\section{Introduction} \label{intro}
Black holes have proven to be an excellent testing ground for theories
of gravity.  In particular, one of the recent exciting developments
in string theory has been the reproduction of many of the macroscopic
black hole properties from the microscopic D-brane picture---for reviews
and references, see e.g.~\cite{horrev,maldrev}.  At the same time, one
of the current puzzles is the failed attempt in~\cite{dps} to
obtain, from a
microscopic calculation,
the macroscopic scattering of a D-string probe off a five-dimensional
supersymmetric black hole carrying the maximum three charges.
Specifically, the interaction that is quadratic in
the charges was reproduced exactly, but the cubic term, which was seen
in~\cite{us} to be a degeneration of a three point interaction, was not
at all reproduced by the microscopic calculation.

In~\cite{us}, the macroscopic scattering of an arbitrary number of
the triply-charged supersymmetric five-dimensional black holes was given.
A proposal
for a microscopic calculation, based on the just-mentioned observation
of the origin of the three-point interaction, was also given.  In this paper,
scattering of supersymmetric four-dimensional black holes carrying four
charges
will be
discussed.  The motivation here rests on the fact that these
non-singular
black holes can be made purely out of D-brane~\cite{kleb,vjlars,vj,fermald}.
(If a supersymmetric four-dimensional black hole has fewer than four charges,
then it will be singular at the horizon.)
In principle, this makes the microscopic structure more
transparent~\cite{kleb,vjlars}.
This is in distinction to the five-dimensional case,
where despite requiring only three charges for non-singularity at the horizon,
there is no U-dual basis in which the charges are pure D-brane; the
usual description is as collections of parallel 5-branes and strings, with
momentum
along the strings.  The difference in four dimensions is due to the additional
internal direction allowing the conditions for preservation of a
supersymmetry to be satisfied by a more general brane configuration.

In section~\ref{construct} we construct and discuss the black hole solution.
The black hole solution that we use is actually familiar from the heterotic
string---see e.g.~\cite{ct,ct2}.  We rederive the
solution in a way that makes explicit its Type II origin; this complements
the discussion in~\cite{cvj}.
In section~\ref{scatter} we give the effective action
that describes the scattering of several of these black holes, and give
a lengthy discussion of its U-dual generalization.  In particular, the
U-dual formulation of the three-point function is rather technical.
While we only explicitly calculate the effective action for
black holes with four charges, we explain at the end of section~\ref{scatter} why
the U-duality invariant formula should hold for arbitrary supersymmetric
black holes including the black holes of~\cite{ct2} that carry five charges.
In section~\ref{conc} we conclude with a discussion of the scattering of
two black
holes.

\section{The Black Hole Solution} \label{construct}
In~\cite{vj}, black holes were constructed purely out of e.g.\ %
several D-4-branes,
intersecting at arbitrary $U(3)$ angles in the compact torus, and a D-0-brane.
In~\cite{kleb,vjlars,fermald},
the black holes were constructed out of e.g.\ orthogonally
intersecting D-3-branes.  However, because we will be doing the macroscopic
calculation, it will be convenient to use neither of these
descriptions in this paper.  Instead, we would like to find an NS-NS
description of the black holes, so we can use the formulas of~\cite{maharana,%
cvetic} for the dimensionally reduced supergravity lagrangian.
