Adopted ABJM localization model on the round three-sphere

Use natural units $c=\mathrm{\hslash }=1$. Fix $r>0$ with units of length and the round oriented sphere ${\mathbb{S}}_r^3=\{y\in {\mathbb{R}}^4:{\sum }_{\alpha =1}^4{y}_{\alpha }^2=r^2\}$ with its induced Riemannian metric, no boundary, and its unique spin structure: ${\mathbb{S}}^3$ is spin, and its spin structures form a torsor for $H^1({\mathbb{S}}^3;{\mathbb{Z}}_2)=0$. Here three-dimensional $\mathcal{N}=m$ counts $m$ independent two-component real (Majorana) supersymmetry generators before Euclidean continuation; their Euclidean spinor parameters are complex. Superconformal means the algebra also contains conformal transformations and the corresponding special-superconformal supersymmetry generators. Adopt the source-specific ABJM model on this background, viewed through one chosen three-dimensional $\mathcal{N}=2$ subalgebra. Its gauge group is $G=U(N)\times U(N)$ with $N\in {\mathbb{Z}}_{\ge 1}$ and supersymmetric Chern–Simons levels $k$ and $-k$, where $k\in \mathbb{Z}\setminus \{0\}$. The global group includes both central $U(1)$ factors. Write ${\mathrm{tr}}_N$ for the trace in the defining representation of either factor; in the Kapustin–Willett–Yaakov convention the level multiplies ${\mathrm{tr}}_N\mathrm{/}(4\pi )$ in the supersymmetric Chern–Simons action.

Write $M={\mathbb{S}}_r^3$ and $g$ for its metric. A spin structure here is a principal $\mathrm{Spin}(3)$ bundle $P\to M$ with a two-to-one equivariant map to the bundle of oriented $g$-orthonormal tangent frames, covering the double covering $\mathrm{Spin}(3)\cong SU(2)\to SO(3)$. Let $\rho :SU(2)\to U({\mathbb{C}}^2)$ be the defining representation and $S=P{\times }_{\rho }{\mathbb{C}}^2$ its complex rank-two spinor bundle. The fiber Hermitian metric is induced by $h(v,w)=v^{†}w$ on ${\mathbb{C}}^2$. The connection ${\mathrm{\nabla }}^S:C^{\mathrm{\infty }}(M,S)\to C^{\mathrm{\infty }}(M,T^{\ast }M\otimes S)$ is induced by the Levi-Civita connection. In a local oriented orthonormal coframe $e^a$, with dual frame $E_a$, choose ${\gamma }_1=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),{\gamma }_2=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right),{\gamma }_3=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)\mathrm{.}$ Thus ${\gamma }_a^{†}={\gamma }_a$, ${\gamma }_a{\gamma }_b+{\gamma }_b{\gamma }_a=2{\delta }_{ab}{I}_2$, and ${\gamma }_a{\gamma }_b={\delta }_{ab}{I}_2+i{ϵ}_{abc}{\gamma }_c$, with ${ϵ}_{123}=1$. Clifford multiplication by a real one-form $\eta$ is $\gamma (\eta )={\sum }_a\eta (E_a){\gamma }_a$; in coordinates ${\gamma }_{\mu }=e_{\mu }^a{\gamma }_a$ and ${\gamma }^{\mu }=g^{\mu \nu }{\gamma }_{\nu }$. If ${\omega }_{ab}$ are the Levi-Civita connection one-forms, then ${\mathrm{\nabla }}^S=d+\frac{1}{4}{\omega }_{ab}{\gamma }_a{\gamma }_b$ in a spin frame. Coordinate indices in the formulas below are contracted with $g$; orthonormal indices with ${\delta }_{ab}$.

For a gauge representation $R$, let $E_R=M\times R$ be its associated bundle on the stated trivial gauge bundle. Spinor fields are smooth sections of $S\otimes E_R$; vector gauginos use the complexified adjoint representation and chiral spinors the indicated bifundamental representation. Fermionic components have odd coefficients in a complex Grassmann algebra (an algebra generated by anticommuting odd variables). Barred fermions are independent sections of the corresponding dual bundle; the dagger on fermions in the displayed actions denotes this independent dual variable, whereas on ordinary commuting spinors it denotes the Hermitian dual. Spinor covariant derivatives use the tensor product of ${\mathrm{\nabla }}^S$ and the specified gauge connection. The $SU(2)$-invariant alternating form is $C=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$; its dual map is $v\mapsto v^{T}C$, with $T$ the ordinary matrix transpose. Contractions of two column spinors use this form. Scalar and auxiliary components are smooth sections of $E_R$, with even coefficients. These conventions type the actions and variations below; all fields are complexified when acted on by the selected odd supersymmetry derivation.

