April 01, 2024
One of the lessons from the recent development in black hole microstate counting, [1], was that the Bekenstein-Hawking entropy can be obtained by extremizing an off-shell quantity known as the entropy function. This idea was turned out to be universal for any observables dual to partition functions of superconformal field theories in diverse dimensions by the AdS/CFT correspondence, [2]. The expressions of these off-shell quantities, namely the”gravitational blocks", [3], were conjectured and proved to reproduce the corresponding observables, \(e.g.\), the Bekenstein-Hawking entropies, central charges, and free energies. Later, the gravitational blocks for solutions with orbifold singularities, \(i.e.\), spindle, were proposed, [4], [5], and proved to calculate the duals of partition functions, [6]–[11]. See [12], [13] also for an approach from string and M-theory.
Recently, for the supergravity solutions with an R-symmetry Killing vector, it was shown that observables dual to partition functions in field theory can be calculated by integrating equivariantly closed forms which are constructed by spinor bilinears of Killing spinors, [14]–[19]. The Berline-Vergne-Atiyah-Bott fixed point formula, [20], [21], was utilized to perform the integrations. In particular, in [19], this method of equivariant localization was applied to gauged supergravity theories and served as a derivation of gravitational blocks for four- and five-dimensional gauged supergravity. In this work, we apply the method of equivariant localization to six- and seven-dimensional gauged supergravity and derive the gravitational blocks for corresponding theories in [5].
In [19], in order to construct the equivariantly closed forms from spinor bilinears of Killing spinors, the structure of respective gauged supergravity theory was exploited, \(e.g.\), the very special geometry of five-dimensional gauged supergravity, [22], and the \(\mathcal{N}=2\) formalism of four-dimensional gauged supergravity. As a foundation of construction, we look out for structures and reveal the structure of very special geometry in six- and seven-dimensional guaged supergravity coupled to a vector multiplet. To the best of our knowledge, it is the first time to formulate six- and seven-dimensional gauged supergravity in the very special geometry. Utilizing the structure of very special geometry, we construct the equivariantly closed forms for the general ansatz of \(AdS_5\times{M}_2\) and \(AdS_4\times{M}_2\) where a Killing vector is assumed to be on a two-dimensional surface, \(M_2\). We derive the gravitational blocks proposed in [5] by performing the integrations of equivariantly closed forms via the Berline-Vergne-Atiyah-Bott fixed point formula. To be specific, we study the cases of \(M_2\) to be a Riemann surface with topological twist and a spindle with two classes of twists.
We closely follow the presentation of [19] as it is well organized and to promote comparison of the studies in diverse dimensions.
In section 2, we construct the equivariantly closed forms and derive the gravitational blocks for M5-branes on a two-dimensional surface, \(M_2\). In section 3, we construct the equivariantly closed forms and derive the gravitational blocks for D4-branes on a two-dimensional surface, \(M_2\). We conclude with some outlook in section 4. The construction of spinor bilinears are presented in appendix 5 and 6.
We consider \(U(1)^2\)-gauged supergravity in seven dimensions, [23], which is a consistent truncation of maximal gauged supergravity in seven dimensions, [24].
In order to study equivariant localization, it is convenient to use the geometric structure of model, [19]. In this section, we present \(U(1)^2\)-gauged supergravity in seven dimensions in the structure of very special geometry, [22].
The bosonic action is given by1 \[\label{sixtheory7} S_7\,=\,\frac{1}{16\pi{G}_N^{(7)}}\int_{M_7}\left(R_7-\mathcal{V}-\mathcal{G}_{ij}\partial_\mu\varphi^i\partial^\mu\varphi^j-\frac{1}{2}G_{IJ}F_{\mu\nu}^IF^{J\mu\nu}\right)\text{vol}_7\,,\tag{1}\] in the mostly plus signature. \(R_7\) is the Ricci scalar and \(\text{vol}_7\) is the volume form. We have two real scalar fields, \(\varphi^1\) and \(\varphi^2\). We introduce the following parametrization of the scalar fields, \[X^I\,=\,\left(e^{\frac{4}{\sqrt{10}}\varphi_2},e^{-\frac{1}{\sqrt{2}}\varphi_1-\frac{1}{\sqrt{10}}\varphi_2},e^{\frac{1}{\sqrt{2}}\varphi_1-\frac{1}{\sqrt{10}}\varphi_2}\right)\,.\] There are two \(U(1)\) gauge fields, \(A^1\) and \(A^2\), and we have \[\label{gaugefd7} A^I\,=\,\left(A^0\equiv0,A^1,A^2\right)\,,\tag{2}\] where, in order to formulate the very special geometry, we have introduced a trivial gauge field, \(A^0\). The metric for the kinetic terms of scalar fields and gauge fields are given, respectively, by \[\mathcal{G}_{ij}\,=\,\frac{1}{2}\text{diag}\left(1,1\right)\,,\] and \[G_{IJ}\,=\,\text{diag}\left(\frac{1}{4\left(X^0\right)^2},\frac{1}{2\left(X^1\right)^2},\frac{1}{2\left(X^2\right)^2}\right)\,.\] The scalar potential is given by \[\mathcal{V}\,\equiv\,\xi_I\xi_J\left(\mathcal{G}^{ij}\partial_iX^I\partial_jX^J-\frac{6}{5}X^IX^J\right)\,,\] where \(\partial_I\equiv\partial_{X^I}\) and \(\partial_i\equiv\partial_{\varphi^i}\). We have the Fayet-Iliopoulos (FI) gauging parameters, \[\xi_I\,=\,\frac{1}{2}\left(m,g,g\right)\,,\] where \(g\) and \(m\) are gauge coupling constant and mass parameter, respectively. In order to have the supersymmetric vacuum, it is required to be \(g=2m\).
The expressions are neatly packaged by introducing a prepotential which satisfies the constraint, \[\label{prepotential7} \mathcal{F}\left(X^I\right)\,\equiv\,X^0\left(X^1X^2\right)^2\,=\,1\,.\tag{3}\] The metrics are obtained from \[G_{IJ}\,=\,-\frac{1}{4}\partial_I\partial_J\log\mathcal{F}\Big|_{\mathcal{F}=1}\,, \qquad \mathcal{G}_{ij}\,=\,\partial_iX^I\partial_jX^JG_{IJ}\Big|_{\mathcal{F}=1}\,,\] and the indices can be raised and lowered by \[X_I\,\equiv\,G_{IJ}X^J\,, \qquad \partial_iX_I\,=\,-G_{IJ}\partial_iX^J\,.\] We also find an expression, \[\label{GIJGIJ7} G^{IJ}\,=\,\mathcal{G}^{ij}\partial_iX^I\partial_jX^J+\frac{4}{5}X^IX^J\,,\tag{4}\] for the inverse of \(G_{IJ}\). Furthermore, we have \(X^IX_I=5/4\).
The supersymmetry variations of the gravitino and gaugino fields are given, respectively, by \[\begin{align} \label{susyvar117} 0\,=&\,\left[\nabla_\mu-\frac{i}{2}\xi_IA^I_\mu+\frac{1}{10}W\Gamma_\mu+\frac{i}{20}X_IF^I_{\nu\rho}\left(\Gamma_\mu\,^{\nu\rho}-8\delta_\mu^\nu\Gamma^\rho\right)\right]\epsilon\,, \notag \\ 0\,=&\,\left[-\frac{i}{2}\mathcal{G}_{ij}\partial_\mu\varphi^j\Gamma^\mu+\frac{i}{2}\partial_iW+\frac{1}{4}\partial_iX_IF^I_{\mu\nu}\Gamma^{\mu\nu}\right]\epsilon\,, \end{align}\tag{5}\] where \(\Gamma_\mu\) are Cliff(1,6) generators. The superpotential is given by \[W\,\equiv\,\xi_IX^I\,,\] and the scalar potential can be expressed in terms of the superpotential, \[\label{potsup7} \mathcal{V}\,=\,\mathcal{G}^{ij}\partial_iW\partial_jW-\frac{6}{5}W^2\,.\tag{6}\] The supersymmetric \(AdS_7\) vacuum of the scalar potential is dual to six-dimensional \(\mathcal{N}=(2,0)\) superconformal field theories.
