Contents

supersymmetry

# Contents

## Idea

Where generally a symmetry is an invariance under a group action (after all the mathematical term “group” is a contraction of group of symmetries), or, infinitesimally, an invariance under a Lie algebra action, in the general sense of the word, a supersymmetry is invariance under a supergroup action or a super Lie algebra action.

By Deligne's theorem on tensor categories it is precisely the context of supersymmetry in which tensor categories over the complex numbers exhibit full Tannaka duality.

In the stricter sense of the word as it originates in theoretical fundamental physics, supersymmetry refers specifically to supergeometric extensions of the isometry group (or its Lie algebra) of the local model of spacetime. For Minkowski spacetime this is the Poincaré group with its Poincaré Lie algebra and their super-extensions are accordingly known as the super Poincaré group and super Poincaré Lie algebra. In physics the latter are often referred to simply as the supersymmetry algebra. An odd element in this super Lie algebra is called a supersymmetry generator and its Hamiltonian is called a supercharge. More generally, if spacetime geometry is taken to be conformal geometry or de Sitter spacetime/anti-de Sitter spacetime, then one considers supergeometric extensions of the conformal group, de Sitter group and anti de Sitter group, respectively, and then these are the relevant “supersymmetries”.

With the focus on spacetime symmetry groups implicitly understood, this implies that supersymmetry is the group theory relevant for super Cartan geometry locally modeled on super Minkowski spacetime or super anti de Sitter spacetime, etc. In terms of physics and in view of the first order formulation of gravity, this means that “locally gauged” supersymmetry is the super-geometric generalization from Einstein gravity to supergravity or, respectively, from conformal field theory to superconformal field theory, etc.

### Global supersymmetry and superparticles

By Wigner-Weyl theory we have in ordinary quantum field theory that unitary representations of the Poincaré group correspond to the particles in the theory. For a globally supersymmetric quantum field theory the Poincaré group here is replaced by the super Poincaré group and accordingly particles are now irreducible representations of this group: the irreducible unitary representations of the super Poincaré group. The new – odd graded – pieces of these representations – called supermultiplets – appearing this way are called the superpartners of the original bosonic particles.

### Local supersymmetry: supergravity

To be distinguished from this global supersymmetry is local supersymmetry (see at gauge group and at Cartan connection): given a gauge theory whose fields are connections with values in the Poincaré Lie algebra – the theory of gravity in its first order formulation – a supersymmetric extension is similarly a gauge theory with fields being connections in the super Poincaré Lie algebra – a theory of supergravity. A gauge transformation in such a theory is called a local supersymmetry transformation.

### The relation between local and global supersymmetry

The distinction between local and global supersymmetry is important when considering supergravity in perturbation theory where all fields are expanded around a fixed spacetime geometry and fixed background gauge fields that form a solution of the Euler-Lagrange equations.

While the theory of supergravity as such has, by definition, local supersymmetry, a solution to it may but need not have any global supersymmetry left. In fact, generically it will not.

To see this, it is maybe helpful to compare with the analogous statement in non-supersymmetric QFT:

the theory of gravity is locally Poincaré-symmetric: in first order formulation it is a Poincaré group gauge theory. Nevertheless, any of its solutions – which is a pseudo-Riemannian manifold – may, but need not, have any Poincaré-symmetry. It does have such a global symmetry for every Killing vector on the spacetime. Such may or may not exist. Generically it does not exist.

The analog of these Killing vectors in supergravity are Killing spinors: covariantly constant spinors (sections of a spinor bundle annihilated by the spin connection‘s covariant derivative). For every such, the background has one global supersymmetry transformation. These may or may not exist. Generically they do not.

The most famous example of this is that of Calabi-Yau manifolds: a standard assumption on phenomenological model building that used to be very popular around the turn of the millenium (but is maybe experimentally ruled out at the time of this writing) is that

1. the spacetime we see around us locally looks like a product $M^4 \times Y^6$ of 4-dimensional Minkowski space times a tiny closed Riemannian manifold $Y^6$ (so tiny that it is not directly observable but manifests itself only by way of its lowest excitation modes that look like different particle species on the remaining $M^4$ – see Kaluza-Klein mechanism );

2. such that on this product space a single covariant constant spinor exists, such that the resulting effective theory on $M^4$ has a single global supersymmetry (“$N=1$ supersymmetry”).

One finds that this condition is solved precisely by $Y^6$ being a Calabi-Yau manifold. For more on this see the corresponding section at heterotic string theory.

But notice that nothing in the theory of 10-dimensional supergravity demands that its solutions have a global supersymmetry left (generically they will not) nor that they factor locally as $M^4 \times Y^6$. All this is an ansatz a phenomenological model . It only says that if we make this ansatz, then $Y^6$ needs to be a Calabi-Yau space. In fact, it turns out to be nontrivial to check that with all the other fields taken into account, such a factor ansatz is indeed consistent. (This problem of “moduli stabilization” is discussed a little bit at landscape of string theory vacua.)

Latest experimental results strongly suggest that this model of global $(N = 1)$-supersymmetry at observable energies is not a description of our phenomenological reality. Still, it could well be that the underlying theory of the world is nevertheless not plain general relativity but supergravity.

### Observed supersymmetry: on the worldline

To appreciate this point it is worthwhile to notice that supersymmetric quantum field theory in fact has been observed in nature, already since a century ago. To see this, one needs to notice how fundamental particles are described by sigma model quantum field theories (see there) on their worldvolume:

the $\sigma$-model action functional for the standard Dirac spinor particle – such as electrons and quarks, the particles that all the matter in the world is made of – just happens to have worldline supersymmetry. This is discussed at spinning particle . Notice that this is true for electrons and quarks in the non-supersymmetric standard model of particle physics: the target space theory is completely void of (global) supersymmetry, and still the worldvolume theory of any fermion is automatically supersymmetric.

