Symmetric Khovanov--Rozansky link homologies

We provide a finite dimensional categorification of the symmetric evaluation of ${\mathfrak{s}\mathfrak{l}}_N$-webs using foam technology. As an output we obtain a symmetric link homology theory categorifying the link invariant associated to symmetric powers of the standard representation of ${\mathfrak{s}\mathfrak{l}}_N$. In addition, the construction is actually made in an equivariant setting. We prove also that there is a spectral sequence from the Khovanov-Rozansky triply graded link homology to the symmetric one and provide along the way a foam interpretation of Soergel bimodules.