On SL(2,R) symmetry in nonlinear electrodynamics theories

Recently, it has been observed that the Noether-Gaillard-Zumino (NGZ) identity holds order by order in α′ expansion in nonlinear electrodynamics theories as Born-Infeld (BI) and Bossard-Nicolai (BN). The nonlinear electrodynamics theory that couples to an axion field is invariant under the SL(2,R) duality in all orders of α′ expansion in the Einstein frame. In this paper we show that there are the SL(2,R) invariant forms of the energy momentum tensors of axion-nonlinear electrodynamics theories in the Einstein frame. These SL(2,R) invariant structures appear in the energy momentum tensors of BI and BN theories at all orders of α′ expansion. The SL(2,R) symmetry appears in the BI and BN Lagrangians as a multiplication of Maxwell Lagrangian and a series of SL(2,R) invariant structures.