A generalised Garfinkle–Vachaspati transform

Abstract

The Garfinkle–Vachaspati transform is a deformation of a metric in terms of a null, hypersurface orthogonal, Killing vector $k^\mu$. We explore a generalisation of this deformation in type IIB supergravity taking motivation from certain studies of the D1–D5 system. We consider solutions of minimal six-dimensional supergravity admitting null Killing vector $k^\mu$ trivially lifted to type IIB supergravity by the addition of four-torus directions. The torus directions provide covariantly constant spacelike vectors $l^\mu$. We show that the original solution can be deformed as $g_{\mu \nu } \rightarrow g_{\mu \nu } + 2 \, \Phi k_{(\mu }l_{\nu )}, \ C_{\mu \nu } \rightarrow C_{\mu \nu } - 2 \, \Phi k_{[\mu }l_{\nu ]},$ provided the two-form supporting the original spacetime satisfies $i_k (dC) = - d k$, and where $\Phi$ satisfies the equation of a minimal massless scalar field on the original spacetime. We show that the condition $i_k (dC) = - d k$ is satisfied by all supersymmetric solutions admitting null Killing vector. Hence all supersymmetric solutions of minimal six-dimensional supergravity can be deformed via this method. As an example of our approach, we work out the deformation on a class of D1–D5–P geometries with orbifolds. We show that the deformed spacetimes are smooth and identify their CFT description. Using Bena–Warner formalism, we also express the deformed solutions in other duality frames.

Appendices

Detailed analysis of the equations of motion

We establish that generalised Garfinkle–Vachaspati transform is a valid solution generating technique via a brute force calculation. We show that the left and the right hand side of the Einstein equations transform in the exactly the same way, thereby establishing that if we start with a solution, we can deform it to a new solution. In our convention, Einstein equations are

$$\begin{aligned} R_{\mu \nu } =\frac{1}{4}F_{\mu \lambda \sigma }{F_\nu }^{\lambda \sigma }, \end{aligned}$$

(A.1)

and matter field equations are

$$\begin{aligned} \nabla _\mu F^{\mu \nu \rho } =0. \end{aligned}$$

(A.2)

The tedious calculations required to show that these equations transform covariantly are organised as follows: in Sect. A.1 the left hand side of the Einstein equations are analysed, in Sect. A.2 the right hand side of the Einstein equations are analysed, and finally in Sect. A.3 matter equations are analysed.

The generalised Garfinkle–Vachaspati transform of the metric is,

$$\begin{aligned} g'_{\mu \nu }=g_{\mu \nu }+\Phi (k_{\mu }l_{\nu }+k_\nu l_\mu ), \end{aligned}$$

(A.3)

where $\Phi$ is a massless scalar on the original background spacetime $g_{\mu \nu }$,

$$\begin{aligned} \Box \Phi =0. \end{aligned}$$

(A.4)

The vector $k^\mu$ appearing in (A.3) is a null Killing vector

$$\begin{aligned} k^\mu k_\mu&= 0,&\nabla _\mu k_\nu +\nabla _\nu k_\mu&=0, \end{aligned}$$

(A.5)

and $l^\mu$ is a unit normalised covariantly constant spacelike (Killing) vector orthogonal to $k^\mu$:

$$\begin{aligned} l^\mu l_\mu&=1,&k^\mu l_\mu&=0,&\nabla _\mu l_\nu&=0. \end{aligned}$$

(A.6)

Furthermore, we also require that the scalar $\Phi$ is compatible with the Killing symmetries,

$$\begin{aligned} k^\mu \nabla _\mu \Phi&=0,&l^\mu \nabla _\mu \Phi&=0, \end{aligned}$$

(A.7)

so that the transformed spacetime $g'_{\mu \nu }$ also has $k^\mu$ and $l^\mu$ as Killing symmetries.

1.1 Left hand side of Einstein equations

The aim of this subsection is to find the transformation of the left hand side of the Einstein equations (A.1). Doing this is straightforward, though somewhat tedious. To compute the change in the Ricci tensor, we essentially need to compute the change in the metric compatible connection and its covariant derivative:

$$\begin{aligned} R'_{\lambda \nu }=R_{\lambda \nu }-{\nabla }_\lambda {{\Omega }^{\mu }}_{\mu \nu } + {\nabla }_{\mu } {{\Omega }^{\mu }}_{\lambda \nu } +{\Omega ^\mu }_{\mu \rho } {\Omega ^\rho }_{\lambda \nu } -{\Omega ^\rho }_{\mu \lambda }{\Omega ^\mu }_{\rho \nu }, \end{aligned}$$

(A.8)

where $\Omega ^\mu _{\lambda \nu }$ is the change in the metric compatible connection

$$\begin{aligned} \Gamma {'}^\mu _{\lambda \nu }=\Gamma ^\mu _{\lambda \nu }+\Omega ^\mu _{\lambda \nu }. \end{aligned}$$

(A.9)

The change in the metric compatible connection is

$$\begin{aligned} \Omega ^\mu _{\lambda \nu } = \frac{1}{2}g'^{\mu \sigma }\left( \nabla _\lambda g'_{\nu \sigma } + \nabla _\nu g'_{\sigma \lambda }- \nabla _\sigma g'_{\nu \lambda }\right) . \end{aligned}$$

(A.10)

We compute various pieces required in Eq. (A.8).

We start by observing that the inverse of the transformed metric (A.3) is simply

$$\begin{aligned} g'^{\mu \nu }=g^{\mu \nu }+\Phi ^2 k^\mu k^\nu -\Phi S^{\mu \nu }. \end{aligned}$$

(A.11)

Next, we introduce the notation,

$$\begin{aligned} S_{\mu \nu }= & {} k_\mu l_\nu + k_\nu l_\mu , \end{aligned}$$

(A.12)

$$\begin{aligned} h_{\mu \nu }= & {} \Phi S_{\mu \nu }, \end{aligned}$$

(A.13)

$$\begin{aligned} n_{\mu \nu }= & {} \nabla _\mu k_\nu -\nabla _\nu k_\mu . \end{aligned}$$

(A.14)

The change in the metric compatible connection, $\Omega ^\mu _{\lambda \nu }$, is conveniently organised in two terms,

$$\begin{aligned} \Omega ^\mu _{\lambda \nu }={\Xi }^\mu _{\lambda \nu }+\frac{1}{2}\left( \Phi ^2 k^\mu k^\alpha -\Phi S^{\mu \alpha }\right) \left( \nabla _\lambda h_{\nu \alpha }+\nabla _\nu h_{\alpha \lambda }-\nabla _\alpha h_{\lambda \nu }\right) , \end{aligned}$$