This can be obtained, for example, via the following series of dualities from
the D-3-brane configuration of~\cite{kleb,vjlars,fermald}:%
\footnote{It is also possible to obtain the NS-NS black hole from
the IIA NS-5-brane, D-6-brane and D-2-brane with momentum configuration
of~\cite{jmphd}.}
\begin{eqnarray} \label{dualities}
\mbox{\scriptsize
\begin{tabular}{ccc}
\begin{tabular}{c|llllllllll}
{IIB} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline
{D-3} & X &   &   &   & X & X & X &   &   &   \\
{D-3} & X &   &   &   &   &   & X & X & X &   \\
{D-3} & X &   &   &   & X &   &   &   & X & X \\
{D-3} & X &   &   &   &   & X &   & X &   & X
\end{tabular}
& {\Large $\stackrel{\mbox{\scriptsize T4,T5}}{\longrightarrow}$} &
\begin{tabular}{c|llllllllll}
{IIB} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline
{D-1} & X &   &   &   &   &   & X &   &   &   \\
{D-5} & X &   &   &   & X & X & X & X & X &   \\
{D-3} & X &   &   &   &   & X &   &   & X & X \\
{D-3} & X &   &   &   & X &   &   & X &   & X
\end{tabular}
\\
\end{tabular}} \nonumber \\ \mbox{ \scriptsize \begin{tabular}{ccc}
& & {\Large $\downarrow$} {S} \\
\begin{tabular}{c|llllllllll}
{IIB} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline
{NS-1}& X &   &   &   &   &   & X &   &   &   \\
{ETN} &   & X & X & X &   &   &   &   &   & X \\
{D-1} & X &   &   &   &   &   &   &   & X &   \\
{D-3} & X &   &   &   & X & X &   & X &   &
\end{tabular}
& {\Large $\stackrel{\mbox{\scriptsize T5,T9}}{\longleftarrow}$} &
\begin{tabular}{c|llllllllll}
{IIB}  & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline
{NS-1} & X &   &   &   &   &   & X &   &   &   \\
{NS-5} & X &   &   &   & X & X & X & X & X &   \\
{D-3}  & X &   &   &   &   & X &   &   & X & X \\
{D-3}  & X &   &   &   & X &   &   & X &   & X
\end{tabular}
\\
\end{tabular} } \\ \mbox{\scriptsize \begin{tabular}{ccc}
{T8,T6} {\Large $\downarrow$} \\
\begin{tabular}{c|llllllllll}
{IIB}  & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline
{mom}  &   &   &   &   &   &   & X &   &   &   \\
{ETN}  &   & X & X & X &   &   &   &   &   & X \\
{D-1}  & X &   &   &   &   &   & X &   &   &   \\
{D-5}  & X &   &   &   & X & X & X & X & X &
\end{tabular}
& {\Large $\stackrel{\mbox{\scriptsize S}} \longrightarrow$} &
\begin{tabular}{c|llllllllll}
{IIB}   & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline
{mom}   &   &   &   &   &   &   & X &   &   &   \\
{ETN}   &   & X & X & X &   &   &   &   &   & X \\
{NS-1}  & X &   &   &   &   &   & X &   &   &   \\
{NS-5}  & X &   &   &   & X & X & X & X & X &
\end{tabular}
\end{tabular} \nonumber
}
\end{eqnarray}
Note that under the T-duality in the 6-direction (T6), the fundamental
string parallel to the 6-direction transformed into a unit of Kaluza-Klein
momentum.  This is just the well-known momentum--winding exchange.
Recalling that the Kaluza-Klein monopole is essentially the product
of time and Euclidean Taub-NUT (ETN)~\cite{kkm1,kkm2},  and that
the NS-5-brane is the magnetic dual of the NS-string (c.f.\ equation~%
(\ref{hsoln10}) below)
the magnetic-dual of this phenomenon is the NS-5-brane--ETN transformation
under the T9 perpendicular to the NS-5-brane~\cite{town} (compare also
with~\cite{cvj}).

Now applying the harmonic function rule for
orthogonally intersecting branes in ten-
dimensions~\cite{tsh,gaunt,argurio}
gives (relabeling $9 \rightarrow 4$ and $6 \rightarrow 9$)
\begin{mathletters} \label{soln10}
\begin{eqnarray}
\label{gsoln10}
ds_{\mbox{\scriptsize} str}^2 & = & \psi_1^{-1} [-dt^2 + dx_9^2
  + \frac{Q_R}{r} (dt -dx_9)^2]
  + \psi_5 \psi_E^{-1} (dx_4 + Q_E (1-\cos \theta) d \phi)^2
\nonumber \\
& & + \psi_5 \psi_E (dr^2 + r^2 d \theta^2 + r^2 \sin^2 \theta d \phi^2)
    + dx_5^2 + dx_6^2 + dx_7^2 + dx_8^2, \\
\label{dilsoln10}
\varphi &=& \frac{1}{2} \ln (\psi_5 \psi_1^{-1}), \\
\label{hsoln10}
H &=& -Q_5 \sin \theta d \theta \wedge d\phi \wedge dx_4
     + \psi_1^{-2} \frac{d \psi_1}{d r} dt \wedge dr \wedge dx_9, \\
\label{psi1soln10}
\psi_1 &=& 1 + \frac{Q_1}{r}, \\
\label{psi5soln10}
\psi_5 &=& 1 + \frac{Q_5}{r}, \\
\label{psiEsoln10}
\psi_E &=& 1 + \frac{Q_E}{r}.
