Higher Dimensional Supersymmetry in 4D Superspace
1/31/2001
359 citations (358 excluding self-citations). A technical paper that became a standard reference for extra-dimensional model building in 4D superspace language.
The Problem
By 2001, extra-dimensional model building was one of the most active areas in theoretical physics. Theories with supersymmetric extra dimensions offered new approaches to the hierarchy problem, gauge coupling unification, and flavor physics. But the calculations were typically done in higher-dimensional notation that was unfamiliar to most particle phenomenologists, who thought in terms of 4D N=1 superspace — the standard language of supersymmetric model building. Every time a phenomenologist wanted to build a model with a 5D or 6D bulk, they had to either learn the higher-dimensional formalism or reinvent the translation piecemeal.
The Key Idea
The paper provides the complete, explicit translation: supersymmetric Yang-Mills theories from 5 through 10 dimensions, written entirely in 4D N=1 superspace. The higher-dimensional fields decompose into 4D superfields with specific transformation properties. The formulation handles interactions between bulk fields and brane-localized matter, orbifold projections that break symmetries at fixed points, localization of chiral fermions, anomaly cancellation via inflow, and super-Chern-Simons terms. The result is a toolkit that lets model builders work in the language they already know while constructing theories in any number of extra dimensions.
Impact
The formalism became the standard language for building realistic grand unified theories in extra dimensions. Orbifold GUT constructions, where the GUT symmetry is broken by boundary conditions at fixed points of the extra dimension, relied heavily on the brane-bulk interaction terms this paper provided. Hall and Nomura’s SO(10) models in 5D (170 citations) and 6D (209 citations) used the superspace dictionary to write gauge-invariant orbifold projections and compute proton decay rates. Burdman, Nomura, and Thaler’s gauge-Higgs unification models (205 citations), where the Higgs is identified as the extra-dimensional component of a gauge field, used the formalism to handle the supersymmetric completion of the gauge-Higgs system. Csáki, Erlich, and Terning’s comprehensive review of proton stability in unified theories (553 citations) treats the paper as a foundational reference for extra-dimensional GUT technology.
The paper also enabled a generation of technical work on the consistency of extra-dimensional field theories: anomaly cancellation on orbifolds (116 citations), the structure of brane-localized Fayet-Iliopoulos terms (137 citations), and the singular behavior of brane kinetic terms (141 citations) were all analyzed using the superspace formulation. The connection to the Little Higgs program runs through this formalism: the moose/theory-space construction is the deconstructed version of the extra dimension, and this paper provides the supersymmetric dictionary that makes the equivalence explicit. With 359 citations and only 1 self-citation, the paper found an audience far beyond what we expected from a technical exercise.
Recollections
This was written in the aftermath of the large extra dimension phase of particle physics that began in 1998. Thomas Gregoire and I were both Nima’s students, and this was our second practice problem. Nima recognized that everyone would try to supersymmetrize the extra-dimensional constructs and it wasn’t obvious how to do it. The challenges came from brane-bulk interactions — how fields living on a brane couple to fields propagating in the bulk.
The key insight was that 4D N=1 superspace constrains the forms of extra-dimensional interactions severely. Once you guarantee the kinetic terms of the bulk fields are correct, there is effectively no freedom in how the supersymmetric interactions can behave. Thomas and I spent most of our time on 5 and 6 dimensions, where the structure is rich and the applications to model building are immediate. The extension to 7 through 10 dimensions went quickly after that. We later discovered that the gods of superspace (Siegel et al.) had done something similar, though without brane-bulk interactions. Embarrassingly, we made some mistakes in the higher dimensions that were later corrected.
The paper had remarkable legs — it continued accumulating citations years after publication. In 2012, I was at the Kavli Institute for Theoretical Physics and talked with Joe Polchinski, who was always gracious and kind to junior scientists. He said he had the paper printed out on his table in his office and reached over to show it to me. He said he really enjoyed the simultaneous rigor and playfulness of it. That meant a great deal coming from him.