Since present Monte Carlo algorithms for lattice QCD may become trapped in a fixed topological charge sector, it is important to understand the effect of calculating at fixed topology. In this work, we show that although the restriction to a fixed topological sector becomes irrelevant in the infinite volume limit, it gives rise to characteristic finite size effects due to contributions from all $theta$-vacua. We calculate these effects and show how to extract physical results from numerical data obtained at fixed topology.
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