SL(2,R) Invariance of Non-Linear Electrodynamics Coupled to An Axion and a Dilaton

Abstract

The most general Lagrangian for non-linear electrodynamics coupled to an axion $a$ and a dilaton $\varphi$ with $SL(2,\mbox{\elevenmsb R})$ invariant equations of motion is $$ -\half\left(\nablaϕ\right)^2 - \half e^{2ϕ}\left(\nabla a\right)^2 + \fraction{1}{4}aF_{μν}\star F^{μν} + L_{\rm inv}(g_{μν},e^{-\frac{1}{2}ϕ}F_{ρσ}) $$ where $L_{\mathrm{inv}}(g_{\mu \nu },F_{\rho \sigma })$ is a Lagrangian whose equations of motion are invariant under electric-magnetic duality rotations. In particular there is a unique generalization of Born-Infeld theory admitting $SL(2,\mbox{\elevenmsb R})$ invariant equations of motion.

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