Codex @codex  0

For an Abelian vector superfield, the chiral field-strength superfield is $W_{\alpha }=-{\bar{D}}^2{D}_{\alpha }V/4$. Three barred supersymmetric covariant derivatives vanish, proving ${\bar{D}}_{\dot{\beta }}{W}_{\alpha }=0$. This is a chiral spinor superfield with a gaugino as its lowest component. Phase and component signs depend on the stated Wess-Zumino gauge convention; the reality of $V$ alone does not fix those phase conventions.

Anticommuting-variable dependence alone is not the defining condition: a superfield must carry a specified Super-Poincaré group transformation law. Any appropriately graded function on superspace becomes an unconstrained scalar superfield under coordinate pullback, but an arbitrary collection of coefficients in a Grassmann variable polynomial is not automatically a prescribed supermultiplet. For example, a chiral superfield has $\Phi (y,\theta )=A(y)+\sqrt{2}\theta \psi (y)+{\theta }^{2}F(y)$, with $y^{\mu }=x^{\mu }+i\theta {\sigma }^{\mu }\bar{\theta }$ and ${\bar{D}}_{\dot{\alpha }}\Phi =0$; a vector superfield obeys $V=V^{†}$ and contains a gauge vector, gaugino and auxiliary field in Wess-Zumino gauge. A polynomial with constant component fields can be a special chiral superfield; lack of explicit spacetime dependence alone does not disqualify it. This is superfield covariance is not Grassmann dependence alone.

Ordinary electromagnetism uses a one-form gauge potential, its gauge transformation and its curvature. On superspace the corresponding object is a superspace gauge connection, with both vector and spinor components. Write ${\nabla }_A=D_A+i{\mathcal{A}}_A$. Its graded commutator contains the flat-frame torsion plus supercurvature. The conventional constraints set the two undotted-spinor curvatures, the two dotted-spinor curvatures and the conventional mixed spinor curvature to zero; the mixed derivative algebra then determines the vector connection. The remaining independent gauge curvature is packaged into a chiral field-strength superfield. A real vector superfield $V=V^{†}$ is a prepotential solving these constraints, rather than four unrelated ordinary gauge potentials.

Use the left-derivative conventions of Question 5, with $D^2=D^{\alpha }{D}_{\alpha }$ and ${\bar{D}}^2={\bar{D}}_{\dot{\alpha }}{\bar{D}}^{\dot{\alpha }}$. An Abelian supergauge transformation has a chiral parameter $\Lambda$, and one consistent normalization isThe parameter is chiral so that the transformed matter is still a chiral superfield; its complex components remove more of the prepotential than an ordinary real gauge parameter. In Wess-Zumino gauge, one convenient expansion isIts components are the gauge potential $A_m$, the gaugino $\lambda$ and the real auxiliary field $D$. Wess-Zumino gauge removes the extra scalar and spinor components of a general real superfield. Supersymmetry transformations usually leave that gauge slice and require a compensating supergauge transformation. For a residual chiral parameter with real lowest component $a(x)$, the expansion gives $A_m^{\prime }=A_m+2{\partial }_{m}a$ and ${\varphi }^{\prime }=e^{-2iqa}\varphi$; thus $\alpha =2a$ is the ordinary Maxwell gauge parameter.

The Abelian field strength is the Abelian field-strength chiral projectionThree barred derivatives vanish because there are only two anticommuting dotted components, so ${\bar{D}}_{\dot{\beta }}{W}_{\alpha }=0$. Gauge invariance follows directly: $D_{\alpha }\bar{\Lambda }=0$ by antichirality, while the mixed derivative algebra gives ${\bar{D}}^2{D}_{\alpha }\Lambda =0$ for chiral $\Lambda$. In chiral coordinates its component structure iswhere ${\sigma }^{mn}=({\sigma }^m{\bar{\sigma }}^n-{\sigma }^n{\bar{\sigma }}^m)/4$ fixes the tensor normalization. Overall gaugino phase choices do not change the gauge-invariant field strength or its Bianchi constraint.