### Conjectured supersymmetry: on spacetime and on the worldsheet

A similar statement as for the spinning particle and its automatic local worldline supersymmetry indeed also holds for the spinning string : while it is a conjecture of string theory which has not been experimentally verified that fundamentally the sigma-model theories that describe the physics around us is defined not on 1-dimensional but on 2-dimensional worldvolumes, it remains a noteworthy fact that the standard action functional for the string sigma model with fermions on the worldsheet just happens to be locally supersymmetric. It is hard to avoid this! And indeed, this was how the abstract notion of supersymmetry in quantum field theory was realized in the west first: people wrote down action functionals for spinning strings and noticed that these happened to have an interesting graded symmetry (see there for references).

So while it is an open conjecture of string theory that the particle worldlines that are experimentally observed are secretly really worldsheets, assuming that this conjecture holds for fermions automatically implies local worldsheet supersymmetry and the superstring.

Now, the effective spacetime target space theories arising from second quantization of the superstring are fairly well understood: these are higher dimensional supergravity theories coupled to the various higher gauge fields that are also in the spectrum of the string (the Kalb-Ramond field and the RR-fields, notably). They are called type II supergravity , heterotic supergravity, etc. All of these are obtained by compactifications of 11-dimensional supergravity.

This means that the assumption of spinning string sigma-models automatically implies that the spacetime QFT that we observe also has local supersymmetry.

Over the decades it has often been suggested that therefore the assumption of spinning strings “suggests” or “favours” the observation of superpartner particles in accelerators. However, this is not so:

in these constructions the particle species seen in accelerators are KK-modes and/or brane-brane open string modes of the compactified locally supersymmetric theory. This means that they are determined by the compactification geometry. Only if that has a global Killing spinor is the effective 4-dimensional theory globally supersymmetric and exhibits superpartners. As was mentioned above, for spacetimes of the form $M^4 \times Y^6$ this is the case precisely if $Y^6$ is a Calabi-Yau manifold.

But this is far from being the generic situation. This is clear qualitatively (a generic solution to the super-Einstein equations will not have a Killing spinor). A more sophisticated and quantitative argument to the same effect is given, for instance, in (DLSW08).

### In fundamental physics

Supersymmetric extensions of quantum field theories have been felt to be compelling in fundamental physics for formal reasons : the simple step from Lie algebras to super Lie algebras

Moreover, since various observables in supersymmetric QFTs are easier to compute than in non-supersymmetric theories, supersymmetric quantum field theory is being used to approximate certain aspects of other QFTs. For instance certain correlators in ordinary Yang-Mills theory coupled to spinors in Yang-Mills theory can be computed using an auxiliary super Yang-Mills theory.

Therefore, if nothing else, supersymmetric quantum field theories constitute a part of the whole space of quantum field theories which is useful for understanding general properties of that space. What is however still missing is any experimental evidence that the world is fundamentally described by a supersymmetric quantum field theory.

## Definition

We start by saying what it means, in generality, to have a supersymmetric extension of an ordinary symmetry, then we specialize to spacetime supersymmetry.

Here we are concerned with symmetry groups that are Lie groups, and we start by considering only the infinitesimal approximation, hence their Lie algebras.

To discuss super-extensions of Lie algebras, recall from geometry of physics – supergeometry the concept of super Lie algebras:

###### Definition

A super Lie algebra is a Lie algebra internal to the symmetric monoidal category $sVect = (Vect^{\mathbb{Z}/2}, \otimes_k, \tau^{super} )$ of super vector spaces. Hence this is

1. a super vector space $\mathfrak{g}$;

2. a homomorphism

$[-,-] \;\colon\; \mathfrak{g} \otimes_k \mathfrak{g} \longrightarrow \mathfrak{g}$

of super vector spaces (the super Lie bracket)

such that

1. the bracket is skew-symmetric in that the following diagram commutes

$\array{ \mathfrak{g} \otimes_k \mathfrak{g} & \overset{\tau^{super}_{\mathfrak{g},\mathfrak{g}}}{\longrightarrow} & \mathfrak{g} \otimes_k \mathfrak{g} \\ {}^{\mathllap{[-,-]}}\downarrow && \downarrow^{\mathrlap{[-,-]}} \\ \mathfrak{g} &\underset{-1}{\longrightarrow}& \mathfrak{g} }$

(here $\tau^{super}$ is the braiding natural isomorphism in the category of super vector spaces)

2. the Jacobi identity holds in that the following diagram commutes

$\array{ \mathfrak{g} \otimes_k \mathfrak{g} \otimes_k \mathfrak{g} && \overset{\tau^{super}_{\mathfrak{g}, \mathfrak{g}} \otimes_k id }{\longrightarrow} && \mathfrak{g} \otimes_k \mathfrak{g} \otimes_k \mathfrak{g} \\ & {}_{\mathllap{[-,[-,-]]} - [[-,-],-] }\searrow && \swarrow_{\mathrlap{[-,[-,-]]}} \\ && \mathfrak{g} } \,.$

Externally this means the following:

###### Proposition

A super Lie algebra according to def. is equivalently

1. a $\mathbb{Z}/2$-graded vector space $\mathfrak{g}_{even} \oplus \mathfrak{g}_{odd}$;

2. equipped with a bilinear map (the super Lie bracket)

