Localization for $\mathcal {N}=2$ Supersymmetric Gauge Theories in Four Dimensions

Abstract

We review the supersymmetric localization of $\mathcal {N} = 2$ theories on curved backgrounds in four dimensions using $\mathcal {N} = 2$ supergravity and generalized conformal Killing spinors. We review some known backgrounds and give examples of new geometries such as local $T^2$-bundle fibrations. We discuss in detail a topological four-sphere with generic $T^2$-invariant metric. This review is a contribution to the special volume on recent developments in $\mathcal {N} = 2$ supersymmetric gauge theory and the 2d-4d relation.

A citation of the form [V:x] refers to article number x in this volume.

Appendices

Appendix 1: Conventions and Useful Identities

1.1 Indices

For the 4d theories with 8 supercharges ($\mathcal {N}=2$ supersymmetry in 4d) we use the notations of the (0, 1) 6d supersymmetric theories under the dimensional reduction. The Table 4 summarizes the index notations.

Table 4 Indices

The symmetrization and anti-symmetrization of tensors

$$\begin{aligned} \begin{aligned} t_{(m_1 \dots m_r)}&= \tfrac{1}{r!} \sum _{\sigma \in \text {Perm}(r)} t_{m_{\sigma (1)}, \ldots m_{\sigma (r)}}\\ t_{[m_1 \dots m_r]}&= \tfrac{1}{r!} \sum _{\sigma \in \text {Perm}(r)} (-1)^{\sigma } t_{m_{\sigma (1)}, \ldots m_{\sigma (r)}} \end{aligned} \end{aligned}$$

(4.25)

1.2 Spinors

The spinors $\lambda$ and $\varepsilon$ in the (0, 1) Euclidean supersymmetric 6d theory are the holomorphic ${\textit{SU(2)}}_\mathrm {R}\simeq {\textit{Sp(1)}}_\mathrm {R}$ doublets of Weyl four-component spinors, of Weyl chirality $+1$, for the 6d Clifford algebra over complex numbers ${\mathbb C}$. We take $\lambda \equiv (\lambda ^i)_{i = 1,2}$ where each $\lambda ^1$ and $\lambda ^2$ is 6d Weyl fermion. In total the spinor $\lambda$ has 8 complex components.⁶

1.3 Clifford Algebra

The $8 \times 8$ complex matrices $\gamma _m$ represent the 6d Clifford algebra

$$\begin{aligned} \{ \gamma _{m}, \gamma _{n} \} = 2 g_{mn} \end{aligned}$$

(4.26)

The chirality operator $\gamma _{*}^{\mathrm {6d}}$ anticommuting with all $\gamma _m$ is

$$\begin{aligned} \gamma _{*} = i \gamma _1 \dots \gamma _{6}; \qquad \{\gamma _{*} , \gamma _m\} = 0; \qquad \gamma _*^2 = 1. \end{aligned}$$

(4.27)

The chirality of the spinors is the eigenvalue of $\gamma _*$. The projection operators that split $\mathcal {S} = \mathcal {S}^{+} \oplus \mathcal {S}^{-}$ are

$$\begin{aligned} \gamma _{\pm } = \tfrac{1}{2} (1 \pm \gamma _{*}),\qquad \varepsilon _{\pm } = \gamma _{\pm } \varepsilon _{\pm } =\pm \gamma _{*} \varepsilon _{\pm } \end{aligned}$$

(4.28)

Explicit form of $\gamma _m$ matrices is not needed, but for concreteness one can recursively define the $\gamma ^{(d)}_m$ matrices of size $2^{d/2} \times 2^{d/2}$ in even dimension d in terms of $\gamma ^{(d-2)}_m$ as follows (see e.g. [60])

$$\begin{aligned} \begin{aligned}&\gamma ^{(d)}_m = \sigma _{3} \otimes \gamma _{m}^{(d-2)}, \quad m \in [1, \dots , d- 2] \\&\gamma ^{(d)}_{d-1} = \sigma _{1} \otimes 1, \qquad \gamma ^{(d)}_{d} = \sigma _{2} \otimes 1 \\&\gamma ^{(d)}_{*} = \sigma _3 \otimes \gamma ^{(d-2)}_{*} \end{aligned} \end{aligned}$$

(4.29)

where $(\sigma _0, \sigma _1,\sigma _2,\sigma _3)$ are the $2 \times 2$ Pauli matrices

$$\begin{aligned} (\sigma _0,\sigma _1,\sigma _2,\sigma _3) = \left( \left( {\begin{matrix} 1 &{} 0 \\ 0 &{} 1 \end{matrix}}\right) , \left( {\begin{matrix} 0 &{} 1 \\ 1 &{} 0 \end{matrix}}\right) , \left( {\begin{matrix} 0 &{} -i \\ i &{} 0 \end{matrix}}\right) , \left( {\begin{matrix} 1 &{} 0 \\ 0 &{} -1 \end{matrix}}\right) \right) . \end{aligned}$$

