Types of quantum field thories
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.
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.
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 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 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
the spacetime we see around us locally looks like a product of 4-dimensional Minkowski space times a tiny closed Riemannian manifold (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 – see Kaluza-Klein mechanism );
such that on this product space a single covariant constant spinor exists, such that the resulting effective theory on has a single global supersymmetry (“ supersymmetry”).
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 . All this is an ansatz a phenomenological model . It only says that if we make this ansatz, then 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 -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.
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 -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.
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 this is the case precisely if 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).
leads to better renormalization properties (indeed the speculation that 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
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.
We start by saying what it means, in generality, to have a supersymmetric extension of an ordinary symmetry, then we specialize to spacetime supersymmetry.
of super vector spaces (the super Lie bracket)
Externally this means the following:
a -graded vector space ;
equipped with a bilinear map (the super Lie bracket)
which is graded skew-symmetric: for two elements of homogeneous degree , , respectively, then
that satisfies the -graded Jacobi identity in that for any three elements of homogeneous super-degree then
for the resulting category of super Lie algebras.
Some obvious but important classes of examples are the following:
These may be called the “abelian” super Lie algebras.
given on the underlying super vector spaces by restriction to the even graded part
Using this we may finally say what a super-extension is supposed to be:
(where is regarded as a super Lie algebra according to example 2)
We now make explicit structure involved in super-extensions of Lie algebras:
a vector space ;
a Lie action of on , hence a Lie algebra homomorphism
from to the endomorphism Lie algebra of ;
a symmetric bilinear map
the pairing is -equivariant in that for all then
the pairing satisfies
for all .
By definition of super-extension, the underlying super vector space of is necessarily of the form
for some vector space .
Moreover the super Lie bracket on restrict to that of when restricted to and otherwise constitutes
a bilinear map
a symmetric bilinear map
The restriction of the super Jacobi identity of to is equivalently the Jacobi identity in and hence is no new constraint;
The restriction of the super Jacobi identity of to says that for and then
This is equivalent to
which means equivalently that is a Lie algebra homomorphism from to the endomorphism Lie algebra of .
The restriction of the super Jacobi identity of to says that for and then
This is exactly the claimed -equivariance of the pairing.
The restriction of the super Jacobi identity of to implies that for all that
and hence that
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 inser into the above cubic condition to obtain a quadratic condition, then polarize once more in .
and with both the action and the pairing (via prop. 2) trivial:
The key example of interest now is going to be this:
an -equivariant symmetric -bilinear pairiing
It turns out that data as in example 4 is given for the Lie algebra version of a real spin representation of the spin group (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. 5) below. These are the standard supersymmetry algebras in the physics literature.
But beware that there are more super-extensions of the Poincaré Lie algebra:
(an exotic super-extension of the Poincaré Lie algebra)
Let and let be an irreducible real spin representation in that dimension. Let be the symmetric spinor pairing as in example 4, but let the action on via the even-odd super-bracket not be the Spin-action of , but the Clifford algebra action
of . Then the condition
from prop. 2 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- rule” of Schray 96, see Baez-Huerta 09, theorem 10.
Now thus defined is clearly not a Lie algebra action and hence fails one of the other conditions in prop. 2, but this is readily fixed: take to be the direct sum of two copies of the Majorana spinor representation and take to map as before, but from to , acting as zero on . This forces the commutator of endomorphisms in the image of to vanish, and hence makes a Lie algebra action of the abelian Lie algebra . Hence we get an “exotic” super-extension of the Poincaré Lie algebra.
By prop. 2 the data in example 4 is sufficient for producing super-extensions (in the sense of def. 2) of Poincaré Lie algebras, namely the super Poincaré Lie algebras. It is however not necessary: example 5 is a super-extension in the sense of def. 2 of the Poincaré Lie algebra which is not a super Poincaré Lie algebra of the form of example 4.
One may add further natural conditions on the super-extension in order to narrow down to the super Poincaré super Lie algebras:
From the assumption alone that is a spin representation and using that the -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 . For and the irreducible Majorana representation this is done in Varadarajan 04, section 3.2.
One may appeal to the Haag–Łopuszański–Sohnius theorem. This does rule out exotic super-extensions, by imposing the additional condition that 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.
equipped with a differential that on generators is the linear dual of the super Lie bracket
and which is extended to by the graded Leibniz rule (i.e. as a graded derivation).
