Rigid supersymmetric background and localization data

Use natural units $c=\mathrm{\hslash }=1$. Let $M$ be a nonempty Hausdorff second-countable smooth four-manifold without boundary, let $g$ be a smooth positive-definite metric on $TM$, and choose an orientation $o$. Coordinates $x^{\mu }$ have units of length, the metric components in length coordinates are dimensionless, and the scalar curvature has units of inverse length squared. A concrete background may instead have a boundary, but then its boundary conditions and the vanishing or treatment of every boundary term are additional data.

Write $SO(4)=\{R\in M_4(\mathbb{R}):R^{\mathsf{T}}R=I,\det R=1\}$. For $x\in M$, an oriented $g$-orthonormal frame is an orientation-preserving linear isometry $e:{\mathbb{R}}^4\to T_{x}M$. The oriented orthonormal frame bundle $F_g^{+}(M)$ is the set of these frames, with projection to $M$ and right action $e\cdot R=e\circ R$. Its smooth principal $SO(4)$-bundle structure is part of the supplied geometry. Use $\mathrm{Spin}(4)=SU(2{)}_{+}\times SU(2{)}_{-}$, where $SU(2)=\{\left(\begin{array}{cc}z& w\\ -\bar{w}& \bar{z}\end{array}\right):z,w\in \mathbb{C},\mathrm{\mid }z{\mathrm{\mid }}^2+\mathrm{\mid }w{\mathrm{\mid }}^2=1\}$ and the two factors act by their defining matrix representations on ${\mathbb{C}}^2$. Identify each factor with the unit quaternions by $\left(\begin{array}{cc}z& w\\ -\bar{w}& \bar{z}\end{array}\right)\leftrightarrow z+w\mathbf{j}$, with ${\mathbf{j}}^2=-1$ and $\mathbf{j}z=\bar{z}\mathbf{j}$; write $\mathbb{H}={\mathrm{span}}_{\mathbb{R}}\{1,i,\mathbf{j},i\mathbf{j}\}\cong {\mathbb{R}}^4$. The fixed standard double-covering homomorphism $\rho :\mathrm{Spin}(4)\to SO(4)$ is $\rho (u_{+},u_{-})q=u_{+}q{u}_{-}^{-1}$; it is surjective with kernel $\{(1,1),(-1,-1)\}$. A Euclidean spin structure is a smooth right principal $\mathrm{Spin}(4)$-bundle $\tilde{F}\to M$ and a smooth map $\lambda :\tilde{F}\to F_g^{+}(M)$ satisfying $\lambda (\tilde{p}s)=\lambda (\tilde{p})\rho (s)$ for all $\tilde{p}\in \tilde{F}$ and $s\in \mathrm{Spin}(4)$, whose restriction to every fibre is a two-sheeted cover. The chiral spinor bundles are the associated bundles $S_{\pm }=\tilde{F}{\times }_{{\pi }_{\pm }}{\mathbb{C}}^2$, where ${\pi }_{\pm }$ are the defining representations of the two $SU(2)$ factors and the quotient convention is $(ps,z)\sim (p,{\pi }_{\pm }(s)z)$. Equip each real associated bundle with the complex structure $[p,z]\mapsto [p,iz]$, so $S_{\pm }$ has complex rank two. A spin connection is a principal connection on $\tilde{F}$ projecting to the Levi-Civita connection of $g$; it induces covariant derivatives on $S_{\pm }$. A spin structure and its connection are supplied data, not conclusions that follow from the metric.

An R-symmetry background consists of a specified finite-dimensional Lie group $K_R$, a smooth right principal $K_R$-bundle $P_R\to M$, a principal connection $A_R$, and a declared smooth representation of $K_R$ on each field bundle. Gauge data consist of a specified compact Lie group $G$, a smooth right principal $G$-bundle $P_G\to M$, its gauge group of smooth bundle automorphisms over ${\mathrm{id}}_M$, and a principal connection $A$. Matter and auxiliary fields are smooth sections of explicitly declared vector bundles associated to $\tilde{F}$, $P_R$, and $P_G$; bosonic and fermionic fields are assigned even and odd parity, respectively. The curvature, global form of $G$, representations, bundle sectors, and reality contours are part of the model data.