We consider the following background, [19], \[\label{ads4m2met7} ds_7^2\,=\,e^{2\lambda}\left[ds_{AdS_5}^2+ds_{M_2}^2\right]\,,\tag{7}\] where \(ds^2_{AdS_5}\) is a metric on \(AdS_5\) of unit radius. The scalar fields and \(\lambda\) are functions on \(M_2\) and \(A^I\) are gauge fields on \(M_2\).
We dimensionally reduce the equations of motion from the action in 1 on the background, 7 , and find that the reduced equations of motion can be obtained from the variations of following two-dimensional action, \(S_7=\frac{\text{vol}_{AdS_5}}{16\pi{G}_N^{(7)}}S_2\), \[\begin{align} \label{2daction7} S_2\,=\,\left.\int_{M_2}\right[&e^{5\lambda}\left(R-20+30\left(\nabla\lambda\right)^2-\mathcal{G}_{ij}\partial_\mu\varphi^i\partial^\mu\varphi^j\right)-e^{7\lambda}\mathcal{V} \notag \\ &-\left.\frac{1}{2}e^{3\lambda}G_{IJ}F_{\mu\nu}^IF^{J\mu\nu}\right]\text{vol}\,, \end{align}\tag{8}\] where \(R\), \(\nabla\) and vol are the Ricci scalar, Levi-Civita connection and volume form on \(M_2\). The Maxwell equations are \[\label{maxwell7} d\left[e^{3\lambda}G_{IJ}\left(*F^J\right)\right]\,=\,0 \qquad \Longrightarrow \qquad d\left[e^{3\lambda}G_{IJ}F_{12}^J\right]\,=\,0\,, \qquad \forall I\,=\,0,1,2\,,\tag{9}\] where we have \(F^I\,=\,F^I_{12}\text{vol}\) with \(F_{12}^I\) a function on \(M_2\). For the warp factor, \(\lambda\), the equation of motion is \[\begin{align} \label{warpeom7} e^{5\lambda}\left[R-20+30\left(\nabla\lambda\right)^2-\mathcal{G}_{ij}\partial_\mu\varphi^i\partial^\mu\varphi^j\right]\,=&\,\frac{7}{5}e^{7\lambda}\mathcal{V}+\frac{3}{10}e^{3\lambda}G_{IJ}F_{\mu\nu}^IF^{J\mu\nu} \notag \\ &+\text{total\,\,derivative}\,. \end{align}\tag{10}\] The trace of Einstein equation is \[\label{traceeinstein7} e^{7\lambda}\mathcal{V}-\frac{1}{2}e^{3\lambda}G_{IJ}F_{\mu\nu}^{I}F^{J\mu\nu}\,=\,-20e^{5\lambda}+\text{total\,\,derivative}\,.\tag{11}\]
From 8 the partially off-shell (POS) action is obtained by imposing 10 , \[S_2|_\text{POS}\,=\,\frac{2}{5}\int_{M_2}\left(e^{7\lambda}\mathcal{V}\text{vol}-e^{3\lambda}G_{IJ}F_{12}^IF^J\right)\,.\] Further imposing 11 we find the on-shell action, \[S_2|_\text{OS}\,=\,-8\int_{M_2}e^{5\lambda}\text{vol}\,.\]
We introduce the Killing spinor on the background, 7 , for the supersymmetry variations, 5 , [19], \[\epsilon\,=\,\vartheta\otimes{e}^{\lambda/2}\zeta\,,\] where \(\vartheta\) is a Killing spinor on \(AdS_5\), \(\zeta\) is a spinor on \(M_2\) and the warp factor is introduced for convenience. Then, as presented in appendix 5, we obtain the supersymmetry equations, 56 .
We introduce the spinor bilinears in \(\zeta\) on \(M_2\), [19], \[\label{sbil7} S\,=\,\zeta^\dagger\zeta\,, \quad P\,=\,\zeta^\dagger\gamma_3\zeta\,, \quad K\,=\,\zeta^\dagger\gamma_{(1)}\zeta\,, \quad \xi^\flat\,=\,-i\zeta^\dagger\gamma_{(1)}\gamma_3\zeta\,,\tag{12}\] where we have \(\gamma_{(n)}=\frac{1}{n!}\gamma_{\mu_1\cdots\mu_n}dx^{\mu_1}\wedge\cdots\wedge{d}x^{\mu_n}\) and the chirality operator on \(M_2\) is \(\gamma_3=-i\gamma_1\gamma_2\). We obtain the spinor bilinear equations in 57 . We find that the Killing vector, \(\xi\), on \(M_2\) is dual to the one-form, \(\xi^\flat\), and we have \[d\xi^\flat\,=\,-2\left(4+PS^{-1}e^\lambda{W}\right)P\text{vol}\,.\]
We found equivariantly closed forms under \(d_\xi=d-\xi\mathbin{\lrcorner}\) in appendix 5, [19], \[\label{phiF7} \Phi^{F^I}\,\equiv\,F^I-X^Ie^\lambda{P}\,,\tag{13}\] and \[\Phi^\text{vol}\,\equiv\,e^{7\lambda}\mathcal{V}\text{vol}-e^{6\lambda}WS\,.\] By employing the Maxwell equations in 36 , we obtain another equivariantly closed form, \[\begin{align} \label{phiS7} \Phi^S\,\equiv&\,\frac{2}{5}\left(\Phi^\text{vol}-e^{3\lambda}{G}_{IJ}F^I_{12}\Phi^{F^I}\right) \notag \\ =&\,\frac{2}{5}\left(e^{7\lambda}\mathcal{V}\text{vol}-e^{3\lambda}G_{IJ}F^I_{12}F^J-e^{6\lambda}WS+e^{4\lambda}F^I_{12}X^JP\right)\,, \end{align}\tag{14}\] where the two-forms are the two-dimensional action in 8 .
As we note that the scalar bilinear, \(S\), is a constant in appendix 5, we choose the normalization to be \(S\equiv1\).
For the spinor bilinears in 12 , we find \(||\xi||^2=||K||^2=S^2-P^2\). At a fixed point, \(p\), where we have \(\xi|_p=0\), we obtain \(P|_p=\pm{S}|_p=\pm1\), and the sign is the chirality of the spinor, \(\zeta\), at the fixed point, \(p\). For \(\xi\ne0\), around the fixed point, \(p\), we may have \(M_2\) to be \(\mathbb{R}^2/\mathbb{Z}_{d_p}\) with orbifold singularities and \(\xi=\epsilon_p\partial_{\phi_p}\) where \(\phi_p\) is a polar coordinate of period, \(2\pi/d_p\). Let us name \(\epsilon_p\) to be the weight of \(\xi\) at the fixed point, \(p\), [19].
Now we employ the Berline-Vergne-Atiyah-Bott fixed point formula, [20], [21], to perform the integrations of equivariantly closed forms in 13 and 14 on \(M_2\), [19]. The magnetic charges are given by \[\label{magch7} \mathfrak{p}^I\,\equiv\,\int_{M_2}\frac{F^I}{2\pi}\,=\,\int_{M_2}\frac{\Phi^{F^I}}{2\pi}\,=\,-\frac{1}{2\pi}\sum_{\text{fixed}\,p}\frac{1}{d_p}\frac{2\pi}{\epsilon_p}\left(X^Ie^\lambda{P}\right)\Big|_p\,,\tag{15}\] and the two-dimensional action is \[\begin{align} \label{twoac7} S_2|_\text{POS}\,=\,\int_{M_2}\Phi^S\,=&\,-\frac{2}{5}\sum_{\text{fixed}\,p}\frac{1}{d_p}\frac{2\pi}{\epsilon_p}\left(e^{6\lambda}WS-e^{4\lambda}G_{IJ}F_{12}^IX^JP\right)\Big|_p \notag \\ =&\,2\sum_{\text{fixed}\,p}\frac{1}{d_p}\frac{2\pi}{\epsilon_p}e^{5\lambda}P\Big|_p\,, \end{align}\tag{16}\] where we employed the algebraic constraint in 59 in the last line and \(P^2|_p=S^2|_p=1\). As we have \(d\xi^\flat|_p=2\epsilon_p\text{vol}\) at a fixed point, we find \[\label{dxidxi7} e^\lambda{W}P|_p\,=\,-\left(4+\epsilon_pP_p\right)\,,\tag{17}\] where we used \(P_p\,=\,\pm1\).