Each $\mathcal{N}=2$ vector multiplet has a Hermitian matrix-valued connection $A_{\mu }$ (or ${\widehat{A}}_{\mu }$), a real Hermitian adjoint scalar $\sigma$ (or $\widehat{\sigma }$), a two-component complex gaugino $\lambda$ (or $\widehat{\lambda }$), and a real auxiliary scalar $D$ (or $\widehat{D}$). In an oriented coordinate chart the first factor's source-normalized action is $S_{\mathrm{C}\mathrm{S},k}=\frac{k}{4\pi }\int d^{3}x{\mathrm{tr}}_N\text{\hspace{0.17em}⁣}({ϵ}^{\mu \nu \rho }(A_{\mu }{\mathrm{\partial }}_{\nu }{A}_{\rho }+\frac{2i}{3}{A}_{\mu }{A}_{\nu }{A}_{\rho })+\sqrt{g}(-{\lambda }^{†}\lambda +2D\sigma ));$ the second factor uses the same expression with $(A,\lambda ,D,\sigma ,k)$ replaced by $(\widehat{A},\widehat{\lambda },\widehat{D},\widehat{\sigma },-k)$. Here ${ϵ}^{\mu \nu \rho }$ is the oriented Levi-Civita density. Write $A=A_{\mu }d{x}^{\mu }$ and $\widehat{A}={\widehat{A}}_{\mu }d{x}^{\mu }$ for the Hermitian connection one-forms, with curvatures $F_A=dA+iA\wedge A$ and $F_{\widehat{A}}=d\widehat{A}+i\widehat{A}\wedge \widehat{A}$. For smooth gauge transformations $U,\widehat{U}:{\mathbb{S}}_r^3\to U(N)$, use $A\mapsto UA{U}^{-1}+idU{U}^{-1},\widehat{A}\mapsto \widehat{U}\widehat{A}{\widehat{U}}^{-1}+id\widehat{U}{\widehat{U}}^{-1}\mathrm{.}$ This convention makes $d+iA$ covariant on a fundamental column and makes each curvature transform by conjugation. In components, $F_{A,\mu \nu }={\mathrm{\partial }}_{\mu }{A}_{\nu }-{\mathrm{\partial }}_{\nu }{A}_{\mu }+i[A_{\mu },A_{\nu }]$. An adjoint field $X$ transforms as $X\mapsto UX{U}^{-1}$ and has covariant derivative $D_{\mu }X={\mathrm{\nabla }}_{\mu }X+i[A_{\mu },X]$; use the hatted connection for the second adjoint factor. Spinor covariant derivatives also contain the Levi-Civita spin connection.

The matter consists of two chiral multiplets $A_a$, $a=1,2$, in $(\mathbf{N},\bar{\mathbf{N}})$ and two $B_{\dot{a}}$, $\dot{a}=1,2$, in $(\bar{\mathbf{N}},\mathbf{N})$. A chiral multiplet means a complex scalar, a two-component complex spinor, and a complex auxiliary scalar in its stated representation. For $(U,\widehat{U})\in G$, the transformations are $A_a\mapsto U{A}_a{\widehat{U}}^{-1}$ and $B_{\dot{a}}\mapsto \widehat{U}{B}_{\dot{a}}{U}^{-1}$; the corresponding rules hold for their spinor and auxiliary components. The covariant derivatives on scalars are $D_{\mu }{A}_a={\mathrm{\partial }}_{\mu }{A}_a+i{A}_{\mu }{A}_a-i{A}_a{\widehat{A}}_{\mu },D_{\mu }{B}_{\dot{a}}={\mathrm{\partial }}_{\mu }{B}_{\dot{a}}+i{\widehat{A}}_{\mu }{B}_{\dot{a}}-i{B}_{\dot{a}}{A}_{\mu }\mathrm{.}$ Spinors also carry the Levi-Civita spin connection. The vector-multiplet fields act on a bifundamental by left multiplication from its fundamental factor minus right multiplication from its antifundamental factor; in particular $\sigma \cdot A_a=\sigma A_a-A_a\widehat{\sigma }$.

