super QCD

#Contents#
* table of contents
{:toc}

## Idea

Super QCD is a [[super Yang-Mills theory]] coupled to [[matter]] which is a [[supersymmetry|supersymmetric]] extension of [[quantum chromodynamics]] ([[QCD]], an ordinary [[Yang-Mills theory]] coupled to matter), possibly as one sector of a [[supersymmetry|supersymmetric]] extension ([[MSSM]]) of the [[standard model of particle physics]]: In super QCD the [[quarks]] and [[gluons]] have [[superpartners]] given by [[squarks]] and [[gluinos]].

## References

### General

See also:

* Wikipedia, _[Super QCD](https://en.m.wikipedia.org/wiki/Super_QCD)

On [[supersymmetry breaking]] from [[super QCD]] to [[QCD]]:

* Francesco Sannino, _From Super QCD to QCD_, AIP Conference Proceedings 452, 245 (1998) ([arXiv:hep-th/9806229](https://arxiv.org/abs/hep-th/9806229))

### Vortex strings

On [[vortex strings]] in $\mathcal{N}=2$ super QCD identified as [[superstrings]] in [[type II string theory]] [[KK-compactification|compactified]] on a [[conifold]]:

* [[Mikhail Shifman]], [[A. Yung]], *Supersymmetric Solitons and How They Help Us Understand Non-Abelian Gauge Theories*, Rev. Mod. Phys. **79** 1139 (2007) [[doi:10.1103/RevModPhys.79.1139](https://doi.org/10.1103/RevModPhys.79.1139), [arXiv:hep-th/0703267](https://arxiv.org/abs/hep-th/0703267)

* [[Mikhail Shifman]], [[A. Yung]], *Supersymmetric Solitons*, Cambridge University Press (2009) [[doi:10.1017/CBO9780511575693](https://doi.org/10.1017/CBO9780511575693)]

* [[Mikhail Shifman]], [[A. Yung]], *Critical String from Non-Abelian Vortex in Four Dimensions*,
Physics Letters B **750** (2015) 416-419
[[arXiv:1502.00683](https://arxiv.org/abs/1502.00683), [doi:10.1016/j.physletb.2015.09.045](https://doi.org/10.1016/j.physletb.2015.09.045)
]

* P. Koroteev, [[Mikhail Shifman]], [[A. Yung]], *Studying Critical String Emerging from Non-Abelian Vortex in Four Dimensions*, Phys.Lett. B **759** (2016) 154-158
[[arXiv:1605.01472](https://arxiv.org/abs/1605.01472), [doi:10.1016/j.physletb.2016.05.075](https://doi.org/10.1016/j.physletb.2016.05.075)
]

and interpreted in [[AdS/QCD]], in the presence of the [[Kalb-Ramond field]]:

* [[A. Yung]], *NS Three-form Flux Deformation for the Critical Non-Abelian Vortex String* [[arXiv:2209.08118](https://arxiv.org/abs/2209.08118)]

[[!redirects super quantum chromodynamics]]


super Riemann surface

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###Context###
#### Supergeometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _super Riemann surface_ is the analog of a [[Riemann surface]] in [[supergeometry]].

## Examples

A [[Riemann surface]] with [[spin structure]] that is also a [[branched cover]] of $\mathbb{C}P^{(1 \vert 1)}$ is canonically endowed with the structure of a super Riemann surface with [[Ramond punctures]] ([Donagi-Witten 15](#DonagiWitten15)). This implies that [[adinkras]] encode certain very special super Riemann surfaces [Doran & Iga & Landweber & Mendez-Diez 13](#DoranIgaLandweberMendez-Diez13).

## Properties

[[!include non-projected super Riemann surface moduli -- section]]

## Related concepts

* [[superstring]]

* [[string scattering amplitude]]

* [[picture changing operator]]

* [[adinkra]]

## References

### General

The concept of super Riemann surfaces originates with the following articles:

* [[Daniel Friedan]], _Notes On String Theory and 2-Dimensional Conformal Field Theory_. 1986

* M.A. Baranov, [[Albert Schwarz]],  _On the Multiloop Contribution to the String Theory_ Int.J.Mod.Phys., A2:1773, 1987

* [[Yuri Manin]], _Critical Dimensions of String Theories and the Dualizing Sheaf on the Moduli Space of (Super) Curves_, Funct.Anal.Appl., 20:244-245,
1987

* [[Steve Giddings]], P. Nelson, _The Geometry of super Riemann surfaces_, Commun. Math. Phys., 116, (1988), 607

See also

* {#DHokerPhong05} [[Eric D'Hoker]], [[Duong Phong]], _Complex geometry and supergeometry_, Current Developments in Mathematics 2005 (2007): 1-40 ([euclid:1223654523](https://projecteuclid.org/euclid.cdm/1223654523))

Discussion of super Riemann surfaces induced by [[supermultiplets]] for $N$-extended $d = 1$ supersymmetry -- via [[adinkra]] symbols, is due to

* {#DoranIgaLandweberMendez-Diez13} [[Charles Doran]], [[Kevin Iga]], [[Greg Landweber]], [[Stefan Méndez-Diez]], _Geometrization of $\mathcal{N}$-Extended 1-Dimensional Supersymmetry Algebras_ ([arXiv:1311.3736](https://arxiv.org/abs/1311.3736))

* {#DoranIgaKostiukMendes-Diez16} [[Charles Doran]], [[Kevin Iga]], Jordan Kostiuk, [[Stefan Méndez-Diez]], _Geometrization of $\mathcal{N}$-Extended 1-Dimensional Supersymmetry Algebras II_ ([arXiv:1610.09983](https://arxiv.org/abs/1610.09983))

[[!include super-moduli of super Riemann surfaces -- references]]

[[!redirects super Riemann surfaces]]

[[!redirects super-Riemann surface]]
[[!redirects super-Riemann surfaces]]

[[!redirects Ramond puncture]]
[[!redirects Ramond punctures]]


super ring

[[!redirects super rings]]

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###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Super-Algebra and Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea ##

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

## Definition ##

A **super ring** is a [[ring]] $R$ with decomposition functions $\mathcal{D}_0:R \to R$ and $\mathcal{D}_1:R \to R$, such that