Here all elements are -bigraded, the first being the cohomological grading in , the second being the super-grading (even/odd).
For two elements of homogeneous bi-degree , respectively, the sign rule is
(See at signs in supergeometry for discussion of this sign rule and of an alternative sign rule that is also in use. )
Given a super Lie algebra , then
that sends a finite dimensional super Lie algebra to its Chevalley-Eilenberg algebra (def. 3) is a fully faithful functor which hence exibits super Lie algebras as a full subcategory of the opposite category of differential-graded algebras.
be a real spin representation of the spin group cover of the Lorentz group in this dimension. Then the -dimensional -supersymmetric super-Minkowski spacetime is the super Lie algebra that is characterized by the fact that its Chevalley-Eilenberg algebra is as follows:
The algebra has generators (as an associative algebra over )
for and subjects to the relations
denotes the -component of the -invariant spinor bilinear pairing that comes with every real spin representation applied to regarded as an -valued form;
summation over paired indices is understood.
This is characterized by the fact that its Chevalley-Eilenberg algebra is as follows:
it is generated from elements
with the super vielbein as before, and with 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 anti-commute with every generator. Finally the differential acts on these generators as follows:
where we are shifting spacetime indicices with the Lorentz metric
We discuss the classification of possible supersymmetry super Lie algebras.
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
|minimal real spin representation||in terms of||supergravity|
|4||4||d=4 N=1 supergravity|
|10||real||16||type II supergravity|
|superconformal super Lie algebra||R-symmetry||brane worldvolume theory|
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.
By realizing the conformal real Lie algebra as a real section of the complexified one is reduced to finding those (finite dimensional) simple super Lie algebras over the complex numbers whose even-graded part extends 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
|vector vector vector|
the only manifest spinor representation of or of appears in the exceptional super Lie algebra , which contains in its even parts acting spinorially on its odd part. This hence gives a superconformal super Lie algebra in dimension , 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
with the spinor representation of being the vector representation of . This we find in the above tables as a summand in the even-graded subalgebra of and of . Hence these are superconformal super Lie algebras in dimension , as shown in the statement.
The other exceptional isomorphism of relevance is
with the spinor representation of being the vector representation of . By the above tables this appears as a summand in the even-graded subalgebra of the super Lie algebra , and so this is the superconformal algebra in dimension .
Finally by triality the vector representation of is isomorphic to its spinor representation. By the above tables this means that is a superconformal algebra in dimension . For details on this see (Shnider 88, last paragraphs)
Further constraints follow from requiring super-unitary representations (Minwalla 98, section 4.3). This restricts for instance the 6d superconformal algebra to and , the latter being (over the reals as ) the symmetry algebra of the 6d (2,0)-superconformal QFT on the worldvolume of the M5-brane.
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.
Historically the first textbook on supersymmetry was
Further physics texts include
More mathematical accounts include
also (Deligne-Freed 99).
More physics-style accounts include
John Terning, Modern Supersymmetry, Oxford Science Publications
especially in the contribution
The appendix there,
I. L. Buchbinder, S. M. Kuzenko, Ideas and methods of supersymmetry and supergravity; or A walk through superspace
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 -algebras, Proc. Symp. Pure Math. 81, AMS, Providence, 2010, pp. 65-80 (arXiv:0909.0551)
Results on the classification of supersymmetry super Lie algebras (including higher dimensions and conformal/de Sitter supersymmetry) includes
Steven Shnider, The superconformal algebra in higher dimensions, Letters in Mathematical Physics November 1988, Volume 16, Issue 4, pp 377-383
Vladimir Dobrev, V.B. Petkova, All Positive Energy Unitary Irreducible Representations of Extended Conformal Supersymmetry, Phys.Lett. B162 (1985) 127-132
Leonardo Castellani, Riccardo D'Auria, Pietro Fré, volume 1, chapter II.2.2 of Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991)
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) (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
The observation that the lightest supersymmetric particle is a natural dark matter candidate goes back to
with review in
Supersymmetry seems to be favored by the Starobinsky model of cosmic inflation (see there for more).
A review of supersymmetry breaking is in
A quantitative analysis showing that locally supersymmetric spacetime theories will generically not exhibit global spacetime supersymmetry is
Pran Nath, Supersymmetry after the Higgs (arXiv:1501.01679)
Remembering that there is a considerable difference between global low energy supersymmetry and local higher energy supersymmetry aka supergravity: