 out of a broader programme of investigating CFTs which 
have been deformed by spatially periodic sources.
The dual gravitational solutions,
known as holographic lattices \cite{Horowitz:2012ky}, provide a natural holographic framework for
studying transport of heat and charge, since the spatially varying sources provide a
mechanism for momentum to dissipate. Holographic lattices are also a powerful framework for
discovering novel holographic ground states with potential applications to condensed matter systems
\cite{Donos:2012js,Donos:2014uba,Gouteraux:2014hca}. The Q lattice construction of
\cite{Donos:2013eha} exploits a global symmetry in the bulk in order to construct holographic lattices by solving a system of ODEs rather than
PDEs. Within the context of the $SO(4)\times SO(4)$ invariant sector of $\mathcal{N}=8$ $D=4$ gauged supergravity,
Q lattice solutions that preserve supersymmetry, hence the name Susy Q, were constructed in \cite{Gauntlett:2018vhk}. 
Associated with the preservation of supersymmetry, it was shown that in the dual deformed ABJM theory the spatially averaged energy 
density is exactly zero.
Another interesting feature of the Susy Q solutions is that they describe boomerang RG flows \cite{Donos:2017ljs}, flowing from 
the $AdS_4$ vacuum in the UV to exactly the
same $AdS_4$ vacuum in the IR \cite{Chesler:2013qla}.

Another special class of known solutions is the $D=11$ Janus-type solutions of \cite{DHoker:2009lky,Bobev:2013yra}, which
includes the special case with $m(\sigma)\sim \delta(\sigma)$. While for generic $m(\sigma)$ a two-dimensional Poincar\'e invariance
is preserved, the Janus solutions are distinguished in preserving two-dimensional conformal invariance. The dual gravitational
solutions can therefore be constructed by foliating the spacetime with $AdS_3$ factors. The solutions
in \cite{DHoker:2009lky} were constructed directly in $D=11$ supergravity, while the solutions in \cite{Bobev:2013yra} of relevance here 
were constructed using the $SO(4)\times SO(4)$ invariant sector of $\mathcal{N}=8$ $D=4$ gauged supergravity.

In this paper we will generalise these solutions by
constructing infinite classes of new supersymmetric solutions of $D=11$ supergravity
that are associated with more general spatial deformations of ABJM theory. 
Remarkably, 
the entire problem reduces to solving
the Helmholtz equation\footnote{More precisely, the Helmholtz equation with imaginary value for the wave-number, also known as the ``homogeneous screened Poisson equation"; see\eqref{helm}} on the complex plane.  For example, the Susy Q solution 
is associated with a solution to the Helmholtz equation on a circular disc that vanishes
on the edge of the disc and with a delta function source at the centre of the disc. This construction
can simply be extended to construct infinite classes of solutions with periodic spatial deformations
that are also boomerang RG flows. 


The plan of the paper is as follows. In section \ref{sec2} we present the BPS equations and show that they reduce to
solving the Helmholtz equation on the complex plane. In order to get a quick feel for the BPS equations
we show how to recover the dielectric RG flow solution, the Janus solution and the Susy Q solution in section \ref{sec3}.
Section \ref{gencoms} contains some general analysis of the BPS equations and section \ref{sec5} concentrates on solutions
associated with boomerang RG flows. 
Section \ref{sec6} concludes with some discussion. 
Appendix \ref{appa} contains some details on the derivation of the BPS equations, in a more general setting,
and appendix \ref{sec:HolographicOnePointFunctions} has some details on the holographic calculation of the sources and VEVs in the dual ABJM theory. Appendix \ref{sec:dist} contains some details of how to deal with solutions with distributional sources as in, for example, the Janus solution.
Finally, appendix \ref{simpsol} discusses some additional simple analytic solutions to the BPS equations, including a discussion
of the sources and VEVs for the Janus solution.



