superpotential

A superpotential is a holomorphic function that appears in certain supersymmetric theories, most prominently four-dimensional N=1 supersymmetric field theories. It is a function W(Φ_i) of chiral superfields Φ_i, and its scalar components φ_i contribute to the F-term part of the Lagrangian. The F-term equations F_i = ∂W/∂φ_i determine many physical features, and the scalar potential includes the F-term contribution V_F = ∑_i |∂W/∂φ_i|^2, together with possible D-term contributions from gauge interactions. The vacua of the theory satisfy F_i = 0 (mod D-term constraints). The superpotential is holomorphic, and, in perturbation theory, it is protected by a non-renormalization theorem: perturbative quantum corrections do not alter W, although non-perturbative effects can generate corrections.

In string theory, superpotentials encode how extra dimensions are compactified and stabilized. A prominent example is the Gukov–Vafa–Witten (GVW) superpotential in type IIB string theory, W_GVW = ∫ G_3 ∧ Ω, which depends on complex structure moduli and the axio-dilaton. Fluxes, D-branes, and instantons can generate or modify superpotentials in various compactifications, providing a mechanism to fix moduli and influence low-energy physics.

In mathematics and related physics contexts, the term also appears in two-dimensional N=(2,2) theories as W, a holomorphic potential in Landau-Ginzburg models. There, W determines the category of B-branes via matrix factorizations and plays a central role in mirror symmetry, connecting Landau-Ginzburg models to nonlinear sigma models on Calabi–Yau manifolds.