We introduce the new scalar fields, \[\label{newsc7} x^I\,\equiv\,-X^Ie^\lambda{P} \qquad \Longrightarrow \qquad \Phi^{F^I}\,=\,F^I+x^I\,.\tag{18}\] With \(\mathcal{F}\left(X^I\right)=1\) in 3 , we obtain \[\mathcal{F}\left(x^I\right)|_p\,=\,x^0\left(x^1x^2\right)^2|_p\,=\,-e^{5\lambda}P|_p\,,\] where we employed \(P^2|_p=1\). Then 16 , 15 and 17 are given, respectively, by \[\label{constraint117} S_2|_\text{POS}\,=\,-2\sum_{\text{fixed}\,p}\frac{2\pi}{d_p\epsilon_p}\mathcal{F}\left(x_p^I\right)\,, \qquad \mathfrak{p}^I\,=\,\sum_{\text{fixed}\,p}\frac{1}{d_p\epsilon_p}x_p^I\,, \qquad \xi_Ix_p^I\,=\,4+\epsilon_pP_p\,,\tag{19}\] where we introduced \(x_p^I\equiv{x}^I|_p\).
Topologically a spindle is the only choice for \(M_2\) if it is compact without boundary and has a \(U(1)\) isometry with fixed points, [25]. The spindle has two fixed points, \(p=\pm\), and we choose \(d_\pm=n_\pm\) with \(n_\pm\ge1\). The Killing vector field on the spindle is given by \[\xi\,=\,b_0\partial_\varphi\,,\] where \(\varphi\) is an azimuthal coordinate on the spindle with period of \(2\pi\). We can write \(\epsilon_+=-b_0/n_+\), \(\epsilon_-=b_0/n_-\). \(P_\pm\) is arbitrary and \(|P_\pm|=1\). Then the constraints in 19 are given by \[\label{constraints227} \mathfrak{p}^I\,=\,-\frac{1}{b_0}\left(x_+^I-x_-^I\right)\,, \qquad \xi_Ix_+^I\,=\,4-\frac{b_0}{n_+}P_+\,, \qquad \xi_Ix_-^I\,=\,4+\frac{b_0}{n_-}P_-\,,\tag{20}\] and, from these, we also obtain \[\label{magneticcon7} \xi_I\mathfrak{p}^I\,=\,\frac{P_+}{n_+}+\frac{P_-}{n_-}\,=\,\frac{n_-P_++n_+P_-}{n_-n_+}\,.\tag{21}\] If we introduce \(\sigma\equiv{P}_+/P_-\), \(\sigma=+1\) is for the case of equal chirality of spinor at two poles of spindle, namely twist. \(\sigma=-1\) is for the opposite chiralities, namely anti-twist, [26]. The off-shell action is given by \[\label{S2pos227} S_2|_\text{POS}\,=\,\frac{4\pi}{b_0}\left[\mathcal{F}\left(x_+^I\right)-\mathcal{F}\left(x_-^I\right)\right]\,.\tag{22}\]
We introduce a new parametrization for the scalar fields with magnetic charges, \[\label{phi0127} \phi_0\,\equiv\,\frac{m}{2}\left(x_\pm^0\pm\mathfrak{p}^0\frac{b_0}{2}\right)\,, \qquad \phi_1\,\equiv\,\frac{g}{2}\left(x_\pm^1\pm\mathfrak{p}^1\frac{b_0}{2}\right)\,, \qquad \phi_2\,\equiv\,\frac{g}{2}\left(x_\pm^2\pm\mathfrak{p}^2\frac{b_0}{2}\right)\,.\tag{23}\] As the gauge field, \(A^0\), is identically zero, 2 , we have \(\mathfrak{p}^0=0\). Accordingly, we choose \(\phi_0=2\). Namely, we have \[\phi_0\,=\,2\,, \qquad \mathfrak{p}^0\,=\,0\,.\] For the choice of \(\phi_0=2\), from 23 and 18 , we find \[2\,=\,\frac{m}{2}\left(-X^0e^\lambda{P}\right)|_\pm\,.\] As \(X^0>0\) and \(e^\lambda>0\), only the solutions in the twist class are allowed, \(P_\pm=-1\), and it is consistent with [26]. We further introduce new parameters for the rest of magnetic charges and \(b_0\), \[\epsilon\,\equiv\,\frac{b_0}{2}\,, \qquad \mathfrak{n}_1\,\equiv\,\frac{g}{2}\mathfrak{p}^1\,, \qquad \mathfrak{n}_2\,\equiv\,\frac{g}{2}\mathfrak{p}^2\,.\] Then, from the constraints in 20 , using the new parameters, we find \[\begin{align} \label{gbcon7} \mathfrak{n}_1+\mathfrak{n}_2\,=&\,\frac{n_++\sigma{n}_-}{n_-n_+}\,, \notag \\ \phi_1+\phi_2-\frac{n_+-\sigma{n}_-}{n_-n_+}\epsilon\,=&\,2\,, \end{align}\tag{24}\] where we keep \(\sigma\equiv{P}_+/P_-\) arbitrary. These are precisely the constraints on the magnetic charges and the variables, \(\phi_i\), for M5-branes wrapped on a spindle in section 5.4 of [5].2 The prepotential is then given by \[\mathcal{F}\left(x_\pm^I\right)\,=\,\frac{4^3}{g^4m}\Big(\left(\phi_1\mp\mathfrak{n}_1\epsilon\right)(\left(\phi_2\mp\mathfrak{n}_2\epsilon\right)\Big)^2\,.\] From 22 , the two-dimensional off-shell action is \[S_2|_\text{POS}\,=\,-\frac{2\pi}{\epsilon}\frac{4^3}{g^3m}\left[\Big(\left(\phi_1+\mathfrak{n}_1\epsilon\right)(\left(\phi_2+\mathfrak{n}_2\epsilon\right)\Big)^2-\Big(\left(\phi_1-\mathfrak{n}_1\epsilon\right)(\left(\phi_2-\mathfrak{n}_2\epsilon\right)\Big)^2\right]\,.\] From this, we recover the gravitational block in [5], \[\label{offfree7} F^-\left(\phi_i,\epsilon;\mathfrak{n}_i\right)\,=\,\frac{9g^4m}{2\,4^7\pi}N^3S_2|_\text{POS}\,.\tag{25}\] By extremizing the gravitational block or the off-shell action subject to the constraints, 24 , we find, [5], \[\begin{align} \label{phistar7} \phi_1^*\,=&\,1+\frac{\left(\mathtt{s}+\mathfrak{n}_1+\mathfrak{n}_2\right)\left[2n_+^2n_-^2\left(\mathfrak{n}_1^2-\mathfrak{n}_2^2\right)+3\left(n_+-\sigma{n}_-\right)^2+n_+^2n_-^2\left(\mathfrak{n}_1-\mathfrak{n}_2\right)\mathtt{s}\right]}{12n_+n_-\left(\sigma-n_+n_-\mathfrak{n}_1\mathfrak{n}_2\right)\left[\mathtt{s}+2\left(\mathfrak{n}_1+\mathfrak{n}_2\right)\right]}\,, \notag \\ \epsilon^*\,=&\,\frac{\left(n_+-\sigma{n}_-\right)\left(\mathtt{s}+\mathfrak{n}_1+\mathfrak{n}_2\right)}{2\left(\sigma-n_+n_-\mathfrak{n}_1\mathfrak{n}_2\right)\left[\mathtt{s}+2\left(\mathfrak{n}_1+\mathfrak{n}_2\right)\right]}\,, \end{align}\tag{26}\] where we have defined \[\mathtt{s}\,\equiv\,\sqrt{7\left(\mathfrak{n}_1^2+\mathfrak{n}_2^2\right)+2\mathfrak{n}_1\mathfrak{n}_2-6\frac{n_+^2+n_-^2}{n_+^2n_-^2}}\,.\] Inserting \(\phi_1^*\) and \(\epsilon^*\) in 25 , we obtain the central charge, \(a\), of dual four-dimensional superconformal field theories, [5], [25], \[F^-\left(\phi_i^*,\epsilon^*;\mathfrak{n}_i\right)\,=\,\frac{3N^3}{8}\frac{\mathfrak{n}_1^2\mathfrak{n}_2^2\left(\mathtt{s}+\mathfrak{n}_1+\mathfrak{n}_2\right)}{\left(\frac{\sigma}{n_+n_-}-\mathfrak{n}_1\mathfrak{n}_2\right)\left[\mathtt{s}+2\left(\mathfrak{n}_1+\mathfrak{n}_2\right)\right]^2}\,,\] for the solutions in the twist class, \(\sigma=+1\).
