A discussion that has been going recently is that supersymmetry corresponds to grading over the sphere spectrum, coming from an insight due to Kapranov.
To formalise such a statement, one needs a generalised notion of grading allowing the sphere spectrum as a possible candidate. Section 2 of Bunke–Nikolaus's Twisted differential cohomology provides such a definition: given a monoidal $\infty$-category $\mathcal{C}$, we define a $\mathcal{C}$-graded(-commutative) $\mathbb{E}_k$-ring to be a lax (symmetric) $\mathbb{E}_k$-monoidal functor $(\mathcal{C},\otimes_{\mathcal{C}},\mathbf{1}_{\mathcal{C}})\to(\mathsf{Sp},\otimes_{\mathbb{S}},\mathbb{S})$.
For $\mathcal{C}$ an ordinary category, we replace $(\mathsf{Sp},\otimes_{\mathbb{S}},\mathbb{S})$ by $(\mathrm{N}_{\bullet}(\mathsf{Ab}),\otimes_{\mathbb{Z}},\mathbb{Z})$. Now, we have a bijection between the following data:
- Lax (symmetric) $\mathbb{E}_k$-monoidal functors $(\mathcal{C},\otimes_{\mathcal{C}},\mathbf{1}_{\mathcal{C}})\to((\mathrm{N}_{\bullet}(\mathsf{Ab}),\otimes_{\mathbb{Z}},\mathbb{Z})$;
- Lax (symmetric) $\mathbb{E}_k$-monoidal functors $(\mathsf{Ho}(\mathcal{C}),\otimes_{\mathcal{C}},\mathbf{1}_{\mathcal{C}})\to(\mathsf{Ab},\otimes_{\mathbb{Z}},\mathbb{Z})$.
so in this case an $\mathbb{S}$-graded(-commutative) ordinary ring corresponds simply to a $\tau_{\leq1}\mathbb{S}$-graded ring. From the description of $\tau_{\leq1}\mathbb{S}$ as a symmetric monoidal category given here, one sees that such an object corresponds to the following definition:
A $\tau_{\leq1}\mathbb{S}$-graded ring is a pair $(R_\bullet,\{\sigma_k\}_{k\in\mathbb{Z}})$ with
- $R_\bullet$ a $\mathbb{Z}$-graded ring (corresponding to $\pi_0(\mathbb{S})\cong\mathbb{Z}$);
- $\{\sigma_k\colon R_k\to R_k\}_{k\in\mathbb{Z}}$ a family of order $2$ automorphisms (corresponding to $\pi_1(\mathbb{S})\cong\mathbb{Z}_2$);
such that, for each $k,\ell\in\mathbb{Z}$, each $a\in R_k$, and each $b\in R_\ell$, we have $$\sigma_{k+\ell}(ab)=\sigma_{k}(a)\sigma_{\ell}(b).$$ Moreover, $R$ is $\tau_{\leq1}\mathbb{S}$-graded commutative if we additionally have $$ ab = \begin{cases} ba &\text{if $\deg(a)\deg(b)$ is even,}\\ \sigma_{\deg(a)+\deg(b)}(ab) &\text{if $\deg(a)\deg(b)$ is odd} \end{cases} $$ for each $a,b\in R_\bullet$.
Similarly, a $\tau_{\leq0}\mathbb{S}\cong\mathbb{Z}_{\mathsf{disc}}$-graded (commutative) ring (corresponding to "$0$-supercommutativity") is just an ordinary (commutative) ring with a $\mathbb{Z}$-gradation. The same discussion goes through replacing $\mathsf{Ab}$ by $\mathsf{Mod}_R$, defining a notion of $\tau_{\leq1}\mathbb{S}$-graded (commutative) $R$-algebras.
Main Question. How far are $\tau_{\leq1}\mathbb{S}$-graded commutative rings from ($\mathbb{Z}$-graded) superalgebras, realising Kapranov's insight?
For instance, they contain each of the following classes of rings, all as full subcategories:
- The category of ordinary commutative rings as those $\tau_{\leq1}\mathbb{S}$-graded rings having everything in degree $0$ and such that $\sigma_k=\mathrm{id}_{R_k}$ for all $k\in\mathbb{Z}$;
- The category of $\mathbb{Z}$-graded supercommutative rings (i.e. satisfying $ab=(-1)^{\deg(a)\deg(b)}ba$) as those $\tau_{\leq1}\mathbb{S}$-graded rings with $\sigma_k(a)=-a$ for each $k\in\mathbb{Z}$ and each $a\in R_k$.
Other questions:
- Has this specific notion appeared before in the literature?
- What are interesting examples of $\tau_{\leq 1}\mathbb{S}$-graded commutative algebras that genuinely do not come from ordinary algebras or $\mathbb{Z}$-graded-commutative algebras?