(4.30)

We use antisymmetric multi-index notations

$$\begin{aligned} \gamma _{m_1 \dots m_r} = \gamma _{[m_1} \dots \gamma _{m_r]}. \end{aligned}$$

(4.31)

and we use underline notation for the multi-index

$$\begin{aligned} \gamma _{\underline{r}} \quad \text {is one of} \quad \gamma _{m_1 \dots m_r} \end{aligned}$$

(4.32)

In the contraction of multi-index we use non-repetitive summation

$$\begin{aligned} A^{\underline{p}}{} B_{\underline{p}} \equiv \sum _{m_{1} < \dots < m_{p}} A^{m_1 \dots m_p} B_{m_1 \dots m_p} = \frac{1}{p!}A^{m_1 \dots m_p} B_{m_1 \dots m_p} \end{aligned}$$

(4.33)

For the forms we use component and slashed notation

(4.34)

Contraction identity

$$\begin{aligned} \gamma ^{m} \gamma _{\underline{r}} \gamma _m = (-1)^{r} (d - 2r) \gamma _{\underline{r}} \end{aligned}$$

(4.35)

Multi-index contraction identity

$$\begin{aligned} \gamma ^{\underline{p}} \gamma _{\underline{r}}\gamma _{\underline{p}} = \Delta (d,r,p) \gamma _{\underline{r}} \end{aligned}$$

(4.36)

with

$$\begin{aligned} \Delta (d,r,p)= (-1)^{p(p-1)/2} (-1)^{rp} \sum _{q=\mathrm {max}(p+r-d,0)}^{\mathrm {min}(r,p)} (-1)^{q} \left( {\begin{array}{c}r\\ q\end{array}}\right) \left( {\begin{array}{c}d-r\\ p-q\end{array}}\right) \end{aligned}$$

(4.37)

The contraction formula and the completeness of $(\gamma _{\underline{p}})_{p \in [0,\dots , d]}$ for $d \in 2 {\mathbb Z}$ in the matrix algebra of $2 ^{d/2} \times 2^{d/2}$ matrices implies the Fierz identity

$$\begin{aligned} (\gamma ^{\underline{r}}) ^{\alpha _1}_{\,\,\alpha _2} (\gamma _{\underline{r}})^{\alpha _3}_{\,\,\,\alpha _4} = \sum _{k=0}^{d} \tilde{\Delta }(d,r,k) (\gamma ^{\underline{k}})^{\alpha _1}_{\,\,\ \alpha _4} (\gamma _{\underline{k}})^{\alpha _3}_{\,\,\, \alpha _2} \end{aligned}$$

(4.38)

where

$$\begin{aligned} \tilde{\Delta }(d,r,k) = (-1)^{ \frac{k(k-1)}{2}} 2^{- \frac{d}{2}} \Delta (d,r,k) \end{aligned}$$

(4.39)

The terms with $k > \frac{d}{2}$ in the Fierz identity are conveniently represented as

$$\begin{aligned} \gamma _{\underline{k}} \gamma _{*} = (-1)^{r(r-1)/2} i^{-n/2} \gamma _{\underline{k}^\vee } \end{aligned}$$

(4.40)

where $\gamma _{\underline{k}^{\vee }}$ is complementary in indices of $\gamma _{\underline{k}}$ with a proper permutation sign. The Fierz identity is

$$\begin{aligned}&(\gamma ^{\underline{l}}) ^{\alpha _1}_{\,\,\alpha _2} (\gamma _{\underline{l}})^{\alpha _3}_{\,\,\,\alpha _4} = \sum _{k=0}^{d/2} \tilde{\Delta }(d,l,k) (\gamma ^{\underline{k}})^{\alpha _1}_{\,\,\ \alpha _4} (\gamma _{\underline{k}})^{\alpha _3}_{\,\,\, \alpha _2}\nonumber \\&\qquad \qquad \qquad \qquad \qquad + (-1)^{d/2} \sum _{k=0}^{d/2 - 1} \tilde{\Delta }(d,l,d- k) (\gamma ^{\underline{k}} \gamma _*)^{\alpha _1}_{\,\,\ \alpha _4} (\gamma _{\underline{k}} \gamma _*)^{\alpha _3}_{\,\,\, \alpha _2}. \end{aligned}$$

(4.41)

This form is useful when applied to the chiral spinors.

1.4 Spinor Bilinears

The spinor representation space $\mathcal {S}$ can be equipped with an invariant complex bilinear form $(, ): \mathcal {S} \otimes \mathcal {S} \rightarrow {\mathbb C}$. In components we write⁷

$$\begin{aligned} ({\eta } \varepsilon ) := \eta ^{\alpha } C_{\alpha \beta } \varepsilon ^{\beta } \end{aligned}$$

(4.42)

where C is a matrix representing the bilinear form.