(A.15)

where the first term ${\Xi }^\mu _{\lambda \nu }$ is the combination that features in the Garfinkle–Vachaspati transform without the spacelike Killing vector $l^\mu$ [19]:

$$\begin{aligned} {\Xi }^\mu _{\lambda \nu }=\frac{1}{2}g^{\mu \alpha }\left( \nabla _\lambda h_{\nu \alpha }+\nabla _\nu h_{\alpha \lambda }-\nabla _\alpha h_{\lambda \nu }\right) . \end{aligned}$$

(A.16)

In order to proceed further we make a convenient definition,

$$\begin{aligned} K^\mu _{\nu \lambda }:=\nabla _\nu S^\mu _\lambda +\nabla _\lambda S^\mu _\nu -\nabla ^\mu S_{\lambda \nu }, \end{aligned}$$

(A.17)

using which it follows that

$$\begin{aligned} {\Xi }^\mu _{\lambda \nu }= \frac{1}{2}\left( (\nabla _\nu \Phi ) S^\mu _\lambda +(\nabla _\lambda \Phi ) S^\mu _\nu -(\nabla ^\mu \Phi ) S_{\nu \lambda }+\Phi K^\mu _{\nu \lambda }\right) , \end{aligned}$$

(A.18)

and therefore,

$$\begin{aligned} \Omega ^\mu _{\lambda \nu }={\Xi }^\mu _{\lambda \nu }-\frac{1}{2}\Phi k^\mu (k_\nu \nabla _\lambda \Phi +k_\lambda \nabla _\nu \Phi ). \end{aligned}$$

(A.19)

The trace of $\Omega ^\mu _{\lambda \nu }$ is easily seen to be zero

$$\begin{aligned} \Omega ^\mu _{\mu \lambda }=0. \end{aligned}$$

(A.20)

As a result the transformation of the Ricci tensor (A.8) simplifies to

$$\begin{aligned} R'_{\lambda \nu }=R_{\lambda \nu }+ {\nabla }_{\mu } {{\Omega }^{\mu }}_{\lambda \nu } -{\Omega ^\rho }_{\mu \lambda }{\Omega ^\mu }_{\rho \nu }. \end{aligned}$$

(A.21)

To compute the right hand side of the above expression, we need to compute ${\nabla }_{\mu } {{\Omega }^{\mu }}_{\lambda \nu }$ and ${\Omega ^\rho }_{\mu \lambda }{\Omega ^\mu }_{\rho \nu }$. We can first show that

$$\begin{aligned}&2\nabla _\mu {\Xi }^\mu _{\lambda \nu }=(\nabla _\mu \nabla _\nu \Phi )S^\mu _\lambda +(\nabla _\mu \nabla _\lambda \Phi )S^\mu _\nu \nonumber \\&\quad -(\nabla ^\mu \Phi )(\nabla _\mu S_{\nu \lambda })+(\nabla _\mu \Phi ) K^\mu _{\nu \lambda } +\Phi (\nabla _\mu K^\mu _{\nu \lambda }) , \end{aligned}$$

(A.22)

where we have used $\nabla _\mu S^\mu _\nu =0$ and the fact that we are deforming the original solution via a massless scalar field (A.4). The first three terms of (A.22) combine to zero,

$$\begin{aligned} (\nabla _\mu \nabla _\nu \Phi )S^\mu _\lambda +(\nabla _\mu \nabla _\lambda \Phi )S^\mu _\nu -(\nabla ^\mu \Phi )(\nabla _\mu S_{\nu \lambda })=0. \end{aligned}$$

(A.23)

In order to simplify (A.22) further we develop some identities. One can easily show that

$$\begin{aligned} K^\mu _{\nu \lambda }= & {} (\nabla _\nu k^\mu -\nabla ^\mu k_\nu )l_\lambda +(\nabla _\lambda k^\mu -\nabla ^\mu k_\lambda ) l_\nu \end{aligned}$$

(A.24)

$$\begin{aligned}= & {} {n_\nu }^\mu l_\lambda + {n_\lambda }^\mu l_\nu . \end{aligned}$$

(A.25)

It then follows that the fourth term of (A.22) simplifies to

$$\begin{aligned} (\nabla _\mu \Phi )K^\mu _{\nu \lambda }= & {} -2 k^\mu \left[ (\nabla _\nu \nabla _\mu \Phi )l_\lambda +(\nabla _\lambda \nabla _\mu \Phi )l_\nu \right] , \end{aligned}$$

(A.26)

where we have used

$$\begin{aligned} (\nabla _\mu \Phi ){n_\nu }^\mu= & {} -2 k^\mu (\nabla _\nu \nabla _\mu \Phi ). \end{aligned}$$

(A.27)

Inserting (A.25) in $(\nabla _\mu K^\mu _{\nu \lambda })$, the last term of (A.22) simplifies to

$$\begin{aligned} \nabla _\mu K^\mu _{\nu \lambda }=-2(\square k_\nu )l_\lambda -2(\square k_\lambda )l_\nu , \end{aligned}$$

(A.28)

where we have also used

$$\begin{aligned} \nabla _\mu {n_\nu }^\mu =-2\square k_\nu . \end{aligned}$$

(A.29)

When the dust settles, we get a simplified expression for Eq. (A.22):

$$\begin{aligned} \nabla _\mu {\Xi }^\mu _{\lambda \nu }=-l_\lambda \left[ k^\mu (\nabla _\nu \nabla _\mu \Phi )+\Phi \square k_\nu \right] -l_\nu \left[ k^\mu (\nabla _\lambda \nabla _\mu \Phi )+\Phi \square k_\lambda \right] . \end{aligned}$$

(A.30)

From (A.19) it then follows that

$$\begin{aligned} 2\nabla _\mu \Omega ^\mu _{\lambda \nu }=2\nabla _\mu {\Xi }^\mu _{\lambda \nu }-\Phi k^\mu \left[ k_\nu (\nabla _\mu \nabla _\lambda \Phi )+k_\lambda (\nabla _\mu \nabla _\nu \Phi )\right] . \end{aligned}$$

(A.31)

