Superspace is thecoordinate space of a theory exhibitingsupersymmetry. In such a formulation, along with ordinary space dimensionsx,y,z, ..., there are also "anticommuting" dimensions whose coordinates are labeled inGrassmann numbers rather than real numbers. The ordinary space dimensions correspond tobosonic degrees of freedom, the anticommuting dimensions tofermionic degrees of freedom.

The word "superspace" was first used byJohn Wheeler in an unrelated sense to describe theconfiguration space ofgeneral relativity; for example, this usage may be seen in his 1973 textbookGravitation.

There are several similar, but not equivalent, definitions of superspace that have been used, and continue to be used in the mathematical and physics literature. One such usage is as a synonym forsuper Minkowski space.^[1] In this case, one takes ordinaryMinkowski space, and extends it with anti-commuting fermionic degrees of freedom, taken to be anti-commutingWeyl spinors from theClifford algebra associated to theLorentz group. Equivalently, the super Minkowski space can be understood as the quotient of thesuper Poincaré algebra modulo the algebra of the Lorentz group. A typical notation for the coordinates on such a space is${\displaystyle (x,\theta ,\overline{\theta })}$ with the overline being the give-away that super Minkowski space is the intended space.

Superspace is also commonly used as a synonym for thesuper vector space. This is taken to be an ordinaryvector space, together with additional coordinates taken from theGrassmann algebra, i.e. coordinate directions that are Grassmann numbers. There are several conventions for constructing a super vector space in use; two of these are described by Rogers^[2] and DeWitt.^[3]

A third usage of the term "superspace" is as a synonym for asupermanifold: a supersymmetric generalization of amanifold. Note that both super Minkowski spaces and super vector spaces can be taken as special cases of supermanifolds.

A fourth, and completely unrelated meaning saw a brief usage in general relativity; this is discussed in greater detail at the bottom.

Several examples are given below. The first few assume a definition of superspace as asuper vector space. This is denoted asR^m|n, theZ₂-graded vector space withR^m as the even subspace andRⁿ as the odd subspace. The same definition applies toC^m|n.

The four-dimensional examples take superspace to be super Minkowski space. Although similar to a vector space, this has many important differences: First of all, it is anaffine space, having no special point denoting the origin. Next, the fermionic coordinates are taken to be anti-commuting Weyl spinors from theClifford algebra, rather than beingGrassmann numbers. The difference here is that the Clifford algebra has a considerably richer and more subtle structure than the Grassmann numbers. So, the Grassmann numbers are elements of theexterior algebra, and the Clifford algebra has an isomorphism to the exterior algebra, but its relation to theorthogonal group and thespin group, used to construct thespin representations, give it a deep geometric significance. (For example, the spin groups form a normal part of the study ofRiemannian geometry,^[4] quite outside the ordinary bounds and concerns of physics.)

The smallest superspace is a point which contains neither bosonic nor fermionic directions. Other trivial examples include then-dimensional real planeRⁿ, which is avector space extending inn real, bosonic directions and no fermionic directions. The vector spaceR^0|n, which is then-dimensional realGrassmann algebra. The spaceR^1|1 of one even and one odd direction is known as the space ofdual numbers, introduced byWilliam Clifford in 1873.

Supersymmetric quantum mechanics withNsupercharges is often formulated in the superspaceR^1|2N, which contains one real directiont identified withtime andN complex Grassmann directions which are spanned by Θ_i and Θ^*_i, wherei runs from 1 toN.

Consider the special caseN = 1. The superspaceR^1|2 is a 3-dimensional vector space. A given coordinate therefore may be written as a triple (t, Θ, Θ^*). The coordinates form aLie superalgebra, in which the gradation degree oft is even and that of Θ and Θ^* is odd. This means that a bracket may be defined between any two elements of this vector space, and that this bracket reduces to thecommutator on two even coordinates and on one even and one odd coordinate while it is ananticommutator on two odd coordinates. This superspace is an abelian Lie superalgebra, which means that all of the aforementioned brackets vanish

${\displaystyle \left[t,t\right]=\left[t,\theta \right]=\left[t,{\theta }^{\ast }\right]=\left\{\theta ,\theta \right\}=\left\{\theta ,{\theta }^{\ast }\right\}=\left\{{\theta }^{\ast },{\theta }^{\ast }\right\}=0}$

where${\displaystyle [a,b]}$ is the commutator ofa andb and${\displaystyle \{a,b\}}$ is the anticommutator ofa andb.