$[-,-] : \mathfrak{g}\otimes_k \mathfrak{g} \to \mathfrak{g}$

which is graded skew-symmetric: for $x,y \in \mathfrak{g}$ two elements of homogeneous degree $\sigma_x$, $\sigma_y$, respectively, then

$[x,y] = -(-1)^{\sigma_x \sigma_y} [y,x] \,,$
3. that satisfies the $\mathbb{Z}/2$-graded Jacobi identity in that for any three elements $x,y,z \in \mathfrak{g}$ of homogeneous super-degree $\sigma_x,\sigma_y,\sigma_z\in \mathbb{Z}_2$ then

$[x, [y, z]] = [[x,y],z] + (-1)^{\sigma_x \cdot \sigma_y} [y, [x,z]] \,.$

A homomorphism of super Lie algebras is a homomorphisms of the underlying super vector spaces which preserves the Lie bracket. We write

$sLieAlg$

for the resulting category of super Lie algebras.

Some obvious but important classes of examples are the following:

###### Example

every $\mathbb{Z}/2$-graded vector space $V$ becomes a super Lie algebra (def. , prop. ) by taking the super Lie bracket to be the zero map

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

These may be called the “abelian” super Lie algebras.

###### Example

Every ordinary Lie algebra becomes a super Lie algebra (def. , prop. ) concentrated in even degrees. This constitutes a fully faithful functor

$LieAlg \hookrightarrow sLieAlg \,.$

which is a coreflective subcategory inclusion in that it has a left adjoint

$LieAlg \underoverset {\underset{ \overset{ \rightsquigarrow}{(-)} }{\longleftarrow}} {\hookrightarrow} {\bot} sLieAlg$

given on the underlying super vector spaces by restriction to the even graded part

$\overset{\rightsquigarrow}{\mathfrak{s}} = \mathfrak{s}_{even} \,.$

Using this we may finally say what a super-extension is supposed to be:

###### Definition

Given an ordinary Lie algebra $\mathfrak{g}$, then a super-extension of $\mathfrak{g}$ is a super Lie algebra $\mathfrak{s}$ (def. , prop. ) equipped with a monomorphism of the form

$i \;\colon\; \mathfrak{g} \hookrightarrow \mathfrak{s}$

(where $\mathfrak{g}$ is regarded as a super Lie algebra according to example )

such that this is an isomorphism on the even part (example )

$\overset{\rightsquigarrow}{i} \;\colon\; \mathfrak{g} \overset{\simeq}{\longrightarrow} \mathfrak{s}_{even} \,.$

We now make explicit structure involved in super-extensions of Lie algebras:

###### Proposition

Given an ordinary Lie algebra $\mathfrak{g}$, then a choice of super-extension $\mathfrak{g} \hookrightarrow \mathfrak{s}$ according to def. is equivalently the following data:

1. a vector space $S$;

2. a Lie action of $\mathfrak{g}$ on $S$, hence a Lie algebra homomorphism

$\rho_{(-)} : \mathfrak{g} \longrightarrow \mathfrak{gl}(S)$

from $\mathfrak{g}$ to the endomorphism Lie algebra of $S$;

3. a symmetric bilinear map

$(-,-) \;\colon\; S \otimes_k S \longrightarrow \mathfrak{g}$

such that

1. the pairing is $\mathfrak{g}$-equivariant in that for all $t \in \mathfrak{g}$ then

$\rho_{t}(-,-) = (\rho_t(-),(-)) + (-,\rho_t(-))$
2. the pairing satisfies

$\rho_{(\phi,\phi)}(\phi) = 0$

for all $\phi \in S$.

###### Proof

By definition of super-extension, the underlying super vector space of $\mathfrak{s}$ is necessarily of the form

$\mathfrak{s}_{even} = \underset{\mathfrak{s}_{even}}{\underbrace{\mathfrak{g}}} \oplus \underset{\mathfrak{s}_{odd}}{\underbrace{S}}$

for some vector space $S$.

Moreover the super Lie bracket on $\mathfrak{s}$ restricts to that of $\mathfrak{g}$ when restricted to $\mathfrak{g} \otimes_{k}\mathfrak{g}$ and otherwise constitutes

1. a bilinear map

$\rho_{(-)}(-) \coloneqq [-,-]\vert_{\mathfrak{s}_{even}\oplus \mathfrak{s}_{odd}} \;\colon\; \mathfrak{g} \otimes_k S \longrightarrow S$
2. a symmetric bilinear map

$(-,-) \coloneqq [-,-]\vert_{\mathfrak{s}_{odd} \oplus \mathfrak{s}_{odd}} \;\colon\; S \otimes_k S \longrightarrow \mathfrak{g} \,.$

This yields the claimed structure. The claimed properties of these linear maps are now just a restatement of the super-Jacobi identity in terms of this data:

1. The restriction of the super Jacobi identity of $\mathfrak{s}$ to $\mathfrak{s}_{even} \otimes_k \mathfrak{s}_{even} \otimes_k \mathfrak{s}_{even}$ is equivalently the Jacobi identity in $\mathfrak{g}$ and hence is no new constraint;

2. The restriction of the super Jacobi identity of $\mathfrak{s}$ to $\mathfrak{s}_{even} \otimes_k \mathfrak{s}_{even} \otimes_k \mathfrak{s}_{odd}$ says that for $t_1,t_2 \in \mathfrak{g}$ and $\phi \in S$ then

$\rho_{t_1}( \rho_{t_2}(\phi) ) = \rho_{[t_1,t_2]}(\phi) + \rho_{t_1}( \rho_{t_2}(\phi) ) \,.$

This is equivalent to

$\rho_{t_1} \circ \rho_{t_2} - \rho_{t_2} \circ \rho_{t_1} = \rho_{[t_1,t_2]}$

which means equivalently that $\rho_{(-)}$ is a Lie algebra homomorphism from $\mathfrak{g}$ to the endomorphism Lie algebra of $S$.