This is one of the pieces that is required to compute the change in the Ricci tensor (A.21). The other piece that is required is $\Omega ^\rho _{\mu \lambda }\Omega ^\mu _{\rho \nu }$. In order to compute this combination, we start by observing that

$$\begin{aligned} 4\Omega ^\rho _{\mu \lambda }\Omega ^\mu _{\rho \nu }&=\left[ 2{\Xi }^\rho _{\mu \lambda } -\Phi k^\rho (k_\mu \nabla _\lambda \Phi +k_\lambda \nabla _\mu \Phi )\right] \left[ 2{\Xi }^\mu _{\rho \nu }-\Phi k^\mu (k_\rho \nabla _\nu \Phi +k_\nu \nabla _\rho \Phi )\right] \end{aligned}$$

(A.32)

$$\begin{aligned}&=4{\Xi }^\rho _{\mu \lambda }{\Xi }^\mu _{\rho \nu }. \end{aligned}$$

(A.33)

The combination ${\Xi }^\rho _{\mu \lambda }{\Xi }^\mu _{\rho \nu }$ is,

$$\begin{aligned} 4{\Xi }^\rho _{\mu \lambda }{\Xi }^\mu _{\rho \nu }= & {} \left[ (\nabla _\mu \Phi )S^\rho _\lambda +(\nabla _\lambda \Phi )S^\rho _\mu -(\nabla ^\rho \Phi )S_{\mu \lambda }+\Phi K^\rho _{\mu \lambda }\right] \nonumber \\&\quad \times \,\left[ (\nabla _\rho \Phi )S^\mu _\nu +(\nabla _\nu \Phi ) S^\mu _\rho -(\nabla ^\mu \Phi )S_{\rho \nu }+\Phi K^\mu _{\rho \nu }\right] . \end{aligned}$$

(A.34)

In order to simplify this further, we use the following non-trivial identities, which can be straightforwardly established:

$$\begin{aligned} S^\rho _\mu K^\mu _{\rho \nu }&=0,&S_{\mu \lambda }K^\mu _{\rho \nu }&=0, \end{aligned}$$

(A.35)

$$\begin{aligned} S^{\mu }_{\nu }K^\rho _{\mu \lambda }&=k_\nu n_\lambda {}^\rho ,&K^\rho _{\mu \lambda }K^\mu _{\rho \nu }&=4(\nabla _\mu k^\rho )(\nabla _\rho k^\mu ) l_\lambda l_\nu . \end{aligned}$$

(A.36)

After all these simplifications, we get

$$\begin{aligned} \Omega ^\rho _{\mu \lambda }\Omega ^\mu _{\rho \nu }= & {} -\frac{1}{2} (\nabla _\rho \Phi )(\nabla ^\rho \Phi )k_\lambda k_\nu -\frac{1}{2}\Phi k^\mu \left[ k_\lambda (\nabla _\mu \nabla _\nu \Phi )+k_\nu (\nabla _\mu \nabla _\lambda \Phi )\right] \nonumber \\&\quad +\,\, \Phi ^2(\nabla _\mu k^\rho )(\nabla _\rho k^\mu ) l_\lambda l_\nu . \end{aligned}$$

(A.37)

Therefore, a final simplified expression for the transformed Ricci tensor is

$$\begin{aligned} R'_{\lambda \nu }= & {} R_{\lambda \nu }-l_\lambda \left[ k^\mu (\nabla _\nu \nabla _\mu \Phi )+\Phi \square k_\nu \right] -l_\nu \left[ k^\mu (\nabla _\lambda \nabla _\mu \Phi )+\Phi \square k_\lambda \right] \nonumber \\&+\,\frac{1}{2}(\nabla _\rho \Phi )(\nabla ^\rho \Phi )k_\lambda k_\nu -\Phi ^2(\nabla _\mu k^\rho )(\nabla _\rho k^\mu ) l_\lambda l_\nu . \end{aligned}$$

(A.38)

In the next subsection we show that the right hand side of the Einstein equations (A.1) also transform in the same way.

1.2 Right hand side of Einstein equations

We start by recalling that under generalised Garfinkle–Vachaspati transform the two-form field transforms as

$$\begin{aligned} C \rightarrow C'=C-\Phi \ k_\mu dx^\mu \wedge l_\nu dx^\nu . \end{aligned}$$

(A.39)

To show that the right hand side of the Einstein equations (A.1) transform in the same way, we require

$$\begin{aligned} k^\mu {F_\mu }^{\nu \rho }=-n^{\nu \rho }, \end{aligned}$$

(A.40)

and

$$\begin{aligned} l^\mu {F_\mu }^{\nu \rho }=0. \end{aligned}$$

(A.41)

As mentioned in the main text, these conditions are satisfied by a large class of solutions of the minimal six-dimensional supergravity embedded in type IIB theory. Introducing the notation

$$\begin{aligned} m_{\mu \nu }= & {} k_\mu l_\nu - k_\nu l_\mu , \end{aligned}$$

(A.42)

we have

$$\begin{aligned} C'_{\mu \nu }= & {} C_{\mu \nu }-\Phi (k_\mu l_\nu -k_\nu l_\mu ) \end{aligned}$$

(A.43)

$$\begin{aligned}= & {} C_{\mu \nu }-\Phi m_{\mu \nu }. \end{aligned}$$

(A.44)

It then simply follows that

$$\begin{aligned} F'_{\mu \nu \rho }= & {} \partial _\mu C_{\nu \rho }+\partial _\rho C_{\mu \nu }+\partial _\nu C_{\rho \mu }-\partial _\mu (\Phi m_{\nu \rho })-\partial _\rho (\Phi m_{\mu \nu })\nonumber \\&-\,\partial _\nu (\Phi m_{\rho \mu }) \end{aligned}$$

(A.45)

$$\begin{aligned}= & {} \partial _\mu C_{\nu \rho }+\partial _\rho C_{\nu \mu }+\partial _\mu C_{\rho \nu }-Q_{\mu \nu \rho }-\Phi P_{\mu \nu \rho } \end{aligned}$$

(A.46)

$$\begin{aligned}= & {} F_{\mu \nu \rho }-Q_{\mu \nu \rho }-\Phi P_{\mu \nu \rho }, \end{aligned}$$

(A.47)

where

$$\begin{aligned} Q_{\mu \nu \rho }= & {} (\partial _\mu \Phi )m_{\nu \rho }+(\partial _\rho \Phi ) m_{\mu \nu }+(\partial _\nu \Phi )m_{\rho \mu }, \end{aligned}$$

