Intersecting Attractors

Abstract

The attractor mechanism [1–4], initially discovered in the context of $\mathcal{N} = 2$ black holes, has been recognized as a universal phenomenon governing any extremal flow in supergravity, i.e. a flow with an AdS horizon. It applies to both BPS and non-BPS black hole solutions in Einstein supergravities [5, 6], ungauged [7] and gauged [8] supergravities with higher derivative interactions and general intersections of brane solutions [9].

Appendix

2.1.1 A.1 Flux/Gauging Dictionary for IIA on SU(3) Structure

Gauged $\mathcal{N} = 2$ supergravities with a scalar potential of the form studied in this paper can be derived by flux compactifications of type II theories on SU(3) and SU(3) ×SU(3) structure manifolds. While we refer to the literature (see e.g. [25, 27, 29–31, 38–40]) for a detailed study of such general $\mathcal{N} = 2$ dimensional reductions and the related issues,⁴ in this appendix we provide a practical dictionary between the 10d and the 4d quantities, with a focus on the scalar potential derived from SU(3) structure compactifications of type IIA. In particular, we illustrate how the expressions one derives for ${V }_{\mathrm{NS}}$ and V _R are consistent with the scalar potential (2.79) studied in the main text.

2.1.1.1 A.1.1 SU(3) Structures and Their Curvature

An SU(3) structure on a 6d manifold M ₆ is defined by a real 2–form J and a complex, decomposable⁵ 3–form Ω, satisfying the compatibility relation $J \wedge \Omega = 0$ as well as the non-degeneracy (and normalization) condition

$$\frac{i} {8}\Omega \wedge \bar{ \Omega }\; =\; \frac{1} {6}J \wedge J \wedge J\; =\; vo{l}_{6}\;\neq \;0\quad \mathrm{everywhere}\,.$$

(A.1)

Ω defines an almost complex structure I, with respect to which is of type (3, 0). In turn, I and J define a metric on M ₆ via g = JI. The latter is required to be positive-definite, and vol ₆ above denotes the associated volume form.

SU(3) structures are classified by their torsion classes ${W}_{i}\,,\,i = 1,\ldots 5$, defined via [46]:

$$\begin{array}{rcl} dJ& =& \frac{3} {2}\mathrm{Im}({\overline{W}}_{1}\Omega ) + {W}_{4} \wedge J + {W}_{3} \\ d\Omega & =& {W}_{1} \wedge J \wedge J + {W}_{2} \wedge J +{ \overline{W}}_{5} \wedge \Omega \,,\end{array}$$

(A.2)

where W ₁ is a complex scalar, W ₂ is a complex primitive (1,1)–form (primitive means ${W}_{2} \wedge J \wedge J = 0$), W ₃ is a real primitive (1,2) + (2,1)–form (primitive $\Leftrightarrow {W}_{3} \wedge J = 0$), W ₄ is a real 1–form, and W ₅ is a complex (1,0)–form.

Reference [47] provides a formula for the Ricci scalar R ₆ in terms of the torsion classes. We will restrict to ${W}_{4} = {W}_{5} = 0$, in which case the formula is

$${R}_{6}\; =\; \frac{1} {2}\left (15\vert {W}_{1}{\vert }^{2} - {W}_{ 2}\lrcorner {\overline{W}}_{2} - {W}_{3}\lrcorner {W}_{3}\right )\,.$$

(A.3)

This can equivalently be expressed as

$${R}_{6}\,vo{l}_{6} = -\frac{1} {2}\left [\,dJ \wedge {_\ast}dJ + d\Omega \wedge {_\ast}d\bar{\Omega } - (dJ \wedge \Omega ) \wedge {_\ast}(dJ \wedge \bar{ \Omega })\,\right ]\,$$

