Space-time transformations of the Born-Infeld gauge field of a D-brane

Recently Gliozzi has shown that, under certain conditions, it is possible to derive the Dirac-Born-Infeld action for an abelian gauge field of a D-brane. Also, the action turns out to be invariant with respect to a non-linear realization of the full Poincar\'e group. A crucial role is played by the transformation properties of the gauge field under the non-linear realization. The aim of this note is to point out that these transformation properties are derived directly from the gauge fixing of the diffeomorphisms of the brane and the necessary compensating reparametrization when one performs a Lorentz transformation.


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In a very recent paper [1] Gliozzi has shown that it is possible to derive the Dirac-Born-Infeld (DBI) action for a gauge field lying on a d-dimensional D-brane, by studying the non-linear representation of the Poincaré group in a D-dimensional Minkowski space in terms of the D − d transverse coordinates of the brane. A crucial point of the derivation lies in the transformation properties of the brane gauge field A a , (a = 0, 1, · · · , d − 1).
According to the author of reference [1]: "Actually it appears to be no specific guiding principle (for deriving the transformations), nevertheless a trial and error method produced a surprisingly simple solution". Then, the author shows that these properties generate a non-linear realization of the Lorentz group on the gauge field. The aim of this brief note is to observe that the transformation properties of the gauge field follow simply from the requirement of a gauge compensating transformation, necessary when one fixes the gauge identifying the longitudinal coordinates with the brane parameters. Notice that the gauge compensating transformation is also required in order to get the correct transformation properties for the transverse coordinates (eq. (1) of [1]).
We assume also the presence on the brane of a Born-Infeld (BI) d-dimensional abelian gauge field, A α (σ β ) transforming as a scalar under the previous transformation On the other hand, under local diffeomorphisms, X µ and A α should transform as a scalar and a vector respectively 3 Now we fix the gauge with respect to local diffeomorphisms by letting Notice that the index α transforms under diffeomorphisms, whereas the index a transforms under the Lorentz group. With this choice of gauge we identify the two kind of indices.
Then, in this gauge The transverse coordinates X i (x a ) describe the long wave-length fluctuations of the brane, that is the Goldstone modes associated to the breaking of the translational invariance along the transverse direction specified by the index i. Of course, also the Lorentz invariance , but this breaking does not produce any further Goldstone boson, see for example [2]. This has to do with the inverse Higgs effect, see [3] and the more recent paper [4]. The Lorentz group SO(1, D − 1) is realized non-linearly in terms of the transverse coordinates X i (x a ). We will show now how this comes about. First of all, let us notice that under transformations of SO(1, d − 1) the coordinates X a transform linearly Then, let us consider Lorentz transformations along the planes (a, i), corresponding to the broken directions. In order to stay in the gauge chosen in (4) we have to perform at the same time a compensating diffeomorphism transformation such that where ǫ a is the compensating diffeomorphism parameter, ensuring that the gauge is preserved. Now, the Lorentz transformation in the plane (a, i) must be supplemented by the compensating diffeomorphism parameterized by ǫ a . For the transverse coordinates we get 4 These transformations have been obtained, for example, in [5][6][7] and discussed in [1]. The identification of the indices a and α implies that the gauge field A a transforms linearly as a vector under SO(1, d − 1) and undergoes an induced Lorentz transformation in the plane (a, i) due to the diffeomorphism necessary to keep the gauge fixed.
This equation is the same as the equation (2) of [1]. Therefore we have shown that the transformation properties of A a have the same geometrical basis as the equation (9)  The transformations (6), (7) in [1] of the world tensors g αβ and F αβ are also understood in the same manner since they transform under diffeomorphisms as and they are invariant under transformations in the plane (a, i). Here the small round (square) parentheses stay for symmetrization (antisymmetrization). Then the gauge fixed (a, i) transformations are We would like also to stress the fact that the transformation of A a could have been obtained thinking to it as it would be a "matter field" by following the standard techniques of the non-linear representations.
This can be seen in a more clear way by considering the general transformation of X µ consisting in a translation plus a generic Lorentz transformation plus a diffeomorphism: where ρ a are the parameters of the translations. To obtain the compensating diffeomorphism, we also require the variation of X a to be zero in the gauge X a (σ) = x a , (a = 0, ..., d − 1) .

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It gives The corresponding general transformations for X i , A a are given by Note that, as expected, the gauge field transforms under longitudinal transformations.
The last second term of δA a in (16) is the rotation of A a and the last term indicates the The infinitesimal transformation (16) can be written as This transformation shows that the BI field is a covariant field of the non-linear realiza- Therefore A a should not be considered as a Goldstone of some new space-time symmetry but rather as a covariant field of the non-linear realization [8,9], of the Poincaré group in D dimensions.