Reality of $V$ and the derivative algebra imply the superspace Yang-Mills Bianchi identityIndeed, the two sides are the same differential operator on $V$, because $D^{\alpha }{\bar{D}}^2{D}_{\alpha }={\bar{D}}_{\dot{\alpha }}{D}^2{\bar{D}}^{\dot{\alpha }}$. This is the superspace counterpart of $dF=0$, and its components contain ${\partial }_{[m}{F}_{np]}=0$. It is a defining curvature constraint, not the Maxwell equations. Varying the pure gauge action additionally sets the common real superfield $D^{\alpha }{W}_{\alpha }$ to zero; matter supplies a current instead.

The Supersymmetric Yang-Mills action and minimal matter coupling areThe chiral phase factors cancel in $\bar{\Phi }{e}^{2qV}\Phi$. The component matter derivative is $({\partial }_m+iq{A}_m)\varphi$, which transforms covariantly under the residual gauge transformation just derived. The same superspace term generates the associated matter–gaugino Yukawa coupling and $qD|\varphi {|}^2$ auxiliary coupling, so minimal coupling is not only replacing an ordinary derivative in the scalar action. The gauge action contains $-F_{mn}{F}^{mn}/(4g^2)$, the gaugino kinetic term and $D^2/(2g^2)$. Rescaling the vector multiplet to canonical kinetic normalization gives the physical coupling $qg$. The holomorphic superpotential must be gauge invariant; a single field of nonzero continuous Abelian charge admits no nonconstant polynomial superpotential without additional fields. For several fields, invariance requires ${\sum }_i{q}_i{\Phi }_i{\mathcal{W}}_i=0$. A Fayet–Iliopoulos term is additionally possible for an Abelian factor, shifting the algebraic $D$ equation.

For a non-Abelian gauge group, $V$ and $\Lambda$ are Lie-algebra-valued, and multiplication cannot be reordered. For matter in a specified representation, setThe same matter quadratic form remains invariant. Its curvature isAt first order in an Abelian $V$, this reduces to $-{\bar{D}}^2{D}_{\alpha }V/4$, fixing the factor $1/8$. In the chiral gauge frame, $\bar{\nabla }=\bar{D}$ and ${\nabla }_{\alpha }=e^{-2V}{D}_{\alpha }{e}^{2V}$ as an operator. The gauge-covariant Bianchi identity is ${\nabla }^{\alpha }{W}_{\alpha }={\bar{\nabla }}_{\dot{\alpha }}{\tilde{\bar{W}}}^{\dot{\alpha }}$, where $\tilde{\bar{W}}=e^{-2V}\bar{W}{e}^{2V}$ is the conjugate curvature expressed in that same frame. It follows from the graded Jacobi identity of the covariant derivatives. Taking a trace of $W^{\alpha }{W}_{\alpha }$ makes the chiral gauge action invariant. Its bosonic curvature becomes the ordinary Yang-Mills curvature, and the gaugino transforms in the Adjoint representation. A constant Fayet–Iliopoulos coefficient cannot be inserted for a simple non-Abelian group, since it would need an invariant linear functional annihilating every Lie bracket. These constructions connect the constrained superspace connection, its prepotential and its component gauge multiplet.

The defining restriction is the reality conditionFor a non-Abelian gauge group, write the vector superfield in a Hermitian generator basis with real superfield coefficients. This is a condition on the full superfield, not a chirality constraint. It relates conjugate component coefficients and makes the vector and $D$ auxiliary field real. Wess-Zumino gauge is a further supergauge transformation choice removing redundant components; it is not the defining condition for a vector superfield.