\end{eqnarray}
\end{mathletters}
Here we have postulated an obvious generalization of the harmonic function
rule to configurations involving the ETN; in particular, the ETN does
not contribute an overall conformal factor, in analogy to the Kaluza-Klein
momentum.
For notational simplicity, only the one-centred black hole has been written;
the generalization to the multi-black hole is almost obvious---see, e.g.~%
\cite{tseyt} for details on multi-centred ETN.  We have also set the
string coupling constant $g=e^{\varphi_\infty}=1$, where the subscript
denotes evaluation at spatial infinity.
The $Q_{\alpha}$s are constants; see also equations~(\ref{normq1})--(%
\ref{normqE}).
It is readily verified that equation~%
(\ref{soln10}) satisfies the equations of motion of the (string-frame)
NS-NS IIB action
\begin{equation} \label{action10}
S = \frac{1}{16 \pi G_{10}}
  \int d^{10}x \sqrt{-g}e^{-2 \varphi} [R + 4 (\nabla \varphi)^2
   - \frac{1}{12} H^2],
\end{equation}
where $G_{10} = 8 \pi^6 g^2 \alpha'^4$ is the ten-dimensional Newton constant.
This action, of course, describes the universal sector of all the
string theories,
and, in fact, the solution of equation~(\ref{soln10}) is not new,
having been discussed in the context of the heterotic string in
e.g.~\cite{ct}.

Dimensional reduction on a $T^6$ now proceeds in the usual way~%
\cite{maharana}.  Of course, the NS-5-brane and the ETN give rise
to magnetic charges in 4-dimensions; it is therefore convenient to
dualize the corresponding vectors, and write the theory in terms of the
magnetic vector potentials and field strengths for which the
Bianchi identity and equation of motion are interchanged, e.g.\ %
$d \tilde{A}^{(2)}_4 \equiv \tilde{F}^{(2)}_4 =
e^{-2 \varphi} G^{44} \star F^{(2)}_4$, using the notation of~%
\cite{maharana}, and tildes to denote magnetic quantities.  The
four-dimensional, Einstein frame
action, and solution for multiple black holes carrying
four charges, are then,
\begin{eqnarray} \label{action4}
S &= & \frac{1}{16 \pi G_4}
   \int d^4x \sqrt{-g} \left\{ R - 2 (\partial_\mu \phi)^2
   - \frac{1}{4} (\partial_\mu \ln G_{44})^2 - \frac{1}{4} (\partial_\mu
   \ln G_{99})^2 - \frac{1}{4} e^{2 \varphi} G^{-1}_{44} (\tilde{F}^{(1)4}_{
   \mu \nu})^2 \right. \nonumber \\
& & \left. - \frac{1}{4} e^{-2 \varphi} G_{99} (F^{(1)9}_{\mu \nu})^2
   - \frac{1}{4} e^{2 \varphi} G_{44} (\tilde{F}^{(2)}_{4 \mu \nu})^2
   - \frac{1}{4} e^{-2 \varphi} G^{-1}_{99} (F^{(2)}_{9 \mu \nu})^2 \right\} ,
\end{eqnarray}
\begin{mathletters} \label{soln4}
\begin{eqnarray} \label{gsoln4}
ds_{\mbox{\scriptsize E}}^2 &=& -(\psi_1 \psi_5 \psi_R \psi_E)^{-\frac{1}{2}}
  dt^2 + (\psi_1 \psi_5 \psi_R \psi_E)^{\frac{1}{2}} d \vec{x}^2, \\
\label{dil4}
\varphi & = & \ln (\psi_1^{-\frac{1}{4}} \psi_5^{\frac{1}{4}}
    \psi_R^{-\frac{1}{4}} \psi_E^{\frac{1}{4}}), \\
