Supersymmetric action ($S=\int d^{4}x[\int d^4\theta K+\int d^2\theta W+h.c.]$)

A supersymmetric action is invariant under supersymmetry up to boundary terms. For ordinary four-dimensional chiral superfields, a full superspace integration of a real Kähler potential and a chiral superspace integration of a superpotential provide the two-derivative ungauged action.

A two-derivative four-dimensional $\mathcal{N}=1$ gauge action is specified by a real gauge-invariant Kähler potential, a gauge-invariant holomorphic superpotential, and a holomorphic symmetric gauge kinetic function. A constant Fayet–Iliopoulos term is additionally allowed for an Abelian gauge factor. Renormalizability restricts the matter kinetic function to a quadratic form, the superpotential to degree at most three, and the gauge kinetic function to a constant invariant tensor.

The gauge kinetic function is a symmetric holomorphic function of chiral superfields multiplying the gauge field-strength F-term. Its real part controls gauge kinetic coefficients and its imaginary part controls topological-angle terms. Its Wilsonian perturbative correction is constrained by one-loop exactness of the holomorphic gauge kinetic function; canonical and one-particle-irreducible gauge couplings require separate normalization and infrared analysis.

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A holomorphic gauge coupling is the coefficient of the gauge kinetic F-term in a local Wilsonian effective action written in holomorphic variables. In a common normalization its real part is $1/g_h^2$. The continuous perturbative shift of its topological angle, together with holomorphy, restricts its beta function to one-loop order. Wave-function renormalization and anomalous canonical field rescalings distinguish it from the physical gauge coupling.

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A classical supersymmetric mechanical model takes its coordinates in a Grassmann algebra: the position is even and the fermionic coordinates are odd. Time differentiation is even, while variation with respect to an odd coordinate uses Grassmann derivatives. For example, moving an odd variation to the left and integrating by parts gives $\delta \int \psi \dot{\psi }dt=2\int \delta \psi \dot{\psi }dt$.

For the real variant $L={\dot{x}}^2/2+U^2/2-{\psi }_1{\dot{\psi }}_1/2+{\psi }_2{\dot{\psi }}_2/2+U^{\prime }{\psi }_1{\psi }_2$, the Euler-Lagrange equations areThe conserved even energy is $E={\dot{x}}^2/2-U^2/2-U^{\prime }{\psi }_1{\psi }_2$ and the odd supercharges are $Q_1=\dot{x}{\psi }_1-U{\psi }_2$, $Q_2=\dot{x}{\psi }_2-U{\psi }_1$. For example, ${\dot{Q}}_i=U^{\prime \prime }{\psi }_1{\psi }_2{\psi }_i=0$, since each odd element squares to zero. Also $d({\psi }_1{\psi }_2)/dt=U^{\prime }({\psi }_2^2+{\psi }_1^2)=0$, which proves $\dot{E}=0$.

In the real Grassmann-valued classical supersymmetric mechanics variant, impose $\dot{x}=U(x)$ and ${\psi }_2=-{\psi }_1$. Then ${\psi }_1{\psi }_2=0$, and differentiating the first-order equation gives $\ddot{x}=U{U}^{\prime }$. The remaining Euler-Lagrange equations reduce to ${\dot{\psi }}_1=-U^{\prime }{\psi }_1$. These solutions have $E=0$ but $Q_1=2U{\psi }_1=-Q_2$, which need not vanish: $d(U{\psi }_1)/dt=U^{\prime }U{\psi }_1-U{U}^{\prime }{\psi }_1=0$.

For $U=1+cx$, one obtains $x(t)=e^{ct}{x}_0+(e^{ct}-1)/c$ and ${\psi }_1(t)=e^{-ct}{\psi }_{10}$ when $c\ne 0$. At $c=0$, the limiting solution is $x(t)=x_0+t$, ${\psi }_1(t)={\psi }_{10}$. In either case $Q_1=2(1+c{x}_0){\psi }_{10}$.

In four spacetime dimensions, power-counting renormalizability restricts an ungauged chiral action to a canonical quadratic Kähler potential, up to transformations, and a superpotential of degree at most three. Terms of larger degree belong to an effective field theory.