\section{BPS equations}\label{sec2}
Our starting point is the $SO(4)\times SO(4)$ invariant sector of $\mathcal{N}=8$ gauged supergravity in $D=4$. 
The bosonic content 
consists of a metric coupled to a complex scalar field, $z$, which parametrises the Poincar\'e disc with $|z|<1$ and
action given by
\begin{equation}\label{eq:Sg2bulk}
S= \int\mathrm{d}^4 x \sqrt{-g}\Big(R - \frac{2}{(1-|z|^2)^2}\partial_\mu z \partial^\mu \bar{z}
+\frac{2(3-|z|^2)}{1-|z|^2}\Big)\,.
\end{equation}
Any solution to the equations of motion gives rise to an exact solution of $D=11$ supergravity
after uplifting on the seven sphere using the formulae in \cite{Cvetic:1999au}. By taking a $\mathbb{Z}_q$ quotient
of the seven-sphere one obtains solutions relevant to ABJM theory.

We are interested in spatial deformations of ABJM theory that depend on one of the two spatial coordinates and
maintains Poincar\'e invariance in the remaining two directions, which we label with coordinates $(t,y)$. 
The ansatz we shall consider is therefore given by
\begin{align}\label{Susyqansatz}
ds^2&=e^{2A}(-dt^2+dy^2)+e^{2V} dx^2 +N^2dr^2\,,
\end{align}
with $A$, $V$, $N$, and the scalar field $z$ all functions of $(x,r)$.
There is some residual coordinate freedom, involving 
$(x,r)$ and maintaining the form of the ansatz \eqref{Susyqansatz}, that we return to below.
The $AdS_4$ vacuum solution
has $e^A=e^V=r$, $N=1/r$ and $z=0$ and uplifts to the maximally supersymmetric $AdS_4\times S^7$ solution. 
We also note that the Susy Q solutions constructed in \cite{Gauntlett:2018vhk} have $z=f(r)e^{ikx}$ with $A$, $V$ and
$N$ all independent of $x$. If, in addition, we set $k=0$ and $A=V$ then we recover the ansatz for the
1/2 supersymmetric and Poincar\'e invariant dielectric RG flows of \cite{Pope:2003jp}.


We demand that the solutions preserve (at least) 1/4 of the supersymmetry. Some details
of this analysis is presented in appendix \ref{appa}, and here we just summarise the main results.
In the $SO(4)\times SO(4)\subset SO(8)$ sector we can decompose the Majorana supersymmetry parameters
into $\epsilon^a_{(M)}$ and $\epsilon^s_{(M)}$, transforming in the ${\bf (4,1)}$ and ${\bf (1,4)}$, respectively.
Using the orthonormal frame $(e^{\hat{t}},e^{\hat{x}},e^{\hat{y}},e^{\hat{r}})= 
(e^A dt, e^V dx, e^A dy, Ndr)$, the Killing spinors 
satisfy the following projections
\begin{align}
\Gamma^{\hat{t}\hat{y}}\epsilon^a_{(M)}&= +\epsilon^a_{(M)}\,,\qquad\left[-\sin\xi\,\Gamma^{\hat{x}}+\cos \xi\,\Gamma^{\hat{r}}\right]\epsilon^a_{(M)} = -\epsilon^a_{(M)} \,,\notag \\
\Gamma^{\hat{t}\hat{y}}\epsilon^s_{(M)}&= -\epsilon^s_{(M)}\,,\qquad 
\left[-\sin\xi\,\Gamma^{\hat{x}}+\cos \xi\,\Gamma^{\hat{r}}\right]\epsilon^s_{(M)} = -\epsilon^s_{(M)}\,,
\end{align}
with the functional dependence given in \eqref{fundepks}.
The associated BPS equations are given by
\begin{align}\label{bpsderxi1}
(1-|z|^2)^{1/2}\left(N^{-1}\partial_r A-ie^{-V}\partial_x A\right)&=e^{i\xi}\,,\notag \\
\left(ie^{-V}\partial_x z+N^{-1}\partial_r z\right)e^{i\xi}+(1-|z|^2)^{1/2}z&=0\,,\notag \\