* for all $a:R$, $a = \mathcal{D}_0(a) + \mathcal{D}_1(a)$
* for all $a:R$, and $b:R$, $\mathcal{D}_0(a + b) = \mathcal{D}_0(a) + \mathcal{D}_0(b)$
* for all $a:R$, and $b:R$, $\mathcal{D}_1(a + b) = \mathcal{D}_1(a) + \mathcal{D}_1(b)$
* for all $a:R$, and $b:R$, $\mathcal{D}_0(a \cdot b) = \mathcal{D}_0(a) \cdot \mathcal{D}_0(b) - \mathcal{D}_1(a) \cdot \mathcal{D}_1(b)$
* for all $a:R$, and $b:R$, $\mathcal{D}_1(a \cdot b) = \mathcal{D}_0(a) \cdot \mathcal{D}_1(b) + \mathcal{D}_1(a) \cdot \mathcal{D}_0(b)$
* for all $a:R$, and $b:R$, $\mathcal{D}_0(1) = 1$
* for all $a:R$, and $b:R$, $\mathcal{D}_1(1) = 1$
* for all $a:R$, $\mathcal{D}_0(\mathcal{D}_0(a)) = \mathcal{D}_0(a)$
* for all $a:R$, $\mathcal{D}_0(\mathcal{D}_1(a)) = 0$
* for all $a:R$, $\mathcal{D}_1(\mathcal{D}_0(a)) = 0$
* for all $a:R$, $\mathcal{D}_1(\mathcal{D}_1(a)) = \mathcal{D}_1(a)$

As a result, the [[image]] of the two decompostion functions $\im(\mathcal{D}_0)$ and $\im(\mathcal{D}_1)$ are rings and there exists an [[abelian group]] [[isomorphism]] $i:V \cong \im(\mathcal{D}_0) \otimes \im(\mathcal{D}_1)$, where $F:Ring \to Ab$ is a forgetful functor and $F(A) \otimes F(B)$ is the [[tensor product of abelian groups]].

The elements of $\im(\mathcal{D}_0)$ are called **even elements** or **bosonic elements**, and the elements of $\im(\mathcal{D}_1)$ are called **odd elements** or **fermionic elements**.

## See also ##

* [[ring]]

* [[differential ring]]

* [[exterior ring]]

* [[Clifford ring]]

* [[graded ring]]

* [[super abelian group]]

* [[super algebra]]

* [[supercommutative ring]]


super smooth set

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#### Geometry
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[[!include higher geometry - contents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--

#Contens#
* table of contents
{:toc}

## Idea

Just as a [[smooth set]] is a [[generalized smooth space]] in [[differential geometry]] modeled as a [[sheaf]] on the [[category]] of [[Cartesian spaces]] with [[smooth functions]] between them, so a **super smooth set** is a sheaf on the category of [[super Cartesian spaces]], being a generalized space in [[supergeometry]].

Since super smooth sets contain [[infinitesimal spaces]], it makes good sense to make this explicit and consider **super formal smooth sets** right away, hence [[sheaves]] on [[super formal Cartesian spaces]]

For details see at _[[geometry of physics -- supergeometry]]_.

## Related concepts

[[!include geometries of physics -- table]]

[[!redirects super smooth sets]]

[[!redirects super formal smooth set]]
[[!redirects super formal smooth sets]]

[[!redirects SuperFormalSmoothSet]]
[[!redirects SuperFormalSmoothSets]]


super smooth topos

<div class="rightHandSide toc">
[[!include supergeometry - contents]]

***

[[!include synthetic differential geometry - contents]]

</div>

#Contents#
* automatic table of contents goes here
{:toc}

#Idea#

The notion of _super smooth topos_ is to that of [[smooth topos]] as [[supergeometry]] is to [[differential geometry]].

The objects of a super smooth topos are generalizations of smooth [[supermanifold]]s that admit besides the usual odd infinitesimals of [[supergeometry]] also the even [[infinitesimal object]]s of [[synthetic differential geometry]].

The study of super smooth toposes is the content of [[synthetic differential supergeometry]]. (See there for references and details for the moment.)

# Definition #

A **super smooth topos** $(\mathcal{T}, R)$ is a [[smooth topos]] together with the refinement of the $k$-algebra structure on $R$ to that of a $k$-[[superalgebra]] structure.

So a super smooth topos is a [[topos]] $\mathcal{T}$ equipped with a [[superalgebra]] object $(R, +, \cdot)$ with even part $R_e$ and odd part $R_o$ etc.

An algebra spectrum object is now an internal object of superalgebra homomorphisms and the condition is that for every super Weil algebra $W = R \oplus m$ we have that $Spec(W) = R SAlg_{\mathcal{T}}(W,R)$ is an [[infinitesimal object]] and that $W \to R^{Spec W}$ is an [[isomorphism]].

This means that essentially all the standard general theory of smooth toposes goes through literally for super smooth toposes, too. The main difference is that a super smooth topos contains more types of [[infinitesimal object]]s.

There is for instance still the standard _even_ infinitesimal interval

$$
  D := D^{1|0} := \{\epsilon \in R_e | \epsilon^2 = 0\}
$$

but there is now also the _odd_ infinitesimal interval

$$
  D^{0|1} := \{\theta \in R_o \}
  \,.
$$

Notice that in the graded commutative algebra $A$ every odd element $\theta$ automatically squares to 0.

> [[Urs Schreiber]]: I'd think that the cominatorial/simplicial definition of [[differential forms in synthetic differential geometry]] applied verbatim in a super smooth topos automatically yields the right/expected notion of differential forms in supergeometry.

# Examples #

Models for super smooth toposes are constructed in

* D. N. Yetter, _Models for synthetic supergeometry_, Cahiers, 29, 2 (1988)

and

* H. Nishimura, _Supersmooth topoi_, International Journal of Theoretical Physics, Volume 39, Number 5 ([journal](http://www.springerlink.com/content/v2573726852x6470/))


super spacetime

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###Context###
#### Gravity
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[[!include gravity contents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[spacetime]] in [[supergeometry]], used in some formulations of [[supergravity]] (see at [[D'Auria-Fre formulation of supergravity]]) and as [[target space]] for super-$p$-brane [[sigma-models]] (see at [[Green-Schwarz action functional]]).

## Examples

* [[super Minkowski spacetime]]

* [[super anti de Sitter spacetime]]

* [[super-vielbein]]

* [[Green-Schwarz action functional]]

## References

Section 7.4 of

* [[Veeravalli Varadarajan]], _[[Supersymmetry for mathematicians]]_, Courant lecture notes in mathematics, American Mathematical Society Providence, R.I 2004
 {#Varadarajan04}

[[!redirects super spacetimes]]

[[!redirects super-spacetime]]
[[!redirects super-spacetimes]]

[[!redirects superspacetime]]
[[!redirects superspacetimes]]


super stack

## Idea

A [[stack]] on the [[site]] of [[supermanifolds]], or similar.

See at

* [[super ∞-groupoid]]

* [[smooth super ∞-groupoid]]

[[!redirects super stacks]]


super tangent bundle

# Super tangent bundles
* table of contents
{: toc}

## Idea

The [[tangent bundle]] of an ordinary [[manifold]] is a [[vector bundle]] whose [[sheaf of sections]] is given by the [[derivation]]s of the [[structure sheaf]]. The same idea applies to a [[supermanifold]] to produce a [[super vector bundle]].

## Definition

The **super tangent bundle** $T X$ of a [[supermanifold]] $X$ is given by the sheaf $U \mapsto Der O_X(U)$.

So a __super tangent vector__ is a [[global section]] of this sheaf of derivations.

