super-embedding formalism

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## Idea

The "doubly supersymmetric geometric approach" ([BPSTV 95](#BPSTV95), [Howe-Sezgin 97](#HoweSezgin97)), also called the _super-embedding approach_ ([Sorokin 99](#Sorokin99)), is a formulation of super-$p$-[[brane]] [[sigma-models]] entirely within [[supergeometry]], where not only the [[target spacetime]] is taken to be a [[supermanifold]], as in [[Green-Schwarz sigma-models]], and not only the [[worldvolume]] is taken to be a supermanifold, as in the [[NSR string]], but where both are taken to be supermanifolds.

> graphics grabbed from [FSS 19c](#FSS19c)

## Properties

### $\kappa$-Symmetry as super-general covariance
 {#KappaSymmetry}

The notorious phenomenon of [[kappa-symmetry]] in [[Green-Schwarz sigma-models]] is revealed by the superembedding approach to be nothing but the odd-graded components of the [[super-diffeomorphism]] [[invariant|invariance]] on the [[worldvolume]], hence: of super-[[general covariance]] ([Sorokin-Tkach-Volkov 89](#SorokinTkachVolkov89), review includes [Sorokin 99, section 4.3](#Sorokin99), [Howe-Sezgin 04, section 4.3](#HoweSezgin04)):

If

1. $X$ denotes a [[superspacetime]] locally modeled on [[super-Minkowski spacetime]] $\mathbb{R}^{d-1,1\vert \mathbf{N}}$

1. $\Sigma$ denotes a [[supermanifold|super]]-[[worldvolume]] of a [[super p-brane]] locally modeled on [[super-Minkowski spacetime]] $\mathbb{R}^{p,1\vert \mathbf{N}/2}$

1. so that a [[sigma-model]] [[field (physics)|field]] configuration for a [[super p-brane]] of shape $\Sigma$ to propagate in $X$ is a morphism of [[supermanifolds]] of the form

   $$
     \array{
        \Sigma && \text{super-worldvolume}
        \\
        \downarrow^{\mathrlap{\phi}} &&\text{sigma-model super-field}&
        \\
        X && \text{super-spacetime}
     }
   $$

then:

1. the postcomposition [[action]] of [[spacetime]] [[super-isometries]] $X \stackrel{\simeq}{\longrightarrow} X$ is in even degree the action of [[spacetime]] [[isometries]] and in odd degree the action of **[[spacetime]] [[supersymmetry]]** on the sigma-model fields;

1. the precomposition action of [[worldvolume]] [[super-diffeomorphism]] $\Sigma \stackrel{\simeq}{\to} \Sigma$ is in even degree the action of bosonic worldvolume [[diffeomorphism]] and in odd degree the action of **$\kappa$-symmetry**:

$$
  \array{
      \Sigma  &\underoverset{\simeq}{\kappa\text{-symmetry}}{\longrightarrow}& \Sigma
      \\
      \downarrow^{\mathrlap{\phi}} && \downarrow^{\mathrlap{\phi'}}
      \\
      X &\underoverset{\simeq}{\text{spacetime supersymmetry}}{\longrightarrow}&
    X
  }
  \,.
$$

Notice here the assumption that the number of odd directions on the [[worldvolume]] is half that of the [[target spacetime]]. This is the default assumption for fundamental [[super p-branes]], and it directly reflects the statement that the corresponding [[black brane]] solutions are $1/2$ [[supergravity]] [[BPS states]].

For example, consider the embedding

$$
  \mathbb{R}^{2,1} \hookrightarrow \mathbb{R}^{10,1}
$$

of 2+1d [[Minkowski spacetime]], thought of as the [[worldvolume]] of a [[membrane]], into 11d Minkowski spacetime, linearly along the coordinate axis. Any such embedding breaks the [[isometry group]] of $\mathbb{R}^{10,1}$ from the 11d [[Poincaré group]] $Iso(10,1)$ to the [[product group]]

$$
  Iso(2,1) \times SO(8)
    \hookrightarrow
  Iso(10,1)
$$

(meaning that this [[subgroup]] is the [[stabilizer subgroup]] of the embedding).

Now consider instead [[super Minkowski spacetime]] $\mathbb{R}^{10,1\vert \mathbf{32}}$ (with $\mathbf{32}$ the irreducible [[Majorana spinor]] representation in 11), hence the local model [[superspace]] for [[super spacetimes]] in [[11-dimensional supergravity]]. We are to ask what subspace of the [[spin representation]] $\mathbf{32}$ preserves the embedding in that the [spinor bilinear pairing](Majorana+spinor#TheSpinorPairingToVectors) $\overline{\psi}_1 \Gamma \psi_2$ on that subspace lands in $\mathbb{R}^{2,1} \hookrightarrow Iso(2,1) \hookrightarrow Iso(10,1)$ ([Sorokin 99, section 5.1](#Sorokin99)). This is found to be the case for a half-dimensional subspace, and hence we may lift the above to a super-embedding of the form

$$
  \label{M2WordvolumeInSpacetime}
  \mathbb{R}^{2,1\vert 8 \otimes \mathbf{2}} \hookrightarrow \mathbb{R}^{10,1\vert \mathbf{32}}
$$

(where now $\mathbf{2}$ is the [[irrep|irreducible]] [[Majorana spinor]] representation in 3d, and $8 \otimes \mathbf{2}$ denotes the [[direct sum]] of 8 copies of it)
such that the induced [[stabilizer group|stabilizer]] [[supergroup]] inside the [[super Poincaré group ]] now is

$$
  Iso(\mathbb{R}^{2,1\vert 8 \otimes \mathbf{2}}) \times Spin(8)
    \hookrightarrow
  Iso(\mathbb{R}^{10,1\vert \mathbf{32}})
  \,.
$$

It is in this sense that the membrane "breaks exactly half the supersymmetry", namely from $\mathbf{32}$ to $8 \otimes \mathbf{2}$.

If one now thinks of this not as inclusions of global spacetimes, but of their super tangent spaces at the points where the membrane sits in spacetime, then this reflects the local structure of $\kappa$-symmetry: the $\kappa$-symmetries are locally generated by the 16 odd dimensions in $Iso(\mathbb{R}^{2,1\vert 8 \otimes \mathbf{2}} )$, being super-translations along the membrane worldvolume.

This explains why $\kappa$-symmetry in [[Green-Schwarz sigma models]] is taken to quotient out precisely half the spinor components, hence why, in the fully super-covariant formulation, one takes the worldvolume of a super $p$-brane in a superspacetime locally modeled on $\mathbb{R}^{d-1,1\vert \mathbf{N}}$ to be $\mathbb{R}^{p,1\vert \mathbf{N}/2}$.
But notice that this is not a mathematical necessity. One may consider the worldvolume instead to have fewer odd directions. This then describes [[sigma models]] for "non-BPS super $p$-branes" (or rather "non-half-BPS" ).

## Brane Lagrangians by relative trivialization

The super-embedding formalism has mostly been used for deriving [[equations of motion]] of [[super p-brane sigma-models]].

But at least for some brane species, also their [[Lagrangian densities]] emerge naturally from the super-embedding, namely as relative trivializations of the brane cocycles as given by the [[schreiber:The brane bouquet|brane scah/brane bouquet]], relative to the superembedding:

[[BraneSuperEmbeddings.jpg:file]]

> graphics grabbed from [HSS 18](#HSS18)

### For strings and membranes
 {#LagrangiansForStringsAndMembranes}

For the [[superstring]] and the [[super-membrane]] the construction of their [[Green-Schwarz sigma-model]] [[Lagrangian densities]] as relative trivialization of their super-cocycles along their super-embeddings is estalished in [Howe-Sezgin 05 (4.72)](#HoweSezgin05), [HSS 18, Prop. 6.10](#HSS18):

> graphics grabbed from [FSS 19c](#FSS19c)

### For the M5-brane

In [FSS 19c](#FSS19c) is offered a proof that combining super-embedding formalism with [[exceptional generalized geometry]], the [[Perry-Schwarz action functional|Perry-Schwarz-type Lagrangian]] for the [[M5-brane]] emerges as the relative trivialization of the super-cocycle of the M5-brane relative to its super-exceptional embedding.

## Related concepts

[[!include worldvolume-target supersymmetry of brane sigma-models]]

## References

### General

Under the name "doubly supersymmetric geometrical approach" the formalism originates in

* {#BPSTV95} [[Igor Bandos]], [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario Tonin]], [[Dmitry Volkov]], _Superstrings and supermembranes in the doubly supersymmetric geometrical approach_, Nucl. Phys. B **446** (1995) 79-118 [[arXiv:hep-th/9501113](https://arxiv.org/abs/hep-th/9501113)]

* {#HoweSezgin97} [[Paul Howe]], [[Ergin Sezgin]], _$D=11$, $p=5$_, Phys.Lett. B394 (1997) 62-66 ([arXiv:hep-th/9611008](https://arxiv.org/abs/hep-th/9611008))

Review is in

* {#Sorokin99} [[Dmitri Sorokin]], _Superbranes and Superembeddings_, Phys. Rept. 329:1-101, 2000 ([arXiv:hep-th/9906142](http://arxiv.org/abs/hep-th/9906142))

* {#Sorokin01} [[Dmitri Sorokin]], _Introduction to the Superembedding Description of Superbranes_, AIP Conference Proceedings 589, 98 (2001) ([arXiv:hep-th/0105102](https://arxiv.org/abs/hep-th/0105102))

where also the term "superembedding approach" is introduced.

Further review:

* [[Igor A. Bandos]], [[Dmitri P. Sorokin]], *Superembedding approach to superstrings and super-$p$-branes*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2301.10668](https://arxiv.org/abs/2301.10668)]

Derivation of [[BPS state|1/2 BPS]] superembedding via [[rational equivariant homotopy theory|rational]] [[ADE-singularity|ADE-]][[equivariant cohomotopy]]:

* {#HSS18} [[John Huerta]], [[Hisham Sati]], [[Urs Schreiber]], Section 4 of _[[schreiber:Equivariant homotopy and super M-branes|Real ADE-equivariant (co)homotopy and Super M-branes]]_, Comm. Math. Phys. 2019 ([arXiv:1805.05987](https://arxiv.org/abs/1805.05987))

### $\kappa$-Symmetry

The [[supergeometry|super-geometric]] interpretation of [[kappa-symmetry]] as the odd-graded part of the action of [[super-diffeomorphism]] on the [[super p-brane]] [[worldvolume]], regarded itself as a [[supermanifold]] was first suggested in

* {#SorokinTkachVolkov89} [[Dmitri Sorokin]], V. Tkach and [[Dmitrij Volkov]], _Superparticles, twistors and Siegel symmetry_, Mod. Phys. Lett. A4 (1989) 901-908 ([spire:271923](http://inspirehep.net/record/271923), [doi:10.1142/S0217732389001064](https://doi.org/10.1142/S0217732389001064))

Review of this perspective includes

* {#Sorokin99} [[Dmitri Sorokin]], _Superbranes and Superembeddings_, Phys.Rept.329:1-101,2000 ([arXiv:hep-th/9906142](http://arxiv.org/abs/hep-th/9906142))

* {#HoweSezgin04} [[Paul Howe]], [[Ergin Sezgin]], section 4.3 of _The supermembrane revisited_, Class.Quant.Grav. 22 (2005) 2167-2200 ([arXiv:hep-th/0412245](http://arxiv.org/abs/hep-th/0412245))

### For the superstring

The [[equations of motion]] for the [[superstring]] have been derived via the [[superembedding approach]] in

* [BPSTV 95, Chapter 4](#BPSTV95)

See also

* [[Igor Bandos]], [[Dmitrij Sorokin]], [[Dmitrij Volkov]], _On the generalized action principle for superstrings and supermembranes_, Phys. Lett. B352:269-275, 1995 ([arXiv:hep-th/9502141](https://arxiv.org/abs/hep-th/9502141))

### For the M2-brane

The [[equations of motion]] for the [[M2-brane]] have been derived via the [[superembedding approach]] in

* [BPSTV 95, Chapter 3](#BPSTV95)

and the [[Lagrangian density]] in

* {#HoweSezgin05} [[Paul Howe]], [[Ergin Sezgin]], _The supermembrane revisited_, Class. Quant. Grav. 22 (2005) 2167-2200 ([arXiv:hep-th/0412245](https://arxiv.org/abs/hep-th/0412245))

