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Action principles of the BF type for diffeomorphism invariant topological field theories living in n-dimensional spacetime manifolds are presented. Their construction is inspired by Cuesta and Montesinos' recent paper where Cartan's first and second structure equations together with first and second Bianchi identities are treated as the equations of motion for a field theory. In opposition to that paper, the current approach involves also auxiliary fields and holds for arbitrary n-dimensional spacetimes. Dirac's canonical analysis for the actions is detailedly carried out in the generic case and it is shown that these action principles define topological field theories, as mentioned. The current formalism is a generic framework to construct geometric theories with local degrees of freedom by introducing additional constraints on the various fields involved that destroy the topological character of the original theory. The latter idea is implemented in two-dimensional spacetimes where gravity coupled to matter fields is constructed out, which has indeed local excitations. .unam.mx end, four different types of auxiliary fields are introduced. The first one φ I is a (n − 2)-form that will be associated to the basis of vielbeins e I ; the second one φ IJ = −φ JI is also a (n − 2)-form that will be associated to the Lorentz connection ω IJ . The other two auxiliary fields ψ I and ψ IJ = −ψ JI are (n − 3)-forms that will be associated to the "torsion" T I and the "curvature" R IJ , respectively. Note that R I J is not the same as R I J [ω]: R IJ is a set of two-forms while R I J [ω] is the curvature of ω I J (an analog comment applies to T I , see Ref.[3] for more details). In this way, for n-dimensional spacetimes M n with n ≥ 3, the field theories studied in this paper are defined by the action principlewhere ω IJ = −ω JI is a Lorentz (or Euclidean) connection valued in the so(n − 1, 1) or so(n) Lie algebra, e I is a basis of one-forms, T I is a set of n two-forms, R IJ = −R JI is a set of n(n − 1)/2 two-forms. The indices I, J, K, . . . , are raised and lowered with the Minkowski (σ = −1) or Euclidean (σ = +1) metric (η IJ ) = diag(σ, +1, +1, . . . , +1) (see Ref. [4] for the canonical analysis of BF theory with structure group SO(3, 1), Refs. [5,6] for alternative action principles for SO(3, 1) BF theory, and Ref.[7] for the study of its symmetries). The equations of motion that follow from the variation of the action (1) with respect to the independent fields arewhere D is the covariant derivative computed with respect to the connection ω I J . In two-dimensional spacetimes M 2 , on the other hand, only Cartan's first and second structure equations are allowed because there is no room for the first and second Bianchi identities, i.e., the terms involving the ψ's in Eq.(1) are not allowed. Consequently, the natural action principle is given bywhich are totally antisymmetric in the free indices. The constraints (16) behave as constraints (10) do for BF theory, i.e., repeatedly applying the operator D a to Eqs. (16)...

In this paper we implement scale anisotropic transformations in the space-time in classical mechanics. The resulting system is consistent with the dispersion relation of gravity at a Lifshitz point recently considered in [9]. Also, we show that our model is a generalization of the conformal mechanics of Alfaro, Fubini and Furlan. For arbitrary z we construct the dynamical symmetries that correspond to the Schroedinger algebra. Furthermore, we obtain the Boltzman distribution for a gas of free particles compatible with anisotropic scaling transformations and compare our result with the corresponding thermodynamics of the recent anisotropic black branes proposed in [8].