(A.4)

as it can be seen recalling (A.2) and computing

$$\begin{array}{rcl} d\Omega \wedge {_\ast}d\bar{\Omega }& =& 12\vert {W}_{1}{\vert }^{2}vo{l}_{ 6} - J \wedge {W}_{2} \wedge {\overline{W}}_{2} \\ & =& \left (12\vert {W}_{1}{\vert }^{2} + {W}_{ 2}\lrcorner {\overline{W}}_{2}\right )vo{l}_{6} \\ dJ \wedge {_\ast}dJ& =& \left (9\vert {W}_{1}{\vert }^{2} + {W}_{ 3}\lrcorner {W}_{3}\right )vo{l}_{6} \\ (dJ \wedge \Omega ) \wedge {_\ast}(dJ \wedge \bar{ \Omega })& =& 36\vert {W}_{1}{\vert }^{2}vo{l}_{ 6}\,.\end{array}$$

(A.5)

2.1.1.2 A.1.2 The Scalar Potential from Dimensional Reduction

The 4d scalar potential receives contributions from both the NSNS and the RR sectors of type IIA supergravity. These are respectively given by

$$\begin{array}{rcl}{ V }_{\mathrm{NS}}& =& \frac{{e}^{2\varphi }} {2\mathcal{V}}{\int \limits }_{{M}_{6}}\left (\,\frac{1} {2}H \wedge {_\ast}H\, - {R}_{6} {_\ast} 1\,\right ) \\ & =& \frac{{e}^{2\varphi }} {4\mathcal{V}}{\int \limits }_{{M}_{6}}\left [\,H \wedge {_\ast}H + dJ \wedge {_\ast}dJ + d\Omega \wedge {_\ast}d\bar{\Omega }\right. \\ & & \left.-(dJ \wedge \Omega ) \wedge {_\ast}(dJ \wedge \bar{ \Omega })\,\right ], \end{array}$$

(A.6)

$$\begin{array}{rcl}{ V }_{\mathrm{R}}\;& =& \;\frac{{e}^{4\varphi }} {2} {\int \limits }_{{M}_{6}}\left (\,{F}_{0}^{2} {_\ast} 1\, +\, {F}_{ 2} \wedge {_\ast}{F}_{2}\, +\, {F}_{4} \wedge {_\ast}{F}_{4}\, +\, {F}_{6} \wedge {_\ast}{F}_{6}\,\right ), \end{array}$$

(A.7)

and the total potential reads $V = {V }_{\mathrm{NS}} + {V }_{\mathrm{R}}$. In (A.6), H is the internal NSNS field-strength, $\mathcal{V} ={ \int \limits }_{{M}_{6}}vo{l}_{6}$, and $\varphi$ is the 4d dilaton ${e}^{-2\varphi } = {e}^{-2\phi }\mathcal{V}$, where we are assuming that the 10d dilaton ϕ is constant along M ₆. The k–forms F _k appearing in expression (A.7) are the internal RR field strengths, satisfying the Bianchi identity $d{F}_{k} - H \wedge {F}_{k-2} = 0$. The F ₆ form can be seen as the Hodge-dual of the F ₄ extending along spacetime, and the term ${F}_{6} \wedge {_\ast}{F}_{6}$ arises in a natural way if one considers type IIA supergravity in its democratic formulation [48].

2.1.1.2.1 A.1.2.1 Expansion Forms

In order to define the mode truncation, we postulate the existence of a basis of differential forms on the compact manifold in which to expand the higher dimensional fields. For a detailed analysis of the relations that these forms need to satisfy in order that the dimensional reduction go through, see in particular [27].