\label{g444}
G_{44} & = & \psi_5 \psi_E^{-1}, \\
\label{g994}
G_{99} & = & \psi_1^{-1} \psi_R, \\
\label{A1soln4}
A^{(2)}_9 & = & \psi_{1}^{-1} dt, \\
\label{A5soln4}
\tilde{A}^{(2)}_4 & = & \psi_{5}^{-1} dt, \\
\label{ARsoln4}
A^{(1)9} & = & \psi_{R}^{-1} dt, \\
\label{AEsoln4}
\tilde{A}^{(1)4} & = & \psi_E^{-1} dt,
\end{eqnarray}
\end{mathletters}
where the $\psi_\alpha$s, $\alpha \in \{1,5,R,E\}$ are harmonic functions,
\begin{mathletters} \label{psiandQ}
\begin{eqnarray} \label{psi}
\psi_\alpha &= &1 + \sum_{a=1}^N \frac{Q_{\alpha a}}{r_a}, \\
\label{normq1}
Q_{1a} &=& \frac{4 G_4 R_9}{\alpha'} n_{1a},\\
\label{normq5}
Q_{5a} &=& \frac{\alpha'}{2 R_4} n_{5a}, \\
\label{normqR}
Q_{Ra} &=& \frac{4 G_4}{R_9} n_{Ra},\\
\label{normqE}
Q_{Ea} &=& \frac{R_4}{2} n_{Ea},
\end{eqnarray}
\end{mathletters}
where the $n_{\alpha a}$ are non-negative integers.
$N$ is the number of black holes, and $\vec{r}_a$ is their positions.
The radii of the internal circles are $R_4,\ldots,R_9$ and
the four-dimensional Newton constant is $G_4 = \frac{g^2 \alpha'^4}%
{R_4 R_5 R_6 R_7 R_8 R_9}$.%
\footnote{For details on deriving the quantization of the charges and
the value of the $D$-dimensional Newton constant, see e.g.~\cite{jmphd}.  In
particular, we obtained the quantum of $Q_{E}$ by T-dualizing the
quantum of $Q_{5}$.}

\section{The Effective Action and U-duality} \label{scatter}
The Manton-type scattering calculation~\cite{man} proceeds exactly as
in~\cite{fe,shir,us}, so we leave out
all the details here.  The result to ${\cal O}(\vec{v}^2)$
is
\begin{eqnarray} \label{result}
S_{\mbox{\scriptsize eff}} &=& \int dt  \left\{ -\sum_a m_a
   + \frac{1}{2} \sum_a m_a \vec{v}_a^2
   + \frac{1}{2 l_p^2} \sum_{\alpha<\beta} \sum_{a,b} Q_{\alpha a} Q_{\beta b}
     \frac{|\vec{v}_a - \vec{v}_b|^2}{r_{ab}} \right. \nonumber \\
& & + \left. \frac{1}{4 l_p^2}
          \sum_{\stackrel{\mbox{\scriptsize $\alpha<\beta$}}
                   {\mbox{\scriptsize $\gamma \neq \alpha, \beta$}}}
     \sum_{a,b,c} Q_{\alpha a} Q_{\beta b} Q_{\gamma c} |\vec{v}_a -
     \vec{v}_b|^2 (\frac{1}{r_{ab} r_{ac}} + \frac{1}{r_{ab}r_{bc}} -
     \frac{1}{r_{ac}r_{bc}}) \right. \nonumber \\
& & \left. + \frac{1}{2 l_p^2}
    \sum_{\stackrel{\mbox{\scriptsize $\alpha<\beta; \gamma<\delta$}}
             {\mbox{\scriptsize $\alpha, \beta, \gamma, \delta$ all distinct}}}
     \sum_{a,b,c,d}
Q_{\alpha a} Q_{\beta b} Q_{\gamma c} Q_{\delta d}
 |\vec{v}_a - \vec{v}_b|^2
 \int d^3x \frac{\vec{r}_a \cdot \vec{r}_b}{4 \pi r_a^3 r_b^3 r_c r_d}
 \right\},
\end{eqnarray}
where saturation of the Bogomol'nyi bound gives~\cite{jmphd}
\begin{equation} \label{bps}
m_a = \frac{1}{l_p^2} (Q_{1a} + Q_{5a} + Q_{Ra} + Q_{Ea}),
\end{equation}
and the four-dimensional Planck constant is $l_p = \sqrt{4 G_4} = \frac{%
g \alpha'^2}{\sqrt{2 R_4 \ldots R_9}}$.

