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Superstrings with Intrinsic Torsion

Jerome P. Gauntlett, Dario Martelli, Daniel Waldram

TL;DR

This paper delivers a systematic classification of static supersymmetric bosonic backgrounds in the NS-NS sector of type II and type I/heterotic string theories by recasting supersymmetry conditions in terms of $G$-structure intrinsic torsion. It demonstrates that the three-form flux $H$ is naturally described by generalised calibrations, corresponding to fivebranes wrapping calibrated cycles, and shows how fibred, higher-dimensional geometries arise from backreacted wrapped branes via Abelian generalised instantons. The results unify canonical geometries with their nine-dimensional fibred extensions, and provide explicit examples in $d=6$ and $d=5$, including ${\cal N}=1,2,3$ type II cases and heterotic compactifications on fibrations over $K3$ or $CY$ manifolds. These insights yield a coherent framework for constructing and analysing new supersymmetric backgrounds, with potential implications for holography and CFT descriptions of wrapped branes. The approach also clarifies vanishing theorems for compact backgrounds and sets the stage for incorporating RR fields and Lorentzian generalisations.

Abstract

We systematically analyse the necessary and sufficient conditions for the preservation of supersymmetry for bosonic geometries of the form R^{1,9-d} \times M_d, in the common NS-NS sector of type II string theory and also type I/heterotic string theory. The results are phrased in terms of the intrinsic torsion of G-structures and provide a comprehensive classification of static supersymmetric backgrounds in these theories. Generalised calibrations naturally appear since the geometries always admit NS or type I/heterotic fivebranes wrapping calibrated cycles. Some new solutions are presented. In particular we find d=6 examples with a fibred structure which preserve N=1,2,3 supersymmetry in type II and include compact type I/heterotic geometries.

Superstrings with Intrinsic Torsion

TL;DR

This paper delivers a systematic classification of static supersymmetric bosonic backgrounds in the NS-NS sector of type II and type I/heterotic string theories by recasting supersymmetry conditions in terms of -structure intrinsic torsion. It demonstrates that the three-form flux is naturally described by generalised calibrations, corresponding to fivebranes wrapping calibrated cycles, and shows how fibred, higher-dimensional geometries arise from backreacted wrapped branes via Abelian generalised instantons. The results unify canonical geometries with their nine-dimensional fibred extensions, and provide explicit examples in and , including type II cases and heterotic compactifications on fibrations over or manifolds. These insights yield a coherent framework for constructing and analysing new supersymmetric backgrounds, with potential implications for holography and CFT descriptions of wrapped branes. The approach also clarifies vanishing theorems for compact backgrounds and sets the stage for incorporating RR fields and Lorentzian generalisations.

Abstract

We systematically analyse the necessary and sufficient conditions for the preservation of supersymmetry for bosonic geometries of the form R^{1,9-d} \times M_d, in the common NS-NS sector of type II string theory and also type I/heterotic string theory. The results are phrased in terms of the intrinsic torsion of G-structures and provide a comprehensive classification of static supersymmetric backgrounds in these theories. Generalised calibrations naturally appear since the geometries always admit NS or type I/heterotic fivebranes wrapping calibrated cycles. Some new solutions are presented. In particular we find d=6 examples with a fibred structure which preserve N=1,2,3 supersymmetry in type II and include compact type I/heterotic geometries.

Paper Structure

This paper contains 13 sections, 199 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. $G$-structures in canonical dimension
  3. Geometries with $\epsilon^+$ Killing spinors in canonical dimension
  4. General geometries with $\epsilon^+$ Killing spinors
  5. Single Killing spinor: $\mathop{\mathrm{\mathit{Spin}}}\nolimits(7)$-structure in $d=9$
  6. Multiple Killing spinors
  7. Explicit examples I
  8. Geometries with both $\epsilon^+$ and $\epsilon^-$ Killing spinors
  9. Explicit examples II
  10. Discussion
  11. Equations of motion
  12. Spinor and $G$-structure conventions
  13. Almost product structures

Figures (1)

  • Figure 1: Special holonomies of manifolds in $d$-dimensions with covariantly constant spinors with respect to either the Levi-Civita connection or a connection with totally anti-symmetric torsion $H$. Only the minimal "canonical" dimension $d$ is presented. The arrows represent the different ways the groups can be embedded in each other.