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

Adopt a classical rigid four-dimensional model on 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 $\eta =\mathrm{diag}(-1,1,1,1)$. Use $c=\mathrm{\hslash }=1$. Fix $G=SU(2):=\{u\in M_2(\mathbb{C}):u^{†}u=I_2,\det u=1\}$, not its quotient $SO(3)$. Its Lie algebra is $\mathfrak{s}\mathfrak{u}(2):=\{X\in M_2(\mathbb{C}):X^{†}=-X,{\mathrm{Tr}}_{2}X=0\}$, with adjoint action ${\mathrm{Ad}}_{u}X=uX{u}^{-1}$. 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 ${\sigma }_3=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$. Choose one smooth right principal bundle $\pi :P\to M$ in the single sector $\mathrm{\Sigma }=\{[P]\}$; its smooth right action is free and transitive on each fibre, it admits equivariant local trivializations, and $[P]$ denotes the nonempty minimum-rank set code of its smooth equivariant bundle-isomorphism type over $M$, in the supplied non-Abelian gauge-data convention. In the anti-Hermitian basis $T_a=-i{\sigma }_a\mathrm{/}2$, use $\kappa (X,Y)=-2{\mathrm{Tr}}_2(XY)$, so $\kappa (T_a,T_b)={\delta }_{ab}$, and a fixed dimensionless coupling $g=g_{\mathrm{Y}\mathrm{M}}>0$. Use ${\sigma }^0=I_2$, ${\sigma }^j:={\sigma }_j$ for $j=1,2,3$, ${\bar{\sigma }}^0=I_2$, ${\bar{\sigma }}^j=-{\sigma }^j$, and ${ϵ}_{12}=1$; there is no theta term in this selected model.

Use the $N=1$ superspace algebra with even coordinates $x^{\mu }$ and odd coordinates ${\theta }^{\alpha },{\bar{\theta }}^{\dot{\alpha }}$, the left derivatives and Berezin measures specified in the superspace definitions, and $D_{\alpha }={\mathrm{\partial }}_{{\theta }^{\alpha }}^L+i{\sigma }_{\alpha \dot{\beta }}^{\mu }{\bar{\theta }}^{\dot{\beta }}{\mathrm{\partial }}_{\mu }$, ${\bar{D}}_{\dot{\alpha }}=-{\mathrm{\partial }}_{{\bar{\theta }}^{\dot{\alpha }}}^L-i{\theta }^{\beta }{\sigma }_{\beta \dot{\alpha }}^{\mu }{\mathrm{\partial }}_{\mu }$. A chiral field obeys ${\bar{D}}_{\dot{\alpha }}\mathrm{\Phi }=0$. In each representation $R$, the dimensionless real vector prepotential $V_R$ transforms by $e^{2g{V}_R}\mapsto e^{i{\mathrm{\Lambda }}_R^{†}}{e}^{2g{V}_R}{e}^{-i{\mathrm{\Lambda }}_R}$ and a chiral matter field transforms by $Q_R\mapsto e^{i{\mathrm{\Lambda }}_R}{Q}_R$, where ${\mathrm{\Lambda }}_R$ is a chiral gauge-parameter superfield. Its represented matrix curvature is ${\mathcal{C}}_{\alpha ,R}=-\frac{1}{4}{\bar{D}}^{\dot{\beta }}{\bar{D}}_{\dot{\beta }}(e^{-2g{V}_R}{D}_{\alpha }{e}^{2g{V}_R})$. Define $\mathrm{ad}(P{)}_{\mathbb{C}}:=(P\times \mathfrak{s}\mathfrak{u}(2{)}_{\mathbb{C}})\mathrm{/}\sim$, where $[pu,X]=[p,{\mathrm{Ad}}_{u}X]$. The connection-normalized vector multiplet consists on shell of the connection $A_{\mu }$, one complex adjoint scalar $\varphi$ which is a smooth section of $\mathrm{ad}(P{)}_{\mathbb{C}}$, and two adjoint Weyl fermions; its $N=1$ decomposition is one vector superfield $V_{\mathrm{a}\mathrm{d}}$ and one adjoint chiral superfield $\mathrm{\Phi }$. The local representatives of $\varphi$ obey ${\varphi }_j={\mathrm{Ad}}_{g_{ij}^{-1}}{\varphi }_i$ when $s_j=s_i{g}_{ij}$. The canonical scalar is ${\varphi }_{\mathrm{c}\mathrm{a}\mathrm{n}}=\varphi \mathrm{/}g$. Here “on shell” means after eliminating algebraic auxiliary fields and imposing the field equations; the full vector superfield also contains the real auxiliary adjoint field $D$.

