Self-Duality of a Topologically Massive Born-Infeld Theory

We consider self-duality in a 2 + 1 dimensional gauge theory containing both the Born-Infeld and the Chern-Simons terms. We introduce a Born-Infeld inspired generalization of the Proca term and show that the corresponding model is equivalent to the Born-Infeld-Chern-Simons model. email: [email protected] email: [email protected] 1 Many years ago Townsend et. al. studied self-duality in gauge theories in 4k − 1 dimensions [1]. In particular, in 2+1 dimensions they considered the Proca equation for the massive gauge field: ∂Fμν +m Aν = 0 , (1) where Fμν = ∂μAν − ∂νAμ . As a consequence of the antisymmetry of the field strength, it follows from above that ∂μA μ = 0, and hence there are two, independent, propagating modes of equal mass. They observed that any gauge field which is proportional to the dual of it’s field strength does satisfy the above equation. In particular any gauge field which satisfy Aμ = 1 2m ǫμνρF νρ , (2) is a solution of the second order Eq. (1). They called Eq. (2) as the self-duality equation. This equation propagates one massive mode instead of two and it can be viewed as a square root of the second order Eq. (1). The self-dual Eq. (2) can be derived from the Lagrangian LP = 1 2 mAμA μ − 1 4 mǫAμFνρ . (3) It is straightforward to see that the above Lagrangian is not gauge invariant. However, interestingly it was soon observed [2] that the above model is equivalent to gauge invariant, topologically massive, electrodynamics characterized by the Lagrangian [3, 4] LM = − 1 4 FμνF μν + 1 4 mǫAμFνρ . (4) The corresponding field equation is ∂μF μν + 1 2 mǫFαβ = 0 , (5) and following [2] it is easily shown that the field Eqs. (5) and (2) are equivalent. In fact, in [2] the authors have even shown the equivalence of the two Lagrangians LP and LM as given by Eqs. (3) and (4) respectively. Long back Born and Infeld proposed [5] a nonlinear generalization to the Maxwell Lagrangian in order to cure the short distance divergence appearing in quantum 2 electrodynamics. Recently it has attracted considerable attention both in field theory, because of it’s remarkable form, as well as in string theory for it is the action which governs the gauge field dynamics of the D-branes [6]. Because of its importance in the open string theory, Gibbons and Rasheed studied various duality invariances of the Born-Infeld theory [7]. In particluar, they have shown that the SO(2) electricmagnetic duality rotation, that appears as a symmetry at the level of equations of motion in the Maxwell theory in four spacetime dimensions, also holds in the Born-Infeld theory. Because of the importance of duality in understanding various non-perturbative aspects of field theroy as well as string theory, the above results have been generalized to nonlinear theories with more then one Abelian gauge field, theories with interacting scalar fields as well as to the supersymmetric theories [8][19] . However most of the discussion about the duality invariance has been restricted to theories in four space time dimensions or more generally to the even dimensional theories. On the other hand, several interesting generalizations of the self-dual ChernSimons-Proca model [1] and its equivalence [2] with the three dimensional massive electrodynamics [3, 4] has been studied in literature. Soon after the work of Deser and Jackiw, it has been realised that the self-duality can also occur in case both the Maxwell as well as the Proca term can simultaneously be incorporated in addition to the Chern-Simons term[20]. The above model has also been used in the study of bosonization in higher dimensions [21, 22]. Recently it has been shown that there exists a unified theory [23] from which the self-dual model[1], the massive electrodynamics [3, 4] as well as the Maxwell-Chern-Simons-Proca systems [20] can be recovered as special cases. However, to the best of our knowledge, the reuslts of Deser and Jackiw have not been generalized to the Born-Infeld theory. The purpose of this note is to consider the generalization of this equivalence in case the Maxwell term is replaced by the celebrated Born-Infeld Lagrangian. Consider the Lagrangian LBI = β √ 1− 1 2β2 FμνF μν + 1 4 mǫAμFνρ . (6) Here we have ignored an irrelevant constant factor proportional to square of the Born-Infeld parameter β which does not contribute to the equation of motion. This Lagrangian reduces to the topologically massive Lagrangian as given by Eq. (4) in the limit β → ∞ when the constant factor is taken into account. The corresponding