The round-sphere component action for one chiral multiplet $(\varphi ,\psi ,F)$ of $R$-charge $\mathrm{\Delta }=\frac{1}{2}$, in the representation determined by its bifundamental type, is $S_{\mathrm{c}\mathrm{h}\mathrm{i}\mathrm{r}\mathrm{a}\mathrm{l}}={\int }_{{\mathbb{S}}_r^3}{d}^{3}x\sqrt{g}((D_{\mu }\varphi {)}^{†}{D}^{\mu }\varphi +\frac{3}{4r^2}{\varphi }^{†}\varphi +i{\psi }^{†}{\gamma }^{\mu }{D}_{\mu }\psi +F^{†}F-{\varphi }^{†}{\sigma }^2\varphi +{\varphi }^{†}D\varphi -{\psi }^{†}\sigma \psi +i{\varphi }^{†}{\lambda }^{†}\psi -i{\psi }^{†}\lambda \varphi )\mathrm{.}$ The representation-valued $\sigma ,D,\lambda$ in this expression act by the first-factor action minus the second-factor action. Sum this action over the four chiral multiplets.

The adopted $\mathcal{N}=2$ superpotential is $W=\frac{2\pi }{k}{ϵ}^{ab}{ϵ}^{\dot{a}\dot{b}}{\mathrm{tr}}_N(A_a{B}_{\dot{a}}{A}_b{B}_{\dot{b}})=\frac{4\pi }{k}{\mathrm{tr}}_N(A_1{B}_1{A}_2{B}_2-A_1{B}_2{A}_2{B}_1),$ where ${ϵ}^{12}=1=-{ϵ}^{21}$ and all other components vanish. Let ${\varphi }^I$ range over all complex matrix entries of the four chiral scalars, and set $W_I=\mathrm{\partial }W\mathrm{/}\mathrm{\partial }{\varphi }^I$ and $W_{IJ}={\mathrm{\partial }}_I{\mathrm{\partial }}_{J}W$. Let spinor indices $\rho ,\tau$ range over $\{1,2\}$ and set ${\epsilon }^{12}=1=-{\epsilon }^{21}$ with all other components zero. The adopted off-shell component interaction is $S_W={\int }_{{\mathbb{S}}_r^3}{d}^{3}x\sqrt{g}({\sum }_I{F}^I{W}_I-\frac{1}{2}{\sum }_{I,J}{\epsilon }^{\rho \tau }{\psi }_{\rho }^I{\psi }_{\tau }^J{W}_{IJ}+{\sum }_I{\bar{F}}_I\overline{W_I}-\frac{1}{2}{\sum }_{I,J}{\epsilon }^{\rho \tau }{\bar{\psi }}_{\rho }^I{\bar{\psi }}_{\tau }^J\overline{W_{IJ}})\mathrm{.}$ Here the barred polynomial has conjugate coefficients and barred field labels; the paired auxiliary variables lie on the chosen conjugate contour, while barred Euclidean fermions are independent Grassmann variables. The displayed $F$-terms are not integrated out. The superpotential is the holomorphic interaction that supplies the chiral auxiliary-field and Yukawa terms. This is the ABJM quartic interaction; the adopted quantum model has the enhanced $\mathcal{N}=6$ superconformal symmetry, while localization uses only the selected $\mathcal{N}=2$ subalgebra. For $\theta \in \mathbb{R}\mathrm{/}(4\pi \mathbb{Z})$, the compact $R$-symmetry circle action on a chiral multiplet of charge $\mathrm{\Delta }$ is $(\varphi ,\psi ,F)\mapsto (e^{i\mathrm{\Delta }\theta }\varphi ,e^{i(\mathrm{\Delta }-1)\theta }\psi ,e^{i(\mathrm{\Delta }-2)\theta }F)$. The superconformal dimension is the weight under dilatations; in this model the scalar weight and scalar $R$-charge are $\mathrm{\Delta }=\frac{1}{2}$, the fermion and auxiliary scaling dimensions are $\mathrm{\Delta }+\frac{1}{2}=1$ and $\mathrm{\Delta }+1=\frac{3}{2}$, and their $R$-charges are $\mathrm{\Delta }-1$ and $\mathrm{\Delta }-2$, respectively. The $4\pi$ period makes the half-integer charges single-valued; equivalently one may use a $2\pi$-periodic angle $\vartheta =\theta \mathrm{/}2$ and integer charges $(1,-1,-3)$. Set all real masses and Fayet–Iliopoulos parameters to zero. A real mass here is the constant scalar in a background flavor vector multiplet; a Fayet–Iliopoulos parameter is the coefficient of the auxiliary $D$ field for an abelian vector multiplet. The level $k$ is dimensionless and $r$ has units of length; $[A_{\mu }]=[\sigma ]=r^{-1}$, $[\lambda ]=r^{-3\mathrm{/}2}$, $[D]=r^{-2}$, $[\varphi ]=r^{-1\mathrm{/}2}$, $[\psi ]=r^{-1}$, and $[F]=r^{-3\mathrm{/}2}$. These dimensions make every displayed action density have units $r^{-3}$.