## Example

On the  [[supermanifold]] $\mathbb{R}^{1|1}$ with its canonical coordinates

$$
  t \in C^\infty(\mathbb{R}^{1|1})^{ev}
$$

$$
  \theta \in C^\infty(\mathbb{R}^{1|1})^{odd}
$$

there is the odd vector field

$$
  D \coloneqq \partial_\theta + \theta \cdot \partial_{t}
$$

whose super Lie bracket with itself vanishes

$$
  [D, D] = 0
  \,.
$$
=--

+-- {: .un_thm}
###### Claim
This odd vector field $D$ is left invariant with respect to the [[supergroup|super translation group]] structure on $\mathbb{R}^{1|1}$.
=--

This means that $Lie(\mathbb{R}^{1|1})$ is free on one odd generator.

[[!redirects super tangent bundle]]
[[!redirects super tangent bundles]]
[[!redirects super tangent bundle of a supermanifold]]
[[!redirects super tangent bundles of supermanifolds]]
[[!redirects super tangent bundle is supermanifold]]

[[!redirects super tangent vector]]
[[!redirects super tangent vectors]]


super vector bundle

## Idea

The concept of _super vector bundle_ is the generalization of that of _[[smooth vector bundle]]_ from [[differential geometry]] to [[supergeometry]]: A super vector bundle is a [[fiber bundle]] over a [[supermanifold]] whose [[fibers]] are [[super vector spaces]] in a compactible way.

[[!redirects super vector bundles]]


super vector space

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###Context###
#### Super-Algebra and Super-Geometry
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[[!include supergeometry - contents]]
=--
#### Linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A **super vector space** is an [[object]] in the non-trivial [[symmetric monoidal category]] structure on the [[monoidal category]] of $\mathbb{Z}/2$-[[graded vector spaces]]: as an object it is just a $\mathbb{Z}/2$-[[graded vector space]], but the [[braiding]] of the underlying [[tensor product of vector spaces]] is taken to be the non-trivial linear map which on elements of homogeneous degree is given by

$$
  \tau^{super}
   \;\colon\;
  v \otimes w \;\mapsto\; (-1)^{deg(v) deg(w)} \, w \otimes v
  \,.
$$

We make this precise as definition \ref{CategoryOfSuperVectorSpaces}  below.

Super vector spaces form the basis of [[superalgebra]] (over [[ground rings]] which are [[fields]]) in direct analogy of how ordinary vector spaces form the basis of ordinary [[algebra]]. For more on this see [below](#StructuresInternalToSuperVectorSpaces), and for yet more see at _[[geometry of physics -- superalgebra]]_.

## Details

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

For $k$ a [[field]], we write $Vect_k$ for the [[category]] whose

* [[object|objects]] are $k$-[[vector spaces]];

* [[morphism|morphisms]] are $k$-[[linear functions]] between these.

When the [[ground field]] $k$ is understood or when its precise nature is irrelevant, we will often notationally suppress it and speak of just the category [[Vect]] of vector spaces.

This is the category inside which [[linear algebra]] takes place.

=--

Of course the category [[Vect]] has some special properties. Not only are its objects "linear spaces", but the whole category inherits linear structure of sorts. This is traditionally captured by the following terminology for **[[additive and abelian categories]]**. Notice that there are several different but equivalent ways to state the following properties (discussed behind the relevant links).

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

Let $\mathcal{C}$ be a [[category]].

1. Say that $\mathcal{C}$ has **[[direct sums]]** if it has [[finite products]] and [[finite coproducts]] and if the canonical comparison  morphism between these is an [[isomorphism]]. We write $V \oplus W$ for the direct sum of two objects of $\mathcal{C}$.

1. Say that $\mathcal{C}$ is an **[[additive category]]** if it has [[direct sums]] and in addition it is [[Ab-enriched category|enriched in abelian groups]], meaning that every [[hom-set]] is equipped with the structure of an [[abelian group]] such that [[composition]] of morphisms is a [[bilinear map]].

1. Say that $\mathcal{C}$ is an **[[abelian category]]** if it is an [[additive category]] and has property that its [[monomorphisms]] are precisely the inclusions of [[kernels]] and its [[epimorphisms]] are precisely the projections onto [[cokernels]].

=--

We also make the following definition of $k$-linear category, but notice that conventions differ as to which extra properties beyond [[Vect]]-[[enriched category|enrichment]] to require on a linear category:

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

For $k$ a [[field]] (or more generally just a [[commutative ring]]), call a [[category]] $\mathcal{C}$ a **$k$-[[linear category]]** if

1. it is an [[abelian category]] (def. \ref{AdditiveAndAbelianCategories});

1. its [[hom-sets]] have the structure of $k$-[[vector spaces]] (generally $k$-[[modules]]) such that [[composition]] of morphisms in $\mathcal{C}$ is a [[bilinear map]]

and the underlying additive [[abelian group]] structure of these [[hom-spaces]] is that of the underlying [[abelian category]].

In other words, a $k$-linear category is an [[abelian category]] with the additional structure of a [[Vect]]-[[enriched category]] (generally $k$[[Mod]]-enriched) such that the underlying [[Ab-enriched category|Ab-enrichment]] according to def. \ref{AdditiveAndAbelianCategories} is obtained from the $Vect$-enrichment under the [[forgetful functor]] $Vect \to Ab$.

A [[functor]] between $k$-linear categories is called a **$k$-[[linear functor]]** if its component functions on [[hom-sets]] are [[linear maps]] with respect to the given $k$-linear structure, hence if it is a [[Vect]]-[[enriched functor]].

=--

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

The category [[Vect]]${}_k$ of [[vector spaces]] (def. \ref{VectorSpaces}) is a $k$-[[linear category]] according to def. \ref{LinearCategory}.

Here the abstract [[direct sum]] is the usual direct sum of [[vector spaces]], whence the name of the general concept.

For $V,W$ two $k$-vector spaces, the vector space structure on the [[hom-set]] $Hom_{Vect}(V,W)$ of [[linear maps]] $\phi \colon V \to W$ is given by "pointwise" multiplication and addition of functions:

$$
  (c_1 \phi_1 + c_2 \phi_2)
  \;\colon\,
   v \;\mapsto\;
   c_1 \phi_1(v) + c_2 \phi_2(v)
$$

for all $c_1, c_2 \in k$ and $\phi_1, \phi_2 \in Hom_{Vect}(V,W)$.

