Four-dimensional $N=4$ SU(2) super-Yang-Mills model

Use the physical action brackets $[U{]}_D=-I_D(U)\mathrm{/}4$ and $[Y{]}_F=I_F(Y)\mathrm{/}2$ from Physical superspace action brackets and canonical chiral components, with raw mathematical coefficient integrals $I_D,I_F$ unchanged. Physical chiral components obey ${\psi }_{\mathrm{r}\mathrm{a}\mathrm{w}}=\sqrt{2}\psi$ and $F_{\mathrm{r}\mathrm{a}\mathrm{w}}=2F$. All physical action and coupling formulas below use these brackets and physical component names; adding Hermitian conjugation applies only to the F action and occurs once.

Use the oriented, time-oriented Minkowski spacetime $M={\mathbb{R}}^{1,3}$ with coordinates $x^{\mu }$ of length, orientation $d{x}^0\wedge d{x}^1\wedge d{x}^2\wedge d{x}^3$, time orientation of increasing $x^0$, and metric $\eta =\mathrm{diag}(-1,1,1,1)$. Let $P\to M$ be one smooth right-principal $SU(2)$ bundle, meaning its smooth right $SU(2)$ action is free and transitive on each fibre, it admits equivariant local trivializations, and its sector $[P]$ is the nonempty minimum-rank set code of its smooth equivariant bundle-isomorphism type over $M$, in the supplied non-Abelian gauge-data convention. The global group is $SU(2):=\{u\in M_2(\mathbb{C}):u^{†}u=I_2,\det u=1\}$, not $SO(3)$; its Lie algebra is $\mathfrak{s}\mathfrak{u}(2):=\{X\in M_2(\mathbb{C}):X^{†}=-X,{\mathrm{Tr}}_{2}X=0\}$ and its adjoint action is ${\mathrm{Ad}}_{u}X=uX{u}^{-1}$. The Pauli matrices are ${\sigma }_1=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$, ${\sigma }_2=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right)$, and ${\sigma }_3=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$. Use $T_a=-i{\sigma }_a\mathrm{/}2$, $\kappa (X,Y)=-2{\mathrm{Tr}}_2(XY)$ with $\kappa (T_a,T_b)={\delta }_{ab}$, $c=\mathrm{\hslash }=1$, and dimensionless coupling $g=g_{\mathrm{Y}\mathrm{M}}>0$. Use ${\sigma }^0=I_2$, ${\sigma }^j:={\sigma }_j$ for these spatial matrices, ${\bar{\sigma }}^0=I_2$, ${\bar{\sigma }}^j=-{\sigma }^j$, and ${ϵ}_{12}=1$, so ${\sigma }^{\mu }{\bar{\sigma }}^{\nu }+{\sigma }^{\nu }{\bar{\sigma }}^{\mu }=-2{\eta }^{\mu \nu }{I}_2$. Define the real adjoint bundle $\mathrm{ad}(P):=(P\times \mathfrak{s}\mathfrak{u}(2))\mathrm{/}\sim$ and its complexification $\mathrm{ad}(P{)}_{\mathbb{C}}:=(P\times \mathfrak{s}\mathfrak{u}(2{)}_{\mathbb{C}})\mathrm{/}\sim$, both with relation $[pu,X]=[p,{\mathrm{Ad}}_{u}X]$; the bar on $\mathfrak{s}\mathfrak{u}(2{)}_{\mathbb{C}}$ is conjugation fixing $\mathfrak{s}\mathfrak{u}(2)$. In the $N=1$ decomposition, $V_{\mathrm{a}\mathrm{d}}$ is the dimensionless real adjoint vector prepotential with $e^{2g{V}_{\mathrm{a}\mathrm{d}}}\mapsto e^{i{\mathrm{\Lambda }}^{†}}{e}^{2g{V}_{\mathrm{a}\mathrm{d}}}{e}^{-i\mathrm{\Lambda }}$, where $\mathrm{\Lambda }$ is a chiral gauge-parameter superfield, and ${\mathrm{\Phi }}_1,{\mathrm{\Phi }}_2,{\mathrm{\Phi }}_3$ are adjoint chiral superfields in connection normalization. Each ${\mathrm{\Phi }}_i$ obeys ${\bar{D}}_{\dot{\alpha }}{\mathrm{\Phi }}_i=0$ and has one complex adjoint scalar, one adjoint Weyl fermion, and an algebraic auxiliary field. In the fixed physical chiral expansion write its scalar component as ${\varphi }_i$, a smooth section of $\mathrm{ad}(P{)}_{\mathbb{C}}$ with local representatives ${\varphi }_{r,\alpha }:U_{\alpha }\to \mathfrak{s}\mathfrak{u}(2{)}_{\mathbb{C}}$ obeying ${\varphi }_{r,\beta }={\mathrm{Ad}}_{g_{\alpha \beta }^{-1}}{\varphi }_{r,\alpha }$ when local sections satisfy $s_{\beta }=s_{\alpha }{g}_{\alpha \beta }$. Define for $r=1,2,3$, with ${\mathrm{i}}^2=-1$