### For the M5-brane

The [[equations of motion]] for the [[M5-brane]] have been derived via the [[superembedding approach]] in

* [Howe-Sezgin 97](#HoweSezgin97)

following the [[superspace]]-computations in

* {#HoweSezginWest97} [[Paul Howe]], [[Ergin Sezgin]], [[Peter West]], _Covariant Field Equations of the M Theory Five-Brane_, Phys. Lett. B399 (1997) 49-59 ([arXiv:hep-th/9702008](https://arxiv.org/abs/hep-th/9702008))

reviewed in

* [Sorokin 99, Section 5.2](#Sorokin99)

Discussion for 3+3-dimensional split:

* Sheng-Lan Ko, [[Dmitri Sorokin]], Pichet Vanichchapongjaroen, _The M5-brane action revisited_, JHEP11(2013)072 ([arXiv:1308.2231](https://arxiv.org/abs/1308.2231))

Claim that combining the super-embedding formalism with [[super-exceptional generalized geometry]], the [[Perry-Schwarz action functional|Perry-Schwarz-type Lagrangian]] for the [[M5-brane]] emerges as the relative trivialization of the super-cocycle of the M5-brane relative to its super-exceptional embedding:

* {#FSS19c} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Super-exceptional embedding construction of the M5-brane|Super-exceptional geometry: origin of heterotic M-theory and super-exceptional embedding construction of M5]]_ ([arXiv:1908.00042](https://arxiv.org/abs/1908.00042))

[[!redirects superembedding formalism]]

[[!redirects super-embedding approach]]
[[!redirects superembedding approach]]

[[!redirects super-embedding construction]]
[[!redirects superembedding construction]]


super-groupoid

[[!redirects supergroupoid]]

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## Definition

A _supergroupoid_ is a [[1-truncated]] [[super ∞-groupoid]].

## Examples

* For $\mathfrak{g}$ a [[super Lie algebra]] and $X$ a [[smooth manifold]], the [[groupoid of Lie-algebra valued forms]] $\Omega^1(X, \mathfrak{g})$ is a super-groupoid: over a given [[superpoint]] $\mathbb{R}^{0|q}$ with function algebra the [[Grassmann algebra]] $\Lambda_q$ it assigns the ordinary [[groupoid of Lie-algebra valued forms]] of the ordinary [[Lie algebra]] $(\mathfrak{g} \otimes \Lambda_q)_{even}$:

  $$
    \Omega^1(X, \mathfrak{g}) : \mathbb{R}^{0|q} \mapsto
    \Omega^1(X, (\mathfrak{g} \otimes \Lambda_q)_{even})
    \,.
  $$


Super-Kamiokande

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## Idea

An [[experiment]] designed for detection and analysis of [[neutrinos]], also [[proton decay]].

* [webpage](http://www-sk.icrr.u-tokyo.ac.jp/sk/index-e.html)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Super-Kamiokande)

## Related entries

* [[LHC]], [[LIGO]], [[EDGES]], [[RHIC]]


super-moduli of super Riemann surfaces -- references

### Super-Moduli space of super Riemann surfaces
 {#ReferencesSuperModuliOfSuperRiemannSurfaces}

On the [[moduli space of super Riemann surfaces|moduli space of]] [[super Riemann surfaces]] (the [[supergeometry|supergeometric]] analog of the [[moduli space of Riemann surfaces]]):

* {#Rabin87} [[Jeffrey Rabin]], _Supermanifolds and Super Riemann Surfaces_,  In: H.C. Lee et. al (eds.) _Super Field Theories_, NATO Science Series (Series B: Physics), vol 160. Springer (1987) ([doi:10.1007/978-1-4613-0913-0_34](https://doi.org/10.1007/978-1-4613-0913-0_34), [pdf](https://www.osti.gov/servlets/purl/6687531))

* {#LeBrunRothstein88} Claude LeBrun, Mitchell Rothstein, _Moduli of super Riemann surfaces_, Comm. Math. Phys. Volume 117, Number 1 (1988), 159-176 ([euclid:cmp/1104161598](https://projecteuclid.org/euclid.cmp/1104161598))

* {#Witten12} [[Edward Witten]], *Notes On Super Riemann Surfaces And Their Moduli*, Pure and Applied Mathematics Quarterly Volume 15 (2019) Number 1 Special Issue on Super Riemann Surfaces and String Theory [[arXiv:1209.2459](http://arxiv.org/abs/1209.2459), [doi:10.4310/PAMQ.2019.v15.n1.a2](https://dx.doi.org/10.4310/PAMQ.2019.v15.n1.a2)]

* {#DonagiWitten15} [[Ron Donagi]], [[Edward Witten]], _Supermoduli Space Is Not Projected_, Proc. Symp. Pure Math. 90 (2015) 19-72 [[arXiv:1304.7798](http://arxiv.org/abs/1304.7798)]

* [Supermoduli Workshop: May 18 – 22, 2015](http://scgp.stonybrook.edu/archives/10356), videos of lecture courses by [[Pierre Deligne]], [[Eric D'Hoker]], [[Ron Donagi]] and [[Edward Witten]]

* {#CodogniViviani17} Giulio Codogni, Filippo Viviani, _Moduli and Periods of Supersymmetric Curves_, Adv. Theor. Math. Phys. 23 (2019) 2, 345-402 ([arXiv:1706.04910](https://arxiv.org/abs/1706.04910), [doi:10.4310/ATMP.2019.v23.n2.a2](https://dx.doi.org/10.4310/ATMP.2019.v23.n2.a2))

* [[Nadia Ott]], [[Alexander A. Voronov]], *The supermoduli space of genus zero SUSY curves with Ramond punctures* [[arXiv:1910.05655](https://arxiv.org/abs/1910.05655)]

* [[Ugo Bruzzo]], Daniel Hernández Ruipérez, *The supermoduli of SUSY curves with Ramond punctures* [[arXiv:1910.12236](https://arxiv.org/abs/1910.12236)]

* [[Nadia Ott]], *The Supermoduli Space of Genus Zero Susy Curves with Ramond Punctures*, University of Minnesota (2020) [[proquest:28094035](https://www.proquest.com/openview/7b7eaefa3b66fdc1eb16eab554d54153/1/advanced)]

* Dimitri Skliros, *Moving NS Punctures on Super Spheres* [[arXiv:2307.06355](https://arxiv.org/abs/2307.06355)]

* {#DonagiOtt23} [[Ron Donagi]], [[Nadia Ott]], *Supermoduli Space with Ramond punctures is not projected* [[arXiv:2308.07957](https://arxiv.org/abs/2308.07957)]

Further discussion of [[supergeometry|supergeometric]] [[Teichmüller space]]:

* [[Robert Penner]], [[Anton Zeitlin]], _Decorated Super-Teichmüller Space_ ([arXiv:1509.06302](https://arxiv.org/abs/1509.06302))

* Ivan C.H. Ip, [[Robert Penner]], [[Anton Zeitlin]], _$\mathcal{N}=2$ Super-Teichmüller Theory_, Advances in Mathematics 336 (2018) 409-454 ([arXiv:1605.08094](https://arxiv.org/abs/1605.08094))

* [[Anton Zeitlin]], _Super-Teichmüller spaces and related structures_ ([arXiv:1811.09939](https://arxiv.org/abs/1811.09939))

In relation to [[fat graphs]]:

* [[Albert S. Schwarz]], [[Anton M. Zeitlin]], *Super Riemann surfaces and fatgraphs* [[arXiv:2307.02706](https://arxiv.org/abs/2307.02706)]


super-orbifold

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## Idea

A _super-orbifold_ is an [[orbifold]] in [[supergeometry]], hence like a [[supermanifold]] but possibly with [[singularities]] of the form of [[fixed points]] of local [[actions]] on the supermanifold by [[finite groups]].

## Examples

### Moduli space of super-Riemann surfaces

Just like the [[moduli space of Riemann surfaces]] is an [[orbifold]], so the [[moduli space of super Riemann surfaces|moduli space of]] [[super-Riemann surfaces]] is a super-orbifold (e.g. [Rabin 87](#Rabin87), [LeBrun-Rothstein 88](#LeBrunRothstein88), [Witten 12](#Witten12), [Codogni-Viviani 17](#CodogniViviani17)).

## References

Super-orbifolds as [[target spaces]] in [[superstring theory]] (orbi-[[superspace]]):

* [[Jaemo Park]], Soo-Jong Reym, _Supertwistor Orbifolds: Gauge Theory Amplitudes and Topological Strings_, JHEP 0412:017, 2004 ([arXiv:hep-th/0411123](https://arxiv.org/abs/hep-th/0411123))

* [[Matthias Gaberdiel]], [[Dan Israel]], [[Eliezer Rabinovici]], _D-branes at multicritical points_, JHEP 0804:086, 2008 [[arXiv:0803.0291](https://arxiv.org/abs/0803.0291)]

and in [[M-theory]]:

* [[John Huerta]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Equivariant homotopy and super M-branes|Real ADE-equivariant (co)homotopy and Super M-branes]]_, Communications in Mathematical Physics 371: 425. (2019) ([arXiv:1805.05987](https://arxiv.org/abs/1805.05987), [doi:10.1007/s00220-019-03442-3](https://doi.org/10.1007/s00220-019-03442-3))

The [[moduli space of super Riemann surfaces]] as a super-orbifold:

* {#Rabin87} [[Jeffrey Rabin]], _Supermanifolds and Super Riemann Surfaces_,  In: H.C. Lee et. al (eds.) _Super Field Theories_, NATO Science Series (Series B: Physics), vol 160. Springer (1987) ([doi:10.1007/978-1-4613-0913-0_34](https://doi.org/10.1007/978-1-4613-0913-0_34), [pdf](https://www.osti.gov/servlets/purl/6687531))

* {#LeBrunRothstein88} Claude LeBrun, Mitchell Rothstein, _Moduli of super Riemann surfaces_, Comm. Math. Phys. Volume 117, Number 1 (1988), 159-176 ([euclid:cmp/1104161598](https://projecteuclid.org/euclid.cmp/1104161598))

* {#Witten12} [[Edward Witten]], _Notes On Super Riemann Surfaces And Their Moduli_, Pure and Applied Mathematics Quarterly Volume 15 (2019) Number 1 Special Issue on Super Riemann Surfaces and String Theory ([arXiv:1209.2459](http://arxiv.org/abs/1209.2459), [doi:10.4310/PAMQ.2019.v15.n1.a2](https://dx.doi.org/10.4310/PAMQ.2019.v15.n1.a2))

* {#CodogniViviani17} Giulio Codogni, Filippo Viviani, _Moduli and Periods of Supersymmetric Curves_, Adv. Theor. Math. Phys. 23 (2019) 2, 345-402 ([arXiv:1706.04910](https://arxiv.org/abs/1706.04910), [doi:10.4310/ATMP.2019.v23.n2.a2](https://dx.doi.org/10.4310/ATMP.2019.v23.n2.a2))

[[!redirects super-orbifolds]]

[[!redirects superorbifold]]
[[!redirects superorbifolds]]

[[!redirects super orbifold]]
[[!redirects super orbifolds]]


super-scheme

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## Idea

The concept of _super-scheme_ is the generalization of that of _[[scheme]]_ from [[commutative algebra]] to [[supercommutative superalgebra]]: where an ordinary [[scheme]] is a space locally modeled on the [[formal dual]] of a [[commutative ring]] (i.e. an [[affine scheme]]), so a superscheme is a space locally modeled on the formal dual of a [[supercommutative algebra]] (i.e. on affine superschemes, def. \ref{AffineSuperSchemes} below).

### Affine superschemes

The following is a detailed and conceptual introduction of the concept of affine super-schemes. This is taken from _[[geometry of physics -- superalgebra]]_.

The key idea of [[supercommutative superalgebra]] is that it is nothing but plain [[commutative algebra]] but "[[internalization|internalized]]" not in ordinary [[vector spaces]], but in [[super vector spaces]]. This is made precise by def. \ref{MonoidsInMonoidalCategory} and ef. \ref{SupercommutativeSuperalgebra} below.

