Supersymmetric field theories and cohomology

Abstract

In this work, the cohomological structure of supersymmetric field theories is studied in a general context. Cohomology problem is defined by using the off-shell structure of the supersymmetry algebra. In order to define a cohomology problem it is crucial to use the two component (Weyl) formalism of the supersymmetry algebra, since each individual component of chiral and anti-chiral supersymmetry generators are nilpotent operators. It is then possible to show that the actions of supersymmetric field theories belong to a class of solutions of this cohomology problem and can be represented as multiple supervariations of some lower dimensional field polynomials due to the fact that the actions of supersymmetric field theories can be viewed as the highest component of respective multiplets. It is also discussed that the aforementioned algebraic structure can be thought as the algebraic source of the well-known non-renormalization theorems of several supersymmetric field theories. As an application of our method, the cohomological structure of Wess-Zumino model is discussed. It is shown that the kinetic, mass and the interaction terms can be constructed from the chiral and/or anti-chiral supervariations of the lower dimensional field' polynomials for off-shell supersymmetry. It is then found out that the -well known non-renormalization theorems of Wess-Zumino model is related to the terms that can only be written as pure chiral or anti-chiral supervariations of (mass) dimension three fields.

Özet

Bu çalışmada, genel anlamda süpersimetrik alan teorilerinin kohomolojik yapısı çalışılmıştır. Kohomoloji problemi süpersimetri cebri kullanılarak tanımlanmıştır. Süpersimetrik alan teorilerinin eylemlerinin ilgili çokluların en üst elemanı olduğu kullanılarak elde edilen bu kohomoloji probleminin çözümleri yardımıyla bu eylemlerin düşük boyutlu alan polinomlarının çoklu süpervaryasyonları olarak temsil edilebileceği gösterilmiştir. Bu yaklaşımın bir uygulaması olarak Wess-Zumino modelinin kohomolojik yapısı incelenmiştir. Elde edilen sonuçlardan modelin eyleminin kinetik, kütle ve etkileşme terimlerinin kabuk-dışı süpersimetrik durum için tam olarak düşük boyutlu alan polinomlarınm kiral ve/veya anti-kiral çoklu süper varyasyonları ile yazılabileceği gösterilmiştir.