Note that the
$a=b$ terms in the multiple sums clearly don't contribute.  Furthermore,
in the triple sum, the singular terms for $a=c$ or $b=c$ cancel, and in
the quadruple sum, the integral converges, even when two or more of
the coordinates coincide.
When one of the charges, say $Q_{Ea}$ vanishes for every black hole,
then equation~(\ref{result}) reduces to the result of~\cite{us} when the
latter is reduced from five to four dimensions, as required by the
arguments of~\cite{myers2}.

We would now like to make equation~(\ref{result}) U-duality invariant.
The U-dual expression for the terms linear and quadratic in the charges
follow exactly as in~\cite{dps,us}.  In particular the mass is
(in a sense elaborated below) already invariant, and the quadratic term
involves the masses and
contraction of two factors of the $E_{7(7)}$ charge vector
$q_{\Lambda a}$ with the inverse of the matrix of moduli, $({\cal M}_\infty^{
-1})^{\Lambda \Sigma}$ (see equation~(\ref{uresult}));
${\cal M}_{\Lambda \Sigma}$ is the matrix which
multiplies the kinetic term for the vector fields in the $E_{7(7)}$
invariant action.
The quartic term is clearly
proportional to the quartic invariant of the U-duality group~$E_{7(7)}$.
However, while in~\cite{us} the cubic term was proportional to the
cubic invariant of the five dimensional U-duality group~$E_{6(6)}$,
we can not directly associate such an interpretation to it in this case,
since $E_{7(7)}$ has no cubic invariant, nor does $E_{6(6)}$ imbed itself
into $E_{7(7)}$ in an intrinsically natural way.
Furthermore, it can be checked that we cannot use the invariants made out
of the matrix of moduli, the charge vectors and the masses to obtain the
cubic term; we can understand this because an expression involving
the matrix of moduli
could not reduce to the moduli independent $E_{6(6)}$ formula.%
\footnote{\label{overallM}
There will actually be overall matrix of moduli factors that
arise during the compactification from five to four dimensions.}

Instead, we recall,
following~\cite{kk,cvetic,feretal}, that there is an intrinsically natural way
of dissecting the $D=4, N=8$ central charge matrix.  Specifically, we
note that for each black hole
the moduli-dependent central charge matrix $\bbox{\sf Z}_a$ can be
$SU(8) \subset E_{7(7)}$
rotated into the form
\begin{equation} \label{ccharge}
\bbox{\sf Z}_a =
\mbox{diag} \{z_{1a},z_{2a},z_{3a},z_{4a}\} \otimes
 \left( \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right),
\end{equation}
with the $z_{\cdot a}$s the (possibly complex%
\footnote{But note that only the overall phase is invariant and not the
individual phases.}%
) ``eigenvalues''.
The largest eigenvalue, which we choose to be $z_{1a}$, is the mass of the
$a$th
black hole, by the BPS condition.
In fact, since
$z_{1a} = l_p^2 m_a$, it, and more technically an $SU(2) \subset
SU(8)$, is singled out.  This was explained in~\cite{feretal} as the
$SU(2)$ corresponding to the supercharges (which transform linearly under
the $SU(8)$ automorphism) for which a complex linear combination annihilates
the state.  This is just the statement that it corresponds,
by the BPS condition, to the
unbroken supersymmetry.

