The reduced phase space of a field theory is the space of its possible initial conditions endowed with a natural symplectic structure. An alternative to Dirac’s method, relying on natural geometric aspects of variational problems, was introduced by Kijowski and Tulczijev. This method also has the advantage of having a natural generalization in the BV context. In this talk, I will explain the method and describe some examples, focusing in particular on the tetradic version of general relativity in four dimensions.

There are two mathematical approaches that are commonly used in the construction of gravity theories : tensorial and Cartan language. It is usually said that they are completely equivalent and that the translation between them should be evident. However, as we show in this work, there are cases where a result in one side is not clearly understood in the other, because the translation is not obvious. This is the case of the Katz procedure, which being constructed in the tensorial language, allows to have a well-defined variational principle and to define finite conserved charges in general relativity. Up to now, it was not known how this method reads in Cartan language, neither how it could be generalized to more general theories (e.g., Einstein-Gauss-Bonnet and Lovelock gravity). In this work we use the Chern-Weil theorem and a topological-algebraic structure called transgression form to provide the translation of the Katz boundary term into the Cartan language. As a consequence, this gives us a guideline to make the generalization of the Katz procedure for a generic Lovelock gravity.

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Strongly coupled theories exhibiting walking dynamics provide a scenario for beyond the Standard Model physics, in which electro-weak symmetry is broken dynamically and the large hierarchy between the electro-weak and Planck energy scales is naturally generated. In this talk, we discuss the construction of walking and multi-scale theories within the context of gauge-gravity duality, focusing in particular on the circumstances under which a parametrically light composite state, the dilaton, is present in the spectrum. The examples we provide come both from bottom-up models as well as top-down models embedded in M-theory and type-IIB supergravity, and include the baryonic branch of the Klebanov-Strassler field theory, as well as an example of a three-dimensional field theory with a strongly coupled IR fixed point.

We discuss the cosmological production and the successive evolution of the electroweak monopole in the standard model, and estimate the remnant monopole density at present universe. We con-firm that, although the electroweak phase transition is of the fi-rst order, it is very mildly fi-rst order.
So, the monopole production arises from the thermal fluctuations of the Higgs -field after the phase transition, not the vacuum bubble collisions during the phase transition. Moreover, while the monopoles are produced copiously around the Ginzburg temperature T_G \simeq 59.6 TeV, most of
them are annihilated as soon as created. This annihilation process continues very long, untill the temperature cools down to about 29.5 MeV. As the result the remnant monopole density in the present universe becomes very small, of 10^-􀀀11 of the critical density, too small to a-ect the standard cosmology and too small comprise a major component of dark matter. We discuss the
physical implications of our results on the ongoing monopole detection experiments, in particular on MoEDAL, IceCube, ANTARES, and Auger.