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Generalized geometry, calibrations and supersymmetry in diverse dimensions

Dieter Lust, Peter Patalong, Dimitrios Tsimpis

TL;DR

This work develops a unified framework linking type II flux backgrounds to D-brane calibrations via generalized geometry. By solving Killing-spinor equations and deriving pure-spinor SUSY conditions across $d=4,6,8,2$, it shows a one-to-one correspondence with generalized calibrations, proving it for $d=6$ and conjecturing it for all even $d$; a complete table of calibration forms and their codimensions emerges, revealing a mod 4 periodicity and the generalized Calabi–Yau constraint on the internal manifold. The results are manifestly invariant under generalized mirror symmetry and offer a powerful dictionary to construct new flux vacua and study brane dynamics within calibrated, supersymmetric backgrounds. Potential extensions include AdS backgrounds and a deeper exploration of the interplay between calibrations, fluxes, and Bianchi identities.

Abstract

We consider type II backgrounds of the form R^{1,d-1} x M^{10-d} for even d, preserving 2^{d/2} real supercharges; for d = 4, 6, 8 this is minimal supersymmetry in d dimensions, while for d = 2 it is N = (2,0) supersymmetry in two dimensions. For d = 6 we prove, by explicitly solving the Killing-spinor equations, that there is a one-to-one correspondence between background supersymmetry equations in pure-spinor form and D-brane generalized calibrations; this correspondence had been known to hold in the d = 4 case. Assuming the correspondence to hold for all d, we list the calibration forms for all admissible D-branes, as well as the background supersymmetry equations in pure-spinor form. We find a number of general features, including the following: The pattern of codimensions at which each calibration form appears exhibits a (mod 4) periodicity. In all cases one of the pure-spinor equations implies that the internal manifold is generalized Calabi-Yau. Our results are manifestly invariant under generalized mirror symmetry.

Generalized geometry, calibrations and supersymmetry in diverse dimensions

TL;DR

This work develops a unified framework linking type II flux backgrounds to D-brane calibrations via generalized geometry. By solving Killing-spinor equations and deriving pure-spinor SUSY conditions across , it shows a one-to-one correspondence with generalized calibrations, proving it for and conjecturing it for all even ; a complete table of calibration forms and their codimensions emerges, revealing a mod 4 periodicity and the generalized Calabi–Yau constraint on the internal manifold. The results are manifestly invariant under generalized mirror symmetry and offer a powerful dictionary to construct new flux vacua and study brane dynamics within calibrated, supersymmetric backgrounds. Potential extensions include AdS backgrounds and a deeper exploration of the interplay between calibrations, fluxes, and Bianchi identities.

Abstract

We consider type II backgrounds of the form R^{1,d-1} x M^{10-d} for even d, preserving 2^{d/2} real supercharges; for d = 4, 6, 8 this is minimal supersymmetry in d dimensions, while for d = 2 it is N = (2,0) supersymmetry in two dimensions. For d = 6 we prove, by explicitly solving the Killing-spinor equations, that there is a one-to-one correspondence between background supersymmetry equations in pure-spinor form and D-brane generalized calibrations; this correspondence had been known to hold in the d = 4 case. Assuming the correspondence to hold for all d, we list the calibration forms for all admissible D-branes, as well as the background supersymmetry equations in pure-spinor form. We find a number of general features, including the following: The pattern of codimensions at which each calibration form appears exhibits a (mod 4) periodicity. In all cases one of the pure-spinor equations implies that the internal manifold is generalized Calabi-Yau. Our results are manifestly invariant under generalized mirror symmetry.

Paper Structure

This paper contains 15 sections, 98 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Generalized geometry
  3. Supersymmetric flux backgrounds
  4. Calibrations
  5. Calibrations and supersymmetry in diverse dimensions
  6. d=4
  7. d=6
  8. d=8
  9. d=2
  10. Conclusions
  11. Spinors and gamma matrices in Euclidean spaces
  12. Explicit solution of the supersymmetry equations in $d=6$
  13. IIA
  14. IIB
  15. Explicit examples

Figures (2)

  • Figure 1: The 'periodic table' of calibration forms for all $d$, each of them corresponding to a background supersymmetry pure-spinor equation. This one-to-one correspondence had been known to hold in the $d=4$ case, and in the present paper is also shown to hold in the $d=6$ case. Based on these results we conjecture it to hold for the $d=2,8$ cases as well.
  • Figure :