The reduced phase space can be considered as the fundamental building block of the analysis of field theories on manifolds with boundary. If the boundary is a Cauchy surface, we can define it to be the space of possible initial conditions. It can be constructed using the (KT) construction. This construction roughly goes as follows: a space of boundary fields together with a closed two-form and some constraint functions are derived from the variation of the action and the Euler-Lagrange equations in the bulk. Then, if the two-form is degenerate and its kernel is regular, it is possible to perform a quotient and obtain a symplectic space, which is called the . On it we define the constraints of the theory deriving them in a suitable way from the Euler-Lagrange equations. One of the reason of the choice of the KT construction is that it is automatically compatible with the cohomological description of the reduced phase space given by the (after Batalin-Fradkin-Vilkovisky). Indeed, if the constraints form a first class system (meaning that the Poisson brackets between them are proportional to the constraints themselves), it is possible to describe the space of functions over the reduced phase space as the zeroth cohomology of a cohomological (i.e., odd and squaring to zero) vector field on a graded manifold constructed out of the geometric phase space and the constraints.

• G. Canepa, A. S. Cattaneo, and M. Schiavina. Boundary structure of general relativity in tetrad variables . Advances in Theoretical and Mathematical Physics 25.2 (2021), pp. 327–377. doi: 10.4310/atmp.2021.v25.n2.a3.
• Giovanni Canepa, Alberto S. Cattaneo, and Manuel Tecchiolli. Gravitational Constraints on a Lightlike boundary. Annales Henri Poincaré (Mar. 2021). doi: 10.1007/s00023-021-01038-z.
• Giovanni Canepa, Alberto S. Cattaneo, and Filippo Fila-Robattino. Boundary structure of gauge and matter fields coupled to gravity (2022). doi: 10.48550/ARXIV.2206.14680.
• Giovanni Canepa, Alberto S. Cattaneo, Filippo Fila-Robattino, and Manuel Tecchiolli. Boundary structure of the standard model coupled to gravity (2023). doi: 10.48550/arXiv.2307.14955.

The is a development of the Faddeev and Popov and BRST constructions and was introduced in 1981 by Batalin and Vilkovisky. Its aim is to solve that arise in gauge theories. In particular, using the path integral quantization, it is not possible to apply to a degenerate Lagrangian the stationary phase formula. Hence this formalism aims at circumventing this problem and allows to define nonetheless the perturbative functional integral through the use of Feynman diagrams. The proposed solution is to work on a graded setup and recover the physically relevant quantities by means of an appropriate cohomology. The , after Batalin, Fradkin and Vilkovisky, is the counterpart of the BV formalism for constrained Hamiltonian systems. In recent years, these two approaches have been linked together to encompass the formulation of field theories with symmetries on manifolds with boundaries and to encode locality in quantum field theory. This combination which goes under the name of , and has been proposed by Cattaneo, Mnev and Reshetikhin. It comes with a novel quantisation scheme, compatible with cutting and gluing on manifolds with boundaries. The classical BV-BFV scheme can be further extended to include theories defined on stratified manifolds with higher codimension strata.

• G. Canepa and M. Schiavina. Fully extended BV-BFV description of General Relativity in three dimensions . Advances in Theoretical and Mathematical Physics 26.3 (2022), pp. 595–642. doi: 10.4310/atmp.2022.v26.n3.a2.
• Giovanni Canepa, Alberto S. Cattaneo, and Michele Schiavina. General Relativity and the AKSZ construction. Communications in Mathematical Physics (Aug. 2021). doi: 10.1007/s00220-021-04127-6.

If we consider a stratified manifold, i.e. roughly speaking a manifold with boundaries, corners, etc., it is possible to associate to a stratum of codimension k a suitable BF^kV theory, together with some compatibility condition between the data of the strata. Such structures are called . In some cases it is possible to induce the BF^kV data of the codimension k stratum from the BF^(k-1)V data of the codimension k-1 stratum. In codimension 1, BFV theories encode the structure of the reduced phase space of the theory at hand. The higher codimension-2 data correspond to a . Upon the choice of a polarization, this structure induces a a structure (this stands for Poisson structure up to coherent homotopies).

• Giovanni Canepa and Alberto S. Cattaneo. Corner Structure of Four-Dimensional General Relativity in the Coframe Formalism (2022). Annales Henri Poincaré. doi: 10.1007/s00023-023-01360-8.

The BV-BFV formalism and functorial topological quantum field theory are both defined on cobordisms. The precise connection between the two theories is roughly built up in three steps. As showed by Costello and Gwilliam, the dual analogue of BV (or, in fact, BF^kV theories) at a classical and quantum level are factorization algebras with value in chain complexes, where the chain operator is constructed out of the differential graded operator Q associated to a BV (or BF^kV) theory. Such factorization algebras are usually locally constant, i.e. satisfying some local to global property. This property is crucial for the second step, since it has also been proved by Lurie that there is a correspondence between a locally constant factorization algebras and En-algebras. As a third step, it is then possible to use factorization homology to construct a functor building a TQFT.

• Giovanni Canepa, Nils Carqueville and Ödül Tetik. Work in Progress.

In general relativity, an IDEAL (Intrinsic, Deductive, Explicit, ALgorithmic) characterization of a reference spacetime metric g 0 consists of a set of tensorial equations T[g]  =  0, constructed covariantly out of the metric g, its Riemann curvature and their derivatives, that are satisfied if and only if g is locally isometric to the reference spacetime metric g 0 . The same notion can be extended to also include scalar or tensor fields, where the equations T[g, φ] = 0 are allowed to also depend on the extra fields φ. The first IDEAL characterization of cosmological FLRW spacetimes, with and without a dynamical scalar (inflaton) field, has been given.

• Giovanni Canepa, Claudio Dappiaggi, and Igor Khavkine. IDEAL characterization of isometry classes of FLRW and inflationary spacetimes. Classical and Quantum Gravity 35.3 (Jan. 2018), p. 035013. issn: 1361-6382. doi: 10.1088/1361-6382/aa9f61.