determined by the ’t Hooft anomaly coefficient for the $U(1)$ global symmetry.

Such a flavoured SQFT can be considered “compact” as long as the Hamiltonian on the circle has sufficiently discrete spectrum for a generic choice of flat $U(1)$ connection on the circle. Then the Witten genus will be well-defined as a meromorphic Jacobi form. If the SQFT is compact even in the un-flavoured sense then the Witten genus will be a holomorphic Jacobi form and admit an expansion in terms of theta functions of index $\ell$, with coefficients which form a vector-valued modular form.

For a sigma model, the calculation of the flavoured Witten genus will involve the equivariant analogues of the characteristic classes involved in the calculation of the standard Witten genus.

The canonical example is a $(0,1)$ sigma model with target ${{\mathds}R}^2$ and a $U(1)$ isometry acting as rotations of ${{\mathds}R}^2$. The corresponding flavoured Witten genus is a meromorphic  Jacobi form of weight $1$ and index $-1$:

$$\label{eqn.R2}
Z_{RR}({{\mathds}R}^2)[\xi;\tau] = \frac{\eta(\tau)^3}{\theta(\xi;\tau)} =\frac{1}{x^{\frac12}-x^{-\frac12}}\prod_{n=1}^\infty \frac{(1-q^n)^2}{(1-x q^n)(1-x^{-1} q^n)}$$

with $q = \exp 2 \pi i \tau$ and $x = \exp 2 \pi i \xi$. The analogue of TMF for the flavoured SQFTs does not appear to be well-studied [@GPPV]. It 