String Theory and M-Theory

4.2 Global world-sheet supersymmetry 113
Superspace
Exercise 4.2 shows that the action (4.2) is invariant under the supersymmetry
transformations. The supersymmetry of component actions, such as
this one, is not manifest. The easiest way to make this symmetry manifest
is by rewriting the action using a superspace formalism. Superspace is
an extension of ordinary space-time that includes additional anticommuting
(Grassmann) coordinates, and superfields are fields defined on superspace.
The superfield formulation entails adding an off-shell degree of freedom to
the world-sheet theory, without changing the physical content. This has the
advantage of ensuring that the algebra of supersymmetry transformations
closes off-shell, that is, without use of the equations of motion.
The superfield formulation is very convenient for making supersymmetry
manifest (and simplifying calculations) in theories that have a relatively
small number of conserved supercharges. The number of supercharges is
two in the present case. When the number is larger than four, as is necessarily
the case for supersymmetric theories when the space-time dimension
is greater than four, a superfield formulation can become very unwieldy or
even impossible.
The super-world-sheet coordinates are given by (σα , θA), where
θ−
θA =
(4.17)
θ+
are anticommuting Grassmann coordinates
{θA, θB} = 0, (4.18)
which form a Majorana spinor. Upper and lower spinor indices need not be
distinguished here, so θ A = θA. Frequently these indices are not displayed.
For the usual bosonic world-sheet coordinates let us define σ 0 = τ and
σ 1 = σ. One can then introduce a superfield Y µ (σ α , θ). The most general
such function has a series expansion in θ of the form
Y µ (σ α , θ) = X µ (σ α ) + ¯ θψ µ (σ α ) + 1
2 ¯ θθB µ (σ α ), (4.19)
where B µ (σ α ) is an auxiliary field whose inclusion does not change the
physical content of the theory. This field is needed to make supersymmetry
manifest. A term with more powers of θ would automatically vanish as a
consequence of the anticommutation properties of the Grassmann numbers
θA. Since ¯ ψθ = ¯ θψ for Majorana spinors, a term linear in θ would be
equivalent to the linear term in ¯ θ appearing above.