Lesson plan
F-term
In theoretical physics, one often analyzes theories with supersymmetry in which F-terms play an important role. In four dimensions, the minimal N=1 supersymmetry may be written using a superspace. This superspace involves four extra fermionic coordinates \theta^1,\theta^2,\bar\theta^1,\bar\theta^2 , transforming as a two-component spinor and its conjugate. Every superfield—i.e. a field that depends on all coordinates of the superspace—may be expanded with respect to the new fermionic coordinates. There exists a special kind of superfields, the so-called chiral superfields, that only depend on the variables \theta but not their conjugates. The last term in the corresponding expansion, namely F \theta^1\theta^2 , is called the F-term. Applying an infinitesimal supersymmetry transformation to a chiral superfield results in yet another chiral superfield whose F-term, in particular, changes by a total derivative. This is significant because then \int{d^4x\, F(x)} is invariant under SUSY transformations as long as boundary terms vanish. Thus F-terms may be used in constructing supersymmetric actions. Manifestly-supersymmetric Lagrangians may also be written as integrals over the whole superspace. Some special terms, such as the superpotential, may be written as integrals over \theta s only.…
Article summary
In theoretical physics, one often analyzes theories with supersymmetry in which F-terms play an important role. In four dimensions, the minimal N=1 supersymmetry may be written using a superspace. This superspace involves four extra fermionic coordinates \theta^1,\theta^2,\bar\theta^1,\bar\theta^2 , transforming as a two-component spinor and its conjugate. Every superfield—i.e. a field that depends on all coordinates of the superspace—may be expanded with respect to the new fermionic coordinates. There exists a special kind of superfields, the so-called chiral superfields, that only depend on the variables \theta but not their conjugates. The last term in the corresponding expansion, namely F \theta^1\theta^2 , is called the F-term. Applying an infinitesimal supersymmetry transformation to a chiral superfield results in yet another chiral superfield whose F-term, in particular, changes by a total derivative. This is significant because then \int{d^4x\, F(x)} is invariant under SUSY transformations as long as boundary terms vanish. Thus F-terms may be used in constructing supersymmetric actions. Manifestly-supersymmetric Lagrangians may also be written as integrals over the whole superspace. Some special terms, such as the superpotential, may be written as integrals over \theta s only.…
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Discussion questions
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