The role of the boundary term in f(Q, B) symmetric teleparallel gravity

In the framework of metric-affine gravity, we consider the role of the boundary term in Symmetric Teleparallel Gravity assuming f(Q, B) models where f is a smooth function of the non-metricity scalar Q and the related boundary term B. Starting from a variational approach, we derive the field equations and compare them with respect to those of f(Q) gravity in the limit of B → 0. It is possible to show that f(Q, B) = f(Q-B) models are dynamically equivalent to f(R) gravity as in the case of teleparallel $f(\tilde{B}-T)$ gravity (where $B\ne \tilde{B}$). Furthermore, conservation laws are derived. In this perspective, considering boundary terms in f(Q) gravity represents the last ingredient towards the Extended Geometric Trinity of Gravity, where f(R), $f(T,\tilde{B})$, and f(Q, B) can be dealt under the same standard. In this perspective, we discuss also the Gibbons-Hawking-York boundary term of General Relativity comparing it with B in f(Q, B) gravity.