Superpotential

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Superpotential is a concept from particle physics' supersymmetry.

[edit] Example of superpotentiality

Let's look at the example of a one dimensional nonrelativistic particle with a 2D (i.e. two state) internal degree of freedom called "spin" (it's not really spin because "real" spin is for 3D particles). Let b be an operator which transforms a "spin up" particle into a "spin down" particle and its adjoint b† transforming a spin down particle into a spin up particle normalized such that the anticommutator {b,b†}=1. And of course, b2=0. Let p be the momentum of the particle and x be its position with [x,p]=i (let's use natural units where \hbar=1). Let W (the superpotential) be an arbitrary differentiable function of x and let the supersymmetric operators

Q_1=\frac{1}{2}\left[(p-iW)b+(p+iW)b^\dagger\right]
Q_2=\frac{i}{2}\left[(p-iW)b-(p+iW)b^\dagger\right]

Note that Q1 and Q2 are self-adjoint. Let the Hamiltonian

H=\{Q_1,Q_1\}=\{Q_2,Q_2\}=\frac{p^2}{2}+\frac{W^2}{2}+\frac{W'}{2}(bb^\dagger-b^\dagger b)

where W' is the derivative of W. Also note that {Q1,Q2}=0. This is nothing other than N=2 supersymmetry.

Let's also call the spin down state "bosonic" and the spin up state "fermionic". This is only in analogy to quantum field theory and should not be taken literally. Then, Q1 and Q2 maps "bosonic" states into "fermionic" states and vice versa.

[edit] Superpotential in dimension 4

In supersymmetry in dimension 4, which might have some connection to the nature, a scalar field in the theory is given as the lowest component of a chiral superfield, which is automatically complex. The complex conjugate of a chiral superfield is an anti-chiral superfield. To obtain the action from a set of superfields, the two choices are

1. Integrate a superfield on the whole superspace spanned by x0,1,2,3 and \theta,\bar\theta

or

2. Integrate a chiral superfield on the chiral half of a superspace, spanned by x0,1,2,3 and θ, not on \bar\theta.

Thus, given a set of chiral superfields, one can take an arbitrary holomorphic function of them, W, one can construct a term in the Lagrangian which is invariant under supersymmetry; W cannot depend on the complex conjugates. The function W is called the superpotential. The fact that W is holomorphic in the chiral superfields is the source of the tractability of supersymmetric theories. Indeed, W is known to receive no perturbative corrections, which is the celebrated perturbative non-renormalization theorem. It is corrected by non-perturbative processes, e.g. by instantons.

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