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We investigate the TTÂ¯-like flows for non-linear electrodynamic theories in D(=2n)-dimensional spacetime. Our analysis is restricted to the deformation problem of the classical free action by employing the proposed TTÂ¯ operator from a simple integration technique. We show that this flow equation is compatible with TTÂ¯ deformation of a scalar field theory in D=2 and of a non-linear Born-Infeld type theory in D=4 dimensions. However, our computation discloses that this kind of TTÂ¯ flow in higher dimensions is essentially different from deformation that has been derived from the AdS/CFT interpretations. Indeed, the gravity that may be exist as a holographic dual theory of this kind of effective Born-Infeld action is not necessarily an AdS space. As an illustrative investigation in D=4, we shall also show that our construction for the TTÂ¯ operator preserves the original SL(2,R) symmetry of a non-supersymmetric Born-Infeld theory, as well as N=2 supersymmetric model. It is shown that the corresponding SL(2,R) invariant action fixes the relationship between the TTÂ¯ operator and quadratic form of the energy-momentum tensor in D=4.
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