3. The restriction of the super Jacobi identity of $\mathfrak{s}$ to $\mathfrak{s}_{even} \otimes_k \mathfrak{s}_{odd} \otimes_k \mathfrak{s}_{odd}$ says that for $t \in \mathfrak{g}$ and $\phi,\psi \in S$ then

$\rho_{t}(\phi,\psi) = (\rho_t(\phi), \psi) + (\phi, \rho_t(\psi)) \,.$

This is exactly the claimed $\mathfrak{g}$-equivariance of the pairing.

4. The restriction of the super Jacobi identity of $\mathfrak{s}$ to $\mathfrak{s}_{odd} \otimes_k \mathfrak{s}_{odd} \otimes_k \mathfrak{s}_{odd}$ implies that for all $\psi \in S$ that

$[\psi,(\psi,\psi)] = [ (\psi,\psi), \psi ] - [\psi, (\psi, \psi)]$

and hence that

$[(\psi,\psi),\psi] = \rho_{(\psi,\psi)}(\psi) = 0 \,.$

Hence it only remains to show that this special case is in fact equivalent to the full odd-odd-odd super Jacobi identity. This follows by polarization: First insert $\psi = \phi_1 + \phi_2$ into the above cubic condition to obtain a quadratic condition, then polarize once more in $\phi_2$.

###### Example

Given an ordinary Lie algebra $\mathfrak{g}$, then for every choice of vector space $V$ there is the trivial super extension (def. ) of $\mathfrak{g}$, with underlying vector space

$\mathfrak{s} \coloneqq \mathfrak{g} \oplus V$

and with both the action and the pairing (via prop. ) trivial:

$\rho = 0$

and

$(-,-) = 0 \,.$

The key example of interest now is going to be this:

###### Example

For $d \in \mathbb{N}$, a super extension (def. ) of the Poincaré Lie algebra $\mathfrak{iso}(\mathbb{R}^{d-1,1})$ (recalled as def. below) which is non-trivial (def. ) is obtained from the following data:

1. a Lie algebra representation $\rho$ of $\mathfrak{so}(d-1,1)$ on some real vector space $S$;

2. an $\mathfrak{so}$-equivariant symmetric $\mathbb{R}$-bilinear pairing $(-,-) \colon S \otimes_k S \to \mathbb{R}^{d-1,1}$

It turns out that data as in example is given for $\rho$ the Lie algebra version of a real spin representation of the spin group $Spin(d-1,1)$ (see this prop). These are introduced and discussed at geometry of physics – supersymmetry in the section Real spin representations.

The super-extensions of the Poincaré Lie algebra induced by real spin representations are called super Poincaré Lie algebras (def. ) below. These are the standard supersymmetry algebras in the physics literature.

But beware that there are more super-extensions of the Poincaré Lie algebra:

###### Example

(an exotic super-extension of the Poincaré Lie algebra)

Let $d \in \{3,4,6,10\}$ and let $S \in Rep_{\mathbb{R}}(Spin(d-1,1))$ be an irreducible real spin representation in that dimension. Let $(-,-) \colon S \otimes S \to \mathbb{R}^{d-1,1}$ be the symmetric spinor pairing as in example , but let the action on $S$ via the even-odd super-bracket not be the Spin-action of $\mathfrak{so}$, but the Clifford algebra action

$\Gamma(-) \;\colon \; \mathbb{R}^{d-1,1} \longrightarrow End(S) \,.$

of $\mathbb{R}^{d-1,1}$. Then the condition

$\Gamma(\psi,\psi)(\psi) = 0$

from prop. does hold: this turns out to be equivalent to the Green-Schwarz superstring cocycle condition in these dimensions, here in its incarnation as the “3-$\psi$ rule” of Schray 96, see Baez-Huerta 09, theorem 10.

Now $\Gamma(-)$ thus defined is clearly not a Lie algebra action and hence fails one of the other conditions in prop. , but this is readily fixed: take $S \coloneqq S_+ \oplus S_-$ to be the direct sum of two copies of the Majorana spinor representation and take $\Gamma(-)$ to map as before, but from $S_+$ to $S_-$, acting as zero on $S_-$. This forces the commutator of endomorphisms in the image of $\Gamma$ to vanish, and hence makes $\Gamma(-)$ a Lie algebra action of the abelian Lie algebra $\mathbb{R}^{d-1,1}$. Hence we get an “exotic” super-extension of the Poincaré Lie algebra.

###### Remark

By prop. the data in example is sufficient for producing super-extensions (in the sense of def. ) of Poincaré Lie algebras, namely the super Poincaré Lie algebras. It is however not necessary: example is a super-extension in the sense of def. of the Poincaré Lie algebra which is not a super Poincaré Lie algebra of the form of example .

One may add further natural conditions on the super-extension in order to narrow down to the super Poincaré super Lie algebras:

1. From the assumption alone that $S$ is a spin representation and using that the $Spin$-equivariant pairing has to take irreducible representations to an irreducible representation, one may in some dimensions already deduce that the pairing has to land in $\mathbb{R}^{d} \hookrightarrow \mathfrak{iso}(\mathbb{R}^{d-1,1})$. For $d = 4$ and $S$ the irreducible Majorana representation this is done in Varadarajan 04, section 3.2.

