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We formulate $f(Q,C)$ gravity and cosmology. Such a construction is based onthe symmetric teleparallel geometry, but apart form the non-metricity scalar$Q$ we incorporate in the Lagrangian the boundary term $C$ of its differenceform the standard Levi-Civita Ricci scalar $\mathring R$. We extract thegeneral metric and affine connection field equations, we apply them at acosmological framework, and adopting three different types of symmetricteleparallel affine connections we obtain the modified Friedmann equations. Aswe show, we acquire an effective dark-energy sector of geometrical origin,which can lead to interesting cosmological phenomenology. Additionally, we mayobtain an effective interaction between matter and dark energy. Finally,examining a specific model, we show that we can obtain the usual thermalhistory of the universe, with the sequence of matter and dark-energy epochs,while the effective dark-energy equation-of-state parameter can bequintessence-like, phantom-like, or cross the phantom-divide during evolution.

Non-metricity with bounday terms: $f(Q,C)$ gravity and cosmology