All operators $\gamma _{\underline{r}}$ are symmetric or antisymmetric with respect to C. The symmetry of $C \gamma _{\underline{r}}$ depends on the dimension d and is summarized in Table 5.

Table 5 Symmetries of $C\gamma _{\underline{r}}$

The entries $s_0 s_1 s_2 s_3$ with $s_{r} = \pm 1$ denote the transposition symmetry of $C\gamma _{\underline{r}}$ for . There are two choices of C denoted by $C_1$ and $C_2$ in the table. In representation (4.29) one can take

$$\begin{aligned} \begin{aligned} C_1 = \cdots \otimes \sigma _1 \otimes \sigma _2 \otimes \sigma _1 \\ C_2 = \cdots \otimes \sigma _2 \otimes \sigma _1 \otimes \sigma _2. \end{aligned} \end{aligned}$$

(4.43)

The bilinear form $C_2$ for spinors in even dimension d can be also used as the bilinear form for spinors in $d+1$-dimensions. For the theories with 8 supercharges in $d=4,5,6$ dimensions we are using C highlighted in the Table 5.

The matrices $C\gamma _{\underline{r}}$ represent bilinear forms on $\mathcal {S}$ valued in r-forms, in other words, for spinors $\eta$ and $\varepsilon$ the

$$\begin{aligned} \omega _{\underline{r}}= ({\eta } \gamma _{\underline{r}} \varepsilon ) \end{aligned}$$

(4.44)

transform covariantly as the rank r form. Since

$$\begin{aligned} \gamma _{*} C = (-1)^{\tfrac{d}{2} } C \gamma _* \end{aligned}$$

(4.45)

it follows that⁸

$$\begin{aligned} \begin{aligned} ({\eta } \gamma _{\underline{r}} \epsilon )&=( {\eta _{+}} \gamma _{\underline{r}} \epsilon _{+}) + ({\eta _{-}} \gamma _{\underline{r}} \epsilon _{-}), \qquad \tfrac{d}{2} + r \in 2 {\mathbb Z}\\ ( { \eta } \gamma _{\underline{r}} \epsilon )&= ({\eta _{-}} \gamma _{\underline{r}} \epsilon _{+}) + ({\eta _{+}} \gamma _{\underline{r}} \epsilon _{-}), \qquad \tfrac{d}{2} + r \in 2 {\mathbb Z}+ 1 \end{aligned} \end{aligned}$$

(4.46)

In $d=6$ the bilinears in the spinors of the same chirality transform as forms of odd rank; while the bilinears in the spinors of opposite chirality transform as forms of even rank.

$$\begin{aligned} d=6: \qquad {\left\{ \begin{array}{ll} ({\varepsilon _+} \gamma _{\underline{r}} \varepsilon _+') \ne 0 \quad \text {only for } r \in \{1,3,5\} \\ ( {\eta _{-}} \gamma _{\underline{r}} \varepsilon _{+}) \ne 0 \quad \text {only for } r \in \{0,2,4,6\} \end{array}\right. } \end{aligned}$$

(4.47)

The bilinear form valued in 1-forms is antisymmetric in $d=6$ for either choice of C. To construct the standard fermionic action $(\lambda \gamma ^m D_m \lambda )$ we need the symmetric 1-form valued bilinear form. For the minimal 6d (0, 1) supersymmetry we introduce a ${\textit{SU(2)}}_{\mathrm {R}}$-doublet of Weyl fermions $(\lambda ^{i})_{i=1,2}$ and then use $C \otimes \epsilon$, where $\epsilon = \epsilon _{ij}$ is the standard $2\times 2$ antisymmetric symbol, as the symmetric bilinear form on the $\mathcal {S^{+}}\otimes {\mathbb C}^{2}$. The resulting 1-form valued bilinear is symmetric and there is a proper fermionic kinetic action

(4.48)

We use the standard antisymmetric $2 \times 2$ tensor $\epsilon _{ij}$ to raise and lower the ${\textit{SU(2)}}_\mathrm {R}$ indices i, j in the pattern $^i{}_i$:

$$\begin{aligned} \begin{aligned}&\lambda ^{i} := \epsilon ^{ij} \lambda _{j}, \qquad \lambda _{j} := \lambda ^{i} \epsilon _{ij}\\&\epsilon ^{ij} \epsilon _{ik} = \delta ^{j}_{k}, \qquad ( {\varepsilon }^{[j} \eta ^{i]}) = \tfrac{1}{2} \epsilon ^{ij} ({\varepsilon }^k \eta _k) \end{aligned} \end{aligned}$$

(4.49)