We take ω₀ = 1 and $\tilde{{\omega }}^{0} = \frac{vo{l}_{6}} {\mathcal{V}}$, and we assume there exist a set of 2–forms ω _a satisfying

$$\,{\omega }_{a} \wedge {_\ast}{\omega }_{b}\, =\, 4\,{g}_{ab}\,vo{l}_{6},\qquad \qquad {\omega }_{a} \wedge {\omega }_{b} = -{d}_{abc}\tilde{{\omega }}^{c}, $$

(A.8)

where ${g}_{ab}$ should be independent of the internal coordinates, d _abc should be a constant tensor, and the dual 4–forms $\tilde{{\omega }}^{a}$ are defined as

$$\tilde{{\omega }}^{a} = - \frac{1} {4\mathcal{V}}{g}^{ab} {_\ast} {\omega }_{ b}\,.$$

(A.9)

From the above relations, we see that

$${\omega }_{a} \wedge \tilde{ {\omega }}^{b}\, =\, -{\delta }_{ a}^{b}\,\tilde{{\omega }}^{0},\qquad \qquad {\omega }_{ a} \wedge {\omega }_{b} \wedge {\omega }_{c} = {d}_{abc}\tilde{{\omega }}^{0}\,.$$

(A.10)

We also assume the existence of a set of 3–forms ${\alpha }_{I},{\beta }^{I}$, satisfying

$${\alpha }_{I} \wedge {\beta }^{J}\, =\, {\delta }_{ I}^{J}\,\tilde{{\omega }}^{0}.$$

(A.11)

Adopting the notation ${\omega }^{\mathbb{A}} = {(\tilde{{\omega }}^{A},{\omega }_{A})}^{T} = {(\tilde{{\omega }}^{0},\tilde{{\omega }}^{a},{\omega }_{0},{\omega }_{a})}^{T}\;$ and $\;{\alpha }^{\mathbb{I}} = {({\beta }^{I},{\alpha }_{I})}^{T}$, we see that the symplectic metrics $\mathbb{C}$ appearing in the main text are here given by

$${\mathbb{C}}_{1}^{\mathbb{I}\mathbb{J}}\, =\, -\int {\alpha }^{\mathbb{I}} \wedge {\alpha }^{\mathbb{J}},\qquad {\mathbb{C}}_{ 2}^{\mathbb{A}\mathbb{B}}\, =\, -\int \langle {\omega }^{\mathbb{A}},{\omega }^{\mathbb{B}}\rangle \,,$$

(A.12)

where the antisymmetric pairing $\langle \,,\,\rangle$ is defined on even forms ρ, σ as $\langle \rho,\sigma \rangle \, =\, {[\lambda (\rho ) \wedge \sigma ]}_{\mathrm{6}}$, with $\,\lambda ({\rho }_{k}) = {(-)}^{\frac{k} {2} }{\rho }_{k}\,$, k being the degree of ρ, and ${[\;\;]}_{6}$ selecting the piece of degree 6.

The basis forms are used to expand Ω as

$$\Omega \, =\, {Z}^{I}{\alpha }_{ I} -{\mathcal{G}}_{I}{\beta }^{I}\, =\, {e}^{-\frac{{K}_{1}} {2} }{\Pi }_{1}^{\mathbb{I}}\,{\alpha }_{\mathbb{I}}\,,$$

(A.13)

and J together with the internal NS 2–form B as:

$$J\, =\, {v}^{a}{\omega }_{ a},\;\;B\, =\, {b}^{a}{\omega }_{ a}\quad \; \Rightarrow \;\quad {e}^{-B-iJ}\, =\, {X}^{A}{\omega }_{ A} -{\mathcal{F}}_{A}\tilde{{\omega }}^{A}\, =\, {e}^{-\frac{{K}_{2}} {2} }{\Pi }_{2}^{\mathbb{A}}{\omega }_{\mathbb{A}}\,,$$