The smooth field domain consists of pairs of smooth connections and the listed smooth associated matter and auxiliary fields on the trivial $U(N)\times U(N)$ bundles. The smooth gauge group is $\mathcal{G}=C^{\mathrm{\infty }}({\mathbb{S}}_r^3,U(N)\times U(N))$, acting by the transformations above. For infinitesimal Hermitian parameters $\alpha ,\widehat{\alpha }$ with $(U,\widehat{U})=(e^{-i\alpha },e^{-i\widehat{\alpha }})$ at first order, write $D_A\alpha =d\alpha +i[A,\alpha ]$ and $D_{\widehat{A}}\widehat{\alpha }=d\widehat{\alpha }+i[\widehat{A},\widehat{\alpha }]$; then $\delta A=D_A\alpha$ and $\delta \widehat{A}=D_{\widehat{A}}\widehat{\alpha }$. The Lorenz gauge-fixing function is $\mathcal{F}(A,\widehat{A})=({\mathrm{\nabla }}^{\mu }{A}_{\mu },{\mathrm{\nabla }}^{\mu }{\widehat{A}}_{\mu }),$ Here $\mathrm{\nabla }$ is the Levi-Civita derivative on the one-form index, applied componentwise to the Lie-algebra values. and its derivative along a gauge orbit is the Faddeev–Popov operator ${\mathcal{M}}_{A,\widehat{A}}(\alpha ,\widehat{\alpha })=({\mathrm{\nabla }}^{\mu }{D}_{A,\mu }\alpha ,{\mathrm{\nabla }}^{\mu }{D}_{\widehat{A},\mu }\widehat{\alpha })\mathrm{.}$ Let $c,\widehat{c}$ be odd Lie-algebra-valued ghost scalars, $\bar{c},\bar{\widehat{c}}$ odd antighosts, and $b,\widehat{b}$ even multiplier scalars; set $[c]=[\widehat{c}]=r^0$ and $[\bar{c}]=[b]=[\bar{\widehat{c}}]=[\widehat{b}]=r^{-1}$, and in the Lorenz limit take $b=i\beta$, $\widehat{b}=i\widehat{\beta }$ with real $\beta ,\widehat{\beta }$. The Faddeev–Popov differential $s$ is the odd derivation $sA=D_{A}c,s\widehat{A}=D_{\widehat{A}}\widehat{c},sc=-\frac{i}{2}[c,c{]}_{\mathrm{g}\mathrm{r}},s\widehat{c}=-\frac{i}{2}[\widehat{c},\widehat{c}{]}_{\mathrm{g}\mathrm{r}},s\bar{c}=b,s\bar{\widehat{c}}=\widehat{b},sb=s\widehat{b}=0,$ where $[X,Y{]}_{\mathrm{g}\mathrm{r}}=XY-(-1{)}^{\mathrm{\mid }X\mathrm{\mid }\mathrm{\mid }Y\mathrm{\mid }}YX$ is the parity-graded matrix bracket; its Jacobi identity gives $s^2=0$. On every other field $s$ is the infinitesimal gauge action with $\alpha ,\widehat{\alpha }$ replaced by $c,\widehat{c}$; in particular $s{A}_a=-ic{A}_a+i{A}_a\widehat{c}$ and $s{B}_{\dot{a}}=-i\widehat{c}{B}_{\dot{a}}+i{B}_{\dot{a}}c$. The gauge-fixing fermion and its conventional BRST part are ${\mathrm{\Psi }}_{\mathrm{g}\mathrm{f}}={\int }_{{\mathbb{S}}_r^3}{d}^{3}x\sqrt{g}{\mathrm{tr}}_N(\bar{c}{\mathrm{\nabla }}^{\mu }{A}_{\mu }+\frac{\xi }{2r}\bar{c}b+\bar{\widehat{c}}{\mathrm{\nabla }}^{\mu }{\widehat{A}}_{\mu }+\frac{\xi }{2r}\bar{\widehat{c}}\widehat{b}),s{\mathrm{\Psi }}_{\mathrm{g}\mathrm{f}}={\int }_{{\mathbb{S}}_r^3}{d}^{3}x\sqrt{g}{\mathrm{tr}}_N(b{\mathrm{\nabla }}^{\mu }{A}_{\mu }+\widehat{b}{\mathrm{\nabla }}^{\mu }{\widehat{A}}_{\mu }+\frac{\xi }{2r}(b^2+{\widehat{b}}^2)-\bar{c}{\mathrm{\nabla }}^{\mu }{D}_{A,\mu }c-\bar{\widehat{c}}{\mathrm{\nabla }}^{\mu }{D}_{\widehat{A},\mu }\widehat{c})\mathrm{.}$ Here $\xi \ge 0$ is a dimensionless gauge parameter; the source uses its Lorenz limit $\xi \to 0$, taken before the multiplier integral. With $b=i\beta$ and $\widehat{b}=i\widehat{\beta }$, integration over the real multiplier fields $\beta ,\widehat{\beta }$ gives the Fourier representations of the two Lorenz delta functionals. These conditions define a local gauge slice; they do not assert a global gauge slice, since a gauge orbit may intersect the Lorenz slice more than once; such repeated intersections are Gribov copies. The ghost determinants omit the kernel of $\mathcal{M}$; infinitesimal stabilizers and residual gauge volumes are treated separately, with the final Cartan reduction retaining the Weyl quotient. The Euclidean contour takes the vector connections and real scalars on their Hermitian/real slices and pairs each matter scalar and auxiliary scalar with its conjugate partner; Euclidean fermions and their barred partners are independent Grassmann variables. The Cartan subalgebras are $\mathfrak{h}=\{\mathrm{diag}(s_1,\dots ,s_N):s_i\in \mathbb{R}\},\widehat{\mathfrak{h}}=\{\mathrm{diag}({\widehat{s}}_1,\dots ,{\widehat{s}}_N):{\widehat{s}}_i\in \mathbb{R}\}\mathrm{.}$ The residual Weyl group $S_N\times S_N$ permutes the corresponding eigenvalues.

