superalgebra

and

supergeometry

# 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 phenomenology? 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 auxioary 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.

## 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$$N$superconformal super Lie algebraR-symmetrybrane worldvolume theory
3$2k+1$$B(k,2) \simeq$ osp$(2k+1/4)$$SO(2k+1)$
3$2k$$D(k,2)\simeq$ osp$(2k/4)$$SO(2k)$M2-brane
4$k+1$$A(3,k)\simeq \mathfrak{sl}(4/k+1)$$U(k+1)$D3-brane
51$F(4)$$SO(3)$
6$k$$D(4,k) \simeq$ osp$(8/2k)$$Sp(k)$M5-brane

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 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

Physics introductions 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.)

### 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):

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. (see the account by (Schwarz))

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:

Revised on September 15, 2016 09:38:18 by Urs Schreiber (193.60.198.37)