(A.48)

$$\begin{aligned} P_{\mu \nu \rho }= & {} \partial _\mu m_{\nu \rho }+\partial _\rho m_{\mu \nu }+\partial _\nu m_{\rho \mu }. \end{aligned}$$

(A.49)

Inserting (A.42) in (A.49) we get,

$$\begin{aligned} P_{\mu \nu \rho }= & {} \partial _\mu (k_\nu l_\rho - k_\rho l_\nu )+\partial _\rho (k_\mu l_\nu - k_\nu l_\mu )+\partial _\nu (k_\rho l_\mu - k_\mu l_\rho ) \end{aligned}$$

(A.50)

$$\begin{aligned}= & {} (\partial _\mu k_\nu -\partial _\nu k_\mu )l_\rho +(\partial _\rho k_\mu -\partial _\mu k_\rho )l_\nu +(\partial _\nu k_\rho -\partial _\rho k_\nu )l_\mu \end{aligned}$$

(A.51)

$$\begin{aligned}= & {} n_{\mu \nu }l_\rho +n_{\rho \mu }l_\nu +n_{\nu \rho }l_\mu . \end{aligned}$$

(A.52)

To compute the transformed right hand side of the Einstein equations, we need to first raise the indices on the three-form field $F_{\mu \nu \lambda }$. Raising the first index we get,

$$\begin{aligned} {F'^\sigma }_{\nu \rho }= & {} g'^{\mu \sigma }F'_{\mu \nu \rho } \end{aligned}$$

(A.53)

$$\begin{aligned}= & {} \left( g^{\mu \sigma }+\Phi ^2 k^\mu k^\sigma -\Phi S^{\mu \sigma }\right) \left( F_{\mu \nu \rho }-Q_{\mu \nu \rho }-\Phi P_{\mu \nu \rho }\right) . \end{aligned}$$

(A.54)

Using the identities,

$$\begin{aligned} k^\mu Q_{\mu \nu \rho }= & {} 0, \end{aligned}$$

(A.55)

$$\begin{aligned} k^\mu P_{\mu \nu \rho }= & {} 0, \end{aligned}$$

(A.56)

$$\begin{aligned} S^{\mu \sigma } F_{\mu \nu \rho }= & {} -l^\sigma n_{\nu \rho }, \end{aligned}$$

(A.57)

$$\begin{aligned} S^{\mu \sigma }Q_{\mu \nu \rho }= & {} k^\sigma \left[ k_\rho (\partial _\nu \Phi ) -k_\nu (\partial _\rho \Phi )\right] , \end{aligned}$$

(A.58)

$$\begin{aligned} S^{\mu \sigma }P_{\mu \nu \rho }= & {} k^\sigma n_{\nu \rho }, \end{aligned}$$

(A.59)

it follows that,

$$\begin{aligned} {F'^\sigma }_{\nu \rho }&= {F^\sigma }_{\nu \rho }-{Q^\sigma }_{\nu \rho }-\Phi {P^\sigma }_{\nu \rho }+\Phi l^\sigma n_{\nu \rho }+\Phi k^\sigma \left[ (\partial _\nu \Phi )k_\rho -(\partial _\rho \Phi )k_\nu \right] . \end{aligned}$$

(A.60)

Similarly raising the second index we get,

$$\begin{aligned} {F'^{\sigma \eta }}_\rho= & {} {g'}^{\eta \nu }{F'^\sigma }_{\nu \rho }\nonumber \\= & {} {F^{\sigma \eta }}_\rho -{Q^{\sigma \eta }}_\rho -\Phi {P^{\sigma \eta }}_\rho +\Phi l^\sigma {n^\eta }_\rho +\Phi k^\sigma \left[ (\partial ^\eta \Phi )k_\rho -(\partial _\rho \Phi )k^\eta \right] \nonumber \\&-\,\Phi l^\eta ({n^\sigma }_\rho )-\Phi k^\eta \left[ (\partial ^\sigma \Phi )-k^\sigma (\partial _\rho \Phi )\right] . \end{aligned}$$

(A.61)

Given the above expressions, it is possible to compute the change in the right hand side of the Einsteins equations. However, it turns out that for various purposes the three-form with all three indices raised is a much easier quantity to work with. We now write an expression for $F'$ with all three indices raised, and then turn to Einstein equations. We have

$$\begin{aligned} F'^{\sigma \eta \alpha }&=g'^{\rho \alpha }{F'^{\sigma \eta }}_\rho \end{aligned}$$

(A.62)

$$\begin{aligned}&=F^{\sigma \eta \alpha }-Q^{\sigma \eta \alpha }-\Phi P^{\sigma \eta \alpha }+\Phi l^\sigma n^{\eta \alpha }+\Phi k^\sigma \left[ (\partial ^\eta \Phi )k^\alpha -(\partial ^\alpha \Phi )k^\eta \right] \nonumber \\&\quad -\,\Phi l^\eta (n^{\sigma \alpha })-\Phi k^\eta \left[ k^\alpha (\partial ^\sigma \Phi )-k^\sigma (\partial ^\alpha \Phi )\right] )+\Phi ^2 k^\alpha k^\rho {F^{\sigma \eta }}_\rho -\Phi S^{\alpha \rho }{F^{\sigma \eta }}_\rho \nonumber \\&\quad +\,\Phi S^{\alpha \rho }{Q^{\sigma \eta }}_\rho +\Phi ^2 S^{\alpha \rho }{P^{\sigma \eta }}_\rho \end{aligned}$$

(A.63)

$$\begin{aligned}&=F^{\sigma \eta \alpha }-Q^{\sigma \eta \alpha }-\Phi (n^{\sigma \eta }l^\alpha +n^{\alpha \sigma }l^\eta +n^{\eta \alpha }l^\sigma ) +\Phi l^\sigma n^{\eta \alpha }\nonumber \\&\quad +\,\Phi k^\sigma \left[ (\partial ^\eta \Phi )k^\alpha -(\partial ^\alpha \Phi ) k^\eta \right] -\Phi l^\eta (n^{\sigma \alpha })-\Phi k^\eta \left[ k^\alpha (\partial ^\sigma \Phi )-k^\sigma (\partial ^\alpha \Phi )\right] )\nonumber \\&\quad +\,\Phi l^\alpha (n^{\sigma \eta }) +\Phi k^\alpha \left[ k^\eta (\partial ^\sigma \Phi )-k^\sigma (\partial ^\eta \Phi )\right] \nonumber \\&=F^{\sigma \eta \alpha }-Q^{\sigma \eta \alpha }+\Phi k^\sigma \left[ (\partial ^\eta \Phi )k^\alpha -(\partial ^\alpha \Phi )k^\eta \right] -\Phi k^\eta \left[ k^\alpha (\partial ^\sigma \Phi )-k^\sigma (\partial ^\alpha \Phi )\right] )\nonumber \\&\quad +\,\Phi k^\alpha \left[ k^\eta (\partial ^\sigma \Phi )-k^\sigma (\partial ^\eta \Phi )\right] \end{aligned}$$

(A.64)

$$\begin{aligned}&=F^{\sigma \eta \alpha }-Q^{\sigma \eta \alpha }, \end{aligned}$$

(A.65)

which is a remarkably simple equation.