One may define functions from this vector space to itself, which are calledsuperfields. The above algebraic relations imply that, if we expand our superfield as apower series in Θ and Θ^*, then we will only find terms at the zeroeth and first orders, because Θ² = Θ^*2 = 0. Therefore, superfields may be written as arbitrary functions oft multiplied by the zeroeth and first order terms in the two Grassmann coordinates

${\displaystyle \mathrm{\Phi }\left(t,\mathrm{\Theta },{\mathrm{\Theta }}^{\ast }\right)=\varphi (t)+\mathrm{\Theta }\mathrm{\Psi }(t)-{\mathrm{\Theta }}^{\ast }{\mathrm{\Phi }}^{\ast }(t)+\mathrm{\Theta }{\mathrm{\Theta }}^{\ast }F(t)}$

Superfields, which are representations of the supersymmetry of superspace, generalize the notion oftensors, which are representations of the rotation group of a bosonic space.

One may then define derivatives in the Grassmann directions, which take the first order term in the expansion of a superfield to the zeroeth order term and annihilate the zeroeth order term. One can choose sign conventions such that the derivatives satisfy the anticommutation relations

${\displaystyle \left\{\frac{\mathrm{\partial }}{\mathrm{\partial }\theta },\mathrm{\Theta }\right\}=\left\{\frac{\mathrm{\partial }}{\mathrm{\partial }{\theta }^{\ast }},{\mathrm{\Theta }}^{\ast }\right\}=1}$

These derivatives may be assembled intosupercharges

${\displaystyle Q=\frac{\mathrm{\partial }}{\mathrm{\partial }\theta }-i{\mathrm{\Theta }}^{\ast }\frac{\mathrm{\partial }}{\mathrm{\partial }t}\text{and}{Q}^{†}=\frac{\mathrm{\partial }}{\mathrm{\partial }{\theta }^{\ast }}+i\mathrm{\Theta }\frac{\mathrm{\partial }}{\mathrm{\partial }t}}$

whose anticommutators identify them as the fermionic generators of asupersymmetry algebra

${\displaystyle \left\{Q,Q^{†}\right\}=2i\frac{\mathrm{\partial }}{\mathrm{\partial }t}}$

wherei times the time derivative is theHamiltonian operator inquantum mechanics. BothQ and its adjoint anticommute with themselves. The supersymmetry variation with supersymmetry parameter ε of a superfield Φ is defined to be

${\displaystyle {\delta }_{ϵ}\mathrm{\Phi }=({ϵ}^{\ast }Q+ϵQ^{†})\mathrm{\Phi }.}$

We can evaluate this variation using the action ofQ on the superfields

${\displaystyle \left[Q,\mathrm{\Phi }\right]=\left(\frac{\mathrm{\partial }}{\mathrm{\partial }\theta }-i{\theta }^{\ast }\frac{\mathrm{\partial }}{\mathrm{\partial }t}\right)\mathrm{\Phi }=\psi +{\theta }^{\ast }\left(F-i\dot{\varphi }\right)+i\theta {\theta }^{\ast }\dot{\psi }.}$

Similarly one may definecovariant derivatives on superspace

${\displaystyle D=\frac{\mathrm{\partial }}{\mathrm{\partial }\theta }-i{\theta }^{\ast }\frac{\mathrm{\partial }}{\mathrm{\partial }t}\text{and}{D}^{†}=\frac{\mathrm{\partial }}{\mathrm{\partial }{\theta }^{\ast }}-i\theta \frac{\mathrm{\partial }}{\mathrm{\partial }t}}$

which anticommute with the supercharges and satisfy a wrong sign supersymmetry algebra

${\displaystyle \left\{D,D^{†}\right\}=-2i\frac{\mathrm{\partial }}{\mathrm{\partial }t}}$.

The fact that the covariant derivatives anticommute with the supercharges means the supersymmetry transformation of a covariant derivative of a superfield is equal to the covariant derivative of the same supersymmetry transformation of the same superfield. Thus, generalizing the covariant derivative in bosonic geometry which constructs tensors from tensors, the superspace covariant derivative constructs superfields from superfields.

Perhaps the most studied concrete superspace inphysics is${\displaystyle d=4,\mathcal{N}=1}$ super Minkowski space${\displaystyle {\mathbb{R}}^{4|4}}$ or sometimes written${\displaystyle {\mathbb{R}}^{1,3|4}}$, which is thedirect sum of four realbosonic dimensions and four realGrassmann dimensions (also known asfermionic dimensions orspin dimensions).^[5]

Insupersymmetricquantum field theories one is interested in superspaces which furnishrepresentations of a Lie superalgebra called asupersymmetry algebra. The bosonic part of the supersymmetry algebra is thePoincaré algebra, while the fermionic part is constructed usingspinors with Grassmann number valued components.

