Classical and quantum supersymmetric vacuum-moduli data

Use natural units $c=\mathrm{\hslash }=1$ in four-dimensional rigid, global $\mathcal{N}=1$ gauge-model data. Let $V$ be a finite-dimensional complex Hermitian vector space of scalar field values, let a compact Lie group $G$ act unitarily on $V$, and fix an orthonormal complex basis $({\varphi }^1,\dots ,{\varphi }^n)$ of $V$. Let $\mathfrak{g}=\mathrm{Lie}(G)$ and fix a positive-definite real inner product on $\mathfrak{g}$ satisfying $⟨{\mathrm{Ad}}_{h}X,{\mathrm{Ad}}_{h}Y{⟩}_{\mathfrak{g}}=⟨X,Y{⟩}_{\mathfrak{g}}$ for all $h\in G$ and $X,Y\in \mathfrak{g}$. Choose an orthonormal basis $(X_1,\dots ,X_r)$ of $\mathfrak{g}$. For the unitary representation $\rho :G\to U(V)$, set $T(X)=-id\rho (X)$, which is Hermitian for each $X\in \mathfrak{g}$. Also choose a gauge coupling $g>0$ and a gauge-invariant holomorphic polynomial $W:V\to \mathbb{C}$. Assume canonical scalar kinetic energy and no Fayet--Iliopoulos term. Write $x=(x^0,x^1,x^2,x^3)\in {\mathbb{R}}^{1,3}$, fix $\eta =\mathrm{diag}(-1,1,1,1)$, sum repeated spacetime indices, and let the spacetime scalar field be a smooth map $\varphi :{\mathbb{R}}^{1,3}\to V$, and let $\mathcal{A}={\mathcal{A}}_{\mu }d{x}^{\mu }$ be a smooth $\mathfrak{g}$-valued connection one-form on the trivial bundle. Use connection normalization: ${\mathrm{\nabla }}_{\mu }\varphi ={\mathrm{\partial }}_{\mu }\varphi +d\rho ({\mathcal{A}}_{\mu })\varphi$ and ${\mathrm{\nabla }}^{\mu }={\eta }^{\mu \nu }{\mathrm{\nabla }}_{\nu }$. A smooth gauge transformation $h:{\mathbb{R}}^{1,3}\to G$ acts by $\varphi \mapsto \rho (h)\varphi$ and ${\mathcal{A}}_{\mu }\mapsto {\mathrm{Ad}}_h{\mathcal{A}}_{\mu }-({\mathrm{\partial }}_{\mu }h)h^{-1}$; then ${\mathrm{\nabla }}_{\mu }\varphi \mapsto \rho (h){\mathrm{\nabla }}_{\mu }\varphi$. The canonical scalar kinetic Lagrangian is therefore

${\mathcal{L}}_{\mathrm{k}\mathrm{i}\mathrm{n}}=-{\sum }_i\overline{({\mathrm{\nabla }}_{\mu }\varphi {)}^i}({\mathrm{\nabla }}^{\mu }\varphi {)}^i,[{\varphi }^i]=[{\mathcal{A}}_{\mu }]=\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s},[W]={\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}}^3\mathrm{.}$

The gauge coupling $g$ is dimensionless in these natural units. Define the auxiliary-field maps on the scalar value space by

$F_i(\varphi )=\frac{\mathrm{\partial }W}{\mathrm{\partial }{\varphi }^i}(\varphi ),D_a(\varphi )=-g{\varphi }^{†}T(X_a)\varphi \mathrm{.}$

The real linear functional $\mathbf{D}(\varphi )\in {\mathfrak{g}}^{\ast }$ is defined by $\mathbf{D}(\varphi )(X)=-g{\varphi }^{†}T(X)\varphi$; thus $D_a(\varphi )=\mathbf{D}(\varphi )(X_a)$. For $\lambda \in {\mathfrak{g}}^{\ast }$, define $\mathrm{\parallel }\lambda {\mathrm{\parallel }}_{{\mathfrak{g}}^{\ast }}^2={\sum }_{a=1}^r\lambda (X_a{)}^2$, the norm from the inner product dual to the fixed inner product on $\mathfrak{g}$; this value is independent of the chosen orthonormal basis. The classical scalar potential is

$V_{\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{e}}(\varphi )={\sum }_{i=1}^n\mathrm{\mid }{F}_i(\varphi ){\mathrm{\mid }}^2+\frac{1}{2}\mathrm{\parallel }\mathbf{D}(\varphi ){\mathrm{\parallel }}_{{\mathfrak{g}}^{\ast }}^2={\sum }_{i=1}^n\mathrm{\mid }{F}_i(\varphi ){\mathrm{\mid }}^2+\frac{1}{2}{\sum }_{a=1}^r{D}_a(\varphi {)}^2\mathrm{.}$

For $h\in G$, define its dual action by $({\mathrm{Ad}}_h^{\ast }\lambda )(X)=\lambda ({\mathrm{Ad}}_{h^{-1}}X)$. Then $\mathbf{D}(h\varphi )={\mathrm{Ad}}_h^{\ast }\mathbf{D}(\varphi )$, and the invariant dual inner product makes this potential gauge-invariant. The quantities $F_i$ and $D_a$ have mass dimension two and $V_{\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{e}}$ has mass dimension four.

A classical supersymmetric vacuum configuration is a constant scalar configuration $\varphi \in V$ for which every $F_i(\varphi )$ and every $D_a(\varphi )$ is zero. Its classical vacuum-moduli set is the topological orbit space

${\mathcal{M}}_{\mathrm{c}\mathrm{l}}=\{\varphi \in V:F_i(\varphi )=0\text{ for all }i,D_a(\varphi )=0\text{ for all }a\}\mathrm{/}G\mathrm{.}$

where two configurations are identified exactly when they lie in the same compact-$G$ orbit, and the set has the quotient topology. The stability convention here retains every such compact-group orbit, including singular orbits. No quotient by the complexified group, geometric-invariant-theory stability criterion, or smooth-manifold structure is included in this definition.

For a specified, already constructed quantum theory $\mathcal{T}$, suppose there is a Hilbert space $\mathcal{H}$, a unitary translation representation $U(a)$, and densely defined supercharges $Q^I$ and their Hilbert adjoints on a common domain $\mathcal{D}$. A quantum supersymmetric vacuum ray is the phase-equivalence class $[\mathrm{\Omega }]$ of a normalized vector $\mathrm{\Omega }\in \mathcal{D}$ satisfying $U(a)\mathrm{\Omega }=\mathrm{\Omega }$ for every spacetime translation $a$ and $Q^I\mathrm{\Omega }=(Q^I{)}^{\ast }\mathrm{\Omega }=0$ for every supercharge $I$. The quantum vacuum-moduli set ${\mathcal{M}}_{\mathrm{q}}(\mathcal{T})$ is the set of these rays, with the topology, if any, supplied by the constructed theory. This definition is conditional on normalizable invariant vacuum vectors and a specified quantum Hilbert-space model. It asserts no path-integral measure, continuum limit, nonperturbative existence, or equality between ${\mathcal{M}}_{\mathrm{c}\mathrm{l}}$ and ${\mathcal{M}}_{\mathrm{q}}(\mathcal{T})$.

Tong, §§4.2.2–4.3, gauge-covariant kinetic terms and vacuum moduli.