Choose a finite-dimensional complex vector space ${\mathcal{H}}_R$ with positive-definite Hermitian form $⟨\cdot ,\cdot {⟩}_R$. A smooth unitary representation ${\rho }_R:SU(2)\to U({\mathcal{H}}_R,⟨\cdot ,\cdot {⟩}_R)$ means a smooth group homomorphism into the group of complex-linear isometries of this Hermitian form. Write ${\rho }_{R\ast }:\mathfrak{s}\mathfrak{u}(2{)}_{\mathbb{C}}\to {\mathrm{End}}_{\mathbb{C}}({\mathcal{H}}_R)$ for the complex-linear extension of its differential at the identity. Its associated vector bundle is the quotient $E_R=P{\times }_{{\rho }_R}{\mathcal{H}}_R=(P\times {\mathcal{H}}_R)\mathrm{/}\sim$, where $[pu,v]=[p,{\rho }_R(u)v]$ for $p\in P$, $u\in SU(2)$, and $v\in {\mathcal{H}}_R$. The hypermultiplet consists of chiral superfields $Q$ and $\tilde{Q}$ whose scalar components are smooth sections of $E_R$ and its dual; the dual action is ${\rho }_{R^{\ast }}(u)\mathrm{\ell }=\mathrm{\ell }\circ {\rho }_R(u^{-1})$, and its Hermitian form is $⟨{\mathrm{\ell }}_1,{\mathrm{\ell }}_2{⟩}_{R^{\ast }}={\sum }_a\overline{{\mathrm{\ell }}_1(e_a)}{\mathrm{\ell }}_2(e_a)$ for any orthonormal basis $(e_a)$ of ${\mathcal{H}}_R$. For a local connection form $A_{\mu ,\alpha }:U_{\alpha }\to \mathfrak{s}\mathfrak{u}(2)$ from the selected Yang--Mills datum, the associated covariant derivatives on local scalar representatives are $D_{\mu }{q}_{\alpha }={\mathrm{\partial }}_{\mu }{q}_{\alpha }+{\rho }_{R\ast }(A_{\mu ,\alpha })q_{\alpha }$ and $D_{\mu }{\tilde{q}}_{\alpha }={\mathrm{\partial }}_{\mu }{\tilde{q}}_{\alpha }+({\rho }_{R^{\ast }}{)}_{\ast }(A_{\mu ,\alpha }){\tilde{q}}_{\alpha }$, where $({\rho }_{R^{\ast }}{)}_{\ast }$ is the complex-linear extension of the differential of the stated dual action. Writing their left-Weyl components as ${\psi }_Q$ and ${\psi }_{\tilde{Q}}$, the pair forms the Dirac field ${\mathrm{\Psi }}_D=({\psi }_Q,\overline{{\psi }_{\stackrel{~}{Q}}})$, with both chiralities in $R$. Each chiral $N=1$ hypermultiplet field has one complex scalar, one Weyl fermion, and one auxiliary field; the pair therefore has two complex scalars, two Weyl fermions, and two auxiliary fields. Write $\mathfrak{s}\mathfrak{u}(2{)}_{\mathbb{C}}=\mathfrak{s}\mathfrak{u}(2){\otimes }_{\mathbb{R}}\mathbb{C}$ and let ${\kappa }_{\mathbb{C}}$ be the complex-bilinear extension of $\kappa$. This form is nondegenerate and invariant under the complexified adjoint action, so for $R=\mathrm{a}\mathrm{d}$ it identifies the dual with the adjoint. On the complexified adjoint define the Hermitian form $h_{\mathrm{a}\mathrm{d}}(X,Y)={\kappa }_{\mathbb{C}}(\bar{X},Y)$, where conjugation is with respect to the real Lie algebra. The map $\mathrm{ȷ}(X)(Y)={\kappa }_{\mathbb{C}}(X,Y)$ identifies this representation isometrically and equivariantly with its dual: in the $\kappa$-orthonormal basis $T_a$, $⟨\mathrm{ȷ}(X),\mathrm{ȷ}(Y){⟩}_{R^{\ast }}=h_{\mathrm{a}\mathrm{d}}(X,Y)={\sum }_a\overline{X^a}{Y}^a$, and invariance of ${\kappa }_{\mathbb{C}}$ gives $\mathrm{ȷ}({\mathrm{Ad}}_{u}X)=\mathrm{ȷ}(X)\circ {\mathrm{Ad}}_{u^{-1}}$. The local invariant pairing is $\tilde{Q}Q$, and $\tilde{Q}\mathrm{\Phi }Q$ means $\tilde{Q}{\rho }_{R\ast }(\mathrm{\Phi })Q$. Under a chiral supergauge transformation $U_R=e^{i{\mathrm{\Lambda }}_R}$, use $Q\mapsto U_{R}Q$, $\tilde{Q}\mapsto \tilde{Q}{U}_R^{-1}$, and $M_R:={\rho }_{R\ast }(\mathrm{\Phi })\mapsto U_R{M}_R{U}_R^{-1}$; the associated mass and cubic bilinears are gauge invariant. The hypermultiplet is canonically normalized, while $\mathrm{\Phi }$ is in the connection normalization above. Write $\mathrm{\Phi }={\sum }_{a=1}^3{\mathrm{\Phi }}^a{T}_a$; "holomorphic" means complex analytic in these complex coordinates on $\mathfrak{s}\mathfrak{u}(2{)}_{\mathbb{C}}$, and ${\mathrm{\partial }}_a$ denotes $\mathrm{\partial }\mathrm{/}\mathrm{\partial }{\mathrm{\Phi }}^a$. The microscopic vector prepotential is the holomorphic, gauge-invariant quadratic function

