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We study boundary conditions for extended topological quantum field theories(TQFTs) and their relation to topological anomalies. We introduce the notion ofTQFTs with moduli level $m$, and describe extended anomalous theories asnatural transformations of invertible field theories of this type. We show howin such a framework anomalous theories give rise naturally to homotopy fixedpoints for $n$-characters on $\infty$-groups. By using dimensional reduction onmanifolds with boundaries, we show how boundary conditions for$n+1$-dimensional TQFTs produce $n$-dimensional anomalous field theories.Finally, we analyse the case of fully extended TQFTs, and show that any fullyextended anomalous theory produces a suitable boundary condition for theanomaly field theory.

Boundary Conditions for Topological Quantum Field Theories, Anomalies and Projective Modular Functors