theory.

Abstract

We explore the connection between super $\mathcal{W}$-algebras ($\mathcal{SW}$-algebras) and $\mathrm{G}$-structures with torsion. The former are realised as symmetry algebras of strings with $\mathcal{N}=(1,0)$ supersymmetry on the worldsheet, while the latter are associated with generic string backgrounds with non-trivial Neveu-Schwarz flux $H$. In particular, we focus on manifolds featuring $\mathrm{Spin}(7)$, $\mathrm{G}_2$, $\mathrm{SU}(2)$, and $\mathrm{SU}(3)$-structures. We compare the full quantum algebras with their classical limits, obtained by studying the commutators of superconformal and $\mathcal{W}$-symmetry transformations, which preserve the action of the $(1,0)$ non-linear $σ$-model. We show that, at first order in the string length scale $\ell_s$, the torsion deforms some of the OPE coefficients corresponding to special holonomy through a scalar torsion class.

$\mathcal{SW}$-algebras and strings with torsion

TL;DR

The paper establishes a concrete link between worldsheet SW-algebras and target-space -structures with torsion in string backgrounds, focusing on NS flux and -compatible geometries. By comparing classical Noether currents from a sigma-model with the quantum SW-algebra data, it shows how scalar torsion classes deform specific OPE coefficients at leading order in the string length , thereby connecting geometric torsion to quantum symmetry data. It systematically analyzes , Spin(7), , SU(2), and SU(3) structures, recovering known special-holonomy algebras in the torsion-free limit and uncovering one-parameter or infinitesimal torsion deformations that encode scalar torsion information. The approach provides a geometric interpretation of SW-algebra couplings and identifies null fields as essential tools to close deformations, offering a framework to study holographic and non-Kähler backgrounds and suggesting directions for higher-order corrections and gauge-bundle interactions. Overall, the work clarifies how intrinsic torsion classes, especially scalar ones, govern the worldsheet symmetry algebras of flux backgrounds and lays groundwork for further exploration of nontrivial geometries in string theory.

Abstract

We explore the connection between super -algebras (-algebras) and -structures with torsion. The former are realised as symmetry algebras of strings with supersymmetry on the worldsheet, while the latter are associated with generic string backgrounds with non-trivial Neveu-Schwarz flux . In particular, we focus on manifolds featuring , , , and -structures. We compare the full quantum algebras with their classical limits, obtained by studying the commutators of superconformal and -symmetry transformations, which preserve the action of the non-linear -model. We show that, at first order in the string length scale , the torsion deforms some of the OPE coefficients corresponding to special holonomy through a scalar torsion class.

Paper Structure

This paper contains 51 sections, 200 equations, 1 figure, 6 tables.

Table of Contents

  1. Introduction
  2. Differential forms conventions
  3. Review of G-structures with torsion
  4. Review of SW-algebras
  5. Classical OPEs from W-symmetries
  6. The (1,0) non-linear sigma-model
  7. Symmetries and classical currents
  8. Superconformal symmetry.
  9. $\mathcal{W}$-symmetry.
  10. From classical commutators to OPEs
  11. $[\delta^{\mathcal{T}}, \delta^{\mathcal{T}}]$: two superconformal transformations.
  12. $[\delta^{\mathcal{T}}, \delta^{\Phi}]$: superconformal and $\mathcal{W}$-transformations.
  13. $[\delta^{\Phi}, \delta^{\Psi}]$: two $\mathcal{W}$-transformations.
  14. $\mathcal{T}_{\text{cl.}} \times \mathcal{T}_{\text{cl.}}$ OPE.
  15. $\mathcal{T}_{\text{cl.}}^{\space} \times \mathcal{J}_{\text{cl.}}^{\Phi}$ OPE.
  16. ...and 36 more sections

Figures (1)

  • Figure 1: Triangle of relationships between the geometry of the string background, the $\mathcal{SW}$-algebra of the worldsheet CFT and the non-linear $\sigma$-model. The dashed line represents the correspondence we want to study, which can be worked out explicitly (perturbatively in the string length scale $\ell_s$) at the level of the classical non-linear $\sigma$-model.