=--

Recall the basic construction of the [[tensor product of vector spaces]]:

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

Given two [[vector spaces]] over some [[field]] $k$,  $V_1, V_2 \in Vect_k$, their [[tensor product of vector spaces]] is the vector space denoted

$$
  V_1 \otimes_k V_2 \;\in\; Vect
$$

whose elements are [[equivalence classes]] of  [[tuples]] of elements $(v_1,v_2)$ with $v_i \in V_i$, for the [[equivalence relation]] given by

$$
  (c v_1 , v_2) \;\sim\; (v_1, c v_2)
$$

$$
  (v_1 + v'_1 , v_2) \; \sim \; (v_1,v_2) + (v'_1, v_2)
$$

$$
  (v_1 , v_2 + v'_2) \; \sim \; (v_1,v_2) + (v_1, v'_2)
$$

More abstractly this means that the [[tensor product of vector spaces]] is the vector space characterized by the fact that

1. it receives a [[bilinear map]]

   $$
     V_1 \times V_2 \longrightarrow V_1 \otimes V_2
   $$

   (out of the [[Cartesian product]] of the underlying sets)

1. any other [[bilinear map]] of the form

   $$
     V_1 \times V_2 \longrightarrow V_3
   $$

   factors through the above bilinear map via a unique [[linear map]]

   $$
     \array{
        V_1 \times V_2 &\overset{bilinear}{\longrightarrow}& V_3
        \\
        \downarrow & \nearrow_{\mathrlap{\exists ! \, linear}}
         \\
         V_1 \otimes_k V_2
     }
   $$

=--

The existence of the [[tensor product of vector spaces]], def. \ref{TensorProductOfVectorSpaces}, equips the category [[Vect]] of vector spaces with extra structure, which is a "[[categorification]]" of the familiar structure of a [[semi-group]]. One also says "[[monoid]]" for [[semi-group]] and therefore [[categories]] equipped with a [[tensor product]] operation are also called  _[[monoidal categories]]_:

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

A **[[monoidal category]]** is a [[category]] $\mathcal{C}$ equipped with

1. a [[functor]]

   $$
      \otimes
        \;\colon\;
      \mathcal{C} \times \mathcal{C}
       \longrightarrow
      \mathcal{C}
   $$

   out of the [[product category]] of $\mathcal{C}$ with itself, called the **[[tensor product]]**,

1. an object

   $$
     1 \in \mathcal{C}
   $$

   called the **[[unit object]]** or **[[tensor unit]]**,

1. a [[natural isomorphism]]

   $$
     a
       \;\colon\;
     ((-)\otimes (-)) \otimes (-)
       \overset{\simeq}{\longrightarrow}
     (-) \otimes ((-)\otimes(-))
   $$

   called the **[[associator]]**,

1. {#MonoidalCategoryUnitors} a [[natural isomorphism]]

   $$
     \ell
       \;\colon\;
     (1 \otimes (-))
       \overset{\simeq}{\longrightarrow}
     (-)
   $$

   called the **[[left unitor]]**, and a natural isomorphism

   $$
     r \;\colon\; (-) \otimes 1 \overset{\simeq}{\longrightarrow} (-)
   $$

   called the **[[right unitor]]**,

such that the following two kinds of [[commuting diagram|diagrams commute]], for all objects involved:

1. **triangle identity**:

   $$
     \array{
        & (x \otimes 1) \otimes y &\stackrel{a_{x,1,y}}{\longrightarrow} & x \otimes (1 \otimes y)
         \\
         & {}_{\rho_x \otimes 1_y}\searrow
         && \swarrow_{1_x \otimes \lambda_y}
         &
        \\
        &&
        x \otimes y
        &&
     }
   $$

1. the **[[pentagon identity]]**:

   $$
     \array{
       && (w \otimes x) \otimes (y \otimes z)
       \\
       & {}^{\mathllap{\alpha_{w \otimes x, y, z}}}\nearrow
        &&
       \searrow^{\mathrlap{\alpha_{w,x,y \otimes z}}}
       \\
       ((w \otimes x ) \otimes y) \otimes z
        && &&
       (w \otimes (x \otimes (y \otimes z)))
       \\
       {}^{\mathllap{\alpha_{w,x,y}} \otimes id_z }\downarrow
        && &&
       \uparrow^{\mathrlap{ id_w \otimes \alpha_{x,y,z} }}
       \\
       (w \otimes (x \otimes y)) \otimes z
         && \underset{\alpha_{w,x \otimes y, z}}{\longrightarrow} &&
       w \otimes ( (x \otimes y) \otimes z )
     }
   $$

=--

As expected, we have the following basic example:

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

For $k$ a [[field]], the category [[Vect]]${}_k$ of $k$-[[vector spaces]] becomes a [[monoidal category]] (def. \ref{MonoidalCategory}) as follows

* the abstract [[tensor product]] is the [[tensor product of vector spaces]] $\otimes_k$ from def. \ref{TensorProductOfVectorSpaces};

* the [[tensor unit]] is the [[field]] $k$ itself, regarded as a 1-dimensional vector space over itself;

* the [[associator]] is the map that on representing [[tuples]] acts as

  $$
    \alpha_{V_{1}, V_2, V_3}
       \;\colon\;
   ((v_1, v_2), v_3) \mapsto (v_1, (v_2,v_3))
  $$

* the left [[unitor]] is the map that on representing [[tuples]] is given by

  $$
    \ell_{V} \colon (k,v) \mapsto k v
  $$

  and the right unitor is similarly given by

  $$
    r_V \colon (v,k) \mapsto k v
    \,.
  $$

That this satisifes the [[pentagon identity]] (def. \ref{MonoidalCategory}) and the left and right unit identities is immediate on representing tuples.

=--

But the point of the abstract definition of [[monoidal categories]] is that there are also more exotic examples. The followig one is just a minimal enrichment of example \ref{VectAsAMonoidalCategory}, and yet it will be important.

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

Let $G$ be a [[group]] (or in fact just a [[monoid]]/[[semi-group]]). A **$G$-[[graded vector space]]** $V$ is a [[direct sum]] of vector spaces labeled by the elements in $G$:

$$
  V = \underset{g \in G}{\oplus} V_g
  \,.
$$

A [[homomorphism]]

$$
  \phi \;\colon\; V \longrightarrow W
$$

of $G$-graded vector spaces is a [[linear map]] that respects this direct sum structure, hence equivalently a [[direct sum]] of [[linear maps]]

$$
  \phi_g \;\colon\; V_g \longrightarrow W_g
$$

for all $g \in G$, such that

$$
  \phi = \underset{g \in G}{\oplus} \phi_g
  \,.
$$

This defines a [[category]], denoted $Vect^G$. Equip this category with a [[tensor product]] which on the underlying vector spaces is just the [[tensor product of vector spaces]] from def. \ref{TensorProductOfVectorSpaces}, equipped with the $G$-grading which is obtained by multiplying degree labels in $G$:

$$
  (V \otimes W)_g
    \;\coloneqq\;
  \underset{{g_1, g_2 \in G} \atop {g_1 g_2 = g}}{\oplus}
   V_{g_1} \otimes_k V_{g_2}
  \,.
$$

The [[tensor unit]] for the tensor product is the ground field $k$, regarded as being in the degree of the [[neutral element]] $e \in G$

$$
  1_g
  \;=\;
  \left\{
    \array{
      k & | g = e
      \\
      0 & | otherwise
    }
  \right.
  \,.
$$

The [[associator]] and [[unitors]] are just those of the monoidal structure on plain vector spaces, from example \ref{VectAsAMonoidalCategory}.