2. One may appeal to the Haag–Łopuszański–Sohnius theorem. This does rule out exotic super-extensions, by imposing the additional condition that $P_a P^a$ remains a Casimir operator after super-extension, and more conditions. These conditions are well motivated from the expected symmetry-behaviour of S-matrices in field theory.

At geometry of physics – supersymmetry in the section supersymmetry from the superpoint we discuss something at least related. The super Poincaré Lie algebras at least in certain dimensions are singled out from a different perspective: they are precisely the result of iterative maximal invariant central extensions of the superpoint.

Now we write out the supersymmetry algebra thus obtained more explicitly. In all of the following it is most convenient to regard super Lie algebras dually via their Chevalley-Eilenberg algebras:

###### Definition

For $\mathfrak{g}$ a super Lie algebra of finite dimension, then its Chevalley-Eilenberg algebra $CE(\mathfrak{g})$ is the super-Grassmann algebra on the dual super vector space

$\wedge^\bullet \mathfrak{g}^\ast$

equipped with a differential $d_{\mathfrak{g}}$ that on generators is the linear dual of the super Lie bracket

$d_{\mathfrak{g}} \;\coloneqq\; [-,-]^\ast \;\colon\; \mathfrak{g}^\ast \to \mathfrak{g}^\ast \wedge \mathfrak{g}^\ast$

and which is extended to $\wedge^\bullet \mathfrak{g}^\ast$ by the graded Leibniz rule (i.e. as a graded derivation).

$\,$

Here all elements are $(\mathbb{Z} \times \mathbb{Z}/2)$-bigraded, the first being the cohomological grading $n$ in $\wedge^\n \mathfrak{g}^\ast$, the second being the super-grading $\sigma$ (even/odd).

For $\alpha_i \in CE(\mathfrak{g})$ two elements of homogeneous bi-degree $(n_i, \sigma_i)$, respectively, the sign rule is

$\alpha_1 \wedge \alpha_2 = (-1)^{n_1 n_2} (-1)^{\sigma_1 \sigma_2}\; \alpha_2 \wedge \alpha_1 \,.$

(See at signs in supergeometry for discussion of this sign rule and of an alternative sign rule that is also in use. )

We may think of $CE(\mathfrak{g})$ equivalently as the dg-algebra of left-invariant super differential forms on the corresponding simply connected super Lie group .

The concept of Chevalley-Eilenberg algebras is traditionally introduced as a means to define Lie algebra cohomology:

###### Definition

Given a super Lie algebra $\mathfrak{g}$, then

1. an $n$-cocycle on $\mathfrak{g}$ (with coefficients in $\mathbb{R}$) is an element of degree $(n,even)$ in its Chevalley-Eilenberg algebra $CE(\mathfrak{g})$ (def. ) which is $d_{\mathbb{g}}$ closed.

2. the cocycle is non-trivial if it is not $d_{\mathfrak{g}}$-exact

3. hene the super-Lie algebra cohomology of $\mathfrak{g}$ (with coefficients in $\mathbb{R}$) is the cochain cohomology of its Chevalley-Eilenberg algebra

$H^\bullet(\mathfrak{g}, \mathbb{R}) = H^\bullet(CE(\mathfrak{g})) \,.$

The following says that the Chevalley-Eilenberg algebra is an equivalent incarnation of the super Lie algebra:

###### Proposition

The functor

$CE \;\colon\; sLieAlg^{fin} \hookrightarrow dgAlg^{op}$

that sends a finite dimensional super Lie algebra $\mathfrak{g}$ to its Chevalley-Eilenberg algebra $CE(\mathfrak{g})$ (def. ) is a fully faithful functor which hence exibits super Lie algebras as a full subcategory of the opposite category of differential-graded algebras.

###### Definition

Let

$d \in \mathbb{N}$

be a spacetime dimension and let

$N \in Rep_{\mathbb{R}}(Spin(d-1,1))$

be a real spin representation of the spin group cover $Spin(d-1,1)$ of the Lorentz group $O(d-1,1)$ in this dimension. Then the $d$-dimensional $N$-supersymmetric super-Minkowski spacetime $\mathbb{R}^{d-1,1|N}$ is the super Lie algebra that is characterized by the fact that its Chevalley-Eilenberg algebra $CE(\mathbb{R}^{d-1,1})$ is as follows:

The algebra has generators (as an associative algebra over $\mathbb{R}$)

$\underset{deg = (1,even)}{\underbrace{e^a}} \;\;\;\; \text{and} \;\;\;\; \underset{deg = (1,odd)}{\underbrace{\psi^\alpha}}$

for $a \in \{0,1,2, \cdots, 9\}$ and $\alpha \in \{1, 2, \cdots dim_{\mathbb{R}}(N)\}$ subjects to the relations

\begin{aligned} e^a \wedge e^b = - e^b \wedge e^a \\ \psi^\alpha \wedge \psi^\beta = + \psi^\beta \wedge \psi^\alpha \\ e^a \wedge \psi^\alpha = - \psi^\alpha \wedge e^a \end{aligned}

(see at signs in supergeometry), and the differential $d_{CE}$ acts on the generators as follows:

\begin{aligned} d_{\mathbb{R}^{d-1,1\vert N}} \; \psi^\alpha & \coloneqq 0 \\ d_{\mathbb{R}^{d-1,1\vert N}} \; e^a & \coloneqq \overline{\psi} \wedge \Gamma^a \psi \\ & \coloneqq \left(C_{\alpha \alpha'} {\Gamma^a}^{\alpha'}{}_\beta\right) \psi^\alpha \wedge \psi^\beta \end{aligned} \,,

where

1. $\overline{\psi} \wedge \Gamma^a \psi$ denotes the $a$-component of the $Spin(d-1,1)$-invariant spinor bilinear pairing $N \otime N \to \mathbb{R}^d$ that comes with every real spin representation applied to $\psi \wedge \psi$ regarded as an $N \otimes N$-valued form;

