Only the Weyl fermions in the charged chiral multiplets contribute to the Abelian triangle gauge anomaly. Their cubic coefficient isand the mixed gauge-gravitational coefficient is $2+0-2=0$. Diagrammatically, the oppositely charged $X_{2+}$ and $X_{2-}$ fermion triangles have equal magnitudes and opposite signs. The neutral gaugino also contributes nothing, so the theory is anomaly free.

Gauge invariance and renormalizability permit the most general superpotentialAfter setting the linear and quadratic parameters to zero, the supersymmetric non-renormalization theorem ensures that perturbative quantum corrections do not regenerate them in the Wilsonian superpotential.

Write the scalar components as $x_{+},x_0,x_{-}$. For generic nonzero $\lambda ,y$, the F-term scalar potential and D-term scalar potential, including a Fayet–Iliopoulos term, areEvery term is nonnegative. The F-flat equations force $x_0=0$ and $x_{+}{x}_{-}=0$. If $\xi >0$, the vacuumalso makes $D=0$; if $\xi <0$, interchange $x_{+}$ and $x_{-}$ and use $|x_{+}{|}^2=-\xi /(2g)$. Thus for nonzero $\xi$ the charged vacuum expectation value spontaneously breaks the gauged $U(1)$ through the Higgs mechanism, but $F_i=D=0$ means supersymmetry remains unbroken. For $\xi =0$, the origin preserves both the gauge symmetry and supersymmetry.

A chiral superfield obeysExamples in the Minimal supersymmetric Standard Model include the quark, lepton and Higgs chiral superfields. Such a multiplet contains a complex scalar $\varphi$, a two-component Weyl spinor $\psi$, and a complex auxiliary field $F$; only the scalar and fermion propagate on shell.

The chiral coordinate $y^{\mu }=x^{\mu }+i\theta {\sigma }^{\mu }\bar{\theta }$ is annihilated in the required combination by ${\bar{D}}_{\dot{\alpha }}$, so the solution has the simple formTaylor-expand each component about $x$. Nilpotence truncates the series, and the Grassmann identitygives the chiral-superfield component expansionTherefore

Fermion parity is defined byIt anticommutes with $Q_{\alpha }$ and ${\bar{Q}}_{\dot{\alpha }}$. At fixed nonzero energy and momentum, choose a supercharge combination $S$ for which $\{S,S^{†}\}=cI$ with $c>0$. Taking the finite-dimensional trace over one supermultiplet givesbecause cyclicity of the trace and anticommutation of $(-1{)}^F$ with $S$ make the two terms cancel. ThereforeThis is boson-fermion degeneracy in a supermultiplet.

If explicit or soft supersymmetry breaking terms are added, the supercharge is no longer a conserved symmetry of the full Hamiltonian and states need not form representations of the supersymmetry algebra at equal energy. The positive anticommutator cannot be replaced by a constant on a purported multiplet, so the supertrace proof and mass degeneracy fail.

Apply parity to the mixed anticommutator. The transformed left-hand side isbecause $|{\eta }_P|=1$. The Pauli-matrix identity represented by this product leaves the temporal matrix unchanged and reverses the three spatial matrices, so it equalsBut a parity transformation acts on momentum as $P^0\mapsto P^0$ and $P^i\mapsto -P^i$. This is exactly the parity transform ofso $\{Q_{\alpha },{\bar{Q}}_{\dot{\beta }}\}=2{\sigma }_{\alpha \dot{\beta }}^{\mu }{P}_{\mu }$ is consistent with the stated transformation law. The arbitrary intrinsic phase cancels.

An internal generator $T_a$ is a Lorentz scalar. The Coleman–Mandula theorem and the graded extension allow it to act nontrivially on supercharges only as an R-symmetry. Since R-symmetries are excluded,

Translation invariance of a conserved global supercharge and the graded Jacobi identities giveLorentz covariance requires the supercharges to transform as Weyl spinors,with the complex-conjugate dotted-spinor relation for ${\bar{Q}}_{\dot{\alpha }}$, up to the sign convention used for the action of generators.

The anticommutator $\{Q_{\alpha },{\bar{Q}}_{\dot{\beta }}\}$ transforms as a Lorentz vector because $(1/2,0)\otimes (0,1/2)=(1/2,1/2)$. The only translation generator with that transformation law is $P_{\mu }$, and a normalization of $Q$ fixesAn equal-chirality anticommutator could only contain an antisymmetric spinor contraction times a central charge, but the anticommutator is symmetric under exchange of the complete supercharges. For one supercharge and no central extension this forcesTogether with the stated Poincare brackets, these are the four-dimensional $\mathcal{N}=1$ Super-Poincare relations.

The Coleman–Mandula theorem says that, under its assumptions on a nontrivial analytic relativistic S-matrix and the particle spectrum, every continuous bosonic symmetry algebra is a direct sum of the Poincare algebra and an internal symmetry algebra. Supersymmetry becomes possible by weakening the assumption that the symmetry algebra is an ordinary Lie algebra: a Z2-graded Lie superalgebra admits odd generators whose bracket is an anticommutator. The Haag–Łopuszański–Sohnius theorem then classifies the allowed extension and leads to the Super-Poincaré algebra.