=--

One advantage of abstracting the concept of a [[monoidal category]] is that it allows to prove general statements uniformly for all kinds of tensor products, familiar ones and more exotic ones. The following lemma \ref{kel1} and remark \ref{CoherenceForMonoidalCategories} are two important such statements.

+-- {: .num_lemma #kel1}
###### Lemma
**([Kelly 64](#Kelly64))**

Let $(\mathcal{C}, \otimes, 1)$ be a [[monoidal category]], def. \ref{MonoidalCategory}. Then the left and right [[unitors]] $\ell$ and $r$ satisfy the following conditions:

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

1. for all objects $x,y \in \mathcal{C}$ the following [[commuting diagram|diagrams commutes]]:

   $$
     \array{
       (1 \otimes x) \otimes y  & &
       \\
       {}^\mathllap{\alpha_{1, x, y}} \downarrow
       & \searrow^\mathrlap{\ell_x \otimes id_y} &
       \\
       1 \otimes (x \otimes y)
       & \underset{\ell_{x \otimes y}}{\longrightarrow} & x \otimes y
     }
     \,;
   $$

   and

   $$
     \array{
       x \otimes (y \otimes 1) & &
       \\
       {}^\mathllap{\alpha^{-1}_{1, x, y}} \downarrow
       & \searrow^\mathrlap{id_x \otimes r_y} &
       \\
       (x \otimes y) \otimes 1
         &
           \underset{r_{x \otimes y}}{\longrightarrow}
         &
       x \otimes y
     }
     \,;
   $$

=--

For **proof** see at _[[monoidal category]]_ [this lemma](monoidal+category#kel1) and [this lemma](monoidal+category#kel2).

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

Just as for an [[associative algebra]] it is sufficient to demand $1 a = a$ and $a 1 = a$ and $(a b) c = a (b c)$ in order to have that expressions of arbitrary length may be re-bracketed at will, so there is a _[[coherence theorem for monoidal categories]]_ which states that all ways of freely composing the [[unitors]] and [[associators]] in a [[monoidal category]] (def. \ref{MonoidalCategory}) to go from one expression to another will coincide. Accordingly, much as one may drop the notation for the bracketing in an [[associative algebra]] altogether, so one may, with due care, reason about monoidal categories without always making all unitors and associators explicit.

(Here the qualifier "freely" means informally that we must not use any non-formal identification between objects, and formally it means that the diagram in question must be in the image of a [[strong monoidal functor]] from a _free_ monoidal category. For example if in a particular monoidal category it so happens that the object $X \otimes (Y \otimes Z)$ is actually _equal_ to $(X \otimes Y)\otimes Z$, then the various ways of going from one expression to another using only associators _and_ this "accidental" equality no longer need to coincide.)

=--

The above discussion makes it clear that a [[monoidal category]] is like a [[monoid]]/[[semi-group]], but "[[categorified]]". Accordingly we may consider additional properties of [[monoids]]/[[semi-groups]] and correspondingly lift them to monoidal categories. A key such property is _[[commutative ring|commutativity]]_. But while for a monoid commutativity is just an extra [[property]], for a [[monoidal category]] it involves choices of commutativity-[[isomorphisms]] and hence is [[stuff, structure and property|extra structure]].
We will see [below](#SuperGroupsAsSuperHopfAlgebras) that this is the very source of [[superalgebra]].

The [[categorification]] of "commutativity" comes in two stages: [[braiding]] and [[symmetric monoidal category|symmetric braiding]].

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

A **[[braided monoidal category]]**, is a [[monoidal category]] $\mathcal{C}$ (def. \ref{MonoidalCategory}) equipped with a [[natural isomorphism]]

$$
  \tau_{x,y} \;\colon\; x \otimes y \to y \otimes x
$$

(for all [[objects]] $x,y in \mathcal{C}$) called the **[[braiding]]**, such that the following two kinds of [[commuting diagram|diagrams commute]] for all [[objects]] involved ("hexagon identities"):

$$
  \array{
   (x \otimes y) \otimes z
   &\stackrel{a_{x,y,z}}{\to}&
   x \otimes (y \otimes z)
   &\stackrel{\tau_{x,y \otimes z}}{\to}&
   (y \otimes z) \otimes x
   \\
   \downarrow^{\tau_{x,y}\otimes Id}
   &&&&
   \downarrow^{a_{y,z,x}}
   \\
   (y \otimes x) \otimes z
   &\stackrel{a_{y,x,z}}{\to}&
   y \otimes (x \otimes z)
   &\stackrel{Id \otimes \tau_{x,z}}{\to}&
   y \otimes (z \otimes x)
  }
$$

and

$$
  \array{
   x \otimes (y \otimes z)
   &\stackrel{a^{-1}_{x,y,z}}{\to}&
   (x \otimes y) \otimes z
   &\stackrel{\tau_{x \otimes y, z}}{\to}&
   z \otimes (x \otimes y)
   \\
   \downarrow^{Id \otimes \tau_{y,z}}
   &&&&
   \downarrow^{a^{-1}_{z,x,y}}
   \\
   x \otimes (z \otimes y)
   &\stackrel{a^{-1}_{x,z,y}}{\to}&
   (x \otimes z) \otimes y
   &\stackrel{\tau_{x,z} \otimes Id}{\to}&
   (z \otimes x) \otimes y
  }
  \,,
$$

where $a_{x,y,z} \colon (x \otimes y) \otimes z \to x \otimes (y \otimes z)$ denotes the components of the [[associator]] of $\mathcal{C}^\otimes$.

=--

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

A **[[symmetric monoidal category]]** is a [[braided monoidal category]] (def. \ref{BraidedMonoidalCategory}) for which the [[braiding]]

$$
   \tau_{x,y} \colon x \otimes y \to y \otimes x
$$

satisfies the condition:

$$
  \tau_{y,x} \circ \tau_{x,y} = 1_{x \otimes y}
$$

for all objects $x, y$

=--

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

In analogy to the [[coherence theorem for monoidal categories]] (remark \ref{CoherenceForMonoidalCategories}, roughly speaking "all diagrams commute") there is a [[coherence theorem for symmetric monoidal categories]] (def. \ref{SymmetricMonoidalCategory}), saying that two parallel morphisms, built freely (see remark \ref{SymmetricMonoidalCategoriesCoherenceTheorem}) from [[associators]], [[unitors]] and [[braidings]], are equal if and only if they correspond to the same [[permutation]] of objects.

=--

Consider the simplest non-trivial special case of $G$-[[graded vector spaces]] from example \ref{GradedVectorSpacesAsAMonoidaCategory}, the case where $G = \mathbb{Z}/2$ is the [[cyclic group of order two]].

