Ruling out a spin-glass phase is a notoriously delicate matter. The existence of a superconformal solution to the Schwinger–Dyson equations in the annealed approximation and its compatibility with the Witten index is only circumstantial evidence that our models are conformal in the IR, and a more direct proof would certainly be welcome.

At finite $N$ the parameter $\alpha$ is quantized such that $\alpha N\in \mathbb{Z}$, but at large $N$, $\alpha$ is essentially continuous.

Indeed let $\ell$ be the inverse of $q/d$ modulo $p/d$, then a rotation $\mathrm{U}(1{)}_F$ by an angle $\alpha =2\pi \ell /p$ acts as $g^d$.

When the transformed time difference ${\tau }^{\prime }(\tau )-{\tau }^{\prime }(0)$ has the opposite sign with respect to $\tau$, one needs to keep track of the extra factor $s{e}^{2\pi \mathcal{E}}$ when the operators commute past each other, see Appendix pp3-s2.

In our conventions $G(\omega )=\int d\tau e^{-i\omega \tau }G(\tau )$.

In our conventions $G(\tau )={\sum }_{n\in \mathbb{Z}+r}G(n)e^{\frac{2\pi in\tau }{\beta }}$ and $G(n)=\frac{1}{\beta }{\int }_0^{\beta }d\tau e^{-\frac{2\pi in\tau }{\beta }}G(\tau )$, where $r=0$ for bosons and $r=1/2$ for fermions.

We have neglected the terms $-p^{-1}\delta (\tau ){\partial }_{\tau }{\tilde{G}}_{\overline{\eta }\eta }(\tau )$ in ${\mathrm{\Sigma }}_{\overline{\eta }\eta }(-\tau )$ and $-p^{-n_{\varphi }}\delta (\tau ){\partial }_{\tau }{\tilde{G}}_{\overline{\varphi }\varphi }(\tau )$ in ${\mathrm{\Sigma }}_{\overline{\varphi }\varphi }(-\tau )$ since we are interested in the equations at $\tau \ne 0$.

We further require $G_{\overline{\varphi }\varphi }>0$, since the sign of the Euclidean $G_{\overline{\varphi }\varphi }$ is the sign of the Lorentzian spectral density which must be non-negative, as we discuss in Appendix app: IR and UV limits. This rules out other exponential solutions of the Schwinger–Dyson equations.

We used the rule $\mathrm{\Theta }(\tau {)}^m\mathrm{\Theta }(-\tau {)}^n\delta (\tau )=2^{-m-n}\delta (\tau )$, namely $\mathrm{\Theta }(0)=\frac{1}{2}$, which follows from using the function $\mathrm{P}\frac{1}{i\omega }+\pi \delta (\omega )$ as the Fourier transform of $\mathrm{\Theta }(\tau )$, and will be consistent with our numerical results.

This is similar to the parity anomaly in 3d theories. For a complex fermion of charge 1, the fermionic Fock space has two states, and if we insist on assigning integer charges then there is no canonical choice and one is forced to break charge conjugation. On the other hand, one could assign charges $\pm \frac{1}{2}$ to the two states in a charge-conjugation invariant fashion, but then the charges are not integer.

The fact that the ground states have a non-vanishing but well-defined R-charge (meaning that they are still eigenvectors of the R-charge operator) means that the R-symmetry is unbroken in those states.

One can sample the interval at either ${\tau }_i=i/M$ or ${\tau }_i=(i+\frac{1}{2})/M$, for $i=0,\dots ,M-1$. For sufficiently high number of points, they give identical solutions within the numerical precision, however we found the latter choice to be numerically more stable. Note that since the model depends on $\beta$ only through $\beta J$ and $\beta \mu$, one can always take $\beta =1$ for numerical purposes.

Since $M$ is even, the truncation is not symmetric around zero for the bosonic Matsubara frequencies: ${\omega }_k^{\mathrm{b}}=-\frac{\pi (M-2)}{\beta },\dots ,0,\dots ,\frac{\pi M}{\beta }$. This creates an apparent difficulty when calculating $\mathrm{\Sigma }(-{\omega }_k)$ for $k=M-1$. We fix this by assuming that the bilocal fields are real, so that $\mathrm{\Sigma }(-{\omega }_k)=\overline{\mathrm{\Sigma }}({\omega }_k)$.

The integrated chemical potential $\int d\tau A_{\tau }$ remains invariant.

When we match $\mathcal{E}$ with $\mu$ using the KMS relation, we implicitly assume that the conformal ansatz holds in the UV, which is not true in the actual model.

The Lagrangian written as ${\mathcal{L}}_{\mathrm{L}}=i{\varphi }^{†}{\partial }_t\varphi +i{\psi }^{†}{\gamma }_t\psi$ is manifestly a 1-form. When the fermion Lagrangian is written simply as ${\mathcal{L}}_{\mathrm{L}}={\psi }^{†}\psi$, the fermion $\psi$ takes values in a square root of the cotangent bundle.

The theory becomes topological if $\eta$, $f$ are an anticommuting scalar and a commuting spinor, respectively.