The key idea then of [[supergeometry]] is to define super-[[spaces]] to be spaces whose [[algebras of functions]] are [[supercommutative superalgebras]]. This is not the case for any "ordinary" space such as a [[topological space]] or a [[smooth manifold]]. But these spaces may be _characterized dually_ via their algebras of functions, and hence it makes sense to generalize the latter.

For [[smooth manifolds]] the [[duality]] statement is the following:

+-- {: .num_prop #EmbeddingOfSmoothManifoldsIntoRAlgebras}
###### Proposition
**([[embedding of smooth manifolds into formal duals of R-algebras]])**

The [[functor]]

$$
  C^\infty(-) \;\colon\; SmoothMfd \longrightarrow Alg_{\mathbb{R}}^{op}
$$

which sends a [[smooth manifold]] ([[finite number|finite]] [[dimension|dimensional]], [[paracompact topological space|paracompact]], [[second countable topological space|second countable]]) to (the [[formal dual]] of) its $\mathbb{R}$-[[associative algebra|algebra]] of [[smooth functions]] is a [[full and faithful functor]].

In other words, for two [[smooth manifolds]] $X,Y$ there is a [[natural bijection]] between the [[smooth functions]] $X \to Y$ and the $\mathbb{R}$-[[associative algebra|algebra]] [[homomorphisms]] $C^\infty(X)\leftarrow C^\infty(Y)$.

=--

A **proof** is for instance in ([Kolar-Slovak-Michor 93, lemma 35.8, corollaries 35.9, 35.10](embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras#KolarSlovakMichor93)).

This says that we may _identify_ [[smooth manifolds]] as the  "[[formal duals]]" of certain [[associative algebras]], namely those in the image of the above full embedding. Accordingly then, any larger class of associative algebras than this may be thought of as the class of [[formal duals]] to a generalized kind of manifold, defined thereby. Given any associative algebra $A$, then we may think of it as representing a space
$Spec(A)$ which is such that it has $A$ as its [[algebra of functions]].

This [[duality]] between certain  [[spaces]] and their [[algebras of functions]] is profound. In [[physics]] it has always been used implicitly, in fact it was so ingrained into theoretical physics that it took much effort to abstract away from [[coordinate system|coordinate functions]] to discover global [[Riemannian geometry]] in the guise of"[[general relativity]]". As mathematics, an early prominent duality theorem is [[Gelfand duality]] (between [[topological spaces]] and [[C*-algebras]]) which served as motivation for the very definition of [[algebraic geometry]], where [[affine schemes]] are nothing but the [[formal duals]] of [[commutative rings]]/[[commutative algebras]]. Passing to [[non-commutative algebras]] here yields [[non-commutative geometry]], and so forth. In great generality this duality between spaces and their function algebras appears as "[[Isbell duality]]" between [[presheaves]] and [[copresheaves]].

In [[supergeometry]] we are concerned with spaces that are formally dual to associative algebras which are "very mildly" non-commutative, namely [[supercommutative superalgebras]]. These are in fact [[commutative algebras]] when viewed internal to [[super vector spaces]]
(def. \ref{SupercommutativeSuperalgebra} below).
The corresponding [[formal dual]] spaces are, depending on some technical details, _[[super schemes]]_ or _[[supermanifolds]]_. In the [[physics]] literature, such spaces are usually just called _[[superspaces]]_.

We now make this precise.

+-- {: .num_defn #MonoidsInMonoidalCategory}
###### Definition

Given a [[monoidal category]] $(\mathcal{C}, \otimes, 1)$ (def \ref{MonoidalCategory}), then a **[[monoid in a monoidal category|monoid internal to]]** $(\mathcal{C}, \otimes, 1)$ is

1. an [[object]] $A \in \mathcal{C}$;

1. a morphism $e \;\colon\; 1 \longrightarrow A$ (called the _[[unit]]_)

1. a morphism $\mu \;\colon\; A \otimes A \longrightarrow A$ (called the _product_);

such that

1. ([[associativity]]) the following [[commuting diagram|diagram commutes]]

   $$
     \array{
       (A\otimes A) \otimes A
         &\underoverset{\simeq}{a_{A,A,A}}{\longrightarrow}&
       A \otimes (A \otimes A)
         &\overset{A \otimes \mu}{\longrightarrow}&
       A \otimes A
       \\
       {}^{\mathllap{\mu \otimes A}}\downarrow
         && &&
       \downarrow^{\mathrlap{\mu}}
       \\
       A \otimes A
         &\longrightarrow&
         &\overset{\mu}{\longrightarrow}&
       A
     }
     \,,
   $$

   where $a$ is the [[associator]] isomorphism of $\mathcal{C}$;

1. {#UnitalityMonoid} ([[unitality]]) the following [[commuting diagram|diagram commutes]]:

   $$
     \array{
       1 \otimes A
         &\overset{e \otimes id}{\longrightarrow}&
       A \otimes A
         &\overset{id \otimes e}{\longleftarrow}&
       A \otimes 1
       \\
       & {}_{\mathllap{\ell}}\searrow
       & \downarrow^{\mathrlap{\mu}} &
       & \swarrow_{\mathrlap{r}}
       \\
       && A
     }
     \,,
   $$

   where $\ell$ and $r$ are the left and right unitor isomorphisms of $\mathcal{C}$.

Moreover, if $(\mathcal{C}, \otimes , 1)$ has the structure of a [[symmetric monoidal category]] (def. \ref{SymmetricMonoidalCategory}) $(\mathcal{C}, \otimes, 1, \tau)$ with symmetric [[braiding]] $\tau$, then a monoid $(A,\mu, e)$ as above is called a **[[commutative monoid in a symmetric monoidal category|commutative monoid in]]** $(\mathcal{C}, \otimes, 1, B)$ if in addition

* (commutativity) the following [[commuting diagram|diagram commutes]]

  $$
    \array{
      A \otimes A
        && \underoverset{\simeq}{\tau_{A,A}}{\longrightarrow} &&
      A \otimes A
      \\
      & {}_{\mathllap{\mu}}\searrow && \swarrow_{\mathrlap{\mu}}
      \\
      && A
    }
    \,.
  $$

A [[homomorphism]] of monoids $(A_1, \mu_1, e_1)\longrightarrow (A_2, \mu_2, f_2)$ is a morphism

$$
  f \;\colon\; A_1 \longrightarrow A_2
$$

in $\mathcal{C}$, such that the following two [[commuting diagram|diagrams commute]]

$$
  \array{
    A_1 \otimes A_1
      &\overset{f \otimes f}{\longrightarrow}&
    A_2 \otimes A_2
    \\
    {}^{\mathllap{\mu_1}}\downarrow && \downarrow^{\mathrlap{\mu_2}}
    \\
    A_1 &\underset{f}{\longrightarrow}& A_2
  }
$$

and

$$
  \array{
    1_{\mathcal{c}} &\overset{e_1}{\longrightarrow}& A_1
    \\
    & {}_{\mathllap{e_2}}\searrow & \downarrow^{\mathrlap{f}}
    \\
    && A_2
  }
  \,.
$$

Write $Mon(\mathcal{C}, \otimes,1)$ for the **[[category of monoids]]** in $\mathcal{C}$, and $CMon(\mathcal{C}, \otimes, 1)$ for its subcategory of commutative monoids.

=--

+-- {: .num_example #MonoidsInVectAreAssociativeAlgebras}
###### Example

A [[monoid object]] according to def. \ref{MonoidsInMonoidalCategory} in the [[monoidal category]] of [[vector spaces]] from example \ref{TensorProductOfVectorSpaces} is equivalently an ordinary [[associative algebra]] over the given [[ground field]]. Similarly a [[commutative monoid]] in $Vect$ is an ordinary [[commutative algebra]]. Moreover, in both cases the [[homomorphisms]] of monoids agree with usual algebra homomorphisms. Hence there are [[equivalences of categories]].

$$
  Mon(Vect_k) \simeq Alg_k
$$

$$
  CMon(Vect_k) \simeq CAlg_k
  \,.
$$

=--

+-- {: .num_example #GradedAlgebras}
###### Example

For $G$ a [[group]], then a **$G$-[[graded algebra|graded]] [[associative algebra]]** is a [[monoid object]] according to def. \ref{MonoidsInVectAreAssociativeAlgebras} in the [[monoidal category]] of $G$-[[graded vector spaces]] from example \ref{GradedVectorSpacesAsAMonoidaCategory}.

$$
  Alg_k^G \simeq Mon(Vect_k^G)
  \,.
$$

This means that a $G$-graded algebra is

1. a $G$-[[graded vector space]] $A = \underset{g\in G}{\oplus} A_g$

1. an [[associative algebra]] structure on the underlying vector space $A$

such that for two elements of homogeneous degree, i.e. $a_1 \in A_{g_1} \hookrightarrow A$ and $a_2 \in A_{g_2} \hookrightarrow A$ then their product is in degre $g_1 g_2$

$$
  a_{g_1} a_{g_2} \in A_{g_1 g_2} \hookrightarrow A
  \,.
$$

=--

Example \ref{MonoidsInVectAreAssociativeAlgebras} motivates the following definition:

+-- {: .num_defn #SupercommutativeSuperalgebra}
###### Definition

A **[[supercommutative superalgebra]]** is a [[commutative monoid]] (def. \ref{MonoidsInMonoidalCategory}) in the [[symmetric monoidal category|symmetric monoidal]] [[category of super vector spaces]] (def. \ref{CategoryOfSuperVectorSpaces}). We write $sCAlg_k$ for the [[category]] of [[supercommutative superalgebras]] with the induced [[homomorphisms]] between them:

$$
  sCAlg_k \;\coloneqq\; CMon(sVect_k)
  \,.
$$

Unwinding what this means, then a [[supercommutative superalgebra]] $A$ is

1. a $\mathbb{Z}/2$-graded associative algebra according to example \ref{GradedAlgebras};

1. such that for any two elements $a, b$ of homogeneous degree, their product satisfies

   $$
     a b \; = \; (-1)^{deg(a) deg(b)}\, b a
     \,.
   $$

=--

+-- {: .num_remark}
###### Remark

In view of def. \ref{SupercommutativeSuperalgebra} we might define a not-necessarily supercommutative superalgebra to be a [[monoid]] (not necessarily commutative) in [[sVect]], and write

$$
  sAlg_k \coloneqq Mon(sVect)
  \,.
$$

However, since the definition of not-necessarily commutative monoids (def. \ref{MonoidsInMonoidalCategory}) does not invoke the [[braiding]] of the ambient [[tensor category]], and since [[super vector spaces]] differ from $\mathbb{Z}/2$-[[graded vector spaces]] only via their [[braiding]] (example \ref{CategoryOfSuperVectorSpaces}), this yields equivalently just the $\mathbb{Z}/2$-graded algebras froom example \ref{GradedAlgebras}:

$$
  sAlg_k \simeq Alg_k^{\mathbb{Z}/2}
  \,.
$$

Hence the heart of [[superalgebra]] is _[[supercommutative superalgebra|super-commutativity]]_.

=--

+-- {: .num_example #GrassmannAlgebra}
###### Example

The [[supercommutative superalgebra]] which is [[free construction|freely generated]] over $k$ from $n$ generators
$\{\theta_i\}_{i = 1}^n$ is the [[quotient]] of the [[tensor algebra]] $T^\bullet \mathbb{R}^n$,
with the generators $\theta_i$ in odd degree, by the ideal generated by the
relations

$$
  \theta_i \theta_j = - \theta_j \theta_i
$$

for all $i,j \in \{1, \cdots, n\}$.

This is also called a _[[Grassmann algebra]]_, in honor of ([Grassmann 1844](#Grassmann1844)), who introduced and
studied the super-sign rule in def. \ref{SupercommutativeSuperalgebra} a century ahead of his time.

We also denote this algebra by

$$
  \wedge^\bullet_{\mathbb{R}}(\mathbb{R}^n)
    \;\in\;
   sCAlg_{\mathbb{R}}
   \,.
$$

=--

+-- {: .num_example #HomotopyCommutativeRingSpectrum}
###### Example

Given a [[homotopy commutative ring spectrum]] $E$ (i.e., via the [[Brown representability theorem]], a [[multiplicative cohomology theory|multiplicative]] [[generalized (Eilenberg-Steenrod) cohomology|generalized cohomology theory]]), then its [[stable homotopy groups]] $\pi_\bullet(E)$ inherit the structure of a super-commutative ring.