+-- {: .num_example #Z2Zgradedvectorspaces}
###### Example

A **$\mathbb{Z}/2$-[[graded vector space]]** is a [[direct sum]] of two vector spaces

$$
  V = V_{even} \oplus V_{odd}
  \,,
$$

where we think of $V_{even}$ as the summand that is graded by the [[neutral element]] in $\mathbb{Z}/2$, and of $V_{odd}$ as being the summand that is graded by the single non-trivial element.

A [[homomorphism]] of $\mathbb{Z}/2$-graded vector spaces

$$
  f \;\colon\; V_1 \longrightarrow V_2
$$

is a [[linear map]] of the underlying vector spaces that respects the grading, hence equivalently a pair of linear maps

$$
  f_{even} \;\colon\; (V_1)_{even} \longrightarrow (V_1)_{even}
$$

$$
  f_{odd} \;\colon\; (V_1)_{odd} \longrightarrow (V_1)_{odd}
$$

between then summands in even degree and in odd degree, respectively:

$$
  f = f_{even} \oplus f_{odd}
  \,.
$$

The [[tensor product]] of $\mathbb{Z}/2$-graded vector space is the [[tensor product of vector spaces]] of the underlying vector spaces, but with the grading obtained from multiplying the original gradings in $\mathbb{Z}/2$. Hence

$$
  (V_1 \otimes V_2)_{even}
    \;\coloneqq\;
  \left((V_1)_{even} \otimes (V_2)_{even}\right)
    \oplus
  \left((V_1)_{odd} \otimes (V_2)_{odd}\right)
$$

and

$$
  (V_1 \otimes V_2)_{odd}
    \;\coloneqq\;
  \left((V_1)_{even} \otimes (V_2)_{odd}\right)
    \oplus
  \left((V_1)_{odd} \otimes (V_2)_{even}\right)
  \,.
$$

As in example \ref{GradedVectorSpacesAsAMonoidaCategory}, this definition makes $\mathbb{Z}/2$ a [[monoidal category]] def. \ref{MonoidalCategory}.

=--

+-- {: .num_prop #TheTwoNontrivialBraidingsOnZ2GradedVectorSpaces}
###### Proposition

There are, up to [[braided monoidal functor|braided monoidal]] [[equivalence of categories]], precisely two choices for a [[symmetric monoidal category|symmetric]] [[braiding]] (def. \ref{SymmetricMonoidalCategory})

$$
  V_1 \otimes V_2
     \stackrel{\tau_{V_1,V_2}}{\longrightarrow}
  V_2 \otimes V_1
$$

on the [[monoidal category]] $(Vect_k^{\mathbb{Z}/2}, \otimes_k)$ of $\mathbb{Z}/2$-[[graded vector spaces]] from def. \ref{Z2Zgradedvectorspaces}:

1. the **trivial braiding** which is the [[natural transformation|natural]] [[linear map]] given on tuples $(v_1,v_2)$ representing an element in $V_1 \otimes V_2$ (according to def. \ref{TensorProductOfVectorSpaces}) by

   $$
     \tau^{triv}_{V_1, V_2} \;\colon\; (v_1,v_2) \mapsto (v_2, v_1)
   $$

1. the **super-braiding** which is the [[natural transformation|natural]] [[linear function]] given on tuples $(v_1,v_2)$ of _homogeneous degree_ (i.e. $v_i \in (V_i)_{\sigma_i} \hookrightarrow V_i$, for $\sigma_i \in \mathbb{Z}/2$) by

   $$
     \tau^{super}_{V_1, V_2}
        \;\colon\;
     (v_1, v_2)
        \mapsto
     (-1)^{deg(v_1) deg(v_2)}
     \,
     (v_2,v_1)
     \,.
   $$

=--

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

For $(\mathcal{C}, \otimes, 1)$ a [[monoidal category]], write

$$
  (Line(\mathcal{C}), \otimes, 1)
    \hookrightarrow
  (\mathcal{C}, \otimes, 1)
$$

for the [[full subcategory]] on those $L \in \mathcal{C}$ which are [[invertible objects]] under the [[tensor product]], i.e. such that there is an object $L^{-1} \in \mathcal{C}$ with $L \otimes L^{-1} \simeq 1$ and $L^{-1} \otimes L \simeq 1$. Since the [[tensor unit]] is clearly in $Line(L)$ (with $1^{-1} \simeq 1$) and since with $L_1, L_2 \in Line(\mathcal{C}) \hookrightarrow \mathcal{C}$ also $L_1 \otimes L_2 \in Line(\mathcal{C})$ (with $(L_1 \otimes L_2)^{-1} \simeq L_2^{-1} \otimes L_1^{-1}$) the [[monoidal category]] structure on $\mathcal{C}$ restricts to $Line(\mathcal{C})$.

Accordingly any [[braiding]] on $(\mathcal{C}, \otimes,1)$ restricts to a braiding on $(Line(\mathcal{C}), \otimes, 1)$. Hence it is sufficient to show that there is an essentially unique non-trivial symmetric braiding on $(Line(\mathcal{C}), \otimes, 1)$, and that this is the restriction of a braiding on $(\mathcal{C}, \otimes, 1)$.

Now $(Line(\mathcal{C}, \otimes , 1))$ is necessarily a [[groupoid]] (the "[[Picard groupoid of a monoidal category|Picard groupoid]]" of $\mathcal{C}$) and in fact is what is called a _[[2-group]]_. As such we may regard it equivalently as a [[homotopy 1-type]] with group structure, and as such it it is equivalent to its [[delooping]]

$$
  B_\otimes Line(\mathcal{C})
$$

regarded as a [[pointed homotopy type]]. (See at _[[looping and delooping]]_).

The [[Grothendieck group]] of $(\mathcal{C}, \otimes, 1)$ is

$$
  \pi_0(Line(\mathcal{C}))
    \simeq
  \pi_1(B Line(\mathcal{C}))
$$

the [[fundamental group]] of the delooping space.

Now a symmetric braiding on $Line(\mathcal{C})$ is precisely the structure that makes it a [[symmetric 2-group]] which is equivalently the structure of a second [[delooping]] $B^2 Line(\mathcal{C})$ (for the braiding) and then a third delooping $B^3 Line(\mathcal{C})$ (for the symmetry), regarded as a [[pointed homotopy type]].

This way we have rephrased the question equivalently as a question about the possible [[k-invariants]] of spaces of this form.

