nLab spectral super-scheme

Idea

The concept of spectral super-scheme is supposed to be the refinement of the concept of super-scheme as one passes to spectral geometry in the sense of derived algebraic geometry over E-infinity rings (E-infinity geometry).

Definition

The following is an argument for a good definition of spectral supergeometry. This was originally motivated from the observation in Kapranov 13 and uses results due to Rezk 09 and Sagave-Schlichtkrull 2011.

Observe that

1. $E_\infty$-geometry is already in itself a higher geometric version of $\mathbb{Z}$-graded supergeometry (in the sense discussed at geometry of physics – superalgebra).

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   At the level of homotopy groups this is the following basic fact:

   For $E$ a homotopy commutative ring spectrum, its stable homotopy groups $\pi_\bullet(E)$ inherit the structure of a $\mathbb{Z}$-graded super-commutative ring (according to this). See this proposition in the section 1-2 Homotopy commutative ring spectra of Introduction to Stable homotopy theory.

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   But more is true: the $E_\infty$-analog of the integers, $\mathbb{Z}$, is the sphere spectrum, $\mathbb{S} \,\simeq\, \Sigma^\infty S^0$, and every $E_\infty$-ring $(E, \cdot)$ is canonically $\mathbb{S}$-graded, in that (Sagave-Schlichtkrull 2011, theorem 1.7-1.8):

   on underlying $E_\infty$-spaces $E_0 \,\coloneqq\, \Omega^\infty(E)$, at least, realized as $\Omega^{\mathcal{J}}(E)$ in SaSc11 (4.4), they are canonically equipped with an $E_\infty$-monoid homomorphism

   $$(E_0, \cdot) \xrightarrow{\;} (\mathbb{S}_0, +)$$

   to the additive $E_\infty$-space underlying the sphere spectrum (traditionally denoted $Q S^0$, which is notation for a construction that yields $\Omega^\infty \Sigma^\infty S^0$).

   The $E_\infty$-monoid homomorphisms of this form are the evident homotopy-theoretic generalization of morphisms of commutative monoids to the additive group $(\mathbb{Z}, +)$ of the integers, and these are evidently equivalent to $\mathbb{Z}$-gradings on the domain monoid.

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   So $E_\infty$-geometry in itself is already a categorified/homotopified version of supergeometry, but of $\mathbb{Z}$-graded supergeometry, not of the proper $\mathbb{Z}/2$-graded supergeometry.

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   (That grading over the sphere spectrum is closely related to superalgebra had been highlighted in Kapranov 2013, but the issue of the difference between homotopified $\mathbb{Z}$-grading compared to homotopified $\mathbb{Z}/2$-grading had been left open.)

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2. But ordinary $\mathbb{Z}/2$-graded supercommutative superalgebra is equivalently $\mathbb{Z}$-graded supercommutative superalgebra over the free even periodic $\mathbb{Z}$-graded supercommutative superalgebra (this prop.).

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3. In view of the first point, the second point has an evident analog in E-∞ geometry:

   The higher/derived analog of an even periodic $\mathbb{Z}$-graded commutative algebra is an E-infinity algebra over an even periodic ring spectrum.

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   That E-infinity algebras over even periodic ring spectra are usefully regarded from the point of view of supercommutative superalgebra was highlighted in Rezk 09, section 2.

Hence it makes sense to say:

Definition. Spectral/$E_\infty$ super-geometry is simply the $E_\infty$-geometry over even periodic ring spectra.

Examples

Spectral superpoint

The ordinary superpoint over some field $k$ is the spectrum of a commutative ring of the graded symmetric algebra on a single odd generator (“graded ring of dual numbers”)

$$\mathbb{A}_k^{0 \vert 1} \;\simeq\; Spec( \,Sym_k (k[1])\, )$$

Accordingly, for $R$ an even periodic ring spectrum, then the spectral superpoint $R^{0 \vert 1}$ should be the spectral scheme given by the spectral symmetric algebra on the suspension spectrum of $R$:

$$\begin{aligned} R^{0 \vert 1} &\coloneqq Spec \left( Sym_R (\Sigma R) \right) \\ & \simeq Spec\left( R \wedge \left( \underset{n \in \mathbb{N}}{\coprod} B \Sigma(n)^{\tau_n} \right)_+ \right) \\ & \simeq Spec\left( R \wedge Sym_{\mathbb{S}}(\Sigma \mathbb{S}) \right) \end{aligned} \,.$$

where on the right we have the Thom space of the vector bundle $\tau_n$ associated to the $\Sigma(n)$-universal principal bundle via the canonical action of $\Sigma(n)$ on $\mathbb{R}^n$ (see also at symmetric group – Classifying space and Thom space).

References

The proposal for spectral super-geometry above invokes observations from

• Charles Rezk, section 2 of The congruence criterion for power operations in Morava E-theory, Homology, Homotopy and Applications, Vol. 11 (2009), No. 2, pp.327-379 (arXiv:0902.2499)

• Steffen Sagave, Christian Schlichtkrull, Diagram spaces and symmetric spectra, Advances in Mathematics, Volume 231, Issues 3–4, October–November 2012, Pages 2116–2193 (arXiv:1103.2764)

The proposal above was originally motivated from the discussion of the sphere spectrum in relation to super algebra highlighted in

• Mikhail Kapranov, Categorification of supersymmetry and stable homotopy groups of spheres, talk at Algebra, Combinatorics and Representation Theory: in memory of Andrei Zelevinsky (1953-2013) (April 2013) [pdf, video:YT]

  Abstract:. The “minimal sign skeleton” necessary to formulate the Koszul sign rule is a certain Picard category, a symmetric monoidal category with all objects and morphisms invertible. It can be seen as the free Picard category generated by one object and corresponds, by Grothendieck‘s dictionary, to the truncation of the spherical spectrum $S$ in degrees 0 and 1, so that $\{\pm 1\}$ appears as the first stable homotopy group of spheres $\pi_{n+1}(S^n)$. This suggest a “higher” or categorified versions of super-mathematics which utilize deeper structure of $S$. The first concept on this path is that of a supersymmetric monoidal category which is categorified version of the concept of a supercommutative algebra.

• Mikhail Kapranov, Supergeometry in mathematics and physics, in Gabriel Catren, Mathieu Anel, (eds.) New Spaces for Mathematics and Physics (arXiv:1512.07042)

• Mikhail Kapranov, Super-geometry, talk at New Spaces for Mathematics & Physics, IHP Paris (Oct-Sept 2015) [video recording]

A closely related suggestion later appears in:

• MathOverflow comment MO:a/40327, Sep 2021