2. hence in components (if $N$ is a Majorana spinor representation, by this prop.):

1. $C = (C_{\alpha \alpha'})$ is the charge conjugation matrix (as discussed at Majorana spinor);

2. $\Gamma^a = ((\Gamma^a)^{\alpha}{}_\beta)$ are the matrices representing the Clifford algebra action on $N$ in the linear basis $\{\psi^\alpha\}_{\alpha = 1}^{dim_{\mathbb{R}}(N)}$

3. summation over paired indices is understood.

That this indeed yields a super Lie algebra follows by the symmetry and equivariance of the bilinear spinor pairing (via this prop.).

There is a canonical Lie algebra action of the special orthogonal Lie algebra

$Lie(Spin(d-1,1)) \simeq \mathfrak{so}(d-1,1)$

on $\mathbb{R}^{d-1,1\vert 1}$. The $N$-supersymmetric super Poincaré Lie algebra $\mathfrak{iso}(\mathbb{R}^{d-1,1\vert N})$ in dimension $d$ is the super Lie algebra which is the semidirect product Lie algebra of this Lie algebra action

$\mathfrak{iso}(\mathbb{R}^{d-1,1\vert N}) = \mathbb{R}^{d-1,1\vert N} \rtimes \mathfrak{so}(d-1,1) \,.$

This is characterized by the fact that its Chevalley-Eilenberg algebra $CE(\mathfrak{iso}(\mathbb{R}^{d-1,1\vert N}))$ is as follows:

it is generated from elements

$\underset{deg = (1,even)}{\underbrace{e^a}} \;\;\;\; and \;\;\;\; \underset{deg = (1,odd)}{\underbrace{\psi^\alpha}} \;\;\;\; and \;\;\;\; \underset{deg = (1,even)}{\underbrace{\omega^{a b} = - \omega^{b a}}}$

with the super vielbein $(e^a, \psi^\alpha)$ as before, and with $\omega^{a b}$ the dual basis of the induced linear basis for vectro space of skew-symmetric matricces underlying the special orthogonal Lie algebra. The commutation relations are as before, together with the relation that the generators $\omega^{a b}$ anti-commute with every generator. Finally the differential $d_{\mathfrak{iso}(\mathbb{R}^{d-1,1\vert N})}$ acts on these generators as follows:

\begin{aligned} d_{\mathfrak{iso}(\mathbb{R}^{d-1,1\vert N})} \; \psi^\alpha & \coloneqq \left(\tfrac{1}{4}\omega^{a b} \Gamma_{a b} \psi \right)^\alpha \\ & \coloneqq \left(\tfrac{1}{4} (\Gamma_{a b})^\alpha{}_{\beta} \right) \omega^{a b} \wedge \psi^\beta \\ d_{\mathfrak{iso}(\mathbb{R}^{d-1,1\vert N})} \; e^a & \coloneqq \overline{\psi} \wedge \Gamma^a \psi - \omega^a{}_b \wedge e^b \\ & \coloneqq \left( C_{\alpha \alpha'} {\Gamma^a}^{\alpha'}{}_\beta \right) \psi^\alpha \wedge \psi^\beta - \omega^a{}_b \wedge e^b \\ \end{aligned} \,,

where we are shifting spacetime indicices with the Lorentz metric

$(\eta_{a b}) \coloneqq diag(-1,1,1,\cdots, 1) \,.$

The canonical maps between these super Lie algebras, dually between their Chevalley-Eilenberg algebras, that send each generator to itself, if present, or to zero if not, constitute the diagram

$\array{ \mathbb{R}^{d-1,1\vert N} &\hookrightarrow& \mathfrak{iso}(\mathbb{R}^{d-1,1\vert N}) \\ && \downarrow \\ && \mathfrak{so}(d-1,1) } \,.$

## Properties

### Classification

We discuss the classification of possible supersymmetry super Lie algebras.

#### Super-Poincaré symmetry

super Poincaré Lie algebras exist for every real spinor representation. See there for more. These come naturally equipped with a symmetric equivariant bilinear pairing of two spinors to a vector, and this constitutes the odd-odd bracket in the super Poincaré Lie algebra.

The classification of minimal real spin representations in Lorentzian signature is as follows