By plugging \(x_-^I=x_+^I+b_0\mathfrak{p}^I\) from 20 into 22 with \(\sigma=+1\), we note that \[\label{spherelim7} \lim_{b_0\rightarrow0}S_2|_\text{POS}\,=\,-4\pi\mathfrak{p}^I\frac{\partial}{\partial{x}^I}\mathcal{F}\left(x^I\right)\,.\tag{27}\] We have \(x_-^I=x_+^I=x^I\) to be constant on \(M_2\). From 21 we find \(\xi_Ix^I=2\). We can extremize 27 with the constraint, \(\xi_Ix^I=2\). In particular, for \(n_+=n_-=1\), as already noticed in [5], we should find the central charge obtained in [27]–[29] for \(AdS_5\times\Sigma_\mathfrak{g}\) solutions with a topological twist on the Riemann surface of genus, \(\mathfrak{g}\), and we have \(\xi_I\mathfrak{p}^I=2P_+\) and \(\sigma=+1\).
We consider \(F(4)\) gauged supergravity, [30], coupled to a vector multiplet, [31], in six dimensions. The model was first presented in [32] and the action was explicitly given in [33]. For the parametrizations and conventions of the model, we follow [5].
In order to study equivariant localization, it is convenient to use the geometric structure of the model, [19]. In this section, we present \(U(1)^2\)-gauged supergravity in six dimensions in the structure of very special geometry, [22].
The bosonic action is given by \[\label{sixtheory} S_6\,=\,\frac{1}{16\pi{G}_N^{(6)}}\int_{M_6}\left(R_6-\mathcal{V}-\mathcal{G}_{ij}\partial_\mu\varphi^i\partial^\mu\varphi^j-\frac{1}{2}G_{IJ}F_{\mu\nu}^IF^{J\mu\nu}\right)\text{vol}_6\,,\tag{28}\] in the mostly plus signature. \(R_6\) is the Ricci scalar and \(\text{vol}_6\) is the volume form. We have two real scalar fields, \(\sigma\) and \(\varphi_2\), \[\varphi^i\,=\,\left(\varphi_1\equiv\sigma,\varphi_2\right)\,.\] We introduce the following parametrization of the scalar fields, \[X^I\,=\,\left(e^{-3\sigma},e^{\sigma+\varphi_2},e^{\sigma-\varphi_2}\right)\,.\] There are two \(U(1)\) gauge fields, \(A^1\) and \(A^2\), and we have \[\label{gaugefd} A^I\,=\,\left(A^0\equiv0,A^1,A^2\right)\,,\tag{29}\] where, in order to formulate the very special geometry, we have introduced a trivial gauge field, \(A^0\). The metric for the kinetic terms of scalar fields and gauge fields are given, respectively, by \[\mathcal{G}_{ij}\,=\,\text{diag}\left(4,1\right)\,,\] and \[G_{IJ}\,=\,\text{diag}\left(\frac{1}{3\left(X^0\right)^2},\frac{1}{2\left(X^1\right)^2},\frac{1}{2\left(X^2\right)^2}\right)\,.\] The scalar potential is given by \[\mathcal{V}\,\equiv\,\xi_I\xi_J\left(\mathcal{G}^{ij}\partial_iX^I\partial_jX^J-\frac{5}{4}X^IX^J\right)\,,\] where \(\partial_I\equiv\partial_{X^I}\) and \(\partial_i\equiv\partial_{\varphi^i}\). We have the Fayet-Iliopoulos (FI) gauging parameters, \[\xi_I\,=\,\left(m,g,g\right)\,,\] where \(g\) and \(m\) are gauge coupling constant and mass parameter, respectively. In order to have the supersymmetric vacuum, it is required to be \(g=3m\).
The expressions are neatly packaged by introducing a prepotential which satisfies the constraint, \[\label{prepotential} \mathcal{F}\left(X^I\right)\,\equiv\,X^0\left(X^1X^2\right)^{3/2}\,=\,1\,.\tag{30}\] The metrics are obtained from \[G_{IJ}\,=\,-\frac{1}{3}\partial_I\partial_J\log\mathcal{F}\Big|_{\mathcal{F}=1}\,, \qquad \mathcal{G}_{ij}\,=\,\partial_iX^I\partial_jX^JG_{IJ}\Big|_{\mathcal{F}=1}\,,\] and the indices can be raised and lowered by \[X_I\,\equiv\,G_{IJ}X^J\,, \qquad \partial_iX_I\,=\,-G_{IJ}\partial_iX^J\,.\] We also find an expression, \[\label{GIJGIJ} G^{IJ}\,=\,\mathcal{G}^{ij}\partial_iX^I\partial_jX^J+\frac{3}{4}X^IX^J\,,\tag{31}\] for the inverse of \(G_{IJ}\). Furthermore, we have \(X^IX_I=4/3\).
The supersymmetry variations of the gravitino and gaugino fields are given, respectively, by \[\begin{align} \label{susyvar11} 0\,=&\,\left[\nabla_\mu-\frac{i}{2}\xi_IA^I_\mu+\frac{1}{8}W\Gamma_\mu+\frac{i}{16}X_IF^I_{\nu\rho}\left(\Gamma_\mu\,^{\nu\rho}-6\delta_\mu^\nu\Gamma^\rho\right)\right]\epsilon\,, \notag \\ 0\,=&\,\left[-\frac{i}{2}\mathcal{G}_{ij}\partial_\mu\varphi^j\Gamma^\mu+\frac{i}{2}\partial_iW+\frac{1}{4}\partial_iX_IF^I_{\mu\nu}\Gamma^{\mu\nu}\right]\epsilon\,, \end{align}\tag{32}\] where \(\Gamma_\mu\) are Cliff(1,5) generators. The superpotential is given by \[W\,\equiv\,\xi_IX^I\,,\] and the scalar potential can be expressed in terms of the superpotential, \[\label{potsup} \mathcal{V}\,=\,\mathcal{G}^{ij}\partial_iW\partial_jW-\frac{5}{4}W^2\,.\tag{33}\]
The supersymmetric \(AdS_6\) vacuum of the scalar potential, which uplifts to the \(AdS_6\times\widetilde{S}^4\) solution, [34], is dual to a class of five-dimensional superconformal field theories, [35], and the free energy is, [36], \[\mathcal{F}_{S^5}\,=\,-\frac{\pi^2}{3}\frac{1}{G_N^{(6)}}\,=\,-\frac{9\sqrt{2}\pi}{5}\frac{N^{5/2}}{\sqrt{8-N_f}}\,.\]
We consider the following background, [19], \[\label{ads4m2met} ds_6^2\,=\,e^{2\lambda}\left[ds_{AdS_4}^2+ds_{M_2}^2\right]\,,\tag{34}\] where \(ds^2_{AdS_4}\) is a metric on \(AdS_4\) of unit radius. The scalar fields and \(\lambda\) are functions on \(M_2\) and \(A^I\) are gauge fields on \(M_2\).