When the ${\textit{SU(2)}}_\mathrm {R}$ indices are omitted, the contraction $^i{}_i$ is assumed

$$\begin{aligned} ({\varepsilon } \gamma _{\underline{r}} \varepsilon ') \equiv ( {\varepsilon }^{i} \gamma _{\underline{r}} \varepsilon '_i) \end{aligned}$$

(4.50)

1.5 $d=6$ Fierz Identities

For $d=6$ and $l=1$ we find

k	0	1	2	3	4	5	6
$\tilde{\Delta }(6,1,k)$	$\tfrac{3}{4}$	$-\tfrac{1}{2}$	$-\tfrac{1}{4}$	0	$-\frac{1}{4}$	$\frac{1}{2}$	$\tfrac{3}{4}^{}$

Notice that $\tilde{\Delta }(6,1,k) = (-1)^k \tilde{\Delta }(6,1,6-k)$. Therefore if we project Fierz identity (4.41) with $\gamma _{+}$ applied to the $\alpha _2$ and $\alpha _4$ indices, we find that terms with even k vanish. In addition the middle term $k = 3$ vanishes too. Finally

$$\begin{aligned} \boxed { (\gamma ^{\underline{1}}) ^{\alpha _1}_{\,\,\alpha _2} (\gamma _{\underline{1}})_{\alpha _3 \alpha _4} = - (\gamma ^{\underline{1}})^{\alpha _1}_{\,\,\ \alpha _4} (\gamma _{\underline{1}})_{ \alpha _3 \alpha _2} \quad \quad \text {projected by } (\gamma _{\pm })^{\alpha _2}_{\,\,\, \alpha _2'} (\gamma _{\pm })^{\alpha _4}_{\,\,\, \alpha _4'} } \end{aligned}$$

(4.51)

A frequently used form of the above identity involves cylic permutation of three $+$-chiral spinor doublets $\varepsilon ^i, \kappa ^i, \lambda ^i$. Taking the sum of

$$\begin{aligned} \begin{aligned} (\varepsilon ^j \gamma _{m} \kappa _j) \gamma ^m \lambda ^i = - (\varepsilon ^j \gamma _m \lambda ^i) \gamma ^m \kappa _j \\ (\lambda ^j \gamma _m \kappa _j) \gamma ^m \varepsilon ^i = - (\lambda ^j \gamma _{m} \varepsilon ^i) \gamma ^m \kappa _j \end{aligned} \end{aligned}$$

(4.52)

we find

$$\begin{aligned} \boxed { (\varepsilon \gamma _{m} \kappa ) \gamma ^m \lambda + (\lambda \gamma _m \kappa ) \gamma ^m \varepsilon + (\varepsilon \gamma _m \lambda ) \gamma ^m \kappa = 0} \end{aligned}$$

(4.53)

Now we consider projection of 6d Fierz identity at $p=1$ on spinors of opposite chirality. Take $+$-chiral doublet $\varepsilon ^i$ and $-$-chiral doublet $\eta ^i$. We find

$$\begin{aligned} \begin{aligned}&({\varepsilon ^{j}} \gamma ^{\underline{1}} \varepsilon _j) \gamma _{\underline{1}} \eta ^i = \tfrac{3}{2} ({\varepsilon ^{j}} \eta ^i) \varepsilon _j - \tfrac{1}{2} ({\varepsilon ^{j}} \gamma ^{\underline{2}} \eta ^i) \gamma _{\underline{2}} \varepsilon _j \\&({\varepsilon ^{j}} \gamma ^{\underline{2}} \eta ^i) \gamma _{\underline{2}} \varepsilon _j = -\tfrac{5}{4} ({\varepsilon ^{j}} \gamma ^{\underline{1}} \varepsilon _j) \gamma _{\underline{1}} \eta ^i \end{aligned} \end{aligned}$$

(4.54)

where at $l=2$ the explicit coefficients in (4.41) are given as follows:

k	0	1	2	3	4	5	6
$\tilde{\Delta }(6,2,k)$	$-\tfrac{15}{8}$	$-\tfrac{5}{8}$	$-\tfrac{1}{8}$	$-\tfrac{3}{8}$	$+\tfrac{1}{8}$	$-\tfrac{5}{8}$	$+\tfrac{15}{8}^{}$

Hence, from the Eq. (4.54) we find another useful 6d Fierz identity

$$\begin{aligned} \boxed { ({\varepsilon ^{j}} \gamma ^{m} \varepsilon _j) \gamma _m \eta ^i = 4 ({\varepsilon ^{j}} \eta ^i) \varepsilon _j, \qquad (\varepsilon = \gamma _{+} \varepsilon , \quad \eta = \gamma _{-} \eta )} \end{aligned}$$

(4.55)