Now we are in position to compute the transformed right hand side of (A.1). Using identities

$$\begin{aligned} -F_{\lambda \alpha \beta }{Q}^{\delta \alpha \beta }-Q_{\lambda \alpha \beta } {F}^{\delta \alpha \beta }= & {} -4\left[ l^\delta (\nabla _\lambda \nabla _\beta \Phi ) +l_\lambda (\nabla ^\delta \nabla _\beta \Phi )\right] k^\beta , \end{aligned}$$

(A.66)

$$\begin{aligned} Q_{\lambda \alpha \beta }{Q}^{\delta \alpha \beta }= & {} 2(\partial _\beta \Phi )(\partial ^\beta \Phi )k_\lambda k^\delta , \end{aligned}$$

(A.67)

$$\begin{aligned} P_{\lambda \alpha \beta }{Q}^{\delta \alpha \beta }= & {} 4 k^\delta k^\alpha (\nabla _\lambda \nabla _\alpha \Phi ), \end{aligned}$$

(A.68)

$$\begin{aligned} P_{\lambda \alpha \beta }F^{\delta \alpha \beta }= & {} 4l_\lambda \square k^\delta , \end{aligned}$$

(A.69)

we get,

$$\begin{aligned} \frac{1}{4}F'_{\lambda \alpha \beta }F'^{\delta \alpha \beta }= & {} \frac{1}{4}F_{\lambda \alpha \beta }F^{\delta \alpha \beta }-[l^\delta (\nabla _\lambda \nabla _\beta \Phi ) +l_\lambda (\nabla ^\delta \nabla _\beta \Phi )]k^\beta \nonumber \\&+\,\frac{1}{2}(\nabla _\beta \Phi )(\nabla ^\beta \Phi )k_\lambda k^\delta +\Phi k^\delta k^\alpha (\nabla _\lambda \nabla _\alpha \Phi )-\Phi l_\lambda \square k^\delta .\quad \end{aligned}$$

(A.70)

From this expression we easily see that $F'_{\lambda \alpha \beta }F'^{\lambda \alpha \beta } = F_{\lambda \alpha \beta }F^{\lambda \alpha \beta } =0$. Moreover,

$$\begin{aligned} \frac{1}{4}g'_{\nu \delta } F'_{\lambda \alpha \beta }F'^{\delta \alpha \beta }= & {} \frac{1}{4}(g_{\nu \delta } +\Phi S_{\nu \delta }) F'_{\lambda \alpha \beta }F'^{\delta \alpha \beta } \end{aligned}$$

(A.71)

$$\begin{aligned}= & {} \frac{1}{4}F_{\lambda \alpha \beta }{F_\nu }^{\alpha \beta }-[l_\nu (\nabla _\lambda \nabla _\mu \Phi ) +l_\lambda (\nabla _\nu \nabla _\mu \Phi )]k^\mu \nonumber \\&+\frac{1}{2}(\nabla _\rho \Phi )(\nabla ^\rho \Phi )k_\lambda k_\nu \nonumber \\&-\,\Phi l_\lambda \square k_\nu -\Phi l_\nu \square k_\lambda +\Phi ^2 l_\lambda l_\nu (\nabla ^\alpha k_\delta )(\nabla _\alpha k^\delta ), \end{aligned}$$

(A.72)

where we have used the identities

$$\begin{aligned} F_{\lambda \alpha \beta }n^{\alpha \beta }= & {} 4\square k_\lambda , \end{aligned}$$

(A.73)

$$\begin{aligned} S_{\nu \delta }\square k^\delta= & {} l_\nu k_\delta \square k^\delta . \end{aligned}$$

(A.74)

We see that the right hand side matches with the left hand side.

1.3 Matter field equations

The matter field equations are

$$\begin{aligned} \nabla _\mu F^{\mu \nu \rho }= & {} 0. \end{aligned}$$

(A.75)

Under the deformation the left hand side of this equation changes as

$$\begin{aligned} \nabla '_\mu F'^{\mu \nu \rho }= & {} \nabla _\mu F'^{\mu \nu \rho }+\Omega ^\mu _{\mu \sigma }F'^{\sigma \nu \rho } +\Omega ^\nu _{\mu \sigma }F'^{\mu \sigma \rho }+\Omega ^\rho _{\mu \sigma } F'^{\mu \nu \sigma } \end{aligned}$$

(A.76)

$$\begin{aligned}= & {} \nabla _\mu F'^{\mu \nu \rho } \end{aligned}$$

(A.77)

$$\begin{aligned}= & {} \nabla _\mu F^{\mu \nu \rho }-\nabla _\mu Q^{\mu \nu \rho }. \end{aligned}$$

(A.78)

The first term in Eq. (A.78) is just the field equations for the background configuration, which is zero. For the second term in (A.78), we have via (A.48)

$$\begin{aligned} Q^{\mu \nu \rho }= & {} g^{\mu \sigma }g^{\nu \eta }g^{\rho \alpha }Q_{\sigma \eta \alpha } \end{aligned}$$

(A.79)

$$\begin{aligned}= & {} (\nabla ^\mu \Phi )m^{\nu \rho } +(\nabla ^\nu \Phi )m^{\rho \mu }+(\nabla ^\rho \Phi )m^{\mu \nu }. \end{aligned}$$

(A.80)

