1 99 8 Supersymmetric Non - Abelian Born - Infeld Theory

Using the natural curvature invariants as building blocks in a superfield construction, we show that the use of a symmetric trace is mandatory if one is to reproduce the square root structure of the non-Abelian Dirac-Born-Infeld Lagrangian in the bosonic sector. We also discuss the BPS relations in connection with our supersymmetry construction. UNLP-FOMEC Investigador CICBA CONICET 1 Dirac-Born-Infeld (DBI) type actions arise in the study of low-energy dynamics of D-branes [1]-[5] (see [6]-[7] for a complete list of references). In the case of superstring theory, one has to deal with a supersymmetric extension of DBI actions and, when a number of D-branes coincide, there is a symmetry enhancement [8] and the Abelian DBI action should be generalized to the non-Abelian case. Several possibilities for extending the Abelian Born-Infeld action to the case of a non-Abelian gauge symmetry have been discussed in the literature [9]-[17]. Basically, they differ in the way the group trace operation is defined. In the string context, a symmetric trace operation as that advocated by Tseytlin [12] seems to be the appropriate one. Among its advantages, one can mention: (i) It eliminates unwelcome odd powers of the curvature, this implying that the field strength F (although possibly large) should be slow varying since F 3 ∼ [D,D]F . With this kind of Abelian approximation (it implies commuting F s) one can make contact with the tree level open string effective action. (ii) It is the only one leading to an action which is linearized by BPS conditions and to equations of motion which coincide with those arising by imposing the vanishing of the β-function for background fields in the open superstring theory [13]-[15]. It should be mentioned, however, that there are some unsolved problems concerning the use of a symmetric trace for the non-Abelian Born-Infeld action. In particular, some discrepancies between the results arising from a symmetrized non-Abelian Born Infeld theory and the spectrum to be expected in brane theories are pointed out in ref. [16]. As noticed in [15], the fact that the symmetric trace is singled out as that leading to BPS relations should be connected with the possibility of supersymmetrizing the Born-Infeld theory. In this respect, we construct in this work the supersymmetric version of the non-Abelian Dirac-Born-Infeld action and discuss the trace issue and the Bogomol’nyi structure of the resulting bosonic sector. Our analysis, close to that developed in [18], extends to the non-Abelian case the results presented for the Abelian Supersymmetric Born-Infeld theory in ref. [19]. As it is well-known, the basic object for constructing supersymmetric