Choose a dimensionless smooth section $u\in C^{\mathrm{\infty }}(M,S)$ with pointwise norm $h(u,u)=1$ satisfying ${\mathrm{\nabla }}_X^{S}u=\frac{i}{2r}\gamma (X^{\mathrm{♭}})u$ for every smooth tangent vector field $X$, where $X^{\mathrm{♭}}=g(X,\cdot )$. This is the chosen normalized Killing spinor; its existence on the round sphere and its choice are part of the adopted source background. In the variations, ${\mathrm{\nabla }}_{\mu }$ on spinors denotes ${\mathrm{\nabla }}_{{\mathrm{\partial }}_{\mu }}^S$. The profile $u$ and parameter $ϵ$ are commuting spinors, while $Q$ is odd. The supersymmetry parameter in the variations below is $ϵ=\sqrt{r}u$, so $[ϵ]=r^{1\mathrm{/}2}$ and ${\mathrm{\nabla }}_{\mu }ϵ=\frac{i}{2r}{\gamma }_{\mu }ϵ$. Put ${\gamma }^{\mu \nu }=\frac{1}{2}[{\gamma }^{\mu },{\gamma }^{\nu }]$. The odd off-shell supersymmetry derivation $Q$ selected by this spinor acts on each vector multiplet, and identically on the hatted one, by $Q{A}_{\mu }=-\frac{i}{2}{\lambda }^{†}{\gamma }_{\mu }ϵ,Q\sigma =-\frac{1}{2}{\lambda }^{†}ϵ,QD=-\frac{i}{2}(D_{\mu }{\lambda }^{†}){\gamma }^{\mu }ϵ+\frac{1}{4r}{\lambda }^{†}ϵ+\frac{i}{2}[{\lambda }^{†},\sigma ]ϵ,Q\lambda =(-\frac{1}{2}{\gamma }^{\mu \nu }{F}_{\mu \nu }-D+i{\gamma }^{\mu }{D}_{\mu }\sigma -\frac{\sigma }{r})ϵ,Q{\lambda }^{†}=0.$ For each chiral multiplet $(\varphi ,\bar{\varphi },\psi ,\bar{\psi },F,\bar{F})$ with $\mathrm{\Delta }=\frac{1}{2}$, it acts by $Q\varphi =0,Q\bar{\varphi }=\bar{\psi }ϵ,Q\psi =(-i{\gamma }^{\mu }{D}_{\mu }\varphi -i\sigma \varphi +\frac{\varphi }{2r})ϵ,Q\bar{\psi }={ϵ}^{T}C\bar{F},QF={ϵ}^{T}C(-i{\gamma }^{\mu }{D}_{\mu }\psi +i\lambda \varphi +i\sigma \psi ),Q\bar{F}=0.$ Here barred chiral variables are the conjugate partners on the scalar/auxiliary contour and independent Euclidean Grassmann variables for fermions; all matrix-valued fields act in the chiral multiplet's stated representation. In the source's selected off-shell complex these transformations have $Q^2=0$. Extend $Q$ by zero on $c,\widehat{c},\bar{c},\bar{\widehat{c}},b,\widehat{b}$. The gauge-covariant BRST differential above then graded-anticommutes with $Q$, and the total odd derivation is $Q_{\mathrm{g}\mathrm{f}}=Q+s$ with $Q_{\mathrm{g}\mathrm{f}}^2=0$.

