String theory duals for mass-deformed $\mathrm{SO}\mathrm{}\left(N\right)\mathrm{}$ and $\mathrm{USp}\mathrm{}\left(2N\right)\mathrm{}$ $\mathcal{N}=4\mathrm{}$ supersymmetric Yang-Mills theories

We generalize the results of Polchinski and Strassler regarding the $\mathcal{N}=1\mathrm{}$ preserving mass deformation of $\mathcal{N}=4\mathrm{}\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ supersymmetric Yang-Mills (SYM) theories and its string theory dual to $\mathrm{SO}\mathrm{}\left(N\right)\mathrm{}$ and $\mathrm{USp}\mathrm{}\left(2N\right)\mathrm{}$ gauge groups. The string theory duals involve 5-branes wrapped on ${\mathbf{RP}}^2.$ In order to match with the field theory classification of vacua, the 3-brane charge carried by such 5-branes must be shifted by a half, and this follows from a conjectured generalization of results of Freed and Witten. Our results provide an elegant physical picture for the classification of classically massive vacua in the mass-deformed $\mathcal{N}=4\mathrm{}$ theories.