Duality-symmetric Action for Electromagnetic Theory with Electric and Magnetic Sources

For the SL(2,$\mathbf{\text{R}}$) duality-invariant generalization of Maxwell electrodynamics in the presence of both electric and magnetic sources, we formulate a local, manifestly duality-symmetric, Zwanziger-type action by introducing a pair of four-potentials $A^{\mu }$ and $B^{\mu }$ in a judicious way. On the two potentials $A^{\mu }$ and $B^{\mu }$ the SL(2,$\mathbf{\text{R}}$) duality transformation acts in a simple linear manner. In quantum theory including charged source fields, this action can be recast as a SL(2,$\mathbf{\text{Z}}$)-invariant action. Also given is a Zwanziger-type action for SL(2,$\mathbf{\text{R}}$) duality-invariant Born-Infeld electrodynamics which can be important for D-brane dynamics in string theory.