We dimensionally reduce the equations of motion from the action in 28 on the background, 34 , and find that the reduced equations of motion can be obtained from the variations of following two-dimensional action, \(S_6=\frac{\text{vol}_{AdS_4}}{16\pi{G}_N^{(6)}}S_2\), \[\begin{align} \label{2daction} S_2\,=\,\left.\int_{M_2}\right[&e^{4\lambda}\left(R-12+20\left(\nabla\lambda\right)^2-\mathcal{G}_{ij}\partial_\mu\varphi^i\partial^\mu\varphi^j\right)-e^{6\lambda}\mathcal{V} \notag \\ &-\left.\frac{1}{2}e^{2\lambda}G_{IJ}F_{\mu\nu}^IF^{J\mu\nu}\right]\text{vol}\,, \end{align}\tag{35}\] where \(R\), \(\nabla\) and vol are the Ricci scalar, Levi-Civita connection and volume form on \(M_2\). The Maxwell equations are \[\label{maxwell} d\left[e^{2\lambda}G_{IJ}\left(*F^J\right)\right]\,=\,0 \qquad \Longrightarrow \qquad d\left[e^{2\lambda}G_{IJ}F_{12}^J\right]\,=\,0\,, \qquad \forall I\,=\,0,1,2\,,\tag{36}\] where we have \(F^I\,=\,F^I_{12}\text{vol}\) with \(F_{12}^I\) a function on \(M_2\). For the warp factor, \(\lambda\), the equation of motion is \[\begin{align} \label{warpeom} e^{4\lambda}\left[R-12+20\left(\nabla\lambda\right)^2-\mathcal{G}_{ij}\partial_\mu\varphi^i\partial^\mu\varphi^j\right]\,=&\,\frac{3}{2}e^{6\lambda}\mathcal{V}+\frac{1}{4}e^{2\lambda}G_{IJ}F_{\mu\nu}^IF^{J\mu\nu} \notag \\ &+\text{total\,\,derivative}\,. \end{align}\tag{37}\] The trace of Einstein equation is \[\label{traceeinstein} e^{6\lambda}\mathcal{V}-\frac{1}{2}e^{2\lambda}G_{IJ}F_{\mu\nu}^{I}F^{J\mu\nu}\,=\,-12e^{4\lambda}+\text{total\,\,derivative}\,.\tag{38}\]
From 35 the partially off-shell (POS) action is obtained by imposing 37 , \[S_2|_\text{POS}\,=\,\frac{1}{2}\int_{M_2}\left(e^{6\lambda}\mathcal{V}\text{vol}-e^{2\lambda}G_{IJ}F_{12}^IF^J\right)\,.\] Further imposing 38 we find the on-shell action, \[S_2|_\text{OS}\,=\,-6\int_{M_2}e^{4\lambda}\text{vol}\,.\] The free energy of dual 3d SCFT. We introduce the trial free energy for \(F\)-maximization, \[\label{finalfree} F^-\,=\,\left(\frac{2\pi^2}{81G_N^{(6)}}g^3m\right)S_2|_\text{POS}\,=\,-\left(\frac{2}{27}g^3m\right)\mathcal{F}_{S^5}S_2|_\text{POS}\,,\tag{39}\]
We introduce the Killing spinor on the background, 34 , for the supersymmetry variations, 32 , [19], \[\epsilon\,=\,\vartheta\otimes{e}^{\lambda/2}\zeta\,,\] where \(\vartheta\) is a Killing spinor on \(AdS_4\), \(\zeta\) is a spinor on \(M_2\) and the warp factor is introduced for convenience. Then, as presented in appendix 6, we obtain the supersymmetry equations, 60 .
We introduce the spinor bilinears in \(\zeta\) on \(M_2\), [19], \[\label{sbil} S\,=\,\zeta^\dagger\zeta\,, \quad P\,=\,\zeta^\dagger\gamma_3\zeta\,, \quad K\,=\,\zeta^\dagger\gamma_{(1)}\zeta\,, \quad \xi^\flat\,=\,-i\zeta^\dagger\gamma_{(1)}\gamma_3\zeta\,,\tag{40}\] where we have \(\gamma_{(n)}=\frac{1}{n!}\gamma_{\mu_1\cdots\mu_n}dx^{\mu_1}\wedge\cdots\wedge{d}x^{\mu_n}\) and the chirality operator on \(M_2\) is \(\gamma_3=-i\gamma_1\gamma_2\). We obtain the spinor bilinear equations in 61 . We find that the Killing vector, \(\xi\), on \(M_2\) is dual to the one-form, \(\xi^\flat\), and we have \[d\xi^\flat\,=\,-2\left(3+PS^{-1}e^\lambda{W}\right)P\text{vol}\,.\]
We found equivariantly closed forms under \(d_\xi=d-\xi\mathbin{\lrcorner}\) in appendix 6, [19], \[\label{phiF} \Phi^{F^I}\,\equiv\,F^I-X^Ie^\lambda{P}\,,\tag{41}\] and \[\Phi^\text{vol}\,\equiv\,e^{6\lambda}\mathcal{V}\text{vol}-e^{5\lambda}WS\,.\] By employing the Maxwell equations in 36 , we obtain another equivariantly closed form, \[\begin{align} \label{phiS} \Phi^S\,\equiv&\,\frac{1}{2}\left(\Phi^\text{vol}-e^{2\lambda}{G}_{IJ}F^I_{12}\Phi^{F^I}\right) \notag \\ =&\,\frac{1}{2}\left(e^{6\lambda}\mathcal{V}\text{vol}-e^{2\lambda}G_{IJ}F^I_{12}F^J-e^{5\lambda}WS+e^{3\lambda}F^I_{12}X^JP\right)\,, \end{align}\tag{42}\] where the two-forms are the two-dimensional action in 35 .
As we note that the scalar bilinear, \(S\), is a constant in appendix 6, we choose the normalization to be \(S\equiv1\).
For the spinor bilinears in 40 , we find \(||\xi||^2=||K||^2=S^2-P^2\). At a fixed point, \(p\), where we have \(\xi|_p=0\), we obtain \(P|_p=\pm{S}|_p=\pm1\), and the sign is the chirality of the spinor, \(\zeta\), at the fixed point, \(p\). For \(\xi\ne0\), around the fixed point, \(p\), we may have \(M_2\) to be \(\mathbb{R}^2/\mathbb{Z}_{d_p}\) with orbifold singularities and \(\xi=\epsilon_p\partial_{\phi_p}\) where \(\phi_p\) is a polar coordinate of period, \(2\pi/d_p\). Let us name \(\epsilon_p\) to be the weight of \(\xi\) at the fixed point, \(p\), [19].
We employ the Berline-Vergne-Atiyah-Bott fixed point formula, [20], [21], to perform the integrations of equivariantly closed forms in 41 and 42 on \(M_2\), [19]. The magnetic charges are given by \[\label{magch} \mathfrak{p}^I\,\equiv\,\int_{M_2}\frac{F^I}{2\pi}\,=\,\int_{M_2}\frac{\Phi^{F^I}}{2\pi}\,=\,-\frac{1}{2\pi}\sum_{\text{fixed}\,p}\frac{1}{d_p}\frac{2\pi}{\epsilon_p}\left(X^Ie^\lambda{P}\right)\Big|_p\,,\tag{43}\] and the two-dimensional action is \[\begin{align} \label{twoac} S_2|_\text{POS}\,=\,\int_{M_2}\Phi^S\,=&\,-\frac{1}{2}\sum_{\text{fixed}\,p}\frac{1}{d_p}\frac{2\pi}{\epsilon_p}\left(e^{5\lambda}WS-e^{3\lambda}G_{IJ}F_{12}^IX^JP\right)\Big|_p \notag \\ =&\,2\sum_{\text{fixed}\,p}\frac{1}{d_p}\frac{2\pi}{\epsilon_p}e^{4\lambda}P\Big|_p\,, \end{align}\tag{44}\] where we employed the algebraic constraint in 63 in the last line and \(P^2|_p=S^2|_p=1\). As we have \(d\xi^\flat|_p=2\epsilon_p\text{vol}\) at a fixed point, we find \[\label{dxidxi} e^\lambda{W}P|_p\,=\,-\left(3+\epsilon_pP_p\right)\,,\tag{45}\] where we used \(P_p\,=\,\pm1\).