(A.14)

where in the last equalities we define ${\alpha }_{\mathbb{I}} = {\mathbb{C}}_{\mathbb{I}\mathbb{J}}{\alpha }^{\mathbb{J}} = {({\alpha }_{I},-{\beta }^{I})}^{T}$ and ${\omega }_{\mathbb{A}} = {\mathbb{C}}_{\mathbb{A}\mathbb{B}}{\omega }^{\mathbb{B}} = {({\omega }_{A},-\tilde{{\omega }}^{A})}^{T}$, and we adopt the symplectic notation defined in (2.76). Here, $({Z}^{I},{\mathcal{G}}_{I})$ and $({X}^{A},{\mathcal{F}}_{A})$ represent the holomorphic sections on the moduli spaces of Ω and B + iJ expanded as above, which (under some conditions [26–28]) indeed exhibit a special Kähler structure, and correspond respectively to the manifolds ${\mathcal{M}}_{1}$ and ${\mathcal{M}}_{2}$ of the main text. Notice that here ${X}^{A} \equiv ({X}^{0},{X}^{a}) \equiv (1,{x}^{a}) = (1,-{b}^{a} - i{v}^{a})$, while ${\mathcal{F}}_{A} = \frac{\partial \mathcal{F}} {\partial {X}^{A}}$, where the cubic holomorphic function $\mathcal{F} = \frac{1} {6}{d}_{abc}\frac{{X}^{a}{X}^{b}{X}^{c}} {{X}^{0}}$ is identified with the prepotential on ${\mathcal{M}}_{2}$. The Kähler potentials on ${\mathcal{M}}_{1}$ and ${\mathcal{M}}_{2}$ are recovered from ${K}_{1} = -\log i\int \Omega \wedge \bar{ \Omega }\,$ and $\,{K}_{2} = -\log \frac{4} {3} \int J \wedge J \wedge J$, the latter yielding the metric g _ab appearing in (A.8). Notice that (A.1) implies ${e}^{-{K}_{1}} = {e}^{-{K}_{2}} = 8\mathcal{V}$.

The matrices $\mathbb{M}$ defined in (2.81) are given by

$${\mathbb{M}}_{1,\mathbb{I}\mathbb{J}}\, =\, -\int {\alpha }_{\mathbb{I}} \wedge {_\ast}{\alpha }_{\mathbb{J}},\qquad \qquad {\mathbb{M}}_{2,\mathbb{A}\mathbb{B}}\, =\, -{\sum \limits }_{k} \int {({e}^{B}{\omega }_{ \mathbb{A}})}_{k} \wedge {_\ast}{({e}^{B}{\omega }_{ \mathbb{B}})}_{k}\,,$$

(A.15)

and from the second relation one finds that the period matrix ${\mathcal{N}}_{2}$ on ${\mathcal{M}}_{2}$ reads

$$\mathrm{Re}{\mathcal{N}}_{AB} = -\left (\begin{array}{cccc} \frac{1} {3}{d}_{abc}{b}^{a}{b}^{b}{b}^{c}&\frac{1} {2}{d}_{abc}{b}^{b}{b}^{c} \\ \frac{1} {2}{d}_{abc}{b}^{b}{b}^{c} & {d}_{ abc}{b}^{c} \end{array} \right ),\quad \mathrm{Im}{\mathcal{N}}_{AB} = -4\mathcal{V}\left (\begin{array}{cc} \frac{1} {4} + {g}_{ab}{b}^{a}{b}^{b}&{g}_{ ab}{b}^{b} \\ {g}_{ab}{b}^{b} & {g}_{ab} \end{array} \right ),$$

(A.16)

which is in agreement with the expression derived from $\mathcal{F}$ via the standard formula [49]

$${N}_{AB}\; =\;{ \overline{\mathcal{F}}}_{AB} + 2i\frac{\mathrm{Im}{\mathcal{F}}_{AD}{X}^{D}\mathrm{Im}{\mathcal{F}}_{BE}{X}^{E}} {{X}^{C}\mathrm{Im}{\mathcal{F}}_{CE}{X}^{E}},\qquad \quad {\mathcal{F}}_{AB} \equiv \frac{{\partial }^{2}\mathcal{F}} {\partial {X}^{A}\partial {X}^{B}}\,.$$

(A.17)