$d$$Spin(d-1,1)$minimal real spin representation $S$$dim_{\mathbb{R}} S\;\;$$V$ in terms of $S^\ast$supergravity
1$\mathbb{Z}_2$$S$ real1$V \simeq (S^\ast)^{\otimes}^2$
2$\mathbb{R}^{\gt 0} \times \mathbb{Z}_2$$S^+, S^-$ real1$V \simeq ({S^+}^\ast)^{\otimes^2} \oplus ({S^-}^\ast)^{\otimes 2}$
3$SL(2,\mathbb{R})$$S$ real2$V \simeq Sym^2 S^\ast$
4$SL(2,\mathbb{C})$$S_{\mathbb{C}} \simeq S' \oplus S''$4$V_{\mathbb{C}} \simeq {S'}^\ast \oplus {S''}^\ast$d=4 N=1 supergravity
5$Sp(1,1)$$S_{\mathbb{C}} \simeq S_0 \otimes_{\mathbb{C}} W$8$\wedge^2 S_0^\ast \simeq \mathbb{C} \oplus V_{\mathbb{C}}$
6$SL(2,\mathbb{H})$$S^\pm_{\mathbb{C}} \simeq S_0^\pm \otimes_{\mathbb{C}} W$8$V_{\mathbb{C}} \simeq \wedge^2 {S_0^+}^\ast \simeq (\wedge^2 {S_0^-}^\ast)^\ast$
7$S_{\mathbb{C}} \simeq S_0 \otimes_{\mathbb{C}} W$16$\wedge^2 S_0^\ast \simeq V_{\mathbb{C}} \oplus \wedge^2 V_{\mathbb{C}}$
8$S_{\mathbb{C}} \simeq S^\prime \oplus S^{\prime\prime}$16${S'}^\ast {S''}^\ast \simeq V_{\mathbb{C}} \oplus \wedge^3 V_{\mathbb{C}}$
9$S$ real16$Sym^2 S^\ast \simeq \mathbb{R} \oplus V \wedge^4 V$
10$S^+ , S^-$ real16$Sym^2(S^\pm)^\ast \simeq V \oplus \wedge_\pm^5 V$type II supergravity
11$S$ real32$Sym^2 S^\ast \simeq V \oplus \wedge^2 V \oplus \wedge^5 V$11-dimensional supergravity

#### Superconformal and super anti de Sitter symmetry

We discuss super Lie algebra extensions of the conformal Lie algebra of $\mathbb{R}^{d-1,1}$ (equivalently the isometry Lie algebra of anti de Sitter space of dimension $d+1$, see also at AdS-CFT.)

##### In dimension 2

Discussion of classification of 2d SCFT algebras includes (Kac 03, section 2).

(…)

##### In dimension $d \gt 2$
###### Proposition

There exist superconformal extensions of the super Poincaré Lie algebra, (besides dimension $\leq 2$) in dimensions 3,4,5,6 as follows (with notation as at super Lie algebra – classification):

$d$$N$superconformal super Lie algebraR-symmetryblack brane worldvolume
superconformal field theory
$\phantom{A}3\phantom{A}$$\phantom{A}2k+1\phantom{A}$$\phantom{A}B(k,2) \simeq$ osp$(2k+1/4)\phantom{A}$$\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/4)\phantom{A}$$\phantom{A}SO(2k)\phantom{A}$M2-brane
3d superconformal gauge field theory
$\phantom{A}4\phantom{A}$$\phantom{A}k+1\phantom{A}$$\phantom{A}A(3,k)\simeq \mathfrak{sl}(4/k+1)\phantom{A}$$\phantom{A}U(k+1)\phantom{A}$D3-brane
4d superconformal gauge field theory
$\phantom{A}5\phantom{A}$$\phantom{A}1\phantom{A}$$\phantom{A}F(4)\phantom{A}$$\phantom{A}SO(3)\phantom{A}$
$\phantom{A}6\phantom{A}$$\phantom{A}k\phantom{A}$$\phantom{A}D(4,k) \simeq$ osp$(8/2k)\phantom{A}$$\phantom{A}Sp(k)\phantom{A}$M5-brane
6d superconformal gauge field theory

(Shnider 88, also Nahm 78, see Minwalla 98, section 4.2)

There exists no superconformal extension of the super Poincaré Lie algebra in dimension $d \gt 6$.

This is due to (Shnider 88), see also (Nahm 78). Review is in (Minwalla 98, section 4.2). See also the references at super p-brane – As part of the AdS-CFT correspondence.

###### Proof (sketch)

By realizing the conformal real Lie algebra $\mathfrak{so}(\mathbb{R}^{d,2})$ as a real section of the complexified $\mathfrak{so}(\mathbb{C}^{d+2})$ one is reduced to finding those (finite dimensional) simple super Lie algebras over the complex numbers whose even-graded part extends $\mathfrak{so}(\mathbb{C}^{d+2})$ and such that the implied representation of that on the odd-graded part contains the spin representation.

The complex finite dimensional simple super Lie algebras have been classified, see at super Lie algebra – Classification. By the tables shown there

$\mathfrak{g}$$\mathfrak{g}_{even}$$\mathfrak{g}_{even}$ rep on $\mathfrak{g}_{odd}$
$B(m,n)$$B_m \oplus C_n$vector $\otimes$ vector
$D(m,n)$$D_m \oplus C_n$vector $\otimes$ vector
$D(2,1,\alpha)$$A_1 \oplus A_1 \oplus A_1$vector $\otimes$ vector $\otimes$ vector
$F(4)$$B_3\otimes A_1$spinor $\otimes$ vector
$G(3)$$G_2\oplus A_1$spinor $\otimes$ vector
$Q(n)$$A_n$adjoint
$\mathfrak{g}$$\mathfrak{g}_{even}$$\mathfrak{g}_{even}$ rep on $\mathfrak{g}_{{-1}}$
$A(m,n)$$A_m \oplus A_n \oplus C$vector $\otimes$ vector $\otimes$ $\mathbb{C}$
$A(m,m)$$A_m \oplus A_n$vector $\otimes$ vector
$C(n)$$\mathbb{C}_{-1} \oplus \mathbb{C}$vector $\otimes$ $\mathbb{C}$

the only manifest spinor representation of $\mathfrak{so}(2k+1) = B_k$ or of $\mathfrak{so}(2k) = D_k$ appears in the exceptional super Lie algebra $F(4)$, which contains $B_3 = \mathfrak{so}(7)$ in its even parts acting spinorially on its odd part. This hence gives a superconformal super Lie algebra in dimension $7-2 = 5$, as shown in the proposition.