See at _[[Introduction to Stable homotopy theory]]_ in the section [1-2 Homotopy commutative ring spectra](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyRingSpectra) [this proposition](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyGroupsOfHomotopyCommutativeRingSpectrum).

=--

The following is an elementary but fundamental fact about the relation between
commutative algbra and supercommutative superalgebra. It is implicit in much of the
literature, but maybe the only place where it has been made explicit before is
([Carchedi-Roytenberg 12, example 3.18](#CarchediRoytenberg12)).

+-- {: .num_prop #InclusionOfCAlgIntosCAlg}
###### Proposition

There is a [[full subcategory]] inclusion

$$
  \array{
     CAlg_k &\hookrightarrow& sCAlg_k
     \\
     = && =
     \\
     CMon(Vect_k) &\hookrightarrow& CMon(sVect_k)
  }
$$

of [[commutative algebras]] (example \ref{MonoidsInVectAreAssociativeAlgebras})
into [[supercommutative superalgebras]] (def. \ref{SupercommutativeSuperalgebra})
induced via prop. \ref{MonoidsPreservedByLaxMonoidalFunctor} from the full inclusion

$$
  i \;\colon\; Vectk \hookrightarrow sVect_k
$$

of [[vector spaces]] (def. \ref{VectorSpaces}) into [[super vector spaces]]
(def. \ref{CategoryOfSuperVectorSpaces}), which is a [[braided monoidal functor]]
by prop. \ref{InclusionOfVectorSpacesIntoSupervectorSpaces}. Hence this regards a
commutative algebra as a superalgebra concentrated in even degree.

This inclusion functor has both a [[left adjoint|left]] [[adjoint functor]] and a [[right adjoint|right]] [[adjoint functor]] ,
(an [[adjoint triple]] exibiting a [[reflective subcategory]] and [[coreflective subcategory]] inclusion, an "[[adjoint cylinder]]"):

$$
  CAlg_k
    \underoverset
      {\underset{(-)_{even}}{\longleftarrow}}
      {\overset{(-)/(-)_{odd}}{\longleftarrow}}
      {\hookrightarrow}
  sCAlg_k
  \,.
$$

Here

1. the [[right adjoint]] $(-)_{even}$ sends a supercommutative superalgebra to its even part $A \mapsto A_{even}$;

1. the [[left adjoint]] $(-)/(-)_{even}$ sends a supercommutative superalgebra to the [[quotient]] by the ideal which is generated by its odd part $A \mapsto A/(A_{odd})$ (hence it sets all elements to zero which may be written as a product such that at least one factor is odd-graded).

=--

+-- {: .proof}
###### Proof

The full inclusion $i$ is evident. To see the [[adjunctions]] observe their characteristic
[[natural bijections]] between [[hom-sets]]:
If $A_{ordinary}$ is an ordinary commutative algebra regarded as a superalgeba $i(A_{ordinary})$ concentrated in even degree,
and if $B$ is any superalgebra,

1. then every super-algebra homomorphism of the form $A_{ordinary} \to B$ must factor
   through $B_{even}$, simply because super-algebra homomorpism by definition respect the $\mathbb{Z}/2$-grading. This gives a natual bijection

   $$
     Hom_{sCAlg_k}(i(A_{ordinary}), B) \simeq Hom_{CAlg_k}(A_{ordinary,B_{even}})
     \,,
   $$

1. every super-algebra homomorphism of the form $B \to i(A_{ordinary})$ must send every odd element of $B$ to 0, again because homomorphism have to respect the $\mathbb{Z}/2$-grading, and since homomorphism of course also preserve products, this means that the entire ideal generated by $B_{odd}$ must be sent to zero, hence the homomorphism must facto through the [[projection]] $B \to B/B_{odd}$, which gives a natural bijection

   $$
     Hom_{sCalg_k}(B, i(A_{ordinary}))
       \simeq
     Hom_{Alg_k}(B/B_{odd}, A_{ordinary})
     \,.
   $$

=--

It is useful to make explicit the following [[formal dual|formally dual]] perspective on [[supercommutative superalgebras]]:

+-- {: .num_defn #Affines}
###### Definition

For $\mathcal{C}$ a [[symmetric monoidal category]], then we write

$$
  Aff(\mathcal{C}) \coloneqq CMon(\mathcal{C})^{op}
$$

for the [[opposite category]] of the [[category of commutative monoids]] in $\mathcal{C}$, according to def. \ref{MonoidsInMonoidalCategory}.

For $R \in CMon(\mathcal{C})$ we write

$$
  Spec(A)
  \in
  Aff(\mathcal{C})
$$

for the same object, regarded in the opposite category. We also call this the **[[affine scheme]]** of $A$. Conversely, for $X \in Aff(\mathcal{C})$, we write

$$
  \mathcal{O}(X) \in CMon(\mathcal{C})
$$

for the same object, regarded in the category of commutative monoids. We also call this the **[[algebra of functions]]** on $X$.

=--

+-- {: .num_defn #AffineSuperSchemes}
###### Definition

For the special case that $\mathal{C} = $ [[sVect]] (def. \ref{CategoryOfSuperVectorSpaces})
in def. \ref{Affines}, then we say that the objects in

$$
  Aff(sVect_k) = scAlg_k^{op} =  CMon(sVect_k)^{op}
$$

are **[[affine super schemes]]** over $k$.

=--

+-- {: .num_example #OrdinaryCAlgAssCAlg}
###### Example

For $A \in CAlg_{\mathbb{R}}$ an ordinary [[commutative algebra]] over $\mathbb{R}$,
then of course this becomes a [[supercommutative superalgebra]] by regarding it
as being concentrated in even degrees. Accordingly, via def. \ref{Affines}, ordinary [[affine schemes]]
[[full subcategory|fully embed]] into [[affine super schemes]] (def. \ref{AffineSuperSchemes})

$$
  Aff(Vect_k)
    \hookrightarrow
  Aff(sVect_k)
  \,.
$$

In particular for $\mathbb{R}^p$ an ordinary [[Cartesian space]], this becomes
an [[affine superscheme]] in even degree, under the above embedding. As such, it is usually
written

$$
  \mathbb{R}^{p \vert 0}
    \in
  Aff(sVect_k)
  \,.
$$

=--

+-- {: .num_example #Superpoint}
###### Example

The formal dual space, according to def. \ref{Affines} (example \ref{Affines}) to a [[Grassmann algebra]] $\wedge^\bullet_{\mathbb{R}}(\mathbb{R}^q)$
(example \ref{GrassmannAlgebra}) is to be thought of as a space which is "so tiny" that
the coefficients of the [[Taylor expansion]] of any real-valued function on it
become "so very small" as to be actually equal to zero, at least after the $q$-th power.

For instance for $q = 2$ then a general element of $\wedge^\bullet_{\mathbb{R}}(\mathbb{R}^q)$
is of the form

$$
  f = a_0 + a_1 \theta_1 + a_2 \theta_2 + a_{12} \theta_1 \theta_2
  \;\;\;\in
  \wedge^\bullet_{\mathbb{R}}(\mathbb{R}^q)
  \,.
$$

for $a_1,a_2, a_{12} \in \mathbb{R}$,
to be compared with the [[Taylor expansion]] of a [[smooth function]] $g \colon \mathbb{R}^2 \to \mathbb{R}$,
which is of the form

$$
  g(x_1, x_2) =
  g(0) + \frac{\partial g}{\partial x_1}(0)\, x_1 + \frac{\partial g}{\partial x_2}(0)\, x_2
    +
    \frac{\partial^2 g}{\partial x_1 \partial x_2}(0) \, x_1 x_2
     +
      \cdots
  \,.
$$

Therefore the [[formal dual]] [[space]] to a [[Grassmann algebra]] behaves like an infinitesimal neighbourhood of a point.
Hence these are also called **[[superpoints]]** and one writes

$$
  \mathbb{R}^{0\vert q}
    \coloneqq
  Spec(\wedge^\bullet_{\mathbb{R}}(\mathbb{R}^q))
  \,.
$$

=--

+-- {: .num_example #SuperCartesianSpace}
###### Example

Combining example \ref{OrdinaryCAlgAssCAlg} with example \ref{Superpoint},
and using prop. \ref{ProductsInAff}, we obtain the  [[affine super schemes]]

$$
  \mathbb{R}^{p \vert q}
    \coloneqq
  \mathbb{R}^{p\vert 0}
    \times
  \mathbb{R}^{0\vert q}
    \simeq
  Spec\left(
     \underbrace{C^\infty(\mathbb{R}^p)}
       \otimes_{\mathbb{R}}
     \wedge^\bullet_{\mathbb{R}} \mathbb{R}^q
  \right)
  \,.
$$

These may be called the **[[super Cartesian spaces]]**.
The play the same role in the theory of [[supermanifolds]] as the ordinary
[[Cartesian spaces]] do for [[smooth manifolds]]. See at _[[geometry of physics -- supergeometry]]_ for more on this.

=--

+-- {: .num_defn #ParityAutomorphism}
###### Definition

Given a [[supercommutative superalgebra]] $A$ (def. \ref{SupercommutativeSuperalgebra}), its **parity involution** is the algebra [[automorphism]]

$$
  par \;\colon\; A \overset{\simeq}{\longrightarrow} A
$$

which on homogeneously graded elements $a$ of degree $deg(a) \in \{even,odd\} = \mathbb{Z}/2\mathbb{Z}$ is multiplication by the degree

$$
  a \mapsto (-1)^{deg(a)}a
  \,.
$$

(e.g. [arXiv:1303.1916, 7.5](http://arxiv.org/abs/1303.1916))

Dually, via def. \ref{Affines}, this means that every [[affine super scheme]] has a canonical [[involution]].

=--

Here are more general and more abstract examples of commutative monoids, which will be useful to make explicit:

+-- {: .num_example #MonoidGivenByTensorUnit}
###### Example

Given a [[monoidal category]] $(\mathcal{C}, \otimes, 1)$ (def. \ref{MonoidalCategory}), then the [[tensor unit]] $1$ is a [[monoid in a monoidal category|monoid in]] $\mathcal{C}$ (def. \ref{MonoidsInMonoidalCategory}) with product given by either the left or right [[unitor]]

$$
  \ell_1 = r_1 \;\colon\; 1 \otimes 1 \overset{\simeq}{\longrightarrow} 1
  \,.
$$

By lemma \ref{kel1}, these two morphisms coincide and define an [[associativity|associative]] product with unit the identity $id \colon 1 \to 1$.

If $(\mathcal{C}, \otimes , 1)$ is a [[symmetric monoidal category]] (def. \ref{SymmetricMonoidalCategory}), then this monoid is a [[commutative monoid in a symmetric monoidal category|commutative monoid]].

=--

+-- {: .num_example #TensorProductOfTwoCommutativeMonoids}
###### Example

Given a [[symmetric monoidal category]] $(\mathcal{C}, \otimes, 1)$ (def. \ref{SymmetricMonoidalCategory}), and given two [[commutative monoid in a symmetric monoidal category|commutative monoids]] $(E_i, \mu_i, e_i)$, $i \in \{1,2\}$ (def. \ref{MonoidsInMonoidalCategory}), then the [[tensor product]] $E_1 \otimes E_2$ becomes itself a commutative monoid with unit morphism

$$
  e \;\colon\;
  1
    \overset{\simeq}{\longrightarrow}
  1 \otimes 1
    \overset{e_1 \otimes e_2}{\longrightarrow}
  E_1 \otimes E_2
$$

(where the first isomorphism is, $\ell_1^{-1} = r_1^{-1}$ (lemma \ref{kel1})) and with product morphism given by

$$
  E_1 \otimes E_2
   \otimes
  E_1 \otimes E_2
   \overset{id \otimes \tau_{E_2, E_1} \otimes id}{\longrightarrow}
  E_1 \otimes E_1 \otimes E_2 \otimes E_2
   \overset{\mu_1 \otimes \mu_2}{\longrightarrow}
  E_1 \otimes E_2
$$

(where we are notationally suppressing the [[associators]] and where $\tau$ denotes the [[braiding]] of $\mathcal{C}$).

