Superspace

Superspace is the coordinate space of a theory exhibiting

degrees of freedom.

The word "superspace" was first used by John Wheeler in an unrelated sense to describe the configuration space of general relativity; for example, this usage may be seen in his 1973 textbook Gravitation.

Informal discussion

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 for

modulo the algebra of the Lorentz group. A typical notation for the coordinates on such a space is ${\displaystyle (x,\theta ,{\bar {\theta }})}$ with the overline being the give-away that super Minkowski space is the intended space.

Superspace is also commonly used as a synonym for the

A third usage of the term "superspace" is as a synonym for a supermanifold: a supersymmetric generalization of a manifold. 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.

Examples

Several examples are given below. The first few assume a definition of superspace as a super vector space. This is denoted as R^m|n, the Z₂-graded vector space with R^m as the even subspace and Rⁿ as the odd subspace. The same definition applies to C^m|n.

The four-dimensional examples take superspace to be

quite outside the ordinary bounds and concerns of physics.)

Trivial examples

The smallest superspace is a point which contains neither bosonic nor fermionic directions. Other trivial examples include the n-dimensional real plane Rⁿ, which is a

in 1873.

The superspace of supersymmetric quantum mechanics

Supersymmetric quantum mechanics with N supercharges is often formulated in the superspace R^1|2N, which contains one real direction t identified with time and N complex Grassmann directions which are spanned by Θ_i and Θ^*_i, where i runs from 1 to N.

Consider the special case N = 1. The superspace R^1|2 is a 3-dimensional vector space. A given coordinate therefore may be written as a triple (t, Θ, Θ^*). The coordinates form a

on two odd coordinates. This superspace is an abelian Lie superalgebra, which means that all of the aforementioned brackets vanish

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

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

One may define functions from this vector space to itself, which are called

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 of t multiplied by the zeroeth and first order terms in the two Grassmann coordinates

${\displaystyle \Phi \left(t,\Theta ,\Theta ^{*}\right)=\phi (t)+\Theta \Psi (t)-\Theta ^{*}\Phi ^{*}(t)+\Theta \Theta ^{*}F(t)}$

Superfields, which are representations of the supersymmetry of superspace, generalize the notion of tensors, 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

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

These derivatives may be assembled into supercharges

${\displaystyle Q={\frac {\partial }{\partial \theta }}-i\Theta ^{*}{\frac {\partial }{\partial t}}\quad {\text{and}}\quad Q^{\dagger }={\frac {\partial }{\partial \theta ^{*}}}+i\Theta {\frac {\partial }{\partial t}}}$

whose anticommutators identify them as the fermionic generators of a supersymmetry algebra

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

where i times the time derivative is the Hamiltonian operator in quantum mechanics. Both Q and its adjoint anticommute with themselves. The supersymmetry variation with supersymmetry parameter ε of a superfield Φ is defined to be

${\displaystyle \delta _{\epsilon }\Phi =(\epsilon ^{*}Q+\epsilon Q^{\dagger })\Phi .}$

We can evaluate this variation using the action of Q on the superfields

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

Similarly one may define covariant derivatives on superspace

${\displaystyle D={\frac {\partial }{\partial \theta }}-i\theta ^{*}{\frac {\partial }{\partial t}}\quad {\text{and}}\quad D^{\dagger }={\frac {\partial }{\partial \theta ^{*}}}-i\theta {\frac {\partial }{\partial t}}}$

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

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

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.

Supersymmetric extensions of Minkowski space

N = 1 super Minkowski space

Perhaps the most studied concrete superspace in physics 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 the direct sum of four real bosonic dimensions and four real Grassmann dimensions (also known as fermionic dimensions or spin dimensions).^[5]

In

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 a

Seiberg–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 many sign 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

${\displaystyle {\bar {\theta }}\ {\stackrel {\mathrm {def} }{=}}\ i\theta ^{\dagger }\gamma ^{0}=-\theta ^{\perp }C}$

where ${\displaystyle C}$ is the charge conjugation matrix, which is defined by the property that when it conjugates a

, the gamma matrix is negated and transposed. The first equality is the definition of ${\displaystyle {\bar {\theta }}}$ while the second is a consequence of the Majorana spinor condition ${\displaystyle \theta ^{*}=i\gamma _{0}C\theta }$. The conjugate spinor plays a role similar to that of ${\displaystyle \theta ^{*}}$ in the superspace ${\displaystyle \mathbb {R} ^{1|2}}$, except that the Majorana condition, as manifested in the above equation, imposes that ${\displaystyle \theta }$ and ${\displaystyle \theta ^{*}}$ are not independent.

In particular we may construct the supercharges

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

which satisfy the supersymmetry algebra

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

where ${\displaystyle P=i\partial _{\mu }}$ 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.

Extended supersymmetry

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 ${\displaystyle \mathbb {R} ^{4|4{\mathcal {N}}}}$.

In general relativity

The word "superspace" is also used in a completely different and unrelated sense, in the book Gravitation by Misner, Thorne and Wheeler. There, it refers to the configuration space of general relativity, and, in particular, the view of gravitation as geometrodynamics, 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 the ADM formalism, as well as ideas surrounding the Hamilton–Jacobi–Einstein equation and the Wheeler–DeWitt equation.

See also

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

Notes

References

• ISBN 978-1-4020-1338-6
  (Second printing)