In the case at hand,~\cite{cvetic,kk}
\begin{mathletters} \label{defzs}
\begin{eqnarray}
\label{defz1}
z_{1a} &=& Q_{1a} + Q_{5a} + Q_{Ra} + Q_{Ea} = l_p^2 m_a, \\
\label{defz2}
z_{2a} &=& Q_{1a} - Q_{5a} + Q_{Ra} - Q_{Ea}, \\
\label{defz3}
z_{3a} &=& Q_{1a} + Q_{5a} - Q_{Ra} - Q_{Ea}, \\
\label{defz4}
z_{4a} &=& Q_{1a} - Q_{5a} - Q_{Ra} + Q_{Ea}.
\end{eqnarray}
\end{mathletters}
It is easily checked that
\begin{eqnarray} \label{cubicu}
\sum_{\alpha \neq \beta \neq \gamma} Q_{\alpha a} Q_{\beta b} Q_{\gamma c}
& = & \frac{1}{16} \left\{ z_{1a} \left(z_{1b} z_{1c} - \sum_{I=2}^4
 z_{Ib} z^*_{Ic}\right) + (\mbox{5 perms}) \right\} \nonumber \\
& &  + \frac{1}{8} \left\{ \mbox{Re} (z_{2a}
   z_{3b} z_{4b}) + (\mbox{5 perms}) \right\},
\end{eqnarray}
where again we have taken into account the fact that for the more general
black holes, the $z_{\cdot a}$ are complex.
Furthermore, if we set $\bbox{\sf Z}_a$ real and traceless, then we can
make contact with the real, traceless
five dimensional central charge matrix.   Specifically,
in this case equation~(\ref{cubicu})
is equivalent (up to a proportionality constant)
to the 5-dimensional $E_{6(6)}$ symmetric invariant,
written in the form~\cite{cvetic}
$\sum_{I=1}^4 z_{Ia} z_{Ib} z_{Ic}$.
In other words, when we restrict to black holes with three charges (or
fewer), then we recover the five-dimensional U-duality invariant formula.

Of course, we still need to convert equation~(\ref{cubicu}) into
a ``U-duality'' invariant formula involving the charge vectors $q_{\Lambda a}$.
The formula won't be truly U-duality invariant because we are decomposing
$E_{7(7)} \supset SU(8) \supset SU(2) \times SU(6)$; it will
only be invariant under the subgroup.%
\footnote{
As the $SU(8)$ is the maximal compact subgroup of $E_{7(7)}$,
this is (almost) the maximal decomposition of $E_{7(7)}$ involving our $SU(2)$
factor.  There is also a possible $U(1)$ factor; however, we have fixed the
$U(1)$ by demanding that $z_{1a}=l_p^2 m_a$, i.e.\ by fixing that $z_{1a}$ be
real.
}
The central charge matrix transforms linearly in the $\bbox{28} \oplus
\bbox{\overline{28}}$
of $SU(8)$;
under the above decomposition~\cite{pat,feretal},
\begin{equation} \label{decomp56}
\bbox{28} \oplus \bbox{\overline{28}} \rightarrow
   (\bbox{1},\bbox{15})
   \oplus (\bbox{1},\bbox{\overline{15}})
   \oplus (\bbox{2},\bbox{6})
   \oplus (\bbox{2},\bbox{\overline{6}})
   \oplus (\bbox{1},\bbox{1})
   \oplus (\bbox{1},\bbox{1})
.
\end{equation}
Clearly, $z_{1a} \in (\bbox{1},\bbox{1})
$ and the other $z_{\cdot a} \in
 (\bbox{1},\bbox{15})
$.  So, the first term of equation~(\ref{cubicu})
is
\begin{mathletters} \label{justifyu}
\begin{equation} \label{firstjustu}
(\bbox{1},\bbox{1})
\otimes \{ [(\bbox{1},\bbox{1})]^2 +
 (\bbox{1},\bbox{15})
\otimes (\bbox{1},\bbox{\overline{15}})
\},
\end{equation}
which indeed contains a singlet, as required.
