Calibrations and T-duality
The most important result is the following theorem: Any spinor field of the form \(\rho^{ev,od}= e^B\wedge\text{Re}([\psi_L\otimes \psi_R]^{ev,od})\) defines a calibration, where \([\psi_L\otimes \psi_R]\) is the complex form naturally associated with the bi-spinor \(\psi_L\otimes \psi_R\in\Delta\times \Delta\), for \(\Delta\) the spinor bundle associated with TM. For such a \(\rho\), the pair \((L,F)\) is calibrated if and only if \[ A(\psi_L)= c{e^{-B}\wedge \tau_{L,F}\over\| e^{-B}\wedge\tau_{L,F}\| G}\cdot\psi_R, \] where \(A\) denotes the so-called charge conjugation operator and \(c\) is a complex constant, ``\(\cdot\)'' is the usual Clifford action of TM on \(\Delta\).
scientific article