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A strict monoidal category referred to as affine Brauer category$\mathcal{AB}$ is introduced over a commutative ring $\kappa$ containingmultiplicative identity $1$ and invertible element $2$. We prove that morphismspaces in $\mathcal{AB}$ are free over $\kappa$. The cyclotomic (or level $k$)Brauer category $\mathcal{CB}^f(\omega)$ is a quotient category of$\mathcal{AB}$. We prove that any morphism space in $\mathcal{CB}^f(\omega)$ isfree over $\kappa$ with maximal rank if and only if the $\mathbf u$-admissiblecondition holds in the sense of (1.30). Affine Nazarov-Wenzl algebras andcyclotomic Nazarov-Wenzl algebras will be realized as certain endomorphismalgebras in $\mathcal{AB}$ and $\mathcal{CB}^f(\omega)$, respectively. We willestablish higher Schur-Weyl duality between cyclotomic Nazarov-Wenzl algebrasand parabolic BGG categories $\mathcal O$ associated to symplectic andorthogonal Lie algebras over the complex field $\mathbb C$. This enables us touse standard arguments in [1,26,27] to compute decomposition matrices ofcyclotomic Nazarov-Wenzl algebras. The level two case was considered by Ehrigand Stroppel in [14].

Affine Brauer category and parabolic category $\mathcal O$ in types $B, C, D$