For a spinor that carries an R-symmetry representation, let $U$ be a finite-dimensional complex vector space and declare its smooth representation $\varrho :K_R\to {\mathrm{G}\mathrm{L}}_{\mathbb{C}}(U)$ and its chiral spin representation ${\pi }_{\pm }:\mathrm{Spin}(4)\to {\mathrm{G}\mathrm{L}}_{\mathbb{C}}({\mathbb{C}}^2)$. The smooth product principal $(\mathrm{Spin}(4)\times K_R)$-bundle $\tilde{F}{\times }_M{P}_R$ has right action $(\tilde{p},p_R)\cdot (s,k)=(\tilde{p}s,p_{R}k)$; its local charts are products of the spin and R charts, and its product connection is the pair of the supplied principal connections. Use the product representation $(s,k)\mapsto {\pi }_{\pm }(s)\otimes \varrho (k)$; the associated spinor/R bundle uses the quotient convention $[(\tilde{p}s,p_{R}k),z]=[(\tilde{p},p_R),({\pi }_{\pm }(s)\otimes \varrho (k))z]$. In simultaneous local sections $({\tilde{s}}_i,r_i)$, let $A_i^S={\tilde{s}}_i^{\ast }{\omega }^S$ and $A_i^R=r_i^{\ast }{A}_R$ be the local forms of the supplied spin and R connections. The associated-covariant-derivative definition gives, for a section with local component ${ϵ}_i:U_i\to {\mathbb{C}}^2\otimes U$, ${\mathrm{\nabla }}_{\mu }{ϵ}_i={\mathrm{\partial }}_{\mu }{ϵ}_i+((d{\pi }_{\pm }{)}_e(A_{i,\mu }^S)\otimes I_U+I_{{\mathbb{C}}^2}\otimes (d\varrho {)}_e(A_{i,\mu }^R)){ϵ}_i\mathrm{.}$ The displayed sum is the differential of the product representation, since differentiating ${\pi }_{\pm }(\exp (tX))\otimes \varrho (\exp (tY))$ at $t=0$ gives $(d{\pi }_{\pm }{)}_e(X)\otimes I_U+I_{{\mathbb{C}}^2}\otimes (d\varrho {)}_e(Y)$. Here each differential is taken at the identity of its group. This is the local expression of the connection induced by the product principal connection and representation; for an R-neutral spinor, $\varrho$ is the trivial representation. The mathematical supplier states this formula for a smooth finite-dimensional complex representation of a finite-dimensional Lie group and proves the local expressions patch; the current associated-covariant-derivative supplier is draft and its proof has not yet received independent acceptance. In $\mathcal{K}({\mathrm{\nabla }}^S,A_R,\mathcal{B})(ϵ)$, ${\mathrm{\nabla }}^S$ and $A_R$ therefore act with these declared representations and conventions, not by an unspecified ordinary derivative.

A rigid supersymmetric background datum is a fixed Riemannian metric, spin structure and spin connection, R-symmetry background, supergravity auxiliary background fields, and a nonzero smooth spinor section $ϵ$ in a declared spinor/R-symmetry representation. It also supplies the exact first-order linear system $\mathcal{K}({\mathrm{\nabla }}^S,A_R,\mathcal{B})(ϵ)=0$ imposed on $ϵ$, where $\mathcal{B}$ denotes the named auxiliary background fields and $\mathcal{K}$ is the operator obtained by setting the chosen supergravity fermion variation to zero. The equation, its signature, gamma-matrix convention, spinor reality conditions, and boundary conditions must be specified separately for each model; no universal Killing-spinor equation or existence statement is built into this definition. “Rigid” means the metric and supergravity fields are fixed external data and are not integrated over.

An off-shell multiplet is a specified ${\mathbb{Z}}_2$-graded collection of bosonic, fermionic, and auxiliary fields together with an odd transformation $Q$ on all of them. An isometry is a diffeomorphism preserving $g$ and $o$; its action on associated bundles must also be supplied. A flavor transformation is the action of a separately named global internal group on the declared field representations. The multiplet closes off shell when $Q^2$ equals an explicitly declared combination of lifted spacetime isometries, gauge transformations, R-symmetry transformations, and flavor transformations on every field, without using Euler–Lagrange equations. A localization package additionally specifies its graded field configuration space $\mathcal{F}$, action $S$, and gauge group $\mathcal{G}$. The ordinary even configuration space ${\mathcal{F}}_{\bar{0}}$ and $\mathcal{G}$ carry supplied differentiable charts modeled on specified real Fréchet spaces: complete metrizable locally convex vector spaces with topology given by declared countable families of seminorms. For smooth section coordinates, use the topology of uniform convergence on compact coordinate subsets of every derivative of every local component; connection coordinates are differences from a supplied reference connection. Boundary or decay restrictions and any other completions must be declared. In these charts $C^1$ means that the directional derivative $d{f}_x(h)={\lim }_{t\to 0}(f(x+th)-f(x))\mathrm{/}t$ exists, is continuous linear in $h$, and is jointly continuous in $(x,h)$; chart transitions and their inverses are $C^1$. Tangent vectors are velocities of $C^1$ curves, identified by their chart derivatives. Require group multiplication, inversion, and the action $a:\mathcal{G}\times {\mathcal{F}}_{\bar{0}}\to {\mathcal{F}}_{\bar{0}}$ to be $C^1$. Put $\mathfrak{g}=T_e\mathcal{G}$ and define the infinitesimal orbit map $R_{\phi }(u)=d(a(\cdot ,\phi ){)}_{e}u\in T_{\phi }{\mathcal{F}}_{\bar{0}}$. Odd fields retain their declared anticommuting coefficients; these ordinary tangent and gauge-slice conditions concern the even configuration space.