That this definition indeed satisfies associativity and commutativity follows from the corresponding properties of $(E_i,\mu_i, e_i)$, and from the hexagon identities for the braiding (def. \ref{BraidedMonoidalCategory}) and from symmetry of the braiding.

Similarly one checks that for $E_1 = E_2 = E$ then the unit maps

$$
  E \simeq E \otimes 1
    \overset{id \otimes e}{\longrightarrow}
  E \otimes E
$$

$$
  E \simeq 1 \otimes E
    \overset{e \otimes 1}{\longrightarrow}
  E \otimes E
$$

and the product map

$$
  \mu
    \;\colon\;
  E \otimes E
    \longrightarrow
  E
$$

and the braiding

$$
  \tau_{E,E}
    \;\colon\;
  E \otimes E
    \longrightarrow
  E \otimes E
$$

are monoid homomorphisms, with $E \otimes E$ equipped with the above monoid structure.

=--

Monoids are preserved by [[lax monoidal functors]]:

+-- {: .num_prop #MonoidsPreservedByLaxMonoidalFunctor}
###### Proposition

Let $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ and $(\mathcal{D}, \otimes_{\mathcal{D}},1_{\mathcal{D}})$ be two [[monoidal categories]] (def. \ref{MonoidalCategory}) and let $F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}$ be a [[lax monoidal functor]] (def. \ref{LaxMonoidalFunctor}) between them.

Then for $(A,\mu_A,e_A)$ a [[monoid in a monoidal category|monoid in]] $\mathcal{C}$ (def. \ref{MonoidsInMonoidalCategory}), its image $F(A) \in \mathcal{D}$ becomes a monoid $(F(A), \mu_{F(A)}, e_{F(A)})$ by setting

$$
  \mu_{F(A)}
   \;\colon\;
  F(A) \otimes_{\mathcal{C}} F(A)
    \overset{}{\longrightarrow}
  F(A \otimes_{\mathcal{C}} A)
    \overset{F(\mu_A)}{\longrightarrow}
  F(A)
$$

(where the first morphism is the structure morphism of $F$) and setting

$$
  e_{F(A)}
    \;\colon\;
  1_{\mathcal{D}}
    \longrightarrow
  F(1_{\mathcal{C}})
    \overset{F(e_A)}{\longrightarrow}
  F(A)
$$

(where again the first morphism is the corresponding structure morphism of $F$).

This construction extends to a functor

$$
  Mon(F)
   \;\colon\;
  Mon(\mathcal{C}, \otimes_{\mathcal{C}}, 1_{\mathcal{C}})
    \longrightarrow
  Mon(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}})
$$

from the [[category of monoids]] of $\mathcal{C}$ (def. \ref{MonoidsInMonoidalCategory}) to that of $\mathcal{D}$.

Moreover, if $\mathcal{C}$ and $\mathcal{D}$ are [[symmetric monoidal categories]] (def. \ref{SymmetricMonoidalCategory}) and $F$ is a [[braided monoidal functor]] (def. \ref{LaxMonoidalFunctor}) and $A$ is a [[commutative monoid]] (def. \ref{MonoidsInMonoidalCategory}) then so is $F(A)$, and this construction extends to a functor

$$
  CMon(F)
   \;\colon\;
  CMon(\mathcal{C}, \otimes_{\mathcal{C}}, 1_{\mathcal{C}})
    \longrightarrow
  CMon(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}})
  \,.
$$

=--

+-- {: .proof}
###### Proof

This follows immediately from combining the associativity and unitality (and symmetry) constraints of $F$ with those of $A$.

=--

## Related concepts

* [[supergroup]]

* [[Deligne's theorem on tensor categories]]

* [[spectral super-scheme]]

## References

General accounts include

* Carmeli-Caston-Fioresi: _Mathematical foundations of supersymmetry_, EMS Series of Lectures in Mathematics, EMS 2011

* [[Mikhail Kapranov]]  [[Eric Vasserot]], _Supersymmetry and the formal loop space_, Advances in Math. 227 (2011), 1078-1128 ([arXiv:1005.4466](https://arxiv.org/abs/1005.4466))

* [[Ugo Bruzzo]], Daniel Hernandez Ruiperez, [[Alexander Polishchuk]], _Notes on fundamental algebraic supergeometry. Hilbert and Picard superschemes_ ([arXiv:2008.00700](https://arxiv.org/abs/2008.00700))

Discussion of [[crystalline cohomology]] of super-schemes:

* Martin Luu, _Crystalline cohomology of superschemes_ ([pdf](http://math.uiuc.edu/~mluu/Luu-crystalline.pdf))

[[!redirects super-schemes]]

[[!redirects super scheme]]
[[!redirects super schemes]]

[[!redirects superscheme]]
[[!redirects superschemes]]

[[!redirects affine super scheme]]
[[!redirects affine super schemes]]
[[!redirects affine super-scheme]]
[[!redirects affine super-schemes]]
[[!redirects affine superscheme]]
[[!redirects affine superschemes]]


super-translation group

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Supergeometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _super-translation group_  is a [[supergroup]] generalization of the [[translation group]], hence of the additive [[Lie group]] $\mathbb{R}^n$. Its underlying [[supermanifold]] is a [[super-Euclidean space]] or [[super-Minkowski spacetime]].

## Definition

Given a [[super Poincaré Lie algebra]] extension $\mathfrak{siso}_N(d-1,1)$ of a [[orthogonal Lie algebra]] $\mathfrak{so}(d-1,1)$ for some [[spin representation]] $N$, then the corresponding _super-translation Lie algebra_ is the [[quotient]]

$$
  \mathbb{R}^{d-1,1\vert N}
  \coloneqq
  \mathfrak{siso}_N(d-1,1)/\mathfrak{o}(d-1,1)
$$

See at _[[super Minkowski spacetime]]_.

The underlying [[super vector space]] of this is

$$
  \mathbb{R}^d \oplus \Pi S
  \,,
$$

where $S$ is the vector space underlying the given [[spin representation]].

The [[super Lie algebra]] structure is mildly non-abelian with the only non-trivial bracket being that between two [[spinors]] and given by the bilinear pairing (the [[charge conjugation matrix]]) between two spinors:

$$
  [\psi, \phi] = \langle \psi, \Gamma^a \phi \rangle t_a
  \,,
$$

where $\{t_a\}$ is a [[basis]] for the translation generators in $\mathbb{R}^d$.

## Properties

### As a central extension of the superpoint

The super-translation Lie algebra is a super-[[Lie algebra extension]] of the abelian [[super Lie algebra]] which is just the [[superpoint]] $\mathbb{R}^{0;N}$ by the $d$ super [[Lie algebra cocycles]]

$$
  (\phi, \psi) \mapsto \langle \phi, \Gamma^a \psi\rangle t_a
  \,,
$$

for $a \in \{1, 2, \cdots, d\}$, where $\{t_a\}$ are the [[basis]] elements of $\mathbb{R}^d$.

This simple but maybe noteworthy fact has been highlighted in the context of the [[brane scan]] in ([CAIP 99, section 2.1](#CAIP99)).

This mechanism plays a role in [[string theory]] when realizing $\mathbb{R}^{11;N=1}$ as a central extension of $\mathbb{R}^{10;N=(1,1)}$, for this formalizes aspects of the idea that [[type IIA string theory]] with a [[D0-brane]] condensate is [[11-dimensional supergravity]]/[[M-theory]] ([FSS 13](#FSS13)).

## Examples

### In dimension 1

The additive group structure on $\mathbb{R}^{1|1}$ is given on [[generalized element]]s in (i.e. in the logic internal to) the [[topos]] of [[sheaf|sheaves]] on the category [[super Cartesian space|SCartSp]] of [[cartesian space|cartesian]] superspaces by

$$
  \mathbb{R}^{1|1} \times \mathbb{R}^{1|1} \to
  \mathbb{R}^{1|1}
$$

$$
  (t_1, \theta_1), (t_2, \theta_2)
  \mapsto
  (t_1 + t_2 + \theta_1 \theta_2, \theta_1 + \theta_2)
  \,.
$$

Recall how the notation works here: by the [[Yoneda embedding]] we have a [[full and faithful functor]]

> [[SDiff]] $\hookrightarrow$ $Fun(SDiff^{op}, Set)$

and we also have the theorem, discussed at [[supermanifold]]s, that maps from some $S \in SDiff$ into $\mathbb{R}^{p|q}$ is given by a tuple of $p$ even section $t_i$ and $q$ odd sections $\theta_j$. The above notation specifies the map of supermanifolds by displaying what map of sets of maps from some test object $S$ it corresponds to under the [[Yoneda embedding]].

Now,  or each $S \in $ [[SDiff]] there is a [[group]] structure on the [[hom-set]] $SDiff(S, \mathbb{R}^{1|1}) \simeq C^\infty(S)^{ev} \times C^\infty(X)^{odd}$ given by precisely the above formula for this given $S$

$$
  \mathbb{R}^{1|1}(S) \times \mathbb{R}^{1|1}(S) \to
  \mathbb{R}^{1|1}(S)
$$

$$
  (t_1, \theta_1), (t_2, \theta_2)
  \mapsto
  (t_1 + t_2 + \theta_1 \theta_2, \theta_1 + \theta_2)
  \,.
$$

where $(t_i, \theta_i) \in C^\infty(S)^{ev} \times C^\infty(S)^{odd}$ etc and where the addition and product on the right takes place in the function [[super algebra]] $C^\infty(S)$.

Since the formula looks the same for all $S$, one often just writes it without mentioning $S$ as above.

## Related concepts

* [[translation Lie algebra]]

* [[super Minkowski spacetime]]

* [[super Poincaré group]]

* [[super Poincaré Lie algebra]]

## References

* [[Veeravalli Varadarajan]], section 7 of _Supersymmetry for mathematicians: An introduction_

Discussion in the context of the [[brane scan]] is in section 2.1 of

* {#CAIP99} C. Chryssomalakos, [[José de Azcárraga]], J.M. Izquierdo, J.C. Pérez Bueno, _The geometry of branes and extended superspaces_, Nucl.Phys.B567:293-330, 2000 ([arXiv:hep-th/9904137](http://arxiv.org/abs/hep-th/9904137))

and more generally in the context of [[schreiber:The brane bouquet]] in

* {#FSS13} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:The brane bouquet|Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields]]_

[[!redirects super-translation groups]]

[[!redirects super-Heisenberg group]]
[[!redirects super-Heisenberg groups]]

[[!redirects super translation Lie algebra]]
[[!redirects super translation Lie algebras]]

[[!redirects super translation super Lie algebra]]
[[!redirects super translation super Lie algebras]]

[[!redirects super translation super-Lie algebra]]
[[!redirects super translation super-Lie algebras]]

[[!redirects translation super-Lie algebra]]
[[!redirects translation super-Lie algebras]]

[[!redirects super-translation super-Lie algebra]]
[[!redirects super-translation super-Lie algebras]]

[[!redirects super translation algebra]]
[[!redirects super translation algebras]]
[[!redirects super-translation algebra]]
[[!redirects super-translation algebras]]
[[!redirects super translation Lie algebra]]
[[!redirects super translation Lie algebras]]
[[!redirects super-translation Lie algebra]]
[[!redirects super-translation Lie algebras]]

[[!redirects super translation group]]
[[!redirects super translation groups]]

[[!redirects translation supergroup]]
[[!redirects translation supergroups]]


super-unitary representation

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#### Representation theory
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## Idea

The generalization of the concept of [[unitary representation]] to [[superalgebra]]/[[supergroups]].

## Examples

* [[super multiplet]]

* [[unitary representation of the super Poincaré group]]

## References

* [[Veeravalli Varadarajan]], _Unitary representations of super Lie groups_, 2006 ([pdf](http://www.math.ucla.edu/~vsv/ur.pdf))

* Hirotoshi Furutsu, T. Hirai, section 1.3 of _Representations of Lie superalgebras, I. Extensions of representations of the even part_, J. Math. Kyoto Univ. 28, 695-749 (1988) ([Euclid](http://projecteuclid.org/euclid.kjm/1250520352))

* Hirotoshi Furutsu, Kyo Nishiyama, _Classification of irreducible super-unitary representations of $\mathfrak{su}(p,q\vert n)$_, Comm. Math. Phys. Volume 141, Number 3 (1991), 475-502. ([Euclid](http://projecteuclid.org/euclid.cmp/1104248389))

[[!redirects super-unitary representations]]

[[!redirects superunitary representation]]
[[!redirects superunitary representations]]


super-vielbein > history

> see _[[super vielbein]]_


supercharge

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* table of contents
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## Idea

A [[conserved charge]] (under [[Noether's theorem]]) of a [[supersymmetry]]
transformation is called a _supercharge_.