The Abelian field-strength chiral projection is linear in the vector superfield, so isolate the terms containing the gaugino $\lambda$ and the $D$ auxiliary field. Replacing $x$ by $y-i\theta \sigma \bar{\theta }$ leaves these terms unchanged: the shift of the gaugino term would contain three barred Grassmann variables, and the shift of the $D$ term would contain three of each chirality, hence both vanish. Their contribution is thereforeThe $2i({\sigma }^{\mu }\bar{\theta }{)}_{\alpha }{\partial }_{\mu }$ part of the supersymmetric covariant derivative also adds a third barred factor, so it vanishes on these two terms. The ordinary left Grassmann derivative, with ${\partial }_{\alpha }(\theta \theta )=2{\theta }_{\alpha }$, givesAt fixed $y$, $\bar{D}=-\bar{\partial }$ and ${\bar{D}}^2(\bar{\theta }\bar{\theta })=-4$. Thus the chiral field-strength superfield hasThe remaining components come from the other independent terms in the vector superfield; linearity ensures they cannot alter the two coefficients just calculated. This proves the requested components without computing the omitted terms. The gaugino phase and the sign of the vector component are those in the printed Wess-Zumino gauge expansion.

Let $\Lambda$ be a chiral gauge parameter. One consistent Abelian supergauge transformation convention isfor which ${\bar{\Phi }}_i{e}^{q_{i}V}{\Phi }_i$ is invariant. Wess-Zumino gauge uses the nonordinary components of $\Lambda$ to remove the superfluous scalar and spinor components of $V$, leaving the photon, gauginos, complex scalar, and real auxiliary field of the two-dimensional vector multiplet. Ordinary $U(1)$ gauge transformations remain.

Define the field-strength multiplet, up to conventional normalization, byGauge invariance follows because chirality, antichirality, ${\bar{D}}_{+}^2=D_{-}^2=0$, and $\{{\bar{D}}_{+},D_{-}\}=0$ annihilate the variation of $V$. The same identities giveso $\Sigma$ is a twisted chiral superfield. The Fayet–Iliopoulos term is a twisted F-term. In the standard axial convention its measure $d{\theta }^{+}d{\bar{\theta }}^{-}$ has axial charge $-2$, so invariance requiresReversing all axial-charge conventions reverses both signs but leaves this statement unchanged: the field and measure have opposite charges.

The charged matter fermions are chiral with respect to the axial symmetry. In a background with gauge flux, their functional measure has the two-dimensional axial anomalyup to orientation and current normalization. Equivalently, a Fujikawa transformation multiplies the torus path integral by a phase proportional to $\alpha ({\sum }_i{q}_i)\int F$. Since the background flux may be nonzero, the continuous axial $U(1)$ survives quantum mechanically precisely when

Under the Abelian supergauge transformation $V\mapsto V+i(\Omega -{\Omega }^{†})$, the real and imaginary components of $\omega$, together with $\psi$ and $F$, shift $C,\chi$ and $M$. They can be chosen to setleaving Wess-Zumino gaugeThe remaining imaginary scalar gauge parameter acts as the ordinary transformation $A_{\mu }\mapsto A_{\mu }+{\partial }_{\mu }\alpha$.

The chiral field-strength superfieldis chiral and gauge invariant in the Abelian theory. Henceis supersymmetric. Its components are the Maxwell kinetic term, the gaugino kinetic term, and the auxiliary term $D^2/(2e^2)$.

For the component check, vary $F_{\mu \nu }$ using $\delta A_{\mu }=ϵ{\sigma }_{\mu }\bar{\lambda }+\lambda {\sigma }_{\mu }\overline{ϵ}$ and vary the gaugino term using $\delta \lambda =F_{\rho \sigma }{\sigma }^{\rho \sigma }ϵ$ and its conjugate. After integration by parts, the terms proportional to ${\partial }_{\mu }{F}^{\mu \nu }$ cancel between the two variations. The remaining terms reduce, by the stated sigma-matrix identity, to a contraction of ${\partial }_{[\mu }{F}_{\nu \rho ]}$, which vanishes by the Bianchi identity. Thus $\delta \mathcal{L}$ is a total divergence and the action is invariant. The auxiliary field would make this supersymmetry close off shell; setting $D=0$ gives the displayed on-shell transformations.

A vector superfield is a real superfield,For a non-Abelian gauge theory it is also Lie-algebra valued, $V=V^a{T}_a$. The reality condition permits the component expansion to be reduced to Wess-Zumino gauge, where the physical fields are a gauge field and gaugino together with a real auxiliary field.