We introduce the new scalar fields, \[\label{newsc} x^I\,\equiv\,-X^Ie^\lambda{P} \qquad \Longrightarrow \qquad \Phi^{F^I}\,=\,F^I+x^I\,.\tag{46}\] With \(\mathcal{F}\left(X^I\right)=1\) in 30 , we obtain \[\mathcal{F}\left(x^I\right)|_p\,=\,x^0\left(x^1x^2\right)^{3/2}|_p\,=\,-e^{4\lambda}|_p\,,\] where we employed \(P^2|_p=1\). Then 44 , 43 and 45 are given, respectively, by \[\label{constraint11} S_2|_\text{POS}\,=\,-2\sum_{\text{fixed}\,p}\frac{2\pi}{d_p\epsilon_p}\mathcal{F}\left(x_p^I\right)P_p\,, \qquad \mathfrak{p}^I\,=\,\sum_{\text{fixed}\,p}\frac{1}{d_p\epsilon_p}x_p^I\,, \qquad \xi_Ix_p^I\,=\,3+\epsilon_pP_p\,,\tag{47}\] where we introduced \(x_p^I\equiv{x}^I|_p\).
Topologically a spindle is the only choice for \(M_2\) if it is compact without boundary and has a \(U(1)\) isometry with fixed points, [5]. The spindle has two fixed points, \(p=\pm\), and we choose \(d_\pm=n_\pm\) with \(n_\pm\ge1\). The Killing vector field on the spindle is given by \[\xi\,=\,b_0\partial_\varphi\,,\] where \(\varphi\) is an azimuthal coordinate on the spindle with period of \(2\pi\). We can write \(\epsilon_+=-b_0/n_+\), \(\epsilon_-=b_0/n_-\). \(P_\pm\) is arbitrary and \(|P_\pm|=1\). Then the constraints in 47 are given by \[\label{constraints22} \mathfrak{p}^I\,=\,-\frac{1}{b_0}\left(x_+^I-x_-^I\right)\,, \qquad \xi_Ix_+^I\,=\,3-\frac{b_0}{n_+}P_+\,, \qquad \xi_Ix_-^I\,=\,3+\frac{b_0}{n_-}P_-\,,\tag{48}\] and, from these, we also obtain \[\label{magneticcon} \xi_I\mathfrak{p}^I\,=\,\frac{P_+}{n_+}+\frac{P_-}{n_-}\,=\,\frac{n_-P_++n_+P_-}{n_-n_+}\,.\tag{49}\] If we introduce \(\sigma\equiv{P}_+/P_-\), \(\sigma=+1\) is for the case of equal chirality of spinor at two poles of spindle, namely twist. \(\sigma=-1\) is for the opposite chiralities, namely anti-twist,[26]. The off-shell action is given by \[\label{S2pos22} S_2|_\text{POS}\,=\,\frac{4\pi}{b_0}\left[\mathcal{F}\left(x_+^I\right)P_+-\mathcal{F}\left(x_-^I\right)P_-\right]\,,\tag{50}\] and we find the final expression of the off-shell free energy from 39 .
We introduce a new parametrization for the scalar fields with magnetic charges, \[\label{phi012} \phi_0\,\equiv\,m\left(x_\pm^0\pm\mathfrak{p}^0\frac{b_0}{2}\right)\,, \qquad \phi_1\,\equiv\,g\left(x_\pm^1\pm\mathfrak{p}^1\frac{b_0}{2}\right)\,, \qquad \phi_2\,\equiv\,g\left(x_\pm^2\pm\mathfrak{p}^2\frac{b_0}{2}\right)\,.\tag{51}\] As the gauge field, \(A^0\), is identically zero, 29 , we have \(\mathfrak{p}^0=0\). Accordingly, we choose \(\phi_0=1\). Namely, we have \[\phi_0\,=\,1\,, \qquad \mathfrak{p}^0\,=\,0\,.\] For the choice of \(\phi_0=2\), from 51 and 46 , we find \[1\,=\,m\left(-X^0e^\lambda{P}\right)|_\pm\,.\] As \(X^0>0\) and \(e^\lambda>0\), only the solutions in the twist class are allowed, \(P_\pm=-1\), and it is consistent with [5]. We further introduce new parameters for the rest of magnetic charges and \(b_0\), \[\epsilon\,\equiv\,\frac{b_0}{2}\,, \qquad \mathfrak{n}_1\,\equiv\,g\mathfrak{p}^1\,, \qquad \mathfrak{n}_2\,\equiv\,g\mathfrak{p}^2\,.\] Then, from the constraints in 48 , using the new parameters, we find \[\begin{align} \label{gbcon} \mathfrak{n}_1+\mathfrak{n}_2\,=&\,\frac{n_++\sigma{n}_-}{n_-n_+}\,, \notag \\ \phi_1+\phi_2-\frac{n_+-\sigma{n}_-}{n_-n_+}\epsilon\,=&\,2\,, \end{align}\tag{52}\] where we keep \(\sigma\equiv{P}_+/P_-\) arbitrary. These are precisely the constraints on the magnetic charges and the variables, \(\phi_i\), for D4-branes wrapped on a spindle in section 5.3 of [5].3 The prepotential is then given by \[\mathcal{F}\left(x_\pm^I\right)\,=\,\frac{1}{g^3m}\Big(\left(\phi_1\mp\mathfrak{n}_1\epsilon\right)(\left(\phi_2\mp\mathfrak{n}_2\epsilon\right)\Big)^{3/2}\,.\] From 50 , the two-dimensional off-shell action is \[S_2|_\text{POS}\,=\,-\frac{2\pi}{\epsilon}\frac{1}{g^3m}\left[\Big(\left(\phi_1+\mathfrak{n}_1\epsilon\right)(\left(\phi_2+\mathfrak{n}_2\epsilon\right)\Big)^{3/2}-\Big(\left(\phi_1-\mathfrak{n}_1\epsilon\right)(\left(\phi_2-\mathfrak{n}_2\epsilon\right)\Big)^{3/2}\right]\,.\] From this, we recover the gravitational block in [5], \[\label{offfree} F^-\left(\phi_i,\epsilon;\mathfrak{n}_i\right)\,=\,\left(\frac{2\sqrt{2}}{15}\frac{N^{5/2}}{\sqrt{8-N_f}}g^3m\right)S_2|_\text{POS}\,.\tag{53}\] By extremizing the gravitational block or the off-shell action subject to the constraints, 52 , we find, [5], \[\begin{align} \label{phistar} \phi_1^*\,=&\,1+\frac{2x\left[\left(n_+-\sigma{n}_-\right)x+n_+n_-\left(\mathfrak{n}_1-\mathfrak{n}_2\right)\right]}{\left(x^2+3\right)\left[\left(n_+-\sigma{n}_-\right)-\left(n_-+\sigma{n}_-\right)x\right]}\,, \notag \\ \epsilon^*\,=&\,\frac{4n_+n_-x^2}{\left(x^2+3\right)\left[\left(n_+-\sigma{n}_-\right)-\left(n_-+\sigma{n}_-\right)x\right]}\,, \end{align}\tag{54}\] where \(x\) is the only solution in the interval \((0,1)\) of the quartic equation, \[\begin{align} \left(n_++\sigma{n}_-\right)^2x^4+4&\left[2n_+^2n_-^2\left(\mathfrak{n}_1-\mathfrak{n}_2\right)^2-3\left(n_+^2+n_-^2-\sigma{n}_+n_-\right)\right]x^2 \notag \\ +12\left(n_+^2-n_-^2\right)&x-9\left(n_+-\sigma{n}_1\right)^2\,=\,0\,. \end{align}\] Inserting \(\phi_1^*\) and \(\epsilon^*\) in 53 , we obtain the free energy of dual three-dimensional superconformal field theories, [5], \[F^-\left(\phi_i^*,\epsilon^*;\mathfrak{n}_i\right)\,=\,\frac{\sqrt{3}\pi}{5}\frac{N^{5/2}}{\sqrt{8-N_f}}\frac{\left[3\left(n_+-\sigma{n}_-\right)\left(x^2+1\right)-\left(n_++\sigma{n}_-\right)x\left(x^2+5\right)\right]^{3/2}}{n_+n_-x\left(x^2+3\right)\left[\left(n_+-\sigma{n}_-\right)-\left(n_++\sigma{n}_-\right)x\right]^{1/2}}\,,\] for the solutions in the twist class, \(\sigma=+1\).