Equivalently, by [[Noether's theorem]], this is an odd symmetry generators in a [[supersymmetry]] [[super Lie algebra]].

## Examples

* The [[Dirac operator]] in [[supersymmetric quantum mechanics]].

* The [[Dirac-Ramond operator]] on the [[superstring]].

## Related concepts

* [[Fredholm operator]], [[Dirac operator]]

* [[index]]

[[!include genera and partition functions - table]]

[[!redirects supercharges]]


supercoherence

The category $\Delta$ has generators (cofaces and codegeneracies) and relations and
a every morphism can be canonically represented via cofaces and codegeneracies. Consequently, for [[simplicial set]]s, hence functors $\Delta^{op}\to Set$ have canonical representation via face and degeneracy maps. We can now look at categorification of this situation to pseudofunctors, say $\Delta^{op}\to Cat$ ([[pseudosimplicial categories]]). While the simplicial identities now hold up to
invertible 2-cells, the canonical representation of general morphisms as sequences of face and degeneracy maps suggest that one should be able to write down the corresponding 2-cells from generating 2-cells corresponding to the quadratic identities, and that different choices of sequences generating 2-cells should be the
same composite 2-cells.

Jardine has proved that the generating 2-cells

$$\array{
\partial_j \partial_i \implies \partial_i \partial_{j+1} & (i \leq j)\\
\partial_i \sigma_j \implies \sigma_{j-1} \partial_i & (i \lt j) \\
\partial_i \sigma_j \implies Id & (i = j, i = j + 1)\\
\partial_i \sigma_j \implies \sigma_j \partial_{i-1} & (i \gt j + 1) \\
\sigma_i \sigma_j \implies \sigma_{j+1} \sigma_i & (i \leq j)
}$$

coming from the pseudosimplicial objects satisfy 17 coherence relations which in turn enable one to reconstruct the 2-cells among more complicated chains of face and degeneracy maps in unique way. To write those, all generating will be denoted by $\alpha$ with some additional superscripts, but for simplicity we skip the superscript as in Jardine's paper (as they are obvious to fill):

...

In particular, this way one can check when one has a pseudosimplicial object coming from a concrete construction by producing just those 2-cells which are generating  and checking the 17 diagrams. He calls the corresponding structure $(X_n, \partial_i, \sigma_j, \alpha)$ of 0-cells, faces, degeneracies and generating 2-cells the supercoherence. They are therefore equivalent to a pseudosimplicial object $X_\bullet$.

* J. F. Jardine, _Supercoherence_, J. Pure Appl. Algebra __75__ (1991), no. 2, 103–194 [MR1138365](http://www.ams.org/mathscinet-getitem?mr=1138365) doi


supercommutative algebra

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## Idea

A _super-commutative algebra_ is a [[commutative algebra]] _[[internalization|internal]]_ to the [[symmetric monoidal category]] of [[super vector spaces]], hence a $\mathbb{Z}/2$-[[graded algebra|graded]] [[associative algebra]] such that for $a, b$ any two elements of homogeneous degree $deg(a), deg(b) \in \mathbb{Z}/2 = \{0,1\}$, we have

$$
  a \cdot b
   \;\;=\;\;
  (-1)^{deg(a) deg(b)}
   \;
  b \cdot a
  \,.
$$

For more see at _[[geometry of physics -- superalgebra]]_.

## Examples

* The [[free construction|free]] supercommutative algebras are the _[[Grassmann algebras]]_.

* [[stable homotopy groups]] of [[homotopy commutative ring spectrum]], see at _[[Introduction to Stable homotopy theory]]_ the section _[1-2 Homotopy commutative ring spectra](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyRingSpectra)_

* [[Jordan superalgebra]]

## Related concepts

* [[supercommutative ring]]

* [[super-scheme]]

* [[commutative differential graded algebra]]

[[!redirects supercommutative algebras]]

[[!redirects super-commutative algebra]]
[[!redirects super-commutative algebras]]

[[!redirects supercommutative superalgebra]]
[[!redirects supercommutative superalgebras]]

[[!redirects super-commutative superalgebra]]
[[!redirects super-commutative superalgebras]]

[[!redirects super-commutative super-algebra]]
[[!redirects super-commutative super-algebras]]

[[!redirects supercommutative super-algebra]]
[[!redirects supercommutative super-algebras]]

[[!redirects graded commutative algebra]]
[[!redirects graded commutative algebras]]

[[!redirects graded-commutative algebra]]
[[!redirects graded-commutative algebras]]

[[!redirects graded commutative ring]]
[[!redirects graded commutative rings]]

[[!redirects graded-commutative ring]]
[[!redirects graded-commutative rings]]


supercommutative ring

[[!redirects supercommutative rings]]

[[!redirects super-commutative ring]]
[[!redirects super-commutative rings]]

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#Contents#
* table of contents
{:toc}

## Idea ##

A supercommutative ring is an $\mathbb{Z}$-[[supercommutative algebra]].

## Definition ##

A **supercommutative ring** is a [[super ring]] $R$, such that

* for all $a:R$, and $b:R$, $\mathcal{D}_0(a) \cdot \mathcal{D}_0(b) = \mathcal{D}_0(b) \cdot \mathcal{D}_0(a)$
* for all $a:R$, and $b:R$, $\mathcal{D}_1(a) \cdot \mathcal{D}_0(b) = \mathcal{D}_0(b) \cdot \mathcal{D}_1(a)$
* for all $a:R$, and $b:R$, $\mathcal{D}_0(a) \cdot \mathcal{D}_1(b) = \mathcal{D}_1(b) \cdot \mathcal{D}_0(a)$
* for all $a:R$, and $b:R$, $\mathcal{D}_1(a) \cdot \mathcal{D}_1(b) = - \mathcal{D}_1(b) \cdot \mathcal{D}_1(a)$

## See also ##

* [[ring]]

* [[differential ring]]

* [[exterior ring]]

* [[graded ring]]

* [[super abelian group]]

* [[supercommutative algebra]]

* [[super ring]]


supercompact cardinal

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# Contents
* table of contents
{: toc}

## Idea

Supercompact cardinals are among the [[large cardinal]]s.

## Definition

For $S$ a [[set]] and $\kappa$ a [[cardinal]], let $P_\kappa(S)$ be the set of [[subset]]s of $S$ of cardinality less than $\kappa$.

For $\lambda$ an [[ordinal]] a cardinal $\kappa$ is called **$\lambda$-supercompact** if $P_\kappa(\lambda)$ admits a [[normal measure]]. It is **supercompact** if it is $\lambda$-supercompact for every $\lambda$.

$\kappa$ being $\lambda$-supercompact is equivalent to there being an [[elementary embedding]]  $j : V \to M$ such that $j(\alpha) = \alpha$ for all $\alpha \lt \kappa$ and $j(\kappa) \gt \lambda$, where $M$ is an [[inner model]] such that $\{f | f : \lambda \to M\} \subset M$, i.e. every $\lambda$-sequence of elements of $M$ is an element of $M$.

## Properties

By invoking [[Vopěnka's principle]] one can make strong statements about the existence of [[reflective subcategories]].  The assumption of supercompact cardinals is much weaker, and accordingly they similarly imply existence of reflective subcategories only under some more additional assumptions.  The following theorems are all from ([BCMR](#BagariaCasacubertaMathiasRosicky)).

+-- {: .num_theorem}
###### Theorem
Suppose there are arbitrarily large supercompact cardinals.  Then if $L$ is a reflection on an accessible category $C$ and the class of $L$-equivalences is $\Sigma_2$-[[Levy hierarchy|definable]], then the $L$-local objects are a small-orthogonality class (so that $L$ is a localization with respect to some [[set]] of morphisms).
=--

+-- {: .num_theorem}
###### Theorem
Suppose there are arbitrarily large supercompact cardinals.  Then any full subcategory of a locally presentable category which is closed under limits and $\Sigma_2$-[[Levy hierarchy|definable]] is reflective.
=--

There is also a generalization to $\Sigma_n$-definability involving [[C(n)-extendible cardinals]]; see [[Vopenka's principle]].

## References

Supercompact cardinals are discussed for instance in

* T. Jech _Set Theory_ The Third Millennium Edition, Revised and Expanded.
Springer Monographs in Mathematics. Springer-Verlag, Berlin, Heidelberg (2003)

* A. Kanamori, _The Higher Infinite: Large Cardinals in Set Theory from Their
Beginnings_ Perspectives in Mathematical Logic. Springer-Verlag, Berlin, Heidelberg (1994)

The relation to refelctive subcategories is discusssed in

* Joan Bagaria, [[Carles Casacuberta]], Adrian Mathias, _Epireflections and supercompact cardinals_  Journal of Pure and Applied Algebra 213 (2009), 1208-1215 ([pdf](http://atlas.mat.ub.es/personals/casac/articles/bcm.pdf))
{#BagariaCasacubertaMathias}

* Joan Bagaria, [[Carles Casacuberta]], Adrian Mathias, [[Jiri Rosicky]] _Definable orthogonality classes in accessible categories are small_, [arXiv](http://arxiv.org/abs/1101.2792)
{#BagariaCasacubertaMathiasRosicky}

[[!redirects supercompact cardinal]]
[[!redirects supercompact cardinals]]
[[!redirects supercompact cardinal number]]
[[!redirects supercompact cardinal numbers]]


superconducting qbits -- references

### Superconducting qbits
 {#SuperconductingQBitsReferences}

On realizing [[qbits]] and [[quantum gates]] (hence [[quantum computation]]) via [[quantum states]] of [[magnetic flux]] through ([[Josephson junctions]] in) [[superconductors]], manipulated via [[electromagnetic radiation|electromagnetic pulses]]:

* M. H. Devoret, A. Wallraff, J. M. Martinis, *Superconducting Qubits: A Short Review* $[$[arXiv:cond-mat/0411174](https://arxiv.org/abs/cond-mat/0411174)$]$

* John Clarke, Frank K. Wilhelm, *Superconducting quantum bits*, Nature **453** (2008) 1031–1042 $[$[doi:10.1038/nature07128](https://doi.org/10.1038/nature07128)$]$

* Jerry Moy Chow, *Quantum Information Processing with Superconducting Qubits* (2010) $[$[pdf](https://rsl.yale.edu/sites/default/files/files/RSL_Theses/jmcthesis.pdf)$]$

* Jay M. Gambetta, Jerry M. Chow, Matthias Steffen, *Building logical qubits in a superconducting quantum computing system*, npj Quantum Information **3** 2 (2017) $[$[doi:10.1038/s41534-016-0004-0](https://doi.org/10.1038/s41534-016-0004-0)$]$

* Morten Kjaergaard et al. *Superconducting Qubits: Current State of Play*, Annual Review of Condensed Matter Physics **11** (2019) 369-395 $[$[doi:10.1146/annurev-conmatphys-031119-050605](https://doi.org/10.1146/annurev-conmatphys-031119-050605)$]$

* He-Liang Huang, Dachao Wu, Daojin Fan, Xiaobo Zhu, *Superconducting Quantum Computing: A Review*, Science China Information Sciences **63** 8 (2020) 1-32 $[$[arXiv:2006.10433](https://arxiv.org/abs/2006.10433), [doi:10.1007/s11432-020-2881-9](https://doi.org/10.1007/s11432-020-2881-9)$]$

* S. Kwon et al., *Gate-based superconducting quantum computing*, Journal of Applied Physics **129** (2021) 041102 $[$[doi:10.1063/5.0029735](https://doi.org/10.1063/5.0029735)$]$

Fine detail of the pulse control:

* M. Werninghaus, D. J. Egger, F. Roy, S. Machnes, F. K. Wilhelm, S. Filipp: *Leakage reduction in fast superconducting qubit gates via optimal control*, npj Quantum Information **7** 14 (2021) $[$[doi:10.1038/s41534-020-00346-2](https://doi.org/10.1038/s41534-020-00346-2)$]$

* M. Carroll, S. Rosenblatt, P. Jurcevic, I. Lauer & A. Kandala. *Dynamics of superconducting qubit relaxation times*, npj Quantum Information **8** 132 (2022) $[$[doi:10.1038/s41534-022-00643-y](https://doi.org/10.1038/s41534-022-00643-y)$]$

* Elisha Siddiqui Matekole, Yao-Lung L. Fang, Meifeng Lin, *Methods and Results for Quantum Optimal Pulse Control on Superconducting Qubit Systems*, 2022 IEEE International Parallel and Distributed Processing Symposium Workshops (2022)  $[$[arXiv:2202.03260](https://arxiv.org/abs/2202.03260), [doi:10.1109/IPDPSW55747.2022.00102](https://doi.org/10.1109/IPDPSW55747.2022.00102)$]$

Corrections due to [[quasiparticle]]-excitations:

* [[Leonid I. Glazman]], [[Gianluigi Catelani]], *Bogoliubov Quasiparticles in Superconducting Qubits*, SciPost Phys. Lect. Notes **31** (2021) [[arXiv:2003.04366](https://arxiv.org/abs/2003.04366), [doi:10.21468/SciPostPhysLectNotes.31](https://doi.org/10.21468/SciPostPhysLectNotes.31)]


superconductivity

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{:toc}

## Idea

(...)