Let $a=0,\dots ,p$ label directions tangent to the D-brane and $i=p+1,\dots ,25$ label transverse directions. The endpoints obey Dirichlet conditions transversely,and the boundary variation of the metric and constant B-field terms gives mixed tangential conditionsup to the orientation sign at the two ends. A worldvolume gauge potential contributes $2\pi {\alpha }^{\prime }{F}_{ab}$ in the same place, so the gauge-invariant condition contains $\mathcal{F}=b+2\pi {\alpha }^{\prime }F$.

From the brane perspective, the pullback of the background B-field is therefore indistinguishable locally from a constant worldvolume electromagnetic field strength, modulo its two-form gauge symmetry and a compensating transformation of the brane gauge field. This is the open-string boundary condition in a B-field; sufficiently general constant $b$ also induces the familiar noncommutative deformation of D-brane endpoint coordinates.

First, the displayed equations are only the one-loop sigma-model beta functions. Higher worldsheet loops generate additional covariant terms with more curvatures, H-fields and derivatives, each accompanied by higher powers of ${\alpha }^{\prime }$. Second, string scattering amplitudes contain momentum corrections from the finite string length and from integrating out the infinite tower of massive string states. Their low-energy expansion produces the same higher-derivative spacetime operators, such as curvature-squared and higher-curvature terms. Both arguments require ${\alpha }^{\prime }$ corrections even though local field redefinitions can move individual correction terms between equations.

Define the leading right-hand side of the dilaton equation byTake a divergence of the metric equation. The contracted Bianchi identity, commutation of covariant derivatives on ${\nabla }_{\nu }\Phi$, the B-field equation ${\nabla }^{\lambda }{H}_{\lambda \mu \nu }=2{\nabla }^{\lambda }\Phi H_{\lambda \mu \nu }$, and the supplied H-field identity giveThe metric equation contracted with ${\nabla }^{\rho }\Phi$ saysSubstitution cancels the curvature, Hessian and H-field terms in ${\nabla }_{\nu }C$, leavingThus $C$ is constant on each connected component. The remaining constant is fixed by the central-charge deficit; it vanishes in the critical 26-dimensional bosonic theory at this order.

The worldsheet metric is a gauge variable, so quantum consistency requires the Weyl transformation to remain non-anomalous. The background fields are couplings of a two-dimensional quantum field theory, and their sigma-model beta functions multiply the trace of the worldsheet stress tensor. Requiring every beta function to vanish gives, at the first nontrivial order in the derivative or ${\alpha }^{\prime }$ expansion, precisely the stated metric, B-field and dilaton equations. They are also the Euler-Lagrange equations of the leading spacetime string-frame effective action

In Euclidean worldsheet signature, the string nonlinear sigma model action can be writtenThe factor of $i$ in the B-field term is absent in Lorentzian signature. Here $G$ is the target metric, $B$ the Kalb–Ramond field, and $\Phi$ the dilaton.

Extract a mode usingIn the radially ordered double contour for the anticommutator, the simple pole $2T(w)/(z-w)$ gives $2L_{m+n}$. Expanding $z^{m+1/2}$ about $w$, the third-order pole contributes one half of its second derivative,The remaining contour is nonzero only for $m+n=0$. Restoring the general leading OPE coefficient $2c/3$ givesThis is the fermionic relation in the N=1 super-Virasoro algebra.

The free equations of motion are $\bar{\partial }\psi =0$ and $\partial \bar{\partial }X=0$, henceThus the superconformal current $G=i\psi \partial X$ is holomorphic. Differentiating the boson OPE gives $\partial X(z)\partial X(w)\sim -1/(z-w{)}^2$. The double contraction in $G(z)G(w)$ is consequently $1/(z-w{)}^3$, while the single contractions combine into twice the full stress tensor. ThereforeThe leading coefficient equals $2c/3=1$ for $c=3/2$.

The holomorphic free boson contributes central charge $c_X=1$, while one real chiral free fermion contributes $c_{\psi }=1/2$. Equivalently, the double contractions in $T(z)T(w)$ giveSince the fourth-order coefficient is $c/2$,

Only the fermionic part of $T$ contracts with $\psi$. Applying Wick theorem to $T_{\psi }(z)=-\frac{1}{2}:\psi \partial \psi :(z)$ and using $\psi (z)\psi (w)\sim 1/(z-w)$ givesThere is no singular OPE with $\bar{T}(\bar{z})$. Thus $\psi$ is a primary operator with

Normal ordering defines a composite field by subtracting all singular self-contractions before bringing its constituent points together. For example,This makes $T$ and $\bar{T}$ well-defined local operators and is equivalent in oscillator language to placing creation operators to the left of annihilation operators.

An operator $\mathcal{O}(w,\bar{w})$ is a primary operator of conformal weights $(h,\tilde{h})$ when its OPEs with the stress tensors arewith no more singular terms. Its scaling dimension and two-dimensional spin are

The momentum integral is replaced by the sum over momentum and winding modes. Combining the zero modes with the oscillator trace givesEquivalently,with the left/right convention of part (iv). The answer is invariant under the T-duality $R\leftrightarrow 1/R$ accompanied by $n\leftrightarrow w$ in these units.