By plugging \(x_-^I=x_+^I+b_0\mathfrak{p}^I\) from 48 into 50 with \(\sigma=+1\), we note that \[\label{spherelim} \lim_{b_0\rightarrow0}S_2|_\text{POS}\,=\,-4\pi\mathfrak{p}^I\frac{\partial}{\partial{x}^I}\mathcal{F}\left(x^I\right)\,.\tag{55}\] We have \(x_-^I=x_+^I=x^I\) to be constant on \(M_2\). From 49 we find \(\xi_Ix^I=2\). We can extremize 55 with the constraint, \(\xi_Ix^I=2\). In particular, for \(n_+=n_-=1\), we should find the extremization performed in [37] which calculates the holographic free energy of \(AdS_4\times\Sigma_\mathfrak{g}\) solutions with a topological twist on the Riemann surface of genus, \(\mathfrak{g}\), [32], [37]–[40], and we have \(\xi_I\mathfrak{p}^I=2P_+\) and \(\sigma=+1\).
In this work, following the work of [19], in six- and seven-dimensional gauged supergravity coupled to a vector multiplet on an ansatz of \(AdS_{5,4}\times{M}_2\), respectively, where \(M_2\) is a two-dimensional surface, we have constructed equivariantly closed forms from the spinor bilinears of Killing spinors. From the integration of equivariantly closed forms via the Berline-Vergne-Atiyah-Bott fixed point formula, we derived the gravitational blocks which were previously conjectured in [5]. For the construction of equivariantly closed forms, we formulated six- and seven-dimensional gauged supergravity theories in the structure of very special geometry.
In order to formulate six- and seven-dimensional gauged supergravity in the very special geometry, it appears to require the introduction of a trivial gauge field, \(A^0\), in \(A^I\,=\,\left(A^0\equiv0,A^1,A^2\right)\) as in 2 and 29 . It would be nice to better understand the structure of special geometry in six and seven dimensions.
The common structure of very special geometry and the similarity of formulations of gauged supergravity in six and seven dimensions here and also in five dimensions in [19] are striking. It may hint a universal structure of gauged supergravity in diverse dimensions which can be utilized to construct solutions and study the geometry of theories. Thus, the study of very special geometry in six- and seven-dimensional gauged supergravity coupled to an arbitrary number of vector multiplets is requested. It would be also interesting to see if there is a structure of very special geometry in four-dimensional theories as well.
A natural extension of the study is to consider the equivariant localization for M5-branes and D4-branes wrapped on four-dimensional manifolds, including orbifolds, [5]–[7], [11], [41]–[43].
This work was supported by the Kumoh National Institute of Technology.
In this appendix we reduce the supersymmetry variations, 5 , on the background of \(AdS_5\times{M}_2\), 7 , by following [19].
We employ the spinor and gamma matrices, [19], \[\epsilon\,=\,\vartheta\otimes{e}^{\lambda/2}\zeta\,, \quad \Gamma_\mathtt{i}\,=\,\beta_\mathtt{i}\otimes\gamma_3\,, \quad \Gamma_{a+2}\,=\,\mathbb{1}\otimes\gamma_a\,,\] where we have frame indices on \(AdS_5\), \(\mathtt{i}=0,1,\ldots,4\), and on \(M_2\), \(a=1,2\) and introduce the chirality operator on \(M_2\), \(\gamma_3=-i\gamma_1\gamma_2\). The Killing spinor equation on \(AdS_5\) is \[\nabla_\mathtt{i}\vartheta\,=\,\frac{1}{2}\beta_\mathtt{i}\vartheta\,.\] Then the supersymmetry variations, 5 , reduce to \[\begin{align} \label{kse117} \nabla_a\zeta\,=&\,\left[\frac{1}{2}\left(1-e^{-\lambda}G_{IJ}X^IF_{12}^J\right)\gamma_a\gamma_3+\frac{i}{2}Q_a\right]\zeta\,, \notag \\ 0\,=&\,\left[\cancel{\partial}\lambda+\frac{1}{5}e^\lambda{W}+\left(1-\frac{1}{5}e^{-\lambda}G_{IJ}X^IF_{12}^J\right)\gamma_3\right]\zeta\,, \notag \\ 0\,=&\,\left[\mathcal{G}_{ij}\cancel{\partial}\varphi^j-e^\lambda\partial_iW+e^{-\lambda}G_{IJ}\partial_iX^IF_{12}^J\gamma_3\right]\zeta\,. \end{align}\tag{56}\] For an arbitrary generator, \(\mathbb{A}\), on \(M_2\), we find the spinor bilinear equations, \[\begin{align} \label{sbileq7} \nabla_a\left(\zeta^\dagger\mathbb{A}\zeta\right)\,=&\,\frac{1}{2}\left(1-e^{-\lambda}G_{IJ}X^IF_{12}^J\right)\zeta^\dagger\left[\mathbb{A},\gamma_a\gamma_3\right]_-\zeta\,, \notag \\ 0\,=\,&\left(\partial_a\lambda\right)\zeta^\dagger\left[\mathbb{A},\gamma^a\right]_\pm\zeta+\frac{1}{5}\left(1\pm1\right)e^\lambda{W}\zeta^\dagger\mathbb{A}\zeta \notag \\ &+\left(1-\frac{1}{5}e^{-\lambda}G_{IJ}X^IF_{12}^J\right)\zeta^\dagger\left[\mathbb{A},\gamma_3\right]_\pm\zeta\,, \notag \\ 0\,=&\,\mathcal{G}_{ij}\left(\partial_a\varphi^j\right)\zeta^\dagger\left[\mathbb{A},\gamma^a\right]_\pm\zeta-\left(1\pm1\right)e^\lambda\partial_i{W}\zeta^\dagger\mathbb{A}\zeta \notag \\ &+e^{-\lambda}G_{IJ}\partial_iX^IF_{12}^J\zeta^\dagger\left[\mathbb{A},\gamma_3\right]_\pm\zeta\,, \end{align}\tag{57}\] where \([\,,\,]_\pm\) are, respectively, commutator and anti-commutator.
The spinor bilinears in 12 , satisfy, [19], \[\xi\mathbin{\lrcorner}\text{vol}\,=\,-K\,, \qquad F_{12}^IK\,=\,-\xi\mathbin{\lrcorner}F^I\,.\] Then the spinor bilinear equations, 61 , reduce to differential equations, \[\begin{align} \label{diffeq7} &dS\,=\,0\,, \qquad dK\,=\,0\,, \qquad d\xi^\flat\,=\,-2\left(4+e^\lambda{W}PS^{-1}\right)P\text{vol}\,, \notag \\ &d\left(e^\lambda{P}\right)\,=\,-\frac{4}{5}G_{JK}X^J\left(\xi\mathbin{\lrcorner}F^K\right)\,, \qquad d\left(e^{6\lambda}S\right)\,=\,\frac{6}{5}e^{7\lambda}W\left(\xi\mathbin{\lrcorner}\text{vol}\right)\,, \notag \\ &dX^I\,=\,-e^{-\lambda}\mathcal{G}^{ij}G_{JK}\partial_iX^I\partial_jX^J\left(\xi\mathbin{\lrcorner}F^K\right)P^{-1} \notag \\ & \qquad =\,-e^\lambda\mathcal{G}^{ij}\partial_iX^I\partial_jW\left(\xi\mathbin{\lrcorner}\text{vol}\right)S^{-1}\,, \end{align}\tag{58}\] and an algebraic equation, \[\label{algecon7} e^{-\lambda}X_IF^I_{12}\,=\,5+e^\lambda{W}PS^{-1}\,.\tag{59}\] Employing 58 and 4 , we find \[\begin{align} d\left(X^Ie^\lambda{P}\right)\,=&\,\left(dX^I\right)e^\lambda{P}+X^Id\left(e^\lambda{P}\right) \notag \\ =&\,-G_{JK}\left(\mathcal{G}^{ij}\partial_iX^I\partial_jX^J+\frac{4}{5}X^IX^J\right)\left(\xi\mathbin{\lrcorner}F^K\right) \notag \\ =&\,-G_{JK}G^{IJ}\left(\xi\mathbin{\lrcorner}F^K\right)\,=\,-\xi\mathbin{\lrcorner}F^I\,, \end{align}\] and the poly-form, \(\Phi^{F^I}\,=\,F^I-X^Ie^\lambda{P}\), is equivariantly closed. Employing 58 and 6 , we find \[\begin{align} d\left(e^{6\lambda}WS\right)\,=&\,\xi_I\left(dX^I\right)e^{6\lambda}S+Wd\left(e^{6\lambda}S\right) \notag \\ =&\,-e^{7\lambda}\left(\mathcal{G}^{ij}\partial_iW\partial_jW-\frac{6}{5}W^2\right)\left(\xi\mathbin{\lrcorner}\text{vol}\right) \notag \\ =&\,-e^{7\lambda}\mathcal{V}\left(\xi\mathbin{\lrcorner}\text{vol}\right)\,, \end{align}\] and the poly-form, \(\Phi^\text{vol}\,=\,e^{7\lambda}\mathcal{V}\text{vol}-e^{6\lambda}WS\), is equivariantly closed.