Finally, we also require the following differential conditions on the basis forms:

$$d{\omega }_{a}\, =\, {e}_{a}^{\mathbb{I}}{\alpha }_{ \mathbb{I}},\qquad \qquad d{\alpha }^{\mathbb{I}}\, =\, {e}_{ a}^{\mathbb{I}}\tilde{{\omega }}^{a},\qquad \qquad d\tilde{{\omega }}^{a}\, =\, 0\,,$$

(A.18)

where the ${e}_{a}^{\mathbb{I}} = ({{e}_{a}}^{I},{e}_{aI})$ are real constants, usually called ‘geometric fluxes’. Defining the total internal NS 3–form as $H = {H}^{\mathrm{fl}} + dB$, and expanding its flux part as

$${H}^{\mathrm{fl}}\, =\, -{{e}_{ 0}}^{I}{\alpha }_{ I} + {e}_{0I}{\beta }^{I}\, \equiv \,-{e}_{ 0}^{\mathbb{I}}{\alpha }_{ \mathbb{I}}\,,$$

(A.19)

with constant ${e}_{0}^{\mathbb{I}}$, we can define ${e}_{A}^{\mathbb{I}} = {({e}_{0}^{\mathbb{I}},{e}_{a}^{\mathbb{I}})}^{T}$, and thus fill in half of the charge matrix $Q$ introduced in (2.77):

$${ Q}_{\mathbb{A}}^{\mathbb{I}} = \left (\begin{array}{c} {e}_{A}^{\mathbb{I}} \\ 0\end{array} \right ).$$

(A.20)

As noticed in [28], more general matrices, involving the ${m}_{A}^{\mathbb{I}}$ charges as well, can be obtained by considering non-geometric fluxes, or SU(3) ×SU(3) structure compactifications. The nilpotency condition d ² = 0 applied to (A.18), together with the Bianchi identity dH = 0, translates into the constraint

$${e}_{A}^{\mathbb{I}}{e}_{ B\mathbb{I}}\, =\, 0\qquad \qquad \qquad \textrm{ with }\;\;{e}_{A\mathbb{I}} = {\mathbb{C}}_{\mathbb{I}\mathbb{J}}{e}_{A}^{\mathbb{J}},$$

(A.21)

which, taking into account (A.20), is consistent with (2.78).

In the following, by using the above relations we recast in turn expressions (A.6) and (A.7) for ${V }_{\mathrm{NS}}$ and ${V }_{\mathrm{R}}$ in terms of 4d degrees of freedom, and show their consistency with (2.79).

2.1.1.2.2 A.1.2.2 Derivation of V _NS

Recalling the expansions of J, H and Ω defined above, using the assumed properties of the basis forms, and adopting the notation introduced in (2.76), one finds

$$\begin{array}{rcl} \int dJ \wedge {_\ast}dJ = -{v}^{a}{v}^{b}\,{e}_{ a}^{\mathbb{I}}\,{\mathbb{M}}_{ 1,\mathbb{I}\mathbb{J}}{e}_{b}^{\mathbb{J}},\quad & & \quad \int H \wedge {_\ast}H = -{b}^{A}{b}^{B}\,{e}_{ A}^{\mathbb{I}}\,{\mathbb{M}}_{ 1,\mathbb{I}\mathbb{J}}{e}_{B}^{\mathbb{J}}\,, \\ \int d\Omega \wedge {_\ast}d\bar{\Omega } = \frac{{e}^{-{K}_{1}}} {4\mathcal{V}} {\Pi }_{1}^{\mathbb{I}}{e}_{ a\mathbb{I}}{g}^{ab}{e}_{ b\mathbb{J}}{\overline{\Pi }}_{1}^{\mathbb{J}},\quad & & \quad \int (dJ \wedge \Omega ) \wedge {_\ast}(dJ \wedge \bar{ \Omega }) \\ & & \quad = \frac{{e}^{-{K}_{1}}} {\mathcal{V}} {\Pi }_{1}^{\mathbb{I}}{e}_{ a\mathbb{I}}{v}^{a}{v}^{b}{e}_{ b\mathbb{J}}{\overline{\Pi }}_{1}^{\mathbb{J}}, \\ \end{array}$$