For this reason, in physical applications one considers an action of the supersymmetry algebra on the four fermionic directions of${\displaystyle {\mathbb{R}}^{4|4}}$ such that they transform as aspinor under the Poincaré subalgebra. In four dimensions there are three distinct irreducible 4-component spinors. There is theMajorana spinor, the left-handedWeyl spinor and the right-handed Weyl spinor. TheCPT theorem implies that in aunitary, Poincaré invariant theory, which is a theory in which theS-matrix is aunitary matrix and the same Poincaré generators act on the asymptotic in-states as on the asymptotic out-states, the supersymmetry algebra must contain an equal number of left-handed and right-handed Weyl spinors. However, since each Weyl spinor has four components, this means that if one includes any Weyl spinors one must have 8 fermionic directions. Such a theory is said to haveextended supersymmetry, and such models have received a lot of attention. For example, supersymmetric gauge theories with eight supercharges and fundamental matter have been solved byNathan Seiberg andEdward Witten, seeSeiberg–Witten gauge theory. However, in this subsection we are considering the superspace with four fermionic components and so no Weyl spinors are consistent with the CPT theorem.

Note: There are manysign conventions in use and this is only one of them.

Therefore, the four fermionic directions transform as a Majorana spinor${\displaystyle {\theta }_{\alpha }}$. We can also form a conjugate spinor

${\displaystyle \overline{\theta }\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}i{\theta }^{†}{\gamma }^0=-{\theta }^{\perp }C}$

where${\displaystyle C}$ is the charge conjugation matrix, which is defined by the property that when it conjugates agamma matrix, the gamma matrix is negated and transposed. The first equality is the definition of${\displaystyle \overline{\theta }}$ while the second is a consequence of the Majorana spinor condition${\displaystyle {\theta }^{\ast }=i{\gamma }_{0}C\theta }$. The conjugate spinor plays a role similar to that of${\displaystyle {\theta }^{\ast }}$ in the superspace${\displaystyle {\mathbb{R}}^{1|2}}$, except that the Majorana condition, as manifested in the above equation, imposes that${\displaystyle \theta }$ and${\displaystyle {\theta }^{\ast }}$ are not independent.

In particular we may construct the supercharges

${\displaystyle Q=-\frac{\mathrm{\partial }}{\mathrm{\partial }\overline{\theta }}+{\gamma }^{\mu }\theta {\mathrm{\partial }}_{\mu }}$

which satisfy the supersymmetry algebra

${\displaystyle \left\{Q,Q\right\}=\left\{\overline{Q},Q\right\}C=2{\gamma }^{\mu }{\mathrm{\partial }}_{\mu }C=-2i{\gamma }^{\mu }{P}_{\mu }C}$

where${\displaystyle P=i{\mathrm{\partial }}_{\mu }}$ is the 4-momentum operator. Again the covariant derivative is defined like the supercharge but with the second term negated and it anticommutes with the supercharges. Thus the covariant derivative of a supermultiplet is another supermultiplet.

It is possible to have${\displaystyle \mathcal{N}}$ sets of supercharges${\displaystyle Q^I}$ with${\displaystyle I=1,\cdots ,\mathcal{N}}$, although this is not possible for all values of${\displaystyle \mathcal{N}}$.

These supercharges generate translations in a total of${\displaystyle 4\mathcal{N}}$ spin dimensions, hence forming the superspace${\displaystyle {\mathbb{R}}^{4|4\mathcal{N}}}$.

The word "superspace" is also used in a completely different and unrelated sense, in the bookGravitation by Misner, Thorne and Wheeler. There, it refers to theconfiguration space ofgeneral relativity, and, in particular, the view of gravitation asgeometrodynamics, an interpretation of general relativity as a form of dynamical geometry. In modern terms, this particular idea of "superspace" is captured in one of several different formalisms used in solving the Einstein equations in a variety of settings, both theoretical and practical, such as in numerical simulations. This includes primarily theADM formalism, as well as ideas surrounding theHamilton–Jacobi–Einstein equation and theWheeler–DeWitt equation.

• Chiral superspace
• Harmonic superspace
• Projective superspace
• Super Minkowski space
• Supergroup
• Lie superalgebra

• Duplij, Steven[in Ukrainian];Siegel, Warren; Bagger, Jonathan, eds. (2005),Concise Encyclopedia of Supersymmetry And Noncommutative Structures in Mathematics and Physics, Berlin, New York:Springer,ISBN 978-1-4020-1338-6 (Second printing)

• Geometry
• Supersymmetry
• General relativity

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