would be very interesting to do so.

### Theories with cylindrical ends

The second way we can enlarge the space of SQFTs which admit some kind of elliptic genus is by considering the SQFT analogue of sigma models on manifolds with an asymptotic boundary region which approaches ${{\mathds}R}^+ \times B$ for some compact manifold $B$. We can formalize this notion by requiring the SQFT ${\mathcal F}$ to be equipped with a local operator $\Phi$ with the following property:

-   Consider the direct product of ${\mathcal F}$ and a free Fermi multiplet, i.e. a free chiral fermion $\lambda$ annihilated by the supercharge.[^12]

-   Deform the product theory by a “fermionic superpotential” $\lambda (\Phi-p)$, i.e. add the terms $\lambda (\bar G_0 \Phi) - (\Phi-p)^2$ to the Lagrangian, where $p \in {{\mathds}R}$ is a parameter.

-   The result is a family of SQFTs ${\mathcal B}_p$ parameterized by a point $p$ in ${{\mathds}R}$. We require ${\mathcal B}_p$ to stabilize to some compact SQFT ${\mathcal B}$ for large positive $p$ and ${\mathcal B}_p$ to spontaneously break supersymmetry for large negative $p$.

Note that in particular the family ${\mathcal B}_p$ built from ${\mathcal F}$ is a nullhomotopy of the compact SQFT ${\mathcal B}$. Conversely, any nullhomotopy 