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We consider a duality between a boson on a ring and a free fermion and showthat they have an anomaly which corresponds to the states transforming underdouble covers of O(2). There are two (in general not isometric) double coversof O(2), known as Pin$_+(2)$ and Pin$_-(2)$. These can in general bedistinguished at the group level by checking whether reflections square to $\pm1$. We show that in irreducible representations in complex Hilbert spaces thecommutation of time reversal and charge conjugation gives another method fordistinguishing Pin$_+(2)$ and Pin$_-(2)$. While we only demonstrate the dualityfor a single fermion, the anomaly is also present in any number of freefermions. For an even number of fermions we show that the two double covers ofO(2n) are isomorphic. For an odd number of fermions we show that thedistinction between irreducible representations of Pin$_+$ and Pin$_-$ canstill be detected by the commutation of time reversal and charge conjugationnamely Pin$_\pm(2n)$ will have $TCT = \pm C$.

The Free Fermion Anomaly and Representations of the Pin Groups