$X^{2r-1}:=\frac{{\varphi }_r+\overline{{\varphi }_r}}{\sqrt{2}},X^{2r}:=\frac{{\varphi }_r-\overline{{\varphi }_r}}{\mathrm{i}\sqrt{2}},{\varphi }_r=\frac{X^{2r-1}+\mathrm{i}{X}^{2r}}{\sqrt{2}}\mathrm{.}$

Thus each $X^I$ is a smooth section of $\mathrm{ad}(P)$, with local components transforming by $X_{\beta }^I={\mathrm{Ad}}_{g_{\alpha \beta }^{-1}}{X}_{\alpha }^I$. The canonical chiral and real scalar fields are ${\mathrm{\Phi }}_{i,\mathrm{c}\mathrm{a}\mathrm{n}}={\mathrm{\Phi }}_i\mathrm{/}g$ and $X_{\mathrm{c}\mathrm{a}\mathrm{n}}^I=X^I\mathrm{/}g$, respectively.

Write $\mathfrak{s}\mathfrak{u}(2{)}_{\mathbb{C}}=\mathfrak{s}\mathfrak{u}(2){\otimes }_{\mathbb{R}}\mathbb{C}$, let ${\kappa }_{\mathbb{C}}$ be the complex-bilinear extension of $\kappa$, set ${\kappa }_H(\bar{X},Y)={\kappa }_{\mathbb{C}}(\bar{X},Y)$, and write $[X,Y]$ for the complexified Lie bracket. The complexified form is nondegenerate and adjoint-invariant. For $U=e^{i\mathrm{\Lambda }}$, each adjoint chiral field transforms by ${\mathrm{\Phi }}_i\mapsto U{\mathrm{\Phi }}_i{U}^{-1}$, so the adjoint cubic in the action is gauge invariant under this law. Use $D_{\alpha }={\mathrm{\partial }}_{{\theta }^{\alpha }}^L+i{\sigma }_{\alpha \dot{\beta }}^{\mu }{\bar{\theta }}^{\dot{\beta }}{\mathrm{\partial }}_{\mu }$ and ${\bar{D}}_{\dot{\alpha }}=-{\mathrm{\partial }}_{{\bar{\theta }}^{\dot{\alpha }}}^L-i{\theta }^{\beta }{\sigma }_{\beta \dot{\alpha }}^{\mu }{\mathrm{\partial }}_{\mu }$. Each ${\mathrm{\Phi }}_i$ is an even chiral superfield satisfying ${\bar{D}}_{\dot{\alpha }}{\mathrm{\Phi }}_i=0$, and the represented adjoint matrix curvature is ${\mathcal{C}}_{\alpha ,\mathrm{a}\mathrm{d}}=-\frac{1}{4}{\bar{D}}^{\dot{\beta }}{\bar{D}}_{\dot{\beta }}(e^{-2g{V}_{\mathrm{a}\mathrm{d}}}{D}_{\alpha }{e}^{2g{V}_{\mathrm{a}\mathrm{d}}})$. The abstract connection-normalized chiral curvature in the action is ${\mathcal{W}}_{\alpha }$, defined by ${\rho }_{R\ast }({\mathcal{W}}_{\alpha })={\mathcal{C}}_{\alpha ,R}\mathrm{/}(2i)$ for every representation $R$. Thus ${\kappa }_{\mathbb{C}}({\mathcal{W}}^{\alpha },{\mathcal{W}}_{\alpha })\mathrm{/}(4g^2)={\mathrm{Tr}}_2({\mathcal{C}}^{\alpha }{\mathcal{C}}_{\alpha })\mathrm{/}(8g^2)$ in the defining representation, before adding its Hermitian conjugate. The adopted connection-normalized $N=1$ action is