1.6 6d (0, 1) Theory Conventions

The spinor $\varepsilon$ is $+$-chiral, the spinor $\eta$ is $-$-chiral

$$\begin{aligned}&\varepsilon = \gamma _* \varepsilon = \gamma _{+} \varepsilon , \qquad&\eta = - \gamma _{*}\eta = \gamma _{-} \eta \end{aligned}$$

(4.56)

The tensor field $T_{\mu \nu a}$ is 6d anti-self-dual, $*_{6d} T = -T$. Useful contraction identities

(4.57)

(4.58)

(4.59)

The Bianchi identity on the field strength

$$\begin{aligned} \begin{aligned}&D_{m} F_{pq} + D_{q} F_{mp} + D_{p} F_{qm} = 0, \qquad \gamma ^{mpq} D_{m} F_{pq} = 0 \end{aligned} \end{aligned}$$

(4.60)

Positive chirality of $\varepsilon \equiv \varepsilon ^{+}$ and negative chirality of $T \equiv T^{-}$ implies

(4.61)

The spin-connection and the metric curvatures

$$\begin{aligned} \begin{aligned}&D_{\mu } v^{\hat{\rho }} = \partial _\mu v^{\hat{\rho }} + \omega ^{\hat{\rho }}{}_{\hat{\sigma }\mu } v^{\hat{\sigma }} \\&R^{\hat{\rho }}{}_{\hat{\sigma }\mu \nu } = [D_{\mu }, D_{\nu }]^{\hat{\rho }}_{\hat{\sigma }}, \qquad R_{\sigma \nu } = R^{\mu }{}_{\sigma \mu \nu }, \qquad R = R^{\mu }{}_{\mu } \end{aligned} \end{aligned}$$

(4.62)

The covariant derivative on spinors, the curvature and the Lichnerowicz formula

(4.63)

where $(V^{\mathrm {R}}_\mu )^{i}{}_{j}$ is the ${\textit{SU(2)}}_{\mathrm {R}}$-connection.

1.7 Supersymmetry Equations

The divergence of the first equation in the system (2.1) implies

(4.64)

which together with Lichnerowicz formula (4.63) produces

(4.65)

and the linear combination with the second equation in (2.1) produces

(4.66)

1.8 The 6d and 4d Spinor Conventions

As in (4.29) we take

$$\begin{aligned} \begin{aligned} \gamma _\mu ^{(6)} = \left( {\begin{matrix} \gamma _\mu ^{(4)} &{} 0 \\ 0 &{} - \gamma _{\mu }^{(4)} \end{matrix}}\right) ,\quad \gamma _5^{(6)} = \left( {\begin{matrix} 0 &{} 1 \\ 1 &{} 0 \end{matrix}}\right) ,\quad \gamma _6^{(6)} = \left( {\begin{matrix} 0 &{}-i \\ i &{} 0 \end{matrix}}\right) \\ \quad C^{(6)} = \left( {\begin{matrix} 0 &{} - c^{(4)}\gamma _{*}^{(4)} \\ - c^{(4)} \gamma _{*}^{(4)} &{} 0 \end{matrix}}\right) ,\quad \varepsilon ^{(6)}_{+} = \left( {\begin{matrix} \varepsilon ^{(4)}_{+} \\ \varepsilon ^{(4)}_{-} \end{matrix}}\right) ,\quad \eta ^{(6)}_{-} = \left( {\begin{matrix} \eta ^{(4)}_{-} \\ - \eta ^{(4)}_{+} \end{matrix}}\right) \end{aligned} \end{aligned}$$

(4.67)

where $\varepsilon _{\pm }^{(4)}$ denote the $\pm$-chiral spinors of the 4d Clifford algebra with respect to $\gamma _{*}^{(4)}$, the $C^{(6)}$ is the bilinear form for the 6d Clifford algebra of type $(-{}-++)$, and $c^{(4)}$ is the bilinear form of the 4d Clifford algebra of type $(-{}-++)$. In these conventions the bilinears computed in 4d and 6d notations agree:

$$\begin{aligned} \begin{aligned} \varepsilon _{+}^{(6)} C^{(6)} \eta _{-}^{(6)}= \varepsilon _{+}^{(4)} c^{(4)} \eta _{+}^{(4)} + \varepsilon _{-}^{(4)} c^{(4)} \eta _{-}^{(4)}\\ \varepsilon _{+}^{(6)} C^{(6)} \gamma _{\mu }^{(6)} \tilde{\varepsilon }_{+}^{(6)}= \varepsilon _{+}^{(4)} c^{(4)} \gamma _{\mu }^{(4)} \tilde{\varepsilon }_{-}^{(4)} + \varepsilon _{-}^{(4)} c^{(4)} \gamma _{\mu }^{(4)}\tilde{\varepsilon }_{+}^{(4)} \end{aligned} \end{aligned}$$

(4.68)