Applying the covariant $\nabla _\mu$ on this expression we find,

$$\begin{aligned} \nabla _\mu Q^{\mu \nu \rho }= & {} (\Box \Phi ) m^{\nu \rho }+(\nabla ^\mu \Phi ) \left[ l^\rho (\nabla _\mu k^\nu )-l^\nu (\nabla _\mu k^\rho )\right] +(\nabla _\mu \nabla ^\nu \Phi )(k^\rho l^\mu - k^\mu l^\rho )\nonumber \\&+\,(\,\nabla _\mu \nabla ^\rho \Phi )(k^\mu l^\nu - k^\nu l^\mu ). \end{aligned}$$

(A.81)

Using

$$\begin{aligned} \Box \Phi= & {} 0, \end{aligned}$$

(A.82)

$$\begin{aligned} l^\mu (\nabla _\mu \nabla ^\nu \Phi )= & {} 0, \end{aligned}$$

(A.83)

$$\begin{aligned} k^\mu (\nabla _\mu \nabla ^\nu \Phi )= & {} (\nabla ^\mu \Phi )(\nabla _\mu k^\nu ), \end{aligned}$$

(A.84)

we get

$$\begin{aligned} \nabla '_\mu F'^{\mu \nu \rho }= & {} \nabla _\mu Q^{\mu \nu \rho }~=~0. \end{aligned}$$

(A.85)

Hence the matter field equations are also satisfied by the transformed configuration.

We have shown that under the generalised Garfinkle–Vachaspati transform, solutions of IIB theory are mapped to solutions of IIB theory.

BW and GMR formalisms

In this appendix, after a brief review of the Gutowski–Martelli–Reall (GMR) and the Bena–Warner (BW) formalisms we relate the two notations. Similar computations were also done in [42,43,44].

1.1 Gutowski–Martelli–Reall formalism

In the GMR formalism [17], we work with minimal six-dimensional supergravity. We follow the notation of appendix A of reference [25]. The bosonic part of this theory consists of metric $g_{\mu \nu }$ and a self-dual three-form $G_{\mu \nu \rho }$. GMR showed that the metric for any supersymmetric solution of minimal 6D supergravity can be written as

$$\begin{aligned} ds^2 = -H^{-1}(dv +\beta )\left( du + \omega + \frac{\mathcal {F}}{2}(dv + \beta )\right) + H h_{mn}dx^m dx^n, \end{aligned}$$

(B.1)

where $h_{mn}$ is a metric on a four-dimensional almost hyper-Kähler base manifold, $\beta$ and $\omega$ are one-forms on this base space, while $\mathcal {F}$ and H are functions on the base space.

In general, the above metric only has

$$\begin{aligned} k = \frac{\partial }{\partial u}, \end{aligned}$$

(B.2)

as the null Killing vector, i.e., $h_{mn}$, $\beta$, $\omega$ $\mathcal {F}$ and H can be v-dependent. However, to compare with the Bena–Warner formalism [15], we must restrict to v-independent solutions. For this case, the six-dimensional field strength G takes the form

$$\begin{aligned} F \ = \ 2G&= \star dH - H^{-1}(dv + \beta ) \wedge \left( \frac{d\omega -\star d\omega }{2}\right) \nonumber \\&+\,H^{-1}\left( du + \omega + \frac{\mathcal {F}}{2}(dv + \beta )\right) \wedge \left( d\beta +H^{-1} (dv + \beta )\wedge dH \right) . \end{aligned}$$

(B.3)

A detailed analysis of the Killing spinor equations shows that the equations of motion then reduce to

$$\begin{aligned} \star d\star d\mathcal {F}- \frac{1}{2}({\mathcal {G}}^{+})^2= & {} 0, \end{aligned}$$

(B.4)

$$\begin{aligned} d\star dH + \frac{d\beta \wedge {\mathcal {G}}^{+} }{2}= & {} 0, \end{aligned}$$

(B.5)

$$\begin{aligned} d\beta - \star d\beta= & {} 0, \end{aligned}$$

(B.6)

$$\begin{aligned} d{\mathcal {G}}^{+}= & {} 0. \end{aligned}$$

(B.7)

In these equations, the Hodge star is with respect to 4-dimensional base metric $h_{\mu \nu }$ and self-dual two-form ${\mathcal {G}}^{+}$ is defined as

$$\begin{aligned} {\mathcal {G}}^{+} = \frac{1}{2H} \left( d\omega + \star d\omega + \mathcal {F}d\beta \right) . \end{aligned}$$

(B.8)

We also note that $\star d\star d\mathcal {F}= -\nabla ^2 \mathcal {F}$ and $({\mathcal {G}}^{+})^2 = ({\mathcal {G}}^{+})^{mn}({\mathcal {G}}^{+})_{mn}$.

1.2 Bena–Warner formalism

Bena and Warner [15] showed that solutions preserving same supersymmetries as those of three charge black holes and black ring can be written in a general form with one forms defined on a four dimensional hyper-Kähler base space. Their formalism is simplest and most symmetric in the M-theory form, with branes intersecting on the six-torus with coordinates $(z_1, \ldots ,z_6)$ as M2(12)–M2(34)–M2(56). We refer the reader to the review [9] for further details on brane-intersection. The metric in eleven-dimensions takes the following symmetrical form,

$$\begin{aligned} ds_{11}^2 = ds_5^2+ ds^2_{\mathrm {T}^6}, \end{aligned}$$

(B.9)

where $ds^2_{\mathrm {T}^6}$ is metric on the six-torus,

$$\begin{aligned} ds^2_{\mathrm {T}^6} = (Z_2 Z_3 Z_1^{-2})^{\frac{1}{3}}(dz_1^2+dz_2^2) +(Z_1Z_3Z_2^{-2})^{\frac{1}{3}}(dz_3^2+dz_4^2) +(Z_1Z_2Z_3^{-2})^{\frac{1}{3}}(dz_5^2+dz_6^2),\nonumber \\ \end{aligned}$$

(B.10)

and $ds_5^2$ is the metric on five-dimensional transverse spacetime,

$$\begin{aligned} ds_5^2=-(Z_1Z_2Z_3)^{-\frac{2}{3}}(dt+\kappa )^2+(Z_1Z_2Z_3)^{\frac{1}{3}} h_{mn}dx^m dx^n, \end{aligned}$$

(B.11)

where $h_{mn}$ is the metric on a 4-dimensional hyper-Kähler base space.

