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A symmetry is (in most classical situations) invariance under a group action, or infinitesimally, 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.
In the stricter sense of the word as it originates in theoretical fundamental physics, supersymmetry refers specifically to Lie superalgebra extensions of the Poincaré Lie algebra – the super Poincaré Lie algebra, often called the supersymmetry algebra in this context. An odd element in this super Lie algebra is called a supersymmetry generator.
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.
To be distinguished from this global supersymmetry is local supersymmetry (see gauge group and links given there for a discussion of local and global gauge groups): 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 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
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 );
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 desription of our phenomenological reality. Still, it could well be that the underlying theory of the world is nevertheless not plain general relativity but supergravity.
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.
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 thes 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 clearl qualitatively (a generic solution to the super-Einstein equations will not have a Killing spinor). A mor sophisticated and quantitative argument to the same extend is for instance given in (DLSW08).
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
make a pure theory of gravity seamlessly incorporate fermions and gauge fields;
leads to better renormalization properties (indeed the speculation that $N=8, D=4$ supergravity is fully renormalizable – in stark contrast to ordinary gravity has more recently been checked up to high loop orders);
produces a wealth of interesting mathematical structures. For instance
Morse theory of a Riemannian manifold can naturally be understood in terms of supersymmetric quantum mechanics for superparticles propagating on that manifold;
the interpretation of quantum field theories in terms of generalized cohomology theories only works for supersymmetric theories (see (1,1)-dimensional Euclidean field theories and K-theory and (2,1)-dimensional Euclidean field theory)
“topological twists” of supersymmetric field theories are a major source of examples of TQFTs, for instance the A-model and the B-model TCFT arise from such twists of 2-dimensional supergravity and there are deep connections between the geometric Langlands correspondence and topologically twisted super Yang-Mills theory.
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.
spinning particle (which happens to have worldline supersymmetry)
Texbook references include
Veeravalli Varadarajan, Supersymmetry for mathematicians: An introduction, Courant lecture notes in mathematics, American Mathematical Society, Providence, R.I 2004
Daniel Freed, Five lectures on supersymmetry AMS (1999)
Antoine Van Proeyen, Tools for supersymmetry (arXiv:hep-th/9910030)
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,
means to sort out various sign issues of relevance in supergeometry and supsersymmetric quantum field theory (see at signs in supergeometry.)
See also
I. L. Buchbinder, S. M. Kuzenko, Ideas and methods of supersymmetry and supergravity; or A walk through superspace
Joseph Polchinski, String theory, volume II, appendix B
The seminal classification of simple supersymmetry super Lie algebras (including higher dimensions and conformal/de Sitter supersymmetry) is due to
For more on this see at super Poincaré Lie algebra.
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
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
A review of supersymmetry breaking is in
A quantitative analysis showing that locally supersymmetric spacetime theories will generically not exhibit global spacetime supersymmetry is