For the explicit form of spinors we use 4d gamma-matrices, the 4d chirality operator and 4d bilinear form in terms of (4.30)

$$\begin{aligned} \begin{aligned}&(\gamma _i, \gamma _4) = (\sigma _2 \otimes \sigma _i, \sigma _1 \otimes \sigma _0), \\&\gamma _{*}^{(4)} = -\gamma _1 \dots \gamma _4 = -\sigma _3 \otimes \sigma _0\\&c^{(4)} = -i \sigma _0 \otimes \sigma _2 \\ \end{aligned} \end{aligned}$$

(4.69)

We decompose

$$\begin{aligned} T_{\mu \nu a} \gamma ^{a} = T_{\mu \nu -} \gamma ^{-} + T_{\mu \nu +} \gamma ^{+} \end{aligned}$$

(4.70)

in terms of

$$\begin{aligned} \begin{aligned}&T_{\mu \nu -} = (T_{\mu \nu 5} - i T_{\mu \nu 6}) \qquad \gamma ^{-} =\tfrac{1}{2} (\gamma ^5 + i \gamma ^6) \qquad \gamma ^- \gamma ^{56}_* = - \gamma ^{-} \\&T_{\mu \nu +} = (T_{\mu \nu 5} + i T_{\mu \nu 6}) \qquad \gamma ^{+} = \tfrac{1}{2} (\gamma ^5 - i \gamma ^6) \qquad \gamma ^+ \gamma ^{56}_* = + \gamma _{+} \\ \end{aligned} \end{aligned}$$

(4.71)

with $\gamma ^{56}_* = - i \gamma _{56}$. Since $T_{\mu \nu a}$ is of negative 6d chirality, the $T_{\mu \nu \pm }$ has $\mp$ 4d chirality. We define

$$\begin{aligned} T_{\mu \nu }^{(4)} \equiv T_{\mu \nu +} - T_{\mu \nu -} = 2 \mathrm {i}T_{\mu \nu 6} \end{aligned}$$

(4.72)

In terms of the 4d spinors the generalized conformal Killing equation (2.1) takes form

$$\begin{aligned} D_{\mu } \varepsilon - \tfrac{1}{16} T^{(4)}_{\rho \sigma } \gamma ^{\rho \sigma } \gamma _\mu \varepsilon = \gamma _\mu \eta \end{aligned}$$

(4.73)

Other 6d-4d notational definitions are

$$\begin{aligned} \begin{aligned} \Phi ^{\pm }&= \tfrac{1}{2} (\Phi ^{5} \mp i \Phi ^{6}),&\quad \Phi _{\pm }&= (\Phi ^{5} \pm i \Phi ^{6})\\ T_{\mu \nu a} \gamma ^{\mu \nu } \Phi ^{a} \varepsilon _{+}^{(6)}&= T_{\mu \nu }^{(4)} \gamma ^{\mu \nu } (\Phi ^{+} \varepsilon _{-}^{(4)} - \Phi ^{-} \varepsilon _{+}^{(4)}), \quad&\Phi ^a \gamma _a \eta&= 2 \Phi ^{-} \eta _{-}^{(4)} - 2 \Phi ^{+} \eta _{+}^{(4)}\\ \end{aligned} \end{aligned}$$

(4.74)

Appendix 2: Supersymmetry Algebra

1.1 The Off-Shell Closure of the Supersymmetry on the Vector Multiplet

Here we explicitly compute $\delta ^2$ on vector multiplet for $\delta$ defined by:

(4.75)

provided that spinors $(\varepsilon , \eta )$ with satisfy generalized conformal Killing equations (2.3).

We find a contribution of several terms in $\delta _{}^2 \lambda ^i$. In the flat space we drop the terms proportional to $D_{\mu } \varepsilon$, $\eta$ and T and find

$$\begin{aligned} \delta _{\text {flat}}^2 \lambda ^i = \delta ( - \tfrac{1}{4} F_{mn} \gamma ^{mn} \varepsilon ^i) + \delta (Y^{i}_{\,\,\, j} \varepsilon ^{j}) \end{aligned}$$

(4.76)

with

(4.77)

and together with the $\delta ( Y^{i}_{\,\,\, j} \varepsilon ^{j})$ we find

(4.78)

Next we account for $D_{\mu } \varepsilon$ and $\eta$ terms, still keeping $T=0$. The transformation would be complete on conformally flat space. The $\delta ^2_{\text {cflat}} \lambda$ acquires new contributions

$$\begin{aligned} \delta ^2_{\text {cflat}} \lambda ^i = \delta ^2_{\text {flat}} \lambda ^i + \mathbf {term_c} \end{aligned}$$