## Properties

[[!include flux quantization in superconductors -- section]]

## Related concepts

* [[Cooper pair]]

* [[Landau-Ginzburg model]]

* [[Josephson junction]]

* [[AdS/CFT in condensed matter physics]]

* [[colour superconductivity]]

## References

### General

Comprehensive survey:

* {#nthu} www.phys.nthu.edu.tw: *Superconductivity*, lecture slides [I:[pdf](http://www.phys.nthu.edu.tw/~spin/course/105S/Chapter%2010%20-1.pdf), II:[pdf](http://www.phys.nthu.edu.tw/~spin/course/106S/Ch10-2.pdf), III:[pdf](http://www.phys.nthu.edu.tw/~spin/course/106S/Ch10-3.pdf), full:[[www.phys.nthu.edu.tw--Superconductivity.pdf:file]]]

Theoretical introduction and review:

* [[Lev Landau]], [[Evgeny Lifshitz]], Section VI of: *Electrodynamics of Continuous Media*, volume 8 of: *[[Course of Theoretical Physics]]*, Pergamon Press (1960, 1984) [[ISBN:9780750626347](https://shop.elsevier.com/books/electrodynamics-of-continuous-media/koch/978-0-08-057060-0), [archive](https://archive.org/details/ost-physics-landaulifshitz-electrodynamicsofcontinuousmedia/mode/2up),  [pdf](https://www.aldebaran.cz/studium/books/1984_Landau-Electrodynamics.pdf)]

* [[Alexander L. Fetter]], [[John Dirk Walecka]], Ch. 13 of: *Quantum theory of many-particle systems*, Mcgraw-Hill (1991); Dover (2003) [[archive.org](https://archive.org/details/quantum-theory-of-many-particle-systems-by-alexander-l.-fetter-john-dirk-walecka-physics-z-lib.org/mode/thumb)]

* {#Chapman00} S. J. Chapman, _A Hierarchy of Models for Type-II Superconductors_, IAM Review Vol. 42, No. 4 (2000), pp. 555-598 ([jstor:2653134](https://www.jstor.org/stable/2653134))

* {#Timm20} [[Carsten Timm]], *Theory of Superconductivity*, lecture notes (2020-2023) [[pdf](https://tu-dresden.de/mn/physik/itp/cmt/ressourcen/dateien/skripte/Skript_Supra.pdf?lang=en), [[Timm-Superconductivity.pdf:file]]]

See also:

* Wikipedia, _[Superconductivity](http://en.wikipedia.org/wiki/Superconductivity)_

* Wikipedia, _[Meissner effect](https://en.wikipedia.org/wiki/Meissner_effect)_

Textbook account in the context of [[quantum materials]] and [[topological insulators]]:

* [[Tudor D. Stanescu]], Section II.5 of: *Introduction to Topological Quantum Matter & Quantum Computation*, CRC Press 2020 ([ISBN:9780367574116](https://www.routledge.com/Introduction-to-Topological-Quantum-Matter--Quantum-Computation/Stanescu/p/book/9780367574116))

Introducing [[Landau-Ginzburg models]] in [[superconductivity]]:

* [[Vitaly Ginzburg]], [[Lev Landau]], _On the Theory of Superconductivity_,  reprinted In: _On Superconductivity and Superfluidity_ Springer (2009) ([doi:1007/978-3-540-68008-6_4](https://doi.org/10.1007/978-3-540-68008-6_4))

Introducing the [[dimer model]]:

* Daniel S. Rokhsar and Steven A. Kivelson, *Superconductivity and the Quantum Hard-Core Dimer Gas*, Phys. Rev. Lett. 61, 2376 – Published 14 November 1988 ([arXiv:10.1103/PhysRevLett.61.2376](https://doi.org/10.1103/PhysRevLett.61.2376))

### Vortices and flux quantization

Original articles:

* {#Abrikosov57a} [[Alexei Abrikosov]], *On the Magnetic properties of superconductors of the second group*, Sov. Phys. JETP **5** (1957) 1174-1182; Zh. Eksp. Teor. Fiz. **32** (1957) 1442-1452 [[spire:9138](https://inspirehep.net/literature/9138), [[Abrikosov-MagneticPropertiesOfSuperconductors.pdf:file]]]

* {#Abrikosov57} [[Alexei Abrikosov]], _The magnetic properties of superconducting alloys_, Journal of Physics and Chemistry of Solids Volume 2, Issue 3, 1957, Pages 199-208 (doi:10.1016/0022-3697(57)90083-5)

Further discussion:

* A. M. Campbell, *Maxwell's Equations in Superconductors*,  IEEE Transactions on Applied Superconductivity **17** 2 (2007) [[doi:10.1109/TASC.2007.900042](https://doi.org/10.1109/TASC.2007.900042)]

* {#BeekmanZaanen11} [[Aron J. Beekman]], [[Jan Zaanen]], *Electrodynamics of Abrikosov vortices: the field theoretical formulation*, Front. Phys. **6** (2011) 357–369 [[doi:10.1007/s11467-011-0205-0](https://doi.org/10.1007/s11467-011-0205-0)]

Lecture notes:

* {#Yu2012} Xiongjie Yu, *Vortices in Type-II Superconductors*, essay for *[Emergent States of Matter](https://guava.physics.uiuc.edu/~nigel/courses/569/Essays_Fall2012/)*, Urbana-Champaign (2012) [[pdf](https://guava.physics.uiuc.edu/~nigel/courses/569/Essays_Fall2012/Files/yu.pdf), [[Yu-VorticesInSuperconductors.pdf:file]]]

* [Timm (2020), pp. 48](#Timm20)

See also

* Wikipedia, _[Abrikosov vortex](https://en.wikipedia.org/wiki/Abrikosov_vortex)_

First [[experiment|experimental]] detection of flux quantization in superconductors:

* Bascom S. Deaver, Jr., William M. Fairbank, _Experimental Evidence for Quantized Flux in Superconducting Cylinders_, Phys. Rev. Lett. 7, 43 – Published 15 (1961) ([doi:10.1103/PhysRevLett.7.43](https://doi.org/10.1103/PhysRevLett.7.43))

* R. Doll, M. Näbauer, _Experimental Proof of Magnetic Flux Quantization in a Superconducting Ring_, Phys. Rev. Lett. 7, 51 (1961) ([doi:10.1103/PhysRevLett.7.51](https://doi.org/10.1103/PhysRevLett.7.51))

More on the experimental detection magnetic flux quantization and vortices:

* {#LoudonMidgley09} J. C. Loudon, P. A. Midgley, *Imaging Flux Vortices in Type II Superconductors with a Commercial Transmission Electron Microscope*, Ultramicroscopy **109** 6 (2009) 700-729 [[arXiv:0807.2401](https://arxiv.org/abs/0807.2401), [doi:10.1016/j.ultramic.2009.01.008](https://doi.org/10.1016/j.ultramic.2009.01.008)]

More theoretically flavored discussion of the flux quantization mechanism:

* {#AlvarezGaume98} [[Luis Alvarez-Gaumé]], Frederic Zamora, Section IV.B of: _Duality in Quantum Field Theory (and String Theory)_, AIP Conference Proceedings 423, 46 (1998) ([arXiv:hep-th/9709180](https://arxiv.org/abs/hep-th/9709180))

On the vortex [[flux tubes]] as dynamical [[vortex string|vortex strings]]:

* [Abrikosov 57a](#Abrikosov57a)

* [[Holger Bech Nielsen]], [[Poul Olesen]], _Vortex-line models for dual strings_, Nuclear Physics B Volume 61, 24 September 1973, Pages 45-61 (doi:10.1016/0550-3213(73)90350-7)

* [[David Tong]], _Quantum Vortex Strings: A Review_, Annals Phys. 324:30-52, 2009 ([arXiv:0809.5060](https://arxiv.org/abs/0809.5060))

On [[superfluidity]]/superconductivity *of* [[anyons]]:

* Yi-Hong Chen, [[Frank Wilczek]], [[Edward Witten]], [[Bertrand Halperin]], *On Anyon Superconductivity*,  International Journal of Modern Physics B **03** 07 (1989) 1001-1067 (reprinted in [Wilczek 1990](#Wilczek90))  $[$[doi:10.1142/S0217979289000725](https://doi.org/10.1142/S0217979289000725), [[CWWH-AnyonSuperfluidity.pdf:file]]$]$

[[!include superconducting qbits -- references]]

[[!include anyons in topological superconductors -- references]]

### Discussion via AdS/CFT

Discussion of [[superconductivity]] via [[AdS/CFT in condensed matter physics]]:

* [[Sean Hartnoll]], [[Christopher Herzog]], [[Gary
 Horowitz]], _Building an AdS/CFT superconductor_, Phys. Rev. Lett. 101:031601, 2008 ([arXiv:0803.3295](https://arxiv.org/abs/0803.3295))

* Alberto Salvio, _Superconductivity, Superfluidity and Holography_ ([arXiv:1301.0201](http://arxiv.org/abs/1301.0201))

* Rong-Gen Cai, Li Li, Li-Fang Li, Run-Qiu Yang, _Introduction to Holographic Superconductor Models_, Sci China-Phys Mech Astron, 2015, 58(6):060401 ([arXiv:1502.00437](https://arxiv.org/abs/1502.00437))

* {#HartnollLucasSachdev16} [[Sean Hartnoll]], [[Andrew Lucas]], [[Subir Sachdev]], Section 6.3 of: _Holographic quantum matter_, MIT Press 2018 ([arXiv:1612.07324](https://arxiv.org/abs/1612.07324), [publisher](https://mitpress.ublish.com/book/holographic-quantum-matter))

* Yiqian Chen, Xiaobo Guo, Peng Wang, *Holographic Superconductors in a Non-minimally Coupled Einstein-Maxwell-scalar Model* ([arXiv:2111.03810](https://arxiv.org/abs/2111.03810))

* Chuan-Yin Xia, Hua-Bi Zeng, Yu Tian, Chiang-Mei Chen, [[Jan Zaanen]], *Holographic Abrikosov lattice: vortex matter from black hole* ([arXiv:2111.07718](https://arxiv.org/abs/2111.07718))

* Dong Wang, Xiongying Qiao, Qiyuan Pan, Chuyu Lai, Jiliang Jing, *Holographic entanglement entropy and complexity for the excited states of holographic superconductors* [[arXiv:2301.00513](https://arxiv.org/abs/2301.00513)]

  > (in relation to [[holographic entanglement entropy]])

### High $T_c$-superconductivity

Via [[topological phases of matter]]:

* M. C. Diamantini, C. A. Trugenberger, V. M. Vinokur, _Topological Nature of High Temperature Superconductivity_ ([arXiv:2009.01763](https://arxiv.org/abs/2009.01763))

[[!redirects superconductor]]
[[!redirects superconductors]]

[[!redirects topological superconductor]]
[[!redirects topological superconductors]]

[[!redirects type I superconductor]]
[[!redirects type I superconductors]]

[[!redirects type II superconductor]]
[[!redirects type II superconductors]]

[[!redirects Meissner effect]]
[[!redirects Meissner-Ochsenfeld effect]]


superconformal geometry

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## Idea

The joint generalization of [[conformal geometry]] and [[supergeometry]].