The second term of equation~(\ref{cubicu}) is
\begin{equation} \label{secondjustu}
[(\bbox{1},\bbox{15})]^3 + [(\bbox{1},\bbox{\overline{15}})]^3,
\end{equation}
\end{mathletters}
and again
each term contains a singlet.%
\footnote{This follows since the $\bbox{15} \in SU(6)$ is an antisymmetric
product of two fundamentals.  The antisymmetric product of six fundamentals
is clearly a singlet; this is the symmetric product of three~$\bbox{15}$s.
}
Note that this explains our choices of complex conjugation on
the right-hand side of equation~%
(\ref{cubicu}); any other polynomial choice that reduces to the left-%
hand side, and treats the $z_{\cdot a}$s and $z^*_{\cdot a}$s symmetrically%
\footnote{This is required since there is no invariant distinction between
the complex representations of $SU(6)$ and their complex conjugates.}%
,
would not be a singlet.  Thus, we have arrived at
equation~(\ref{cubicu}) uniquely.

So, to write down a more invariant expression we decompose the
integer-valued $E_{7(7)}$ charge vector $q_{\Lambda a}$.
More precisely, since we were working with the
central charge matrix, which is moduli dependent, it is convenient
to raise the index using the matrix of moduli:
\begin{equation} \label{raiseq}
q^{\Lambda}_a \equiv ({\cal M}_\infty^{-\frac{1}{2}})^{\Lambda \Sigma}
   q_{\Sigma a}.
\end{equation}
Then we can decompose $q^{\Lambda}_a$ as
$\{m_a, q^{A}_a, q^{\bar{A}}_a,\ldots\}$ where $m_a = l_p^{-2} z_{1 a}$
has been used for the $(\bbox{1},\bbox{1})$;
the index $A,\bar{A}=1,\ldots,15$ labels respectively
the $\bbox{15},
\bbox{\overline{15}} \in SU(6)$;
and the ellipses denote the representations that have not been included.
Then, we finally have the U-duality invariant version of
equation~(\ref{result}).
\begin{eqnarray} \label{uresult}
S_{\mbox{\scriptsize eff}} & = &
  \int dt \left\{ - \sum_a m_a
  +\frac{1}{2} \sum_a m_a \vec{v}_a^2
  +\frac{1}{2}
       \sum_{a<b} (l_p^2 m_a m_b - q_{\Lambda a} ({\cal M}_\infty^{-1})^{
       \Lambda \Sigma}
       q_{\Sigma b}) \frac{|\vec{v}_a - \vec{v}_b|^2}{
       r_{ab}} \right. \nonumber \\
& & \left.
  +\frac{3}{32} \sum_{a<b} \sum_c \left[l_p^4 m_a m_b m_c - \frac{l_p^2}{6}
       (m_a q^{A}_{b} \delta_{A \bar{A}} q^{\bar{A}}_{c}
           + \mbox{5 perms})
      + l_p d_{(6)ABC}
           q^{A}_{a} q^{B}_{b} q^{C}_{c} \right. \right. \nonumber \\
& & \left. \left. \hspace{1.5cm}
      + l_p d^*_{(6)\bar{A}\bar{B}\bar{C}}
      q^{\bar{A}}_a q^{\bar{B}}_b q^{\bar{C}}_c \right]
      |\vec{v}_a-\vec{v}_b|^2
      \left[ \frac{1}{r_{ab} r_{ac}} +
      \frac{1}{r_{ab} r_{bc}} - \frac{1}{r_{ac}r_{bc}} \right] \right.
      \nonumber \\
& & \left.
  + \frac{l_p^2}{4} \sum_{a<b} \sum_{c,d} d^{\Lambda \Sigma \Gamma \Pi}
      q_{\Lambda a} q_{\Sigma b} q_{\Gamma c} q_{\Pi d}
      |\vec{v}_a - \vec{v}_b|^2 \int d^3x \frac{\vec{r}_a \cdot
      \vec{r}_b}{4 \pi r_a^3 r_b^3 r_c