In this appendix we reduce the supersymmetry variations, 32 , on the background of \(AdS_4\times{M}_2\), 34 , by following [19].
We employ the spinor and gamma matrices, [19], \[\epsilon\,=\,\vartheta\otimes{e}^{\lambda/2}\zeta\,, \quad \Gamma_\mathtt{i}\,=\,\beta_\mathtt{i}\otimes\gamma_3\,, \quad \Gamma_{a+2}\,=\,\mathbb{1}\otimes\gamma_a\,,\] where we have frame indices on \(AdS_4\), \(\mathtt{i}=0,1,2,3\), and on \(M_2\), \(a=1,2\) and introduce the chirality operator on \(M_2\), \(\gamma_3=-i\gamma_1\gamma_2\). The Killing spinor equation on \(AdS_4\) is \[\nabla_\mathtt{i}\vartheta\,=\,\frac{1}{2}\beta_\mathtt{i}\vartheta\,.\] Then the supersymmetry variations, 32 , reduce to \[\begin{align} \label{kse11} \nabla_a\zeta\,=&\,\left[\frac{1}{2}\left(1-e^{-\lambda}G_{IJ}X^IF_{12}^J\right)\gamma_a\gamma_3+\frac{i}{2}Q_a\right]\zeta\,, \notag \\ 0\,=&\,\left[\cancel{\partial}\lambda+\frac{1}{4}e^\lambda{W}+\left(1-\frac{1}{4}e^{-\lambda}G_{IJ}X^IF_{12}^J\right)\gamma_3\right]\zeta\,, \notag \\ 0\,=&\,\left[\mathcal{G}_{ij}\cancel{\partial}\varphi^j-e^\lambda\partial_iW+e^{-\lambda}G_{IJ}\partial_iX^IF_{12}^J\gamma_3\right]\zeta\,. \end{align}\tag{60}\] For an arbitrary generator, \(\mathbb{A}\), on \(M_2\), we find the spinor bilinear equations, \[\begin{align} \label{sbileq} \nabla_a\left(\zeta^\dagger\mathbb{A}\zeta\right)\,=&\,\frac{1}{2}\left(1-e^{-\lambda}G_{IJ}X^IF_{12}^J\right)\zeta^\dagger\left[\mathbb{A},\gamma_a\gamma_3\right]_-\zeta\,, \notag \\ 0\,=\,&\left(\partial_a\lambda\right)\zeta^\dagger\left[\mathbb{A},\gamma^a\right]_\pm\zeta+\frac{1}{4}\left(1\pm1\right)e^\lambda{W}\zeta^\dagger\mathbb{A}\zeta \notag \\ &+\left(1-\frac{1}{4}e^{-\lambda}G_{IJ}X^IF_{12}^J\right)\zeta^\dagger\left[\mathbb{A},\gamma_3\right]_\pm\zeta\,, \notag \\ 0\,=&\,\mathcal{G}_{ij}\left(\partial_a\varphi^j\right)\zeta^\dagger\left[\mathbb{A},\gamma^a\right]_\pm\zeta-\left(1\pm1\right)e^\lambda\partial_i{W}\zeta^\dagger\mathbb{A}\zeta \notag \\ &+e^{-\lambda}G_{IJ}\partial_iX^IF_{12}^J\zeta^\dagger\left[\mathbb{A},\gamma_3\right]_\pm\zeta\,, \end{align}\tag{61}\] where \([\,,\,]_\pm\) are, respectively, commutator and anti-commutator.
The spinor bilinears in 40 , satisfy, [19], \[\xi\mathbin{\lrcorner}\text{vol}\,=\,-K\,, \qquad F_{12}^IK\,=\,-\xi\mathbin{\lrcorner}F^I\,.\] Then the spinor bilinear equations, 61 , reduce to differential equations, \[\begin{align} \label{diffeq} &dS\,=\,0\,, \qquad dK\,=\,0\,, \qquad d\xi^\flat\,=\,-2\left(3+e^\lambda{W}PS^{-1}\right)P\text{vol}\,, \notag \\ &d\left(e^\lambda{P}\right)\,=\,-\frac{3}{4}G_{JK}X^J\left(\xi\mathbin{\lrcorner}F^K\right)\,, \qquad d\left(e^{5\lambda}S\right)\,=\,\frac{5}{4}e^{6\lambda}W\left(\xi\mathbin{\lrcorner}\text{vol}\right)\,, \notag \\ &dX^I\,=\,-e^{-\lambda}\mathcal{G}^{ij}G_{JK}\partial_iX^I\partial_jX^J\left(\xi\mathbin{\lrcorner}F^K\right)P^{-1} \notag \\ & \qquad =\,-e^\lambda\mathcal{G}^{ij}\partial_iX^I\partial_jW\left(\xi\mathbin{\lrcorner}\text{vol}\right)S^{-1}\,, \end{align}\tag{62}\] and an algebraic equation, \[\label{algecon} e^{-\lambda}X_IF^I_{12}\,=\,4+e^\lambda{W}PS^{-1}\,.\tag{63}\] Employing 62 and 31 , we find \[\begin{align} d\left(X^Ie^\lambda{P}\right)\,=&\,\left(dX^I\right)e^\lambda{P}+X^Id\left(e^\lambda{P}\right) \notag \\ =&\,-G_{JK}\left(\mathcal{G}^{ij}\partial_iX^I\partial_jX^J+\frac{3}{4}X^IX^J\right)\left(\xi\mathbin{\lrcorner}F^K\right) \notag \\ =&\,-G_{JK}G^{IJ}\left(\xi\mathbin{\lrcorner}F^K\right)\,=\,-\xi\mathbin{\lrcorner}F^I\,, \end{align}\] and the poly-form, \(\Phi^{F^I}\,=\,F^I-X^Ie^\lambda{P}\), is equivariantly closed. Employing 62 and 33 , we find \[\begin{align} d\left(e^{5\lambda}WS\right)\,=&\,\xi_I\left(dX^I\right)e^{5\lambda}S+Wd\left(e^{5\lambda}S\right) \notag \\ =&\,-e^{6\lambda}\left(\mathcal{G}^{ij}\partial_iW\partial_jW-\frac{5}{4}W^2\right)\left(\xi\mathbin{\lrcorner}\text{vol}\right) \notag \\ =&\,-e^{6\lambda}\mathcal{V}\left(\xi\mathbin{\lrcorner}\text{vol}\right)\,, \end{align}\] and the poly-form, \(\Phi^\text{vol}\,=\,e^{6\lambda}\mathcal{V}\text{vol}-e^{5\lambda}WS\), is equivariantly closed.