superpotentials

A superpotential is a holomorphic function that encodes interactions in supersymmetric theories. In four-dimensional N=1 supersymmetric quantum field theories, the superpotential W is a gauge-invariant function of chiral superfields and enters the Lagrangian through F-terms. The scalar potential and the Yukawa couplings of scalars and fermions are determined by W via derivatives with respect to the scalar components of the superfields. Specifically, the scalar potential contains terms proportional to |∂W/∂φ_i|^2, while fermion masses and interactions arise from the second and higher derivatives of W. The structure of W is constrained by gauge invariance and holomorphy (dependence only on the fields, not their complex conjugates).

A key property of supersymmetric theories is the non-renormalization of the superpotential in perturbation theory: W does not receive perturbative quantum corrections, though non-perturbative effects (such as instantons or gaugino condensation) can modify it. This makes W a robust organizer of low-energy couplings and vacua.

In string theory, superpotentials arise in compactifications and play a central role in moduli stabilization. For example, in Type IIB flux compactifications, the Gukov-Vafa-Witten superpotential W = ∫ G_3 ∧ Ω depends on complex structure moduli and the axio-dilaton, generating a potential that fixes moduli. Similar constructions occur in other corners of string theory, including heterotic and F-theory contexts, often encoding non-perturbative effects.

Beyond field theory and string theory, the term appears in supersymmetric quantum mechanics and in mathematics, where a superpotential functions as a potential in a Landau-Ginzburg model or as part of constructions related to mirror symmetry and the study of singularities.