But other spinor representations may still disguise as vector representations of other Lie algebras under one of the exceptional isomorphisms. These exist only in low dimensions, and hence to conclude the proof it is sufficient to just list all candidates.

First there is the exceptional isomorphism

$\mathfrak{so}(5) \simeq \mathfrak{sp}(2) = C_2$

with the spinor representation of $\mathfrak{so}(5)$ being the vector representation of $\mathfrak{sp}(2) = C_2$. This we find in the above tables as a summand in the even-graded subalgebra of $B(m,2)$ and of $D(m,2)$. Hence these are superconformal super Lie algebras in dimension $5-2 = 3$, as shown in the statement.

The other exceptional isomorphism of relevance is

$\mathfrak{so}(6)\simeq \mathfrak{su}(4) = A_3$

with the spinor representation of $\mathfrak{so}(6)$ being the vector representation of $\mathfrak{su}(4) = A_3$. By the above tables this appears as a summand in the even-graded subalgebra of the super Lie algebra $A(3,k)$, and so this is the superconformal algebra in dimension $6-2 = 4$.

Finally by triality the vector representation of $\mathfrak{so}(8) = D_4$ is isomorphic to its spinor representation. By the above tables this means that $D(4,k)$ is a superconformal algebra in dimension $8-2 = 6$. For details on this see (Shnider 88, last paragraphs)

###### Remark

Further constraints follow from requiring super-unitary representations (Minwalla 98, section 4.3). This restricts for instance the 6d superconformal algebra to $D(4,1) = \mathfrak{osp}(8|2)$ and $D(4,1) = \mathfrak{osp}(8,4)$, the latter being (over the reals as $\mathfrak{so}(8^\ast|4) = \mathfrak{osp}(6,2|4)$) the symmetry algebra of the 6d (2,0)-superconformal QFT on the worldvolume of the M5-brane.

### Extensions

In special dimensions supersymmetry super Lie algebras have polyvector extensions by brane charges, called for instance the type II supersymmetry algebra for the case of type II supergravity or the M-theory supersymmetry algebra for the case of 11-dimensional supergravity. These in turn are the truncations of super L-infinity algebra extensions, related to the supergravity Lie 3-algebra, supergravity Lie 6-algebra etc. See also at extended supersymmetry and extended super Minkowski spacetime.

## References

### Textbooks and lectures

Historically the first textbook on supersymmetry was

Further physics texts include

More mathematical accounts include

also (Deligne-Freed 99).

More physics-style accounts include

Discussion with an eye towards non-perturbative effects such as in AdS-CFT includes

A fair bit of detail on supersymmetry and of supergravity with an eye towards their role in string theory is in the collection

especially in the contribution

The appendix there,

• Sign manifesto (pdf)

means to sort out various sign issues of relevance in supergeometry and supsersymmetric quantum field theory (see at signs in supergeometry.)

Discussion of the relation between supersymmetry and division algebras includes

• Jörg Schray, The general classical solution of the superparticle, Class. Quant. Grav. 13 (1996), 27–38. (arXiv:hep-th/9407045)

• John Baez, John Huerta, Division algebras and supersymmetry I, in R. Doran, G. Friedman and Jonathan Rosenberg (eds.), Superstrings, Geometry, Topology, and $C*$-algebras, Proc. Symp. Pure Math. 81, AMS, Providence, 2010, pp. 65-80 (arXiv:0909.0551)

and for superconformal symmetry:

• Jerzy Lukierski, Francesco Toppan, Generalized Space-time Supersymmetries, Division Algebras and Octonionic M-theory, Phys.Lett. B539 (2002) 266-276 (arXiv:hep-th/0203149)

### Classification

Results on the classification of supersymmetry super Lie algebras (including higher dimensions and conformal/de Sitter supersymmetry) includes

Review includes

For more on this see at super Poincaré Lie algebra.

### History

The notion of Poincaré supersymmetry was found in parallel by two groups in the 1970s (separated and isolated at that time by “Cold War” nuisances) (see the account by (Schwarz)):

André Neveu, Pierre Ramond and John Schwarz wrote down in 1971 the system called the spinning string – a 2-dimensional quantum field theory with fermions and notice that it just so happens to have an extra graded extension of 2-dimensional Poincaré symmetry.

Around the same time Golfand and Likhtman in Russia wrote down the super Poincaré Lie algebra in four dimensions. This then motivated Julius Wess and Zumino to study supersymmetric QFTs in four dimensions.

An account of the history of the development of supersymmetry is in

### Supersymmetry in the standard model of particle physics

A nontechnical survey of the idea of supersymmetry in the standard model of particle physics including the hierarchy problem and the naturality question is in

### Supersymmetry in the standard model of cosmology

The observation that the lightest supersymmetric particle is a natural dark matter candidate goes back to

• John Ellis, J.S. Hagelin, Dimitri V. Nanopoulos, Keith Olive, M. Srednicki Supersymmetric relics from the Big Bang, Nuclear Physics B 238[2]: 453-76, 1984 (SPIRE)

with review in

Supersymmetry seems to be favored by the Starobinsky model of cosmic inflation (see there for more).

### Supersymmetry breaking

A review of supersymmetry breaking is in

A quantitative analysis showing that locally supersymmetric spacetime theories will generically not exhibit global spacetime supersymmetry is

### Experimental searches

Remembering that there is a considerable difference between global low energy supersymmetry and local higher energy supersymmetry aka supergravity:

Last revised on July 10, 2018 at 07:55:30. See the history of this page for a list of all contributions to it.