[Paper Review] Supersymmetry, a Biased Review

Ulf Lindström|ArXiv.org|Apr 2, 2002

Noncommutative and Quantum Gravity Theories35 references3 citations

TL;DR

This paper provides a physics-oriented introduction to supersymmetry, focusing on its geometric structure through superspace and superfields, and establishes a deep connection between N=1 and N=2 supersymmetric nonlinear sigma models and complex geometry. It demonstrates how tri-holomorphic isometries in hyperkähler target spaces allow for a gauging procedure that leads to the hyperkähler quotient construction, yielding new target space geometries such as Calabi metrics via the Kähler potential derived from moment map constraints.

ABSTRACT

This set of lectures contain a brief review of some basic supersymmetry and its representations, with emphasis on superspace and superfields. Starting from the Poincaré group, the supersymmetric extensions allowed by the Coleman-Mandula theorem and its generalisation to superalgebras, the Haag, Lopuszanski and Sohnius theorem, are discussed. Minkowski space is introduced as a quotient space and Superspace is presented as a direct generalization of this. The focus is then shifted from a general presentation to the relation between supersymmetry and complex geometry as manifested in the possible target space geometries for N=1 and N=2 supersymmetric nonlinear sigma models in four dimensions. Gauging of isometries in nonlinear sigma models is discussed for these cases, and the quotient construction is described.

Motivation & Objective

• To present a self-contained, physics-oriented introduction to supersymmetry and its representations, particularly in four dimensions.
• To clarify the geometric structure of supersymmetry by introducing superspace as a generalization of Minkowski space.
• To establish the connection between supersymmetry and complex geometry, especially in the context of nonlinear sigma models with N=1 and N=2 supersymmetry.
• To describe the gauging of isometries in N=1 and N=2 nonlinear sigma models and its realization through the hyperkähler quotient construction.
• To derive the Kähler potential of the quotient target space from moment map constraints and Fayet-Iliopoulos terms.

Proposed method

• Using the Poincaré algebra and the Coleman-Mandula–Haag–Lopuszanski–Sohnius theorems to motivate the existence of supersymmetry algebras.
• Introducing superspace as a quotient of the Poincaré group and constructing superfields as representations of the super-Poincaré algebra.
• Analyzing N=1 and N=2 nonlinear sigma models with Kähler and hyperkähler target spaces, respectively, using superfield formalism.
• Implementing the gauging of isometries via Fayet-Iliopoulos terms and moment map constraints, leading to the hyperkähler quotient construction.
• Deriving the effective Kähler potential on the quotient space from the extremization of the action with respect to vector and chiral superfields.
• Solving the moment map constraints using gauge choices and redefinitions to express the resulting Lagrangian in terms of the quotient geometry.

Experimental results

Research questions

• RQ1 How does supersymmetry extend the Poincaré algebra, and what are the implications for the structure of spacetime and fields?
• RQ2 What is the role of superspace and superfields in formulating supersymmetric field theories in four dimensions?
• RQ3 How do N=1 and N=2 nonlinear sigma models relate to complex and hyperkähler geometry in their target spaces?
• RQ4 What conditions allow for the gauging of isometries in supersymmetric nonlinear sigma models, and how does this lead to quotient constructions?
• RQ5 What is the explicit form of the Kähler potential on the hyperkähler quotient target space, and how does it generalize known models like CP^n?

Key findings

• The hyperkähler quotient construction for N=2 nonlinear sigma models yields a new Kähler potential expressed as a function of the moment map constraints, specifically √(c² + 4M₊M₋) − c ln(c + √(c² + 4M₊M₋)) + c ln M₊.
• The resulting target space geometry is hyperkähler and generalizes the Calabi metric, extending the known Kähler potential of CP^n models.
• The gauging of abelian tri-holomorphic isometries in the N=2 model leads to a consistent action with Fayet-Iliopoulos terms, invariant under local supersymmetry and gauge symmetry.
• The moment map constraints Φₐ₋Φₐ₊ = b and the V-equation ∂V(…)=c are solved via gauge fixing Φₙ₊₁₊ = Φₙ₊₁₋ = φ, reducing the system to a lower-dimensional effective theory.
• The final Lagrangian density is written solely in terms of the invariants M₊ and M₋, with the Kähler potential derived from the extremization of the action with respect to the vector and chiral superfields.
• The construction explicitly realizes the hyperkähler quotient as a physical mechanism to generate new supersymmetric nonlinear sigma models with rich geometric structure.