  We prove quadratic estimates for complex perturbations of Dirac-type
operators, and thereby show that such operators have a bounded functional
calculus. As an application we show that spectral projections of the
Hodge--Dirac operator on compact manifolds depend analytically on $L_\infty$
changes in the metric. We also recover a unified proof of many results in the
Calder\'on program, including the Kato square root problem and the boundedness
of the Cauchy operator on Lipschitz curves and surfaces.