The gauge-fixing term used for localization is the combined insertion $S_{\mathrm{g}\mathrm{f}}=Q_{\mathrm{g}\mathrm{f}}{\mathrm{\Psi }}_{\mathrm{g}\mathrm{f}}=s{\mathrm{\Psi }}_{\mathrm{g}\mathrm{f}}-{\int }_{{\mathbb{S}}_r^3}{d}^{3}x\sqrt{g}{\mathrm{tr}}_N(\bar{c}{\mathrm{\nabla }}^{\mu }Q{A}_{\mu }+\bar{\widehat{c}}{\mathrm{\nabla }}^{\mu }Q{\widehat{A}}_{\mu })\mathrm{.}$ The sign follows from the left graded Leibniz rule and the odd parity of each antighost; $Q$ annihilates antighosts and multipliers. The displayed $s{\mathrm{\Psi }}_{\mathrm{g}\mathrm{f}}$ alone is not the insertion asserted to be $Q_{\mathrm{g}\mathrm{f}}$-closed. With the supplied off-shell algebra, $Q_{\mathrm{g}\mathrm{f}}{S}_{\mathrm{g}\mathrm{f}}=Q_{\mathrm{g}\mathrm{f}}^2{\mathrm{\Psi }}_{\mathrm{g}\mathrm{f}}=0$. Thus the completed gauge-fixed action, rather than its BRST part alone, is the one used in the localization deformation. This is the combined gauge-fixing prescription in Kapustin–Willett–Yaakov §3.1, equations (3.10)–(3.13) in the current PDF (the equations numbered (30)–(33) in earlier numbering). Their extra supersymmetry variation is retained here explicitly; no unproved equality of two infinite-dimensional integrals is used to discard it.

