form, acted on by the real Clifford algebra $\operatorname{Cliff}(n,{{\mathds}R})$. Modules for $\operatorname{Cliff}(n,{{\mathds}R})$ represent degree-$n$ classes in oriented K-theory.

Non-compact SQFTs {#sec.noncompact}
-----------------

The discussion in §\[sec.anomaly\] implicitly assumed that the SQFT in question was “compact” — for instance, that its Hamiltonian had sufficiently discrete spectrum — in order for its elliptic genus and K-theory class to be well-defined. There are at least two distinct ways one can enlarge the space of SQFTs which admit some kind of elliptic genus.

### Flavoured-compact theories {#subsec.flavoured-compact}

The first way is to consider SQFTs equipped with a continuous global symmetry, say $U(1)$ for simplicity, and define a “flavoured” elliptic genus as a partition function on a torus equipped with a flat $U(1)$ connection. The flat connection can be parameterized by a point $\xi$ in the elliptic curve $E_\tau$ of parameter $\tau$. As long as the current sits in a standard $(0,1)$ multiplet, there is a superpartner of the anti-holomorphic part of the current which should guarantee the independence of the partition function on the anti-holomorphic part $\bar \xi$ of the connection, leaving a holomorphic dependence on $\xi$. As a consequence, the flavoured Witten genus is a Jacobi form, of weight $\frac{n}{2}$ and index $\ell$ 