A diffeomorphism anomaly in every dimension

Field-theoretic pure gravitational anomalies only exist in 4k+2 dimensions. However, canonical quantization of non-field-theoretic systems may give rise to diffeomorphism anomalies in any number of dimensions. I present a simple example, where a higher-dimensional generalization of the Virasoro algebra arises upon quantization. The Polyakov action in string theory is nothing but 1+1-dimensional gravity coupled to scalar fields [2]. It thus seems reasonable to expect that canonical quantization of 3+1-dimensional gravity should proceed along similar lines. We know from string theory that quantization of the coordinate fields gives rise to a conformal anomaly, eventually to be cancelled by ghosts. Analogously, one may expect that quantization of the fields alone gives rise to an anomaly in the diffeomorphism constraint. At first sight, this idea appears to be ruled out by two no-go theorems: 1. The diffeomorphism algebra in N dimensions has no central extension, except when N = 1 [7]. 2. There are no pure gravitational anomalies in 3+1 dimensions. However, these no-go theorems can be evaded. The extension (13) that generalizes the Virasoro algebra to several dimensions is not a central one, except in one dimension. In general, it is an extension of vect(N), the algebra of vector fields (diffeomorphism algebra) in N dimensions, by its modules of closed (N −1)-forms. A closed zero-form is a constant function is the trivial module, so the Virasoro extension is central when N = 1, but not otherwise. The claim that there are no diffeomorphism anomalies in four dimensions is simply false. What is true and has been proven is that there are no fieldtheoretic diffeomorphism anomalies [1]. Non-field-theoretic diffeomorphism