${\mathcal{L}}_{N=4}=\frac{1}{g^2}\{\frac{1}{4}([{\kappa }_{\mathbb{C}}({\mathcal{W}}^{\alpha },{\mathcal{W}}_{\alpha }){]}_F+\mathrm{h}\mathrm{.}\mathrm{c}\mathrm{.})+{\sum }_{i=1}^3[{\kappa }_H({\bar{\mathrm{\Phi }}}_i,e^{2g{V}_{\mathrm{a}\mathrm{d}}}{\mathrm{\Phi }}_i){]}_D+([\sqrt{2}{\kappa }_{\mathbb{C}}({\mathrm{\Phi }}_1,[{\mathrm{\Phi }}_2,{\mathrm{\Phi }}_3]){]}_F+\mathrm{h}\mathrm{.}\mathrm{c}\mathrm{.})\}\mathrm{.}$

Here $[Y{]}_F=I_F(Y)\mathrm{/}2$ and $[Z{]}_D=-I_D(Z)\mathrm{/}4$ are the normalized physical brackets; the raw mathematical measures retain their coefficient-extraction meaning. In canonical fields, the cubic superpotential coefficient is $\sqrt{2}g$, so its Yukawa coupling equals the gauge coupling. On shell, meaning after eliminating algebraic auxiliary fields and imposing the field equations, the physical content is one gauge connection, six real adjoint scalars $X^I$ ($I=1,\dots ,6$), and four adjoint Weyl fermions ${\lambda }^A$ ($A=1,\dots ,4$). The scalar potential in connection normalization is

$V_{\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{a}\mathrm{r}}=\frac{1}{2g^2}{\sum }_{1\le I<J\le 6}\kappa ([X^I,X^J],[X^I,X^J]);$

for canonical fields $X_{\mathrm{c}\mathrm{a}\mathrm{n}}^I=X^I\mathrm{/}g$ this is $\frac{g^2}{2}{\sum }_{I<J}\kappa ([X_{\mathrm{c}\mathrm{a}\mathrm{n}}^I,X_{\mathrm{c}\mathrm{a}\mathrm{n}}^J],[X_{\mathrm{c}\mathrm{a}\mathrm{n}}^I,X_{\mathrm{c}\mathrm{a}\mathrm{n}}^J])$. With the $N=2$ model's $N=1$ decomposition, this is the $R=\mathrm{a}\mathrm{d}$, $m=0$ hypermultiplet specialization after rescaling its two canonical hypermultiplet chiral fields by $g$ to the connection normalization of ${\mathrm{\Phi }}_2$ and ${\mathrm{\Phi }}_3$.

Adopt rigid four-dimensional $\mathcal{N}=4$ supersymmetry, with four independent left-Weyl supercharges and sixteen real components, as part of this physical model postulate. The action's extended invariance and on-shell closure are not proved in this item. This is a classical action with smooth fields in local bundle trivializations and compactly supported admissible variations; no path-integral measure, regulator, quantum continuum theory, or empirical result is asserted. Use $c=\mathrm{\hslash }=1$: $g$ is dimensionless, connection-normalized scalar fields have units ${\mathrm{m}}^{-1}$, and canonical fields have mass dimension one.

D'Hoker and Phong, §§2.4 and 2.9–2.10

Martin, §§3.4, 4.5 and 4.9

Four-dimensional $N=2$ SU(2) vector and hypermultiplet model

Gauge invariance of the adjoint cubic superpotential

Rigid $N=1$ SU(2) gauge-matter superspace action

Classical pure SU(2) Yang–Mills model

Four-dimensional $N=1$ non-Abelian vector superfield in connection normalization

Four-dimensional $N=1$ chiral superfield on flat superspace

Four-dimensional $N=1$ superspace Grassmann algebra and Berezin integration

Four-dimensional real extended super-Poincare algebra

Physical superspace action brackets and canonical chiral components