With the radius restored, define the positive Hermitian inner product on the direct sum of the two matrix algebras by $⟨(X,\widehat{X}),(Y,\widehat{Y}){⟩}_{+}={\mathrm{tr}}_N(X^{†}Y)+{\mathrm{tr}}_N({\widehat{X}}^{†}\widehat{Y})$. The source's positive gauge-sector deformation uses this form, independent of the signed Chern–Simons levels, and the odd gauge-invariant functional $V_{\mathrm{v}\mathrm{e}\mathrm{c}}={\int }_{{\mathbb{S}}_r^3}{d}^{3}x\sqrt{g}({\mathrm{tr}}_N((Q\lambda {)}^{†}\lambda )+{\mathrm{tr}}_N((Q\widehat{\lambda }{)}^{†}\widehat{\lambda }))\mathrm{.}$ For the chiral-sector deformation, the unit-sphere source functional is multiplied by $r^{-1}$ so that the restored-radius functional below is dimensionless; this positive rescaling leaves its zero locus unchanged: $V_{\mathrm{m}\mathrm{a}\mathrm{t}}=\frac{1}{r}{\int }_{{\mathbb{S}}_r^3}{d}^{3}x\sqrt{g}{\sum }_I((Q{\psi }_I{)}^{†}{\psi }_I+{\psi }_I^{†}(Q{\bar{\psi }}_I{)}^{†})\mathrm{.}$ The sum over all chiral components is gauge invariant. The source adds the $Q_{\mathrm{g}\mathrm{f}}$-exact terms $t{Q}_{\mathrm{g}\mathrm{f}}(V_{\mathrm{v}\mathrm{e}\mathrm{c}}+V_{\mathrm{m}\mathrm{a}\mathrm{t}})$ for a dimensionless parameter $t\ge 0$, together with the completed gauge-fixing term $S_{\mathrm{g}\mathrm{f}}=Q_{\mathrm{g}\mathrm{f}}{\mathrm{\Psi }}_{\mathrm{g}\mathrm{f}}$. On the source localization locus, flat connections are gauge equivalent to zero on the stated trivial bundles over simply connected ${\mathbb{S}}^3$, matter and fermion fields vanish, and $\sigma =\mathrm{diag}(s_1,\dots ,s_N),\widehat{\sigma }=\mathrm{diag}({\widehat{s}}_1,\dots ,{\widehat{s}}_N),D=-\sigma \mathrm{/}r,\widehat{D}=-\widehat{\sigma }\mathrm{/}r\mathrm{.}$ The corresponding one-loop factors are the source's Gaussian boson/fermion/ghost determinant ratios about this locus. For the positive-spectrum determinant convention, fix a smooth finite-rank Hermitian complex bundle $E\to M$, with a fixed smooth compatible connection, and the Hilbert space $\mathcal{H}=L^2(M,E)$ with inner product $⟨f,g⟩={\int }_M{h}_E(f,g)d{\mathrm{vol}}_g$. Let $L_0:C^{\mathrm{\infty }}(M,E)\to C^{\mathrm{\infty }}(M,E)$ be a differential operator of positive integer order $m_L$ with smooth coefficients. In a local expression $L_0={\sum }_{\mathrm{\mid }\alpha \mathrm{\mid }\le m_L}{a}_{\alpha }(x){\mathrm{\partial }}^{\alpha }$, its principal symbol is ${\sigma }_{m_L}(L_0)(x,\xi )={\sum }_{\mathrm{\mid }\alpha \mathrm{\mid }=m_L}{a}_{\alpha }(x)(i\xi {)}^{\alpha }$; ellipticity means this endomorphism of $E_x$ is invertible for every nonzero real cotangent vector $\xi \in T_x^{\ast }M$. Use the realization $Lf=L_{0}f$ in the distributional sense with domain $\mathcal{D}(L)=H^{m_L}(M,E)\subset \mathcal{H}$, where $H^{m_L}$ is the completion of smooth sections in the norm squared ${\sum }_{j=0}^{m_L}{\int }_M\mathrm{\mid }{\mathrm{\nabla }}^{j}f{\mathrm{\mid }}^{2}d{\mathrm{vol}}_g$. Require that $L$ is densely defined and closed, that its Hilbert-space adjoint has $\mathcal{D}(L^{\ast })=\mathcal{D}(L)$ and $L^{\ast }=L$, and that $⟨f,Lf⟩\ge c_L\mathrm{\parallel }f{\mathrm{\parallel }}^2$ for all $f\in \mathcal{D}(L)$ for some $c_L>0$. These are the self-adjointness and strict positivity hypotheses. Require an orthonormal eigenbasis with positive eigenvalues ${\nu }_j$ of finite multiplicity tending to infinity, listed with multiplicity and with no kernel; no spectral-existence theorem is being inferred from terminology. Assume ${\sum }_j{\nu }_j^{-s_0}<\mathrm{\infty }$ for some real $s_0>0$. With $[L]=r^{-m_L}$, set $\tilde{L}=r^{m_L}L$ so that its eigenvalues ${\tilde{\nu }}_j=r^{m_L}{\nu }_j$ are dimensionless. Define ${\zeta }_{\tilde{L}}(s)={\sum }_j{\tilde{\nu }}_j^{-s}$ initially for $\mathrm{Re}s>s_0$, using the real logarithm of each positive eigenvalue; when it has a meromorphic continuation holomorphic at $s=0$, set ${\det }_{\zeta }\tilde{L}=\exp (-{\zeta }_{\tilde{L}}^{\mathrm{\prime }}(0))$. For an operator with a finite-dimensional kernel, the same convention on its orthogonal complement defines a primed determinant provided the remaining positive spectrum and continuation satisfy the stated hypotheses. A primed determinant omits the kernel only; its zero modes are treated separately, with the finite Cartan modes integrated and gauge stabilizer modes removed by the residual quotient. The source's phase prescription for fermionic or other nonpositive/complex fluctuation operators and for the oscillatory Chern–Simons Gaussian is part of the cited calculation; the positive-spectrum formula above is not applied to those operators.