Gauge fixing supplies a $C^1$ map $f:U\to E_f$ from a declared open $U\subseteq {\mathcal{F}}_{\bar{0}}$ to a specified real Fréchet space, and a supplied $C^1$ submanifold $\mathcal{S}=f^{-1}(0)$ with $T_{\phi }\mathcal{S}=\ker d{f}_{\phi }$. In the sign convention used here the Faddeev–Popov operator is ${\mathcal{M}}_{\phi }=-d{f}_{\phi }\circ R_{\phi }:\mathfrak{g}\to E_f$. At each $\phi \in \mathcal{S}$, specify a residual zero-mode space ${\mathfrak{h}}_{\phi }=\ker {\mathcal{M}}_{\phi }$ and a closed complement ${\mathfrak{g}}_{\mathrm{r}\mathrm{e}\mathrm{d},\phi }$, with $\mathfrak{g}={\mathfrak{h}}_{\phi }\oplus {\mathfrak{g}}_{\mathrm{r}\mathrm{e}\mathrm{d},\phi }$. Local transversality modulo these residual modes means the required splitting $T_{\phi }{\mathcal{F}}_{\bar{0}}=T_{\phi }\mathcal{S}\oplus R_{\phi }({\mathfrak{g}}_{\mathrm{r}\mathrm{e}\mathrm{d},\phi })$ and that ${\mathcal{M}}_{\phi }:{\mathfrak{g}}_{\mathrm{r}\mathrm{e}\mathrm{d},\phi }\to E_f$ is a continuous linear bijection with continuous inverse. These are supplied chartwise hypotheses, not consequences of an inverse-function theorem in arbitrary Fréchet spaces; no global slice is asserted. For connection gauge action $R_{A}u=-D_{A}u$ and $f(A)=d_{A_{\ast }}^{\ast }(A-A_{\ast })$, this convention gives ${\mathcal{M}}_A=d_{A_{\ast }}^{\ast }{D}_A$. A Faddeev–Popov zero mode lies in $\ker {\mathcal{M}}_{\phi }$; a connection stabilizer satisfies $D_{A}u=0$, and residual zero modes need not all stabilize the connection. The determinant of the reduced operator requires a separately specified regularization; no infinite-dimensional determinant is supplied by differentiability alone. The package supplies that determinant prescription and a ghost complex (graded ghost, antighost, and multiplier fields with an odd BRST differential). The package also specifies its integration contour $\mathcal{C}$, boundary conditions, and either an actual measure or a regulator map from field configurations to regulated spaces ${\mathcal{F}}_{\mathrm{\Lambda }}$ with measures ${\mu }_{\mathrm{\Lambda }}$. If a measure is only written formally as $[D\mathrm{\Phi }]$, this notation is not a constructed measure. Applying a $Q$-exact deformation $S_t=S+tQV$ further requires an explicit assumption or proof of $Q$-invariance of the regulated measure and contour, absence or treatment of anomalies, justified differentiation under the integral, and vanishing or controlled boundary terms. This definition supplies no such proof.

For a four-dimensional gauge model in these units, $g_{\mathrm{Y}\mathrm{M}}$ is dimensionless, $[A_{\mu }]=[\mathrm{\Phi }]={\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}^{-1}$, $[F_{\mu \nu }]={\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}^{-2}$, and a fermion field has mass dimension $3\mathrm{/}2$; the supercharge has mass dimension $1\mathrm{/}2$ and its spinor parameter has mass dimension $-1\mathrm{/}2$. Every additional mass, radius, equivariant parameter, and suppressed constant must be assigned dimensions in the concrete item. No metric is asserted to admit this data, and finite-dimensional equivariant identities alone do not construct a continuum field-space measure.

Festuccia and Seiberg, §2 and §4

Pestun, §§2 and 4.1–4.3

Peelaers, §1

Smooth manifolds and their smooth charts

Riemannian metric and riemannian manifold

The frame bundle of a smooth manifold

Oriented real bundles and oriented frame bundles

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

Smooth principal $G$-bundle

Principal g bundle and associated fiber bundle

Principal connections and vertical curvature

Associated covariant derivative and local representation formula

Associated vector bundles are well-defined

Complex structure on an associated bundle by realification

A Riemannian metric has a unique Levi-Civita connection on the cotangent bundle