Now in the case at hand, $Line(Vect^{\mathbb{Z}/2})$ has precisely two [[isomorphism classes]] of objects, namely the [[ground field]] $k$ itself, regarded as being in even degree and regarded as being in odd degree. We write $k^{1\vert 0}$ and $k^{0 \vert 1}$ for these, respectively. By the rules of the tensor product of [[graded vector spaces]] we have

$$
  k^{1\vert 0} \otimes_k k^{1\vert 0} \simeq k^{1\vert 0}
$$

$$
  k^{1\vert 0} \otimes_k k^{0\vert 1} \simeq k^{0\vert 1}
$$

and

$$
  k^{0 \vert 1} \otimes_k k^{0 \vert 1} \simeq k^{1 \vert 0}
  \,.
$$

In other words

$$
  \pi_0(Line(Vect^{\mathbb{Z}/2})) \simeq \mathbb{Z}/2
  \,.
$$

Now under the above homotopical identification the non-trivial braiding is identified with the elements

$$
  1 =
  k^{1 \vert 0}
     \simeq
  k^{0\vert 1} \otimes_k k^{0 \vert 1}
    \stackrel{\tau^{super}_{k^{0\vert 1}, k^{0 \vert 1}}}{\longrightarrow}
  k^{0\vert 1} \otimes_k k^{0\vert 1}
    \simeq
  k^{1 \vert 0}
  = 1
$$

Due to the [[symmetric monoidal category|symmetry]] condition (def. \ref{SymmetricMonoidalCategory}) we have

$$
  (\tau^{super}_{k^{0\vert 1}, k^{0 \vert 1}})^2 = id
$$

which implies that

$$
  \tau^{super}_{k^{0\vert 1}, k^{0 \vert 1}}
  \in
  \{+ id, -id\}
  \,.
$$

Therefore for classifying just the symmetric braidings, it is sufficient to restrict the [[hom-spaces]] in $Line(Vect^{\mathbb{Z}/2})$ from being either $k$ or empty, to [[hom-sets]] being $\mathbb{Z}/2 = \{+1-1\} \hookrightarrow k$ or empty. Write $\widetilde{Line}(sVect)$ for the resulting [[2-group]].

In conclusion then the [[equivalence classes]] of possible [[k-invariants]] of $B^3 Line(sVect)$, hence the possible symmetric braiding on $Line(Vect^{\mathbb{Z}/2})$ are in the degree-4 [[ordinary cohomology]] of the [[Eilenberg-MacLane space]] $K(\mathbb{Z}/2,3)$ with [[coefficients]] in $\mathbb{Z}/2$. One finds (...)

$$
  H^4(K(\mathbb{Z}/2, 3), \mathbb{Z}/2)
  \;\simeq\;
  \mathbb{Z}/2
  \,.
$$

=--

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

The [[symmetric monoidal category]] (def. \ref{SymmetricMonoidalCategory})

* whose underlying [[monoidal category]] is that of $\mathbb{Z}/2$-[[graded vector spaces]] (example \ref{Z2Zgradedvectorspaces}) and

* whose [[braiding]] (def. \ref{BraidedMonoidalCategory}) is the unique non-trivial symmetric grading $\tau^{super}$ from prop. \ref{TheTwoNontrivialBraidingsOnZ2GradedVectorSpaces}

is called the **[[category of super vector spaces]]**:

$$
  sVect_k
    \;\coloneqq\;
  (Vect_k^{\mathbb{Z}/2}, \otimes = \otimes_k, 1 = k, \tau = \tau^{super} )
  \,.
$$

=--

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

The non-full symmetric monoidal subcategory

$$
  (\widetilde{Line}(sVect), \otimes_k, k, \tau^{super})
$$

of

$$
  (Line(sVect) , \otimes_k, k, \tau^{super})
    \hookrightarrow
  (sVect, \otimes_k, k, \tau^{super})
$$

(on the two objects $k^{1\vert 0} $ and $k^{0\vert 1}$ and with [[hom-sets]] restricted to $\{+1,-1\} \subset k$, as in the proof of prop. \ref{TheTwoNontrivialBraidingsOnZ2GradedVectorSpaces}) happens to be the [[truncated object of an (infinity,1)-category|1-truncation]] of the [[looping]] of the [[sphere spectrum]] $\mathbb{S}$, regarded as a group-like [[E-infinity space]] ("[[abelian infinity-group]]")

$$
  (\widetilde{Line}(sVect), \otimes, k, \tau^{super})
  \;\simeq\;
  \trunc_1 \Omega \mathbb{S}
  \,.
$$

It has been suggested (in [Kapranov 15](super+algebra#Kapranov15)) that this and other phenomena are evidence that in the wider context of [[homotopy theory]]/[[stable homotopy theory]] super-grading (and hence [[superalgebra]]) is to be regarded as but a shadow of grading in [[higher algebra]] over the [[sphere spectrum]]. Notice that the [[sphere spectrum]] is just the analog of the group of [[integers]] in [[stable homotopy theory]].

=--

## Properties

### Structures internal to super-vector spaces
 {#StructuresInternalToSuperVectorSpaces}

By [[internalization|internalizing]] [[algebra]] and [[geometry]] in the category $sVect$ of super vector spaces, one obtains the corresponding [[superalgebra]] and [[supergeometry]].

For example

* [[monoids]] in $sVect$ are [[super algebras]] and, more interestingly, [[commutative monoids]] in $sVect$ are [[supercommutative superalgebra]];

* hence for example [[commutative Hopf algebras]] in $sVect$ are equivalently [[supercommutative Hopf algebras]], which are the [[formal duals]] of affine algebraic [[supergroups]];

* [[Lie algebras]] in $sVect$ are [[super Lie algebras]];

* [[manifolds]] modeled on objects of $sVect$ are [[supermanifolds]]

* etc.

By the above definition, any structure in $sVect$ works just like the corresponding structure in [[Vect]], but with a sign inserted whenever two odd-graded symbols are interchanged. For more on this see also at _[[signs in supergeometry]]_.

### Deligne's theorem on tensor categories

[[Deligne's theorem on tensor categories]] says that all suitable [[tensor categories]] of subexponential growth have a [[fiber functor]] to $sVect$ and are equivalent to [[categories of representations]] of affine algebraic [[supergroups]].

## Related entries

* [[supertrace]]

* [[superdeterminant]]

* [[super vector bundle]]

* [[chain complex of super vector spaces]]

* [[model structure on chain complexes of super vector spaces]]

* [[super module]]

* [[superalgebra]], [[supercommutative superalgebra]], [[differential graded-commutative superalgebra]],

  [[model structure on differential graded-commutative superalgebras]]

* [[geometry of physics -- supergeometry]]

## References

* {#Varadarajan04} [[Veeravalli Varadarajan]], section 3.1 of  _[[Supersymmetry for mathematicians]]: An introduction_, Courant lecture notes in mathematics, American Mathematical Society Providence, R.I 2004 ([doi:10.1090/cln/011](http://dx.doi.org/10.1090/cln/011))

* {#Westra09} [[Dennis Westra]], section 3 of _Superrings and supergroups_, 2009 ([pdf](http://www.mat.univie.ac.at/~michor/westra_diss.pdf))

[[!redirects SVect]]
[[!redirects sVect]]
[[!redirects SuperVect]]

[[!redirects super vector space]]
[[!redirects super vector spaces]]
[[!redirects supervector space]]
[[!redirects supervector spaces]]
[[!redirects super-vector space]]
[[!redirects super-vector spaces]]

[[!redirects category of super vector spaces]]
[[!redirects category of super-vector spaces]]
[[!redirects category of supervector spaces]]


super vertex operator algebra

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[supersymmetry|supersymmetric]] version of a [[vertex operator algebra]]; the local data of a [[2d SCFT]].