The M-theory three-form potential ${\mathcal {A}}$ for this class of solutions can be written in terms of three one-form potentials $A^{(I)}$ on the five-dimensional spacetime,

$$\begin{aligned} {\mathcal {A}}=A^{(1)}\wedge dz_1\wedge dz_2+A^{(2)}\wedge dz_3\wedge dz_4+A^{(3)}\wedge dz_5\wedge dz_6, \end{aligned}$$

(B.12)

which in turn take the form,

$$\begin{aligned} A^{(I)} = -\frac{(dt+\kappa )}{Z_I}+\omega _I, \end{aligned}$$

(B.13)

where $\kappa$ and $\omega _I$ are one-forms on the four-dimensional base space while $Z_I$ are functions on the base space. These functions and one-forms are determined by the BW equations [15]:

$$\begin{aligned} d\omega _{I}= & {} \star d\omega _I, \end{aligned}$$

(B.14)

$$\begin{aligned} d\kappa + \star d\kappa= & {} Z_I d\omega _I, \end{aligned}$$

(B.15)

$$\begin{aligned} \nabla ^2 Z_I= & {} \frac{1}{2}|\epsilon _{IJK} | \star (d\omega _J \wedge d\omega _K), \end{aligned}$$

(B.16)

where the Hodge star is with respect to the four-dimensional base metric $h_{mn}$.

To compare with the GMR formalism, we convert from the M-theory form to the type IIB D1–D5–P form using dualities and dimensional reduction (later we will truncate to six-dimensional minimal supergravity). Performing a dimensional reduction along the $z_6$-direction we can go from M-theory to type-IIA theory with the metric of a D2(12)–D2(34)–F1(5) brane intersection. The resulting IIA metric in the string frame is,

$$\begin{aligned} ds_{10}^2=- & {} \frac{1}{Z_3\sqrt{Z_1Z_2}} (dt+\kappa )^2+\sqrt{Z_1Z_2}h_{mn}dx^m dx^n \nonumber \\&+\,\sqrt{\frac{Z_2}{Z_1}}(dz_1^2+dz_2^2)+\sqrt{\frac{Z_1}{Z_2}} (dz_3^2+dz_4^2)+\frac{\sqrt{Z_1Z_2}}{Z_3}dz_5^2, \end{aligned}$$

(B.17)

with IIA dilaton,

$$\begin{aligned} e^{2\phi }=\frac{\sqrt{Z_1Z_2}}{Z_3}, \end{aligned}$$

(B.18)

and with three-form RR field,

$$\begin{aligned} C_{\mu z_1 z_2}= & {} A^{(1)}_\mu , \end{aligned}$$

(B.19)

$$\begin{aligned} C_{\mu z_3 z_4}= & {} A^{(2)}_\mu , \end{aligned}$$

(B.20)

and two-form NS–NS B-field,

$$\begin{aligned} B_{\mu z_5}= & {} A^{(3)}_\mu . \end{aligned}$$

(B.21)

Next we need to perform T-dualities along $z_3,z_4$ and $z_5$ directions to get D5(12345)–D1(5)–P(5) system. We recall the T-duality rules for a duality along z-direction:

$$\begin{aligned} G'_{zz}= & {} \frac{1}{G_{zz}}, \end{aligned}$$

(B.22)

$$\begin{aligned} G'_{\mu z}= & {} \frac{B_{\mu z}}{G_{zz}}, \end{aligned}$$

(B.23)

$$\begin{aligned} G'_{\mu \nu }= & {} G_{\mu \nu }-\frac{G_{\mu z}G_{\nu z}-B_{\mu z} B_{\nu z}}{G_{zz}}, \end{aligned}$$

(B.24)

$$\begin{aligned} B'_{\mu z}= & {} \frac{G_{\mu z}}{G_{zz}}, \end{aligned}$$

(B.25)

$$\begin{aligned} B'_{\mu \nu }= & {} B_{\mu \nu }-\frac{B_{\mu z}G_{\nu z} -G_{\mu z}B_{\nu z}}{G_{zz}}, \end{aligned}$$

(B.26)

$$\begin{aligned} e^{2 \phi '}= & {} \frac{e^{2\phi }}{G_{zz}}, \end{aligned}$$

(B.27)

$$\begin{aligned} {C'}^{(n)}_{\mu \ldots \nu \alpha z}= & {} {C}^{(n-1)}_{\mu \ldots \nu \alpha }-(n-1)\frac{{C}^{(n-1)}_{[\mu \ldots \nu | z}G_{|\alpha ]z}}{G_{zz}}, \end{aligned}$$

(B.28)

$$\begin{aligned} C'^{(n)}_{\mu \ldots \nu \alpha \beta }= & {} C^{(n+1)}_{\mu \ldots \nu \alpha \beta z}+nC^{(n-1)}_{[\mu \ldots \nu \alpha } B_{\beta ]z}+n (n-1)\frac{{C}^{(n-1)}_{[\mu \ldots \nu | z}B_{|\alpha |z}G_{|\beta ]z}}{G_{zz}}. \end{aligned}$$

(B.29)

We perform the required dualities in two steps. Performing T-dualities along $z_3,z_4$ directions we get the following fields:

$$\begin{aligned} ds_{10}^2=- & {} \frac{1}{Z_3\sqrt{Z_1Z_2}}(dt+\kappa )^2+\sqrt{Z_1Z_2}h_{mn}dx^m dx^n \nonumber \\&+\,\sqrt{\frac{Z_2}{Z_1}}(dz_1^2+dz_2^2+dz_3^2+dz_4^2) +\frac{\sqrt{Z_1Z_2}}{Z_3}dz_5^2, \end{aligned}$$

(B.30)

$$\begin{aligned} e^{2\phi }= & {} \frac{Z_2^{3/2}}{Z_3\sqrt{Z_1}}, \end{aligned}$$

(B.31)

$$\begin{aligned} C^{(5)}_{\mu z_1 z_2 z_3 z_4}= & {} A^{(1)}_\mu , \qquad C^{(1)}_\mu = -A^{(2)}_\mu , \qquad B_{\mu z_5}=A^{(3)}_\mu . \end{aligned}$$

(B.32)

Now doing T-duality along $z_5$-direction, we get our required D1–D5–P configuration. The IIB dilaton reads:

$$\begin{aligned} e^{2\phi }=\frac{Z_2}{Z_1}, \end{aligned}$$

(B.33)

and the metric takes the form,

$$\begin{aligned} ds_{10}^2= & {} -\frac{1}{Z_3\sqrt{Z_1Z_2}}(dt+\kappa )^2+\sqrt{Z_1Z_2}h_{mn}dx^m dx^n \nonumber \\&+\,\frac{Z_3}{\sqrt{Z_1Z_2}}\left( dz_5+A^{(3)}_\mu dx^\mu \right) ^2+\sqrt{\frac{Z_2}{Z_1}}\left( dz_1^2 +dz_2^2+ dz_3^2 + d z_4^2\right) , \qquad \qquad \end{aligned}$$

