# On conformal powers of the Dirac operator on spin manifolds

## Matthias Fischmann

Address: Eduard Čech Institute, Mathematical Institute of Charles University, Sokolovská 83, Praha 8 - Karlín, Czech Republic

E-mail: fischmann@karlin.mff.cuni.cz

Abstract: The well known conformal covariance of the Dirac operator acting on spinor fields does not extend to its powers in general. For odd powers of the Dirac operator we derive an algorithmic construction in terms of associated tractor bundles computing correction terms in order to achieve conformal covariance. These operators turn out to be formally (anti-) self-adjoint. Working out this algorithm we recover explicit formula for the conformal third and present a conformal fifth power of the Dirac operator. Finally, we will present polynomial structures for the first examples of conformal powers in terms of first order differential operators acting on the spinor bundle.

AMSclassification: primary 53A30; secondary 53C27.

Keywords: conformal and spin geometry, conformal powers of the Dirac operator, conformal covariance, tractor bundle, tractor D-operator.

DOI: 10.5817/AM2014-4-237