(4.79)

where

$$\begin{aligned} \begin{aligned}&\mathbf {term_c} = - \tfrac{1}{4} ({\lambda } \gamma _q \gamma _\mu \eta ) \gamma ^{\mu q} \varepsilon ^i + \tfrac{1}{2} ({\lambda }\gamma _a \varepsilon ) \gamma ^a \eta ^i \mathop {=}\limits ^{ (4.53)\,\mathrm {on}\,\,\gamma _q{}\gamma ^q} \\&\qquad = -\tfrac{1}{2} ({\varepsilon } \gamma _q \lambda ) \gamma ^q \eta + \tfrac{1}{4} ({\eta } \gamma _{\mu q} \varepsilon ) \gamma ^{\mu q} \lambda + ({\eta } \varepsilon ) \lambda + ({\eta }\lambda ) \varepsilon \end{aligned} \end{aligned}$$

(4.80)

Then we expand the middle term in 4d indices

$$\begin{aligned} \tfrac{1}{4} ({\eta } \gamma _{\mu q} \varepsilon ) \gamma ^{\mu q} \lambda = + \tfrac{1}{8} ({\eta } \gamma _{\mu \nu } \varepsilon ) \gamma ^{\mu \nu } \lambda + \tfrac{1}{8} ({\eta } \gamma _{p q} \varepsilon ) \gamma ^{p q} \lambda - \tfrac{1}{8} ({\eta } \gamma _{ab} \varepsilon ) \gamma ^{ab} \lambda \end{aligned}$$

(4.81)

and apply Fierz identity (4.38) to the first and the last term in (4.80) to find

$$\begin{aligned} \begin{aligned} -\tfrac{1}{2} ({\varepsilon }^j \gamma _q \lambda _j) \gamma ^q \eta ^i&= - \tfrac{3}{4} ({\varepsilon }^j \eta ^i) \lambda _j + \tfrac{1}{8} ({\varepsilon }^j \gamma _{pq} \eta ^{j}) \gamma ^{pq} \lambda _j \\ ({\eta }^j \lambda _j) \varepsilon ^i&= \tfrac{1}{4} ({\eta }^{j} \varepsilon ^{i}) \lambda _j - \tfrac{1}{8} ({\eta }^{j} \gamma _{pq} \varepsilon ^{i}) \gamma ^{pq} \lambda _j \end{aligned} \end{aligned}$$

(4.82)

All $\gamma _{pq} \gamma ^{pq}$ terms are cancelled using (4.49) and the scalar terms are simplified as

$$\begin{aligned} ({\eta } \varepsilon ) \lambda - \tfrac{3}{4} ({\varepsilon }^j \eta ^i) \lambda _j+ \tfrac{1}{4} ({\eta }^{j} \varepsilon ^{i}) \lambda _j = \tfrac{3}{ 4} ({\eta } \varepsilon ) \lambda + ({\eta }^{(i} \varepsilon ^{j)}) \lambda _j \end{aligned}$$

(4.83)

and the contribution from the non-flat but conformally flat terms is

$$\begin{aligned} \mathbf {term_c} = + \tfrac{1}{8} ({\eta } \gamma _{\mu \nu } \varepsilon ) \gamma ^{\mu \nu } \lambda - \tfrac{1}{8} ({\eta } \gamma _{ab} \varepsilon ) \gamma ^{ab} \lambda + \tfrac{3}{ 4} ({\eta } \varepsilon ) \lambda + ({\eta }^{(i} \varepsilon ^{j)}) \lambda _j \end{aligned}$$

(4.84)

Then we compute the T-terms in

$$\begin{aligned} \delta ^2 \lambda ^i = \delta ^2_{\text {cflat}} \lambda ^i + \mathbf {term_T} \end{aligned}$$

(4.85)

and find

(4.86)

The $\mathbf {term_T}$ can be combined with the $\mathbf {term_c}$:

$$\begin{aligned} \tfrac{1}{8} (\eta \gamma _{\mu \nu } \varepsilon ) \gamma ^{\mu \nu } \lambda + \tfrac{1}{32} T_{\mu \nu a} ({\varepsilon } \gamma ^a \varepsilon ) \gamma ^{\mu \nu } \lambda = \tfrac{1}{16} D_{\mu } (\varepsilon \gamma _\nu \varepsilon ) \gamma ^{\mu \nu } \lambda \end{aligned}$$

(4.87)

so that finally

$$\begin{aligned} \delta _{\varepsilon ,\eta }^2 \lambda ^i = \tfrac{1}{4} ({\varepsilon }\gamma ^m \varepsilon ) D_m \lambda ^i + \tfrac{1}{16} D_{\mu } (\varepsilon \gamma _\nu \varepsilon ) \gamma ^{\mu \nu } \lambda ^i - \tfrac{1}{8} ({\eta } \gamma _{ab} \varepsilon ) \gamma ^{ab} \lambda ^i + \tfrac{3}{ 4} ({\eta } \varepsilon ) \lambda ^i + ({\eta }^{(i} \varepsilon ^{j)}) \lambda _j \end{aligned}$$

(4.88)

The variation $\delta ^2_{\varepsilon , \eta } Y^{ij}$ and $\delta ^2_{\varepsilon ,\eta } A_m$ are computed similarly.

