I will present work done in collaboration with David Holmes. 

We construct a derived pushforward of the rth root of the universal line bundle over the Picard stack of genus g prestable curves carrying a line bundle. We prove a number of basic properties, and give a formula in terms of standard tautological generators. After pullback, our formula recovers formulae of Mumford, of Pagani-Ricolfi-van Zelm and my Grothendieck-Riemann-Roch formula for r-spin curves. We apply these constructions to prove a conjecture expressing the double ramification cycle and several generalizations.