Supersymmetry Breaking in Wess-Zumino Models

  • John Z. Imbrie
Part of the NATO ASI Series book series (NSSB, volume 234)


A combination of index theory [1,2] and multiphase analysis [3,4] should lead to an understanding of the vacuum structure of the weakly coupled two-dimensional Wess-Zumino model in infinite volume. The supersymmetric interaction is |V′(φ)|2 + V″(φ)\(\bar{\psi }\) ψ, where for the N = 1 model φ,ψ are real, while for the N = 2 model φ,ψ are ocmplex. We take V to be a polynomial of degree n¡ and scale it as V(Ф) → λ−2 V(λФ) with λ small. If V′has n−1 distinct zeros, then |V′|2 has n−1 minima at zero. At small λ multiphase expansion should be possible, leading as in [4] to a construction of the stable vacua. The index theorems proven in [5] should help determine the number of stable vacua. For example, in the N = 2 model the periodic partition function is equal to n−1, so the free energy must be zero and so supersymmetry is unbroken. A careful analysis of the expansion should lead to the further conclusion that all n−1 possible vacua are stable thermodynamically [3]. These conclusions must be modified in the N = 1 case because the Pffaffian in the measure of alternates in sign from one minimum to the next.


  1. 1.
    Jaffe, A., Lesniewski, A., these proceedings.Google Scholar
  2. 2.
    Witten, E.: Nucl. Phys. B202, 253 (1982).MathSciNetADSCrossRefGoogle Scholar
  3. 3.
    Borgs, C., Imbrie, J.Z.: Multiphase Analysis in Quantum Field Theory, in preparation.Google Scholar
  4. 4.
    Imbrie, J.Z.: Commun. Math. Phys. 82, 261 (1981) and 82, 305 (1981).MathSciNetGoogle Scholar
  5. 5.
    Jaffe, A., Lesniewski, A.. Weitzman, J.: Commun Math. Phys. 112,75 (1987). The Loop Space S′→R and Supersymmetric Quantum Fields, to appear.Google Scholar

Copyright information

© Plenum Press, New York 1990

Authors and Affiliations

  • John Z. Imbrie
    • 1
  1. 1.Departments of Mathematics and PhysicsHarvard UniversityCambridgeUSA

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