1.2 The Invariance of the Lagrangian

The 4d $\mathcal {N}=2$ supersymmetric Lagrangian for vector multiplet in curved background for vanishing fermionic fields of Weyl multiplet is proportional to (2.11)

$$\begin{aligned} \tfrac{1}{2} F_{mn} F^{mn} + {\lambda }^i \gamma ^m D_{m} \lambda _i&+ (\tfrac{1}{6}R + M) \Phi _a \Phi ^a - 2 Y_{ij} Y^{ij} -F^{\mu \nu } T_{\mu \nu a} \Phi ^a \nonumber \\ {}&+ \tfrac{1}{4} T_{\mu \nu a} T^{\mu \nu b} \Phi ^a \Phi _b \end{aligned}$$

(4.89)

The trace is implicitly implied in all terms. To check the invariance under (2.8) we first consider the flat background with $D_{\mu } \varepsilon = 0, \eta = 0, T=0, M =0$. After that we will add the variational terms in conformally flat background, and finally we will add the remaining T-terms. We find modulo total derivative

(4.90)

that all terms add to zero. In conformally flat background the new terms appear in the variation of fermionic kinetic term and the coupling of scalars to the curvature

$$\begin{aligned} \begin{aligned}&\delta _{\text {cflat}} ((\tfrac{1}{6} R+M) \Phi _a \Phi ^a) = (\tfrac{1}{6}R + M) ({\lambda } \gamma ^a \Phi _a \varepsilon ) \\&\delta _{\text {cflat}} ( {\lambda } \gamma ^m D_m \lambda ) = \delta _{\text {flat}} ( {\lambda } \gamma ^m D_m \lambda ) + \mathbf {term_c} \end{aligned} \end{aligned}$$

(4.91)

where

(4.92)

so all terms in (4.91) cancel when added together.

Next we consider the remaining T-terms for a generic background. We set

$$\begin{aligned} \mathcal { L} = \mathcal {L}_{\text {cflat}} + \mathcal {L}_{T} \end{aligned}$$

(4.93)

where

$$\begin{aligned} \mathcal {L}_{T} = -F^{\mu \nu } T_{\mu \nu a} \Phi ^a + \tfrac{1}{4} T_{\mu \nu a} T^{\mu \nu b} \Phi ^a \Phi _b \end{aligned}$$

(4.94)

and we find

(4.95)

In the variation of the fermionic action the new terms are

$$\begin{aligned} \delta ( {\lambda } \gamma ^m D_m \lambda ) = \delta _{\text {cflat}} ( {\lambda } \gamma ^m D_m \lambda ) + \mathbf {term_{T1}} + \mathbf { term_{T2}} \end{aligned}$$

(4.96)

where $\mathbf {term_{T1}}$ comes from T-terms in generalized conformal Killing equation (2.3) and $\mathbf {term_{T2}}$ comes from the T-term in the variation $\delta _{\varepsilon , \eta } \lambda$ (2.8)

(4.97)

Then we find

(4.98)

and

(4.99)

Using (4.61) all TF terms cancel between (4.99) and (4.95) and finally the $[DT]\Phi$ terms in (4.95), (4.97), (4.99) cancel as well as

$$\begin{aligned} \begin{aligned}&(4.95): \qquad (\lambda \gamma ^{\nu } \varepsilon ) [D^{\mu } T_{\mu \nu a}] \Phi ^a \\&(4.97): \qquad \tfrac{1}{2} (D^{\mu } T_{\mu \nu a} ) \Phi _b ({\lambda } \gamma ^{b} \gamma ^{\nu a} \varepsilon ) = - \tfrac{1}{2} ({\lambda } \gamma ^{\nu } \varepsilon ) D^{\mu } T_{\mu \nu a} \Phi ^a + \tfrac{1}{2} ({\lambda } \gamma ^{\nu ab} \varepsilon ) \Phi ^b D^{\mu } T_{\mu \nu a} \\&(4.99): \qquad -\tfrac{1}{4} {\lambda } D_{\rho } (T_{\mu \nu a}) \Phi ^a \gamma ^{\rho } \gamma ^{\mu \nu } \varepsilon = - \tfrac{1}{2} ({\lambda } \gamma ^{\mu } \varepsilon ) D^{\mu } T_{\mu \nu a} \Phi ^a - \tfrac{1}{4} ({\lambda } \gamma ^{\mu \nu \rho } \varepsilon ) \Phi ^a D_{\rho } T_{\mu \nu a} \end{aligned} \end{aligned}$$

(4.100)