Form of scalar potential in SUSY/SUGRA $\sigma$-models

Question

In supersymmetry or supergravity, textbooks always show that one can define a Kähler potential $K=K(\phi^i,(\phi^i)^\ast)$ and an holomorphic superpotential $W=W(\phi_i)$ such that the scalar potential is given by (up to some normalisation)

$$ V=\left|\frac{\partial W}{\partial \phi_i}\right|^2~~\text{(SUSY)}\qquad V=e^{-K}\left( |D_iW|^2-3|W|^2\right)~~\text{(SUGRA)}. $$

with $D_i=\partial_i+\partial_i K$ the Kähler covariant derivative.

Now, consider a $\sigma$-model $\Sigma\to \mathcal{M}$ $$ \mathcal{L}=\frac{1}{2}g_{ij}(\partial_\mu \phi^i)(\partial^\mu \phi^i)^\ast-V(\phi^i,(\phi^i)^\ast)+\text{(higher spins)} $$

If the sigma model is supersymmetric, it is a consequence of Berger theorem that $\mathcal{M}$ is a Kähler manifold (because SUSY restricts the holonomy of $\mathcal{M}$, see for instance Cecotti's wonderful book Supersymmetric Field Theories) and thus the metric is kähler and $g_{ij}=\partial_i\overline{\partial}_j K$ (locally), proving that there exists a Kähler potential (In the case of SUGRA, the target manifold is Hodge Kähler and thus the same result holds).

The question now is: Is there a way to prove only with similar geometric arguments that the scalar potentials take the form above? It is fairly easy to prove those result "brute force", but I'm looking for something more elegant.

Cecotti proves from Morse theory that in SQM we indeed must have a superpotential, because it is the only way (except if $\mathcal{M}$ has Killing vectors) to deform the superalgebra. But even if the same reasoning holds in higher dimensions, I fail to see way the SUGRA case should be different (and why it explicitely depends on the Kähler potential).

This post imported from StackExchange Physics at 2016-11-16 11:18 (UTC), posted by SE-user Bulkilol

Answer 1

Let me first mention that I think that the answer is in Cecotti's book, even if quite scattered: in Section 6.8, a general relation is proved relating the scalar potential in a 4d SUGRA theory and the supersymetric variations of the fermionic fields of the theory. These supersymmetric variations and the Lagrangian of the theory are related in Section 8.6. Finally, Section 10.2.2 expresses the consistency constraints and, combined with the two above points, gives the result.

Some general comments:

1) In the rigid SUSY case, $W$ is an holomorphic function and the scalar potential is $|dW|^2$. In SUGRA, things are different: $W$ is an holomorphic section of the line bundle $L^{-1}$, where $L$ is the holomorphic line bundle on $\mathcal{M}$ coming from the Hodge-Kähler structure. In particular, it does no longer make sense to naively derive $W$ and to take the norm of the derivative. To derive $W$, one needs a connection on the line bundle, but it is exactly what $D_i$ is. Similarly, to take the norm, one needs an hermitian metric on the line bundle, but it is exactly what $e^{-K}$ is. I think that shows that $e^{-K}|D_i W|^2$ is the most natural analogue of $|dW|^2$.

2) Another difference between rigid SUSY and SUGRA is the presence of the gravitational multiplet: graviton, gravitino. The extra term $-3|W|^2$ in the scalar potential exactly comes from this additional multiplet.