${\mathcal{F}}_{\mathrm{U}\mathrm{V}}(\mathrm{\Phi })=\frac{i}{2g^2}{\kappa }_{\mathbb{C}}(\mathrm{\Phi },\mathrm{\Phi }),\mathrm{Im}({\mathcal{F}}_{ab})=\frac{1}{g^2}{\kappa }_{ab}>0.$

Here ${\mathcal{F}}_{ab}={\mathrm{\partial }}_a{\mathrm{\partial }}_b{\mathcal{F}}_{\mathrm{U}\mathrm{V}}$ in the basis $T_a$. With the normalized physical action brackets, its $N=1$ vector action is

${\mathcal{L}}_{\mathrm{v}\mathrm{e}\mathrm{c}}=\frac{1}{4g^2}([{\kappa }_{\mathbb{C}}({\mathcal{W}}^{\alpha },{\mathcal{W}}_{\alpha }){]}_F+\mathrm{h}\mathrm{.}\mathrm{c}\mathrm{.})+\frac{1}{g^2}[{\kappa }_H(\bar{\mathrm{\Phi }},e^{2g{V}_{\mathrm{a}\mathrm{d}}}\mathrm{\Phi }){]}_D,$

where ${\mathcal{W}}_{\alpha }$ is the abstract connection-normalized Lie-algebra-valued chiral curvature, with ${\rho }_{R\ast }({\mathcal{W}}_{\alpha })={\mathcal{C}}_{\alpha ,R}\mathrm{/}(2i)$; hence its gauge F-term coefficient also equals ${\mathrm{Tr}}_2({\mathcal{C}}^{\alpha }{\mathcal{C}}_{\alpha })\mathrm{/}(8g^2)$ in the defining representation, before adding its Hermitian conjugate, and ${\kappa }_H(\bar{X},Y)={\kappa }_{\mathbb{C}}(\bar{X},Y)$. For a superspace expression $X$, $[X{]}_F=I_F(X)\mathrm{/}2$ and $[X{]}_D=-I_D(X)\mathrm{/}4$, using the raw generator order with the normalized physical brackets. Thus the vector kinetic term has exactly the connection-normalized Yang-Mills coefficient $-\kappa (F\wedge {\ast }_{\eta }F)\mathrm{/}(2g^2)$.

The adopted renormalizable one-hypermultiplet action is

${\mathcal{L}}_{N=2}={\mathcal{L}}_{\mathrm{v}\mathrm{e}\mathrm{c}}+[Q^{†}{e}^{2g{V}_R}Q+{\tilde{Q}}^{†}{e}^{2g{V}_{R^{\ast }}}\tilde{Q}{]}_D+([\sqrt{2}\tilde{Q}\mathrm{\Phi }Q+m\tilde{Q}Q{]}_F+\mathrm{h}\mathrm{.}\mathrm{c}\mathrm{.}),$

with complex mass parameter $m$ of dimension ${\mathrm{m}}^{-1}$. In canonical normalization ${\mathrm{\Phi }}_{\mathrm{c}\mathrm{a}\mathrm{n}}=\mathrm{\Phi }\mathrm{/}g$, the cubic coefficient is $\sqrt{2}g$, the same gauge coupling that appears in the covariant derivative. The ultraviolet prepotential above is quadratic; a low-energy effective prepotential may instead be non-polynomial and valid only on a specified electric/magnetic patch. This postulate adopts no such low-energy solution or duality claim.

The model includes rigid four-dimensional $\mathcal{N}=2$ supersymmetry: “rigid” means its spinor parameters are constant in spacetime, and $\mathcal{N}=2$ means two independent left-Weyl supercharges, hence eight real components. Its invariance is an adopted model premise, not a deduction in this item. The $N=1$ superspace expression is a field/action packaging; the full hypermultiplet transformations close only after imposing the physical field equations, and the finite $N=1$ auxiliary fields are eliminated in this on-shell description. Fields are smooth in local trivializations and admissible variations are smooth and compactly supported; actions are taken over relatively compact regions or when the spacetime integral converges. No path-integral measure, regulator, quantum theory, or continuum existence claim is included. In mass-dimension notation $[x]=-1$, $[\theta ]=-\frac{1}{2}$, $[\mathrm{\Phi }]=[Q]=[\tilde{Q}]=1$, $[{\mathcal{W}}_{\alpha }]=3\mathrm{/}2$, $[m]=1$, and $[g]=0$; equivalently scalar fields and mass parameters have units ${\mathrm{m}}^{-1}$.

D'Hoker and Phong, §§2.4, 2.9–2.10 and 3.5–3.7

Martin, §§3.4, 4.5 and 4.9

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

Gauge invariance of hypermultiplet mass and adjoint couplings

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