In the Kapustin–Willett–Yaakov Euclidean convention, let ${\mathcal{S}}_{\mathrm{C}\mathrm{S}}$ be the sum of the two real functionals displayed above, let $S_{\mathrm{c}\mathrm{h}\mathrm{i}\mathrm{r}\mathrm{a}\mathrm{l}}$ be the sum of the four chiral kinetic actions, and let $S_W$ be the off-shell interaction just specified. Define the Euclidean action and weight by $S_E=S_{\mathrm{c}\mathrm{h}\mathrm{i}\mathrm{r}\mathrm{a}\mathrm{l}}+S_W-i{\mathcal{S}}_{\mathrm{C}\mathrm{S}},e^{-S_E}=e^{i{\mathcal{S}}_{\mathrm{C}\mathrm{S}}-S_{\mathrm{c}\mathrm{h}\mathrm{i}\mathrm{r}\mathrm{a}\mathrm{l}}-S_W}\mathrm{.}$ At the stated localization locus this gives $i{\mathcal{S}}_{\mathrm{C}\mathrm{S}}=-i\pi k({\sum }_i(r{s}_i{)}^2-{\sum }_i(r{\widehat{s}}_i{)}^2),$ so the real Cartan integral has the oscillatory classical phase used in the Example. In particular, the real functional $S_{\mathrm{C}\mathrm{S}}$ enters the Euclidean weight with the displayed factor $+i$.

Before gauge fixing, the orbit-normalized functional integral is only the formal expression $Z_{\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}}=\frac{1}{\mathrm{Vol}(\mathcal{G})}{\int }_{\mathcal{C}}[D\mathrm{\Phi }{]}_{\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}}{e}^{-S_E[\mathrm{\Phi }]},$ where $\mathrm{Vol}(\mathcal{G})$ is a formal normalization of the gauge-orbit volume, not a defined finite number, and $\mathcal{C}$ is the source Euclidean contour. The local Lorenz-gauge representation is symbolically the integral with the gauge-fixing delta functional $\delta [\mathcal{F}]$ and $\det {}^{\mathrm{\prime }}{\mathcal{M}}_{A,\widehat{A}}$, whose conventional multiplier/ghost part is $s{\mathrm{\Psi }}_{\mathrm{g}\mathrm{f}}$. In the localization representation this is completed to $S_{\mathrm{g}\mathrm{f}}=Q_{\mathrm{g}\mathrm{f}}{\mathrm{\Psi }}_{\mathrm{g}\mathrm{f}}$, including the antighost–supersymmetry term displayed above; the multiplier constraint and residual quotient are retained. The orbit-volume division is implemented once. No quotient notation $\mathcal{C}\mathrm{/}\mathcal{G}$ or second volume divisor is added. Residual stabilizers and the Weyl quotient are treated separately as specified above. The source calculation assumes that its gauge-fixed measure and contour are $Q_{\mathrm{g}\mathrm{f}}$-invariant and that the deformation introduces no anomalous Jacobian (a multiplicative change of the formal measure under $Q_{\mathrm{g}\mathrm{f}}$) or surface contribution at the boundary of field space. These are adopted formal field-theory inputs: this postulate does not construct an infinite-dimensional measure, prove convergence of the localization limit, or assert a theorem for arbitrary three-dimensional $\mathcal{N}=2$ models.

Kapustin, Willett, and Yaakov, §§2–4

Mariño, §§2.1–2.4 and 6.2–6.5