where we define ${b}^{A} = (-1,{b}^{a})$. Plugging this into (A.6), we get the NSNS contribution to V, expressed in a 4d language:

$${V }_{\mathrm{NS}} = -\frac{{e}^{2\varphi }} {4\mathcal{V}}\left [{X}^{A}{e}_{ A}^{\mathbb{I}}\,{\mathbb{M}}_{ 1,\mathbb{I}\mathbb{J}}{e}_{B}^{\mathbb{J}}\,{\overline{X}}^{B} -\frac{{e}^{-{K}_{1}}} {4\mathcal{V}} {\Pi }_{1}^{\mathbb{I}}{e}_{ a\mathbb{I}}({g}^{ab} - 4{v}^{a}{v}^{b}){e}_{ b\mathbb{J}}{\overline{\Pi }}_{1}^{\mathbb{J}}\right ]\,.$$

(A.22)

Recalling (A.16), noticing that $\frac{1} {4\mathcal{V}}({g}^{ab} - 4{v}^{a}{v}^{b}) = -{(\mathrm{Im}{\mathcal{N}}_{ 2})}^{-1\,ab} - 4{e}^{{K}_{2}}({X}^{a}{\overline{X}}^{b} +{ \overline{X}}^{a}{X}^{b})$, and recalling that ${e}^{-{K}_{1}} = {e}^{-{K}_{2}} = 8\mathcal{V}$, we conclude that (A.22) is consistent with (2.79).

2.1.1.2.3 A.1.2.3 Derivation of V _R

We consider the modified field-strengths ${G}_{k} \equiv {\left [{e}^{-B}F\right ]}_{k}$, which satisfy the Bianchi identity $d{G}_{k} - {H}^{\mathrm{fl}} \wedge {G}_{k-2} = 0$, and we define the expansions

$${G}_{0}\, =\, {p}^{0},\qquad \;\;\qquad {G}_{ 2}\, =\, {p}^{a}{\omega }_{ a},\qquad \;\;\qquad {A}_{3}\, =\, {\xi }^{\mathbb{I}}{\alpha }_{ \mathbb{I}}$$

$${G}_{4}\, =\, {G}_{4}^{\mathrm{fl}} + d{A}_{ 3}\, =\, ({q}_{a} - {e}_{a\mathbb{I}}{\xi }^{\mathbb{I}})\tilde{{\omega }}^{a},\qquad {G}_{ 6}\, =\, {G}_{6}^{\mathrm{fl}} - {H}^{\mathrm{fl}} \wedge {A}_{ 3}\, =\, ({q}_{0} - {e}_{0\mathbb{I}}\,{\xi }^{\mathbb{I}})\tilde{{\omega }}^{0}.$$

The Bianchi identities then amount just to the following constraint among the charges

$${p}^{A}{e}_{ A}^{\mathbb{I}}\, =\, 0\,,$$

(A.23)

which, recalling (A.20), gives the last equality in (2.78). Then the integral in (A.7) reads

$$\sum \limits_{k} \int {F}_{k} \wedge {_\ast}{F}_{k}\, =\,\sum \limits_{k} \int {({e}^{B}G)}_{ k} \wedge {_\ast}{({e}^{B}G)}_{ k}\, =\, {(c +\widetilde{ Q}\xi )}^{T}{\mathbb{M}}_{ 2}(c +\widetilde{ Q}\xi )\,,$$

(A.24)

where for the second equality we use (A.15), and here ${(c +\widetilde{ Q}\xi )}^{\mathbb{A}}\, =\, {({p}^{A},\,{q}_{A} - {e}_{A\mathbb{I}}{\xi }^{\mathbb{I}})}^{T}$. The expression for V _R we obtain is therefore consistent with (2.79).