Generalized geometry, calibrations and supersymmetry in diverse dimensions

TL;DR

This paper explores the relationship between supersymmetry equations and D-brane calibrations in diverse dimensions, revealing a periodic pattern and invariance under mirror symmetry, with explicit solutions for certain cases.

Contribution

It establishes a one-to-one correspondence between supersymmetry equations and D-brane calibrations across multiple dimensions, generalizing known results and listing calibration forms.

Findings

Confirmed the correspondence in d=6 by explicit solutions.

Identified a (mod 4) periodicity in calibration forms.

Showed internal manifolds are generalized Calabi-Yau under pure-spinor equations.

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.