## Classification

See at _[Supersymmetry -- Classification of superconformal algebras](supersymmetry#ClassificationSuperconformal)_

## Related concepts

* [[supersymmetry]]

* [[superconformal super Lie group]]

* [[superconformal field theory]]

## References

### General

* {#Nahm78} [[Werner Nahm]], _[[Supersymmetries and their Representations]]_, Nucl.Phys. B135 (1978) 149 ([spire](https://inspirehep.net/record/120988/), [pdf](http://cds.cern.ch/record/132743/files/197709213.pdf))

* {#Shnider88} [[Steven Shnider]], _The superconformal algebra in higher dimensions_, Letters in Mathematical Physics November 1988, Volume 16, Issue 4, pp 377-383

* [[Andree Lischewski]], _Conformal superalgebras via tractor calculus_, Class.Quant.Grav. 32 (2015) 015020 ([spire](https://inspirehep.net/record/1334198), [arXiv:1408.2238](http://arxiv.org/abs/1408.2238))

* _Conformal and superconformal symmetries_ ([pdf](http://bolvan.ph.utexas.edu/~vadim/Classes/2009f/SCA.pdf))

### In dimension $6 = 5+1$

Discussion of the superconformal symmetry group, the [[orthosymplectic supergroup]] $OSp(8^\ast,2N)$, includes

* [[Sergio Ferrara]], Emery Sokatchev, _Representations of superconformal algebras in the $AdS_{7/4}/CFT_{6/3}$ correspondence_, J.Math.Phys. 42 (2001) 3015-3026 ([arXiv:hep-th/0010117](http://arxiv.org/abs/hep-th/0010117))

* Murat Gunaydin, Seiji Takemae, _Unitary Supermultiplets of $OSp(8^*|4)$ and the $AdS_7/CFT_6$ Duality_, Nucl.Phys.B578:405-448,2000; Erratum-ibid.B697:399-402,2004 ([arXiv:hep-th/9910110](http://arxiv.org/abs/hep-th/9910110))

[[!redirects superconformal]]

[[!redirects superconformal symmetry]]
[[!redirects superconformal symmetries]]


superconformal multiplet

#Contents#
* table of contents
{:toc}

## Idea

[[super multiplets]] for [[superconformal groups]].

## Supersingletons

There are special such superconformal multiplet, which have no analog as super Poincare multiplet, called _supesingletons_, first observe by ([Dirac 63](#Dirac63))

## References

* {#Dirac63} [[P. A. M. Dirac]], _A remarkable representation of the 3+2 de Sitter group_, J. Math. Phys. 4 (1963) 901.

* [[Christian Fronsdal]], _Dirac supermultiplet_, Phys. Rev. D **26** (1982) 1988 [[doi:10.1103/PhysRevD.26.1988](https://doi.org/10.1103/PhysRevD.26.1988)]

* [[Moshé Flato]], [[Christian Fronsdal]], _Quantum field theory of singletons. The rac_, J. Math. Phys. **22** (1981) 1100 [[doi:10.1063/1.524993](https://doi.org/10.1063/1.524993)]

* [[Murat Günaydin]], _Singleton and doubleton supermultiplets of space-time supergroups and infinite spin superalgebras, in [[Mike Duff]], [[Christopher Pope]] and [[Ergin Sezgin]] (eds.) Proceedings of the 1989 Trieste Conference "Supermembranes and Physics in 2 + 1 Dimensions",  World Scientific (1990)

Discussion of the [[perturbation theory]] of [[Green-Schwarz sigma-models]] around clssical solutions stretched along asymptotic boundaries of [[anti de Sitter spacetimes]]:

* [[Mike Duff]], _Anti-de Sitter space, branes, singletons, superconformal field theories and all that_, Int.J.Mod.Phys.A14:815-844,1999 ([arXiv:hep-th/9808100](https://arxiv.org/abs/hep-th/9808100))

[[!redirects superconformal multiplets]]

[[!redirects Dirac supermultiplet]]
[[!redirects Dirac supermultiplets]]

[[!redirects supersingleton]]
[[!redirects supersingletons]]


superconformal super Lie group

## Idea

A _superconformal super Lie group_ is a [[Lie group]] [[analogy|analog]] of a [[super Lie algebra]] which extends the [[Lie algebra]] of a [[conformal group]], hence the [[Lie integration]] of a [[superconformal super Lie algebra]]. See at _[Supersymmetry -- Classification -- Superconformal symmetry](supersymmetry#ClassificationSuperconformal)_.

## Related concepts

* [[super Poincare group]]

* [[super anti de Sitter group]], [[super anti de Sitter spacetime]]

* [[superconformal multiplet]]

[[!include table of orthogonal groups and related]]

[[!redirects superconformal group]]
[[!redirects superconformal groups]]

[[!redirects superconformal super Lie algebra]]

[[!redirects super anti de Sitter group]]


superconformal symmetry -- table

| $d$ | $N$ | [[superconformal super Lie algebra]] | [[R-symmetry]] | [[black brane]] [[worldvolume]]
 [[superconformal field theory]]
 via [[AdS-CFT]] |
|-----|-----|--|---|--|
| $\phantom{A}3\phantom{A}$  |  $\phantom{A}2k+1\phantom{A}$ | $\phantom{A}B(k,2) \simeq $ [[osp]]$(2k+1 \vert 4)\phantom{A}$ | $\phantom{A}SO(2k+1)\phantom{A}$ | |
|  $\phantom{A}3\phantom{A}$   | $\phantom{A}2k\phantom{A}$ | $\phantom{A}D(k,2)\simeq $ [[osp]]$(2k \vert 4)\phantom{A}$ | $\phantom{A}SO(2k)\phantom{A}$ | [[M2-brane]]
 [[D=3 SYM]]
 [[BLG model]]
 [[ABJM model]] |
|  $\phantom{A}4\phantom{A}$   | $\phantom{A}k+1\phantom{A}$  | $\phantom{A}A(3,k)\simeq \mathfrak{sl}(4 \vert k+1)\phantom{A}$ | $\phantom{A}U(k+1)\phantom{A}$ | [[D3-brane]]
 [[D=4 N=4 SYM]]
 [[D=4 N=2 SYM]]
 [[D=4 N=1 SYM]] |
|  $\phantom{A}5\phantom{A}$   |  $\phantom{A}1\phantom{A}$  | $\phantom{A}F(4)\phantom{A}$ | $\phantom{A}SO(3)\phantom{A}$  | [[D4-brane]]
 [[D=5 SYM]] |
|  $\phantom{A}6\phantom{A}$   | $\phantom{A}k\phantom{A}$ | $\phantom{A}D(4,k) \simeq$ [[osp]]$(8 \vert 2k)\phantom{A}$ | $\phantom{A}Sp(k)\phantom{A}$  | [[M5-brane]]
 [[D=6 N=(2,0) SCFT]]
 [[D=6 N=(1,0) SCFT]] |

([Shnider 88](supersymmetry#Shnider88), also [Nahm 78](supersymmetry#Nahm78), see [Minwalla 98, section 4.2](supersymmetry#Minwalla98))


superconnection

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## Idea

The notion of _superconnection_ generalizes the notion of [[connection on a bundle]] from the [[internalization|context]] of [[differential geometry]] to that of [[supergeometry]].

An [[connection on a bundle|ordinary connection on a vectorbundle]] is given by a suitable [[functor]] $P_1(X) \to Vect$ on the [[path groupoid]] of some [[manifold]] $X$ -- its [[parallel transport]] functor. Here a path is a smooth map $I \to X$ from an [[interval]] $I_t = [0,t] \subset \mathbb{R}^1$ to $X$. A _superconnection_ is more generally given by a functor on _superpaths_ in $X$, where a superpath is a map on superintervals $I_{t,theta} \subset \mathbb{R}^{1|1}$.

## Definition

...

## Push-forward

### Idea

There is a natural notion of push-forward of superconnections along maps $\pi : Y \to X$ of [[manifold]]s whose [[fibers]] are [[compact space|compact]] [[spin structure|spin manifold]]s. Under this push-forward the different components of a superconnection mix. In particular, the push-forward of an ordinary connection in this sense is in general a superconnection.

The push-forward of superconnections corresponds to (...details...) the push-forward in [[topological K-theory]] and [[differential K-theory]]. Bismut famously originally found a superconnection formula for the [[Chern character]] of a pushed [[K-theory]] class. See the references below.

### Details

Let $E \to Y$ be a Hermitian $\mathbb{Z}_2$-graded [[vector bundle]] of finite rank with superconnection $\nabla = \nabla^E + \omega$ with ordinary connection part $\nabla^E$.

The push-forward of $E$ along $\pi$ is the $\mathbb{Z}_2$-graed  vector bundle $\pi_* E \to X$ of infinite rank whose fiber over $x \in X$ is the space of [[section]]s of the [[tensor product]] of the spin bundle over $Y_x$ and $E_y$

$$
  (\pi_* E)_x = \Gamma(\mathbb{S}(Y/X)_x \otimes E_x)
  \,.
$$

The pushed connection $\pi_* \nabla$ on $\pi_* E$ is given by

$$
  \pi_* \nabla = D^\pi(\nabla^E) + \nabla^{\pi}_{spin}\otimes Id + Id \otimes \nabla^E
  + \frac{1}{4}c^\pi(T^\pi) + \pi_* \omega
 \,.
$$

### Example: push-forward of ordinary connection to point

So in particular when $X = {*}$ is the [[point]] and $\nabla = \nabla^E$ is an ordinary connection, we find that the push-forward of an ordinary connection on a vector bundle $E$ on a Riemannian spin manifold $Y$ to the point is the [[Dirac operator]] $D(\nabla^E)$ acting on the space of [[section]]s of $E$ and regarded as the odd endomorphism-valued 0-form part of a superconnection on the point.

By Dumitrescu's formula for the parallel transport of a superconnection the parallel transport of this $\pi_*\nabla$ along the ordinary interval $I_{t,0}$ of length $t$ is the endomorphism

$$
  e^{-t D(\nabla^E)^2} : \Gamma(E) \to \Gamma(E)
  \,.
$$

This happens to be the (Euclidean) [[quantum mechanics]] time evolution operator for the [[sigma-model]] given by the spinning particle on $Y$ charged under the connection $\nabla$.

## Related concepts

* [[super parallel transport]]

* [[Bismut superconnection]]

## References

The geometric interpretation of superconnections in terms of parallel transport along superpaths is due to

* [[Florin Dumitrescu]], _Superconnections and parallel transport_ ([arXiv](http://arxiv.org/abs/0711.2766))

The algebraic formulation of superconnections as differential operators on the algebra of differential forms with values in endomorphisms of a $\mathbb{Z}_2$-graded [[vector bundle]] is much older, due to

* Daniel Quillen, _Superconnections and the Chern character_  Topology, 24(1):89–95, 1985.

There the notion of a superconnection was introduced as a means to encode the difference of the chern characters of two vector bundles, motivated from [[topological K-theory]].

This was extended to the parameterized ("families") version in

* Jean-Michel Bismut, _The Atiyah-Singer index theorem for families of Dirac operators: Two heat equation proofs_

Bismut also showed that under the push-forward in [[topological K-theory]] superconnections naturally appear even if one starts with just an ordinary [[connection]].

This statement is generalized to a complete notion of push-forward of superconnections from vector bundles on a space $Y$ to vector bundles un a space $X$ along maps $\pi : Y \to X$ in

* [[Alexander Kahle]], _Superconnections and index theory_ ([arXiv](http://arxiv.org/abs/0810.0820), [talk notes](http://ncatlab.org/nlab/show/Oberwolfach+Workshop,+June+2009+--+Strings,+Fields,+Topology#kahle))

More on [[Chern-Weil theory]] of superconnections is in

* Sylvie Paycha, Simon Scott, _Chern-Weil forms associated with superconnections_ ([pdf](http://math.univ-bpclermont.fr/~paycha/articles/PaychaScottfinalproceedings.pdf))