## Examples

* [[chiral differential operators]]

## Related concepts

* [[super Virasoro algebra]]

* [[2-spectral triple]]

* [[automorphism of a super vertex operator algebra]]

## References

### General

* [[Edward Frenkel]], [[David Ben-Zvi]], _Vertex algebras and algebraic curves_, 2nd Ed., American Mathematical Society, Providence, RI, 2004.

In relation to [[conformal nets]]:

* [[Sebastiano Carpi]], [[Tiziano Gaudio]], [[Robin Hillier]], *From vertex operator superalgebras to graded-local conformal nets and back* [[arXiv:2304.14263](https://arxiv.org/abs/2304.14263)]

[[!include elliptic genera as partition functions -- references]]

[[!redirects super vertex operator algebras]]

[[!redirects vertex operator super-algebra]]
[[!redirects vertex operator super-algebras]]

[[!redirects super vertex algebra]]
[[!redirects super vertex algebras]]

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

[[!redirects sVOA]]


super vielbein

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _super vielbein_ is the analogue of a _[[vielbein]]_ in [[supergeometry]].

## The canonical super vielbein

Let $N$ be a [[real spin representation]] and let $(x^a, \theta^\alpha)$ be the canonical [[coordinates]] on the [[supermanifold]] $\mathbb{R}^{d-1,1\vert N}$ underlying the [[super Minkowski spacetime]] super translation group. Then the canonical super vielbein is the $\mathbb{R}^{d-1,1\vert N}$-valued [[super differential form]] with components

* $\psi^\alpha \coloneqq \mathbf{d} \theta^\alpha$.

* $e^a \coloneqq \mathbf{d} x^a + \overline{\theta} \Gamma^a \mathbf{d} \theta$.

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

## Related concepts

* [[super Cartan geometry]]

* [[generalized vielbein]]

* [[super-torsion]]

* [[geometry of physics -- supersymmetry]]

[[!redirects super vielbeins]]

[[!redirects super-vielbein]]
[[!redirects super-vielbeins]]

[[!redirects supervielbein]]
[[!redirects supervielbeins]]

[[!redirects super-vielbein field]]
[[!redirects super-vielbein fields]]

[[!redirects supervielbein field]]
[[!redirects supervielbein fields]]


super Yang-Mills theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
#### $\infty$-Chern-Weil theory
+--{: .hide}
[[!include infinity-Chern-Weil theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A  [[supersymmetry|supersymmetric]] extension of plain [[Yang-Mills theory]].

## Properties

### Classification

The existence of super Yang-Mills (SYM) theories of a certain number of [[supersymmetries]] in a certain [[dimension]] of [[spacetime]] is linked to the existence of certain [[Lie algebra cohomology|cocycles]] on the [[super Poincaré Lie algebra]] (those that also govern the [[brane scan]]). These in turn are closely related to the [[normed division algebras]]. See ([ABDH 13](#ABDH13)). See also at _[[division algebra and supersymmetry]]_.

This classification may be paired with that for [[supergravity]] theories, to yield a [[magic pyramid]] of super [[Einstein-Yang-Mills theories]].

## Examples

* [[4d superconformal gauge field theory]]

  * [[D=4 N=1 super Yang-Mills theory]]

  * [[D=4 N=2 super Yang-Mills theory]]

  * [[D=4 N=4 super Yang-Mills theory]]

  * [[topologically twisted D=4 super Yang-Mills theory]]

* [[3d superconformal gauge field theory]]

  * [[D=3 N=1 super Yang-Mills theory]]

  * [[D=3 N=2 super Yang-Mills theory]]

  * [[D=3 N=4 super Yang-Mills theory]]

* [[D=5 super Yang-Mills theory]]

* [[D=6 super Yang-Mills theory]]

* [[D=7 super Yang-Mills theory]]

* [[D=10 super Yang-Mills theory]]

## Related concepts

* [[super QCD]]

* [[supergravity]]

* [[MSSM]]

* [[holomorphic block]]

## References

Original articles (see also at _[Supersymmetry -- References -- History](supersymmetry#ReferencesHistory)_):

* [[Sergio Ferrara]], [[Bruno Zumino]], _Supergauge invariant Yang-Mills theories_, Nuclear Physics B Volume 79, Issue 3, 25 September 1974, Pages 413-421 (doi:10.1016/0550-3213(74)90559-8)

* {#SalamStrathdee74} [[Abdus Salam]], [[John Strathdee]], _Super-symmetry and non-Abelian gauges_, Physics Letters B Volume 51, Issue 4, 19 August 1974, Pages 353-355 (doi:10.1016/0370-2693(74)90226-3)

A general introduction is for instance

* [[Eric D'Hoker]], D. H. Phong, _Lectures on Supersymmetric Yang-Mills Theory and Integrable Systems_ ([arXiv:hep-th/9912271](http://arxiv.org/abs/hep-th/9912271))

as well as various of the lectures in the collection

* [[Pierre Deligne]] et al., _[[Quantum Fields and Strings]]_

The [[deformation theory]] of SYM theories is discussed in

* {#SchwarzMovshev} [[Albert Schwarz]], [[Mikhail Movshev]], _Supersymmetric Deformations of Maximally Supersymmetric Gauge Theories_ ([arXiv:0910.0620](http://arxiv.org/abs/0910.0620))

Special properties of [[scattering amplitudes]] are discussed for instance in

* [[Nima Arkani-Hamed]], Jacob L. Bourjaily, Freddy Cachazo, Alexander B. Goncharov, Alexander Postnikov, Jaroslav Trnka, _Scattering Amplitudes and the Positive Grassmannian_ ([arXiv:1212.5605](http://arxiv.org/abs/1212.5605))

The [[chiral rings]] of SYM are discussed in

* Freddy Cachazo, [[Michael Douglas]], [[Nathan Seiberg]], [[Edward Witten]]. _Chiral Rings and Anomalies in Supersymmetric Gauge Theory_, JHEP 0212:071,2002 ([arXiv:hep-th/0211170](http://arxiv.org/abs/hep-th/0211170))

Classification in terms of [[division algebra and supersymmetry]] / [[magic pyramid]] is in

* {#ABDH13} A. Anastasiou, [[Leron Borsten]], [[Mike Duff]], L. J. Hughes, S. Nagy, _Super Yang-Mills, division algebras and triality_ ([arXiv:1309.0546](http://arxiv.org/abs/1309.0546))

On application to problems in non-supersymmetric [[Yang-Mills theory]] ([[confinement]] etc.) and to [[QCD]]:

* [PO discussion](http://www.physicsoverflow.org/24797/conceivable-supersymmetric-theory-understand-traditional)

[[!redirects super Yang-Mills theories]]

[[!redirects SYM]]
[[!redirects SYM theory]]
[[!redirects SYM theories]]

[[!redirects supersymmetric Yang-Mills theory]]