(B.34)

together with the associated RR-field components,

$$\begin{aligned} C^{(6)}= & {} A^{(1)}_\mu dx^\mu \wedge dx^1\wedge dx^2\wedge dx^3\wedge dx^4\wedge dx^5 \nonumber \\&+\,A^{(1)}_\mu A^{(3)}_\nu dx^\mu \wedge dx^\nu \wedge dx^1\wedge dx^2\wedge dx^3\wedge dx^4, \nonumber \\ C^{(2)}= & {} -A^{(2)}_\mu dx^\mu \wedge dx^5- A^{(2)}_\mu A^{(3)}_\nu dx^\mu \wedge dx^\nu . \end{aligned}$$

(B.35)

We can dualize the 6-form potential to get a 2-form potential. This is a tedious step. Fortunately, we do not need to do this electromagnetic duality. Comparing metric (B.34) to the GMR form, we obtain a complete dictionary between the GMR and the BW variables. Using this dictionary we can convert the GMR form of the field strength (B.3) into the BW variables. We expect the electromagnetic duality to give the same result.

Since GMR formalism is for minimal six-dimensional supergravity, in order to compare the above configuration with the GMR form we must set $Z_1=Z_2$. In that case, the dilaton vanishes $e^{2\phi }=1$. Inserting $A^{(3)}_\mu dx^\mu$ from (B.13) in metric (B.34) we get,

$$\begin{aligned} ds_{10}^2=- 2 Z_1^{-1}(dt + \kappa )(dz_5 +\omega _3) + Z_3 Z_1^{-1}(d z_5+\omega _3 )^2+Z_1 h_{mn}dx^m dx^n + ds_{\mathrm {T}_4}^2, \end{aligned}$$

(B.36)

where

$$\begin{aligned} ds_{\mathrm {T}_4}^2 = dz_1^2 +dz_2^2+ dz_3^2 + d z_4^2, \end{aligned}$$

(B.37)

is the metric on the four-torus. To match with the GMR form (B.1), we identify

$$\begin{aligned}&z_5 ~=~v, \quad Z_1 ~=~H, \nonumber \\&Z_3 ~=~1-\frac{\mathcal {F}}{2},\quad \omega _3 ~=~ \beta , \nonumber \\&\kappa ~=~\frac{\beta + \omega }{2}, \quad t~=~ \frac{u+v}{2}. \end{aligned}$$

(B.38)

Using the identification (B.38) in the GMR field strength (B.3), we get

$$\begin{aligned} G= & {} \frac{1}{2} \star dZ_1-\frac{1}{4Z_1} (dz_5 +\omega _3)\wedge [d\kappa -\star d\kappa ]\nonumber \\&+\frac{1}{2Z_1}\left[ (dt+\kappa )-\frac{Z_3}{2}(dz_5+\omega _3)\right] \wedge d\omega _3 \nonumber \\&-\frac{1}{2Z_1^2}(dz_5+\omega _3)\wedge (dt+\kappa ) \wedge dZ_1, \end{aligned}$$

(B.39)

which using the BW equations of motion simplifies to

$$\begin{aligned} 2G= & {} \star dZ_1 + d \left[ (dz_5+\omega _3)]\wedge \left( \frac{dt +\kappa }{Z_1} -\omega _1 \right) \right] + \omega _1 \wedge d\omega _3. \end{aligned}$$

(B.40)

The RR field strength in ten dimensions is normalised as $F = 2G$, with the associated 2-form field

$$\begin{aligned} C=-\left[ \left( \frac{dt+\kappa }{Z_1}-\omega _1\right) \wedge (dz_5+\omega _3)\right] + \sigma , \end{aligned}$$

(B.41)

where an explicit expression for $\sigma$ cannot be obtained in general. It satisfies,

$$\begin{aligned} d\sigma = \star dZ_1+\omega _1\wedge d\omega _3. \end{aligned}$$

(B.42)

One can easily check that the three form $\star dZ_1+\omega _1\wedge d\omega _3$ appearing on the right hand side of equation (B.42) is exact due to BW equations of motion for $Z_1$.

1.3 Relation between GMR and BW

Now that we have a simple dictionary (B.38) we can easily relate BW and GMR equations of motion. On the GMR side, we look at v-independent solutions while on the BW side we consider solutions with $Z_1 =Z_2$ and $\omega _1 =\omega _2$.

We consider BW equations and using the dictionary transform them into GMR equations. Consider BW equation (B.15),

$$\begin{aligned} d\kappa + \star d\kappa = 2 Z_1 d\omega _1 + Z_3d\omega _3. \end{aligned}$$

(B.43)

Rewriting this equation using dictionary (B.38), we have

$$\begin{aligned} 2 d\omega _1= & {} \frac{1}{Z_1}\left( d\kappa + \star d\kappa - Z_3d\omega _3\right) \end{aligned}$$

(B.44)

$$\begin{aligned}= & {} \frac{1}{2H}\left( d\omega + \star d\omega +2 (1 - Z_3)d\beta \right) = \frac{1}{H}\left( d\omega + \star d\omega + \mathcal {F}d\beta \right) = {\mathcal {G}}^+,\nonumber \\ \end{aligned}$$

(B.45)

where we have used the fact that $d\beta =d\omega _3$ is self dual, cf. (B.14). It then immediately follows that $d{\mathcal {G}}^+ =0$, which is one of the GMR equations, cf. (B.7). Similarly, from the BW scalar equations (B.16) for $Z_1$ we have,

$$\begin{aligned} \nabla ^2 Z_1 =\nabla ^2 H = - \star d \star d H = \star (d\omega _3 \wedge d\omega _2) = \star \left( \frac{d\beta \wedge {\mathcal {G}}^{+} }{2}\right) , \end{aligned}$$

(B.46)

which implies (B.5). Similarly,

$$\begin{aligned} \nabla ^2 Z_3 =-\frac{1}{2}\nabla ^2 \mathcal {F}= \star (d\omega _1 \wedge d\omega _2) = \star \left( \frac{{\mathcal {G}}^+ \wedge {\mathcal {G}}^{+} }{4}\right) , \end{aligned}$$

(B.47)

which implies (B.4).