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

TL;DR

The paper investigates how non-linear electrodynamics coupled to an axion and a dilaton can preserve SL(2,R) duality in four dimensions. It shows that if the electromagnetic sector satisfies an SO(2) duality, the combined system admits SL(2,R) invariance of the equations of motion (though not of the action), provided the Lagrangian takes a constrained form with a function tildeL of the scaled field strength. A key result is that the EM Lagrangian must be expressible as L = R - 2 Lambda - 1/2 (grad φ)^2 - 1/2 e^{2φ} (grad a)^2 + 1/4 a F star F + tildeL(barF) plus a constant, with barF = e^{-1/2 φ} F, and a duality-consistent relation between tildeL and the auxiliary fields, yielding a unique Born-Infeld–like SL(2,R) invariant extension. The invariance extends to the axion/dilaton equations and the energy-momentum tensor, and an explicit generalized Born-Infeld Lagrangian is provided; these results reveal a family of SL(2,R) invariant theories built from any SO(2) duality–invariant EM sector, distinct from previous string-theory constructions.

Abstract

The most general Lagrangian for non-linear electrodynamics coupled to an axion $a$ and a dilaton $φ$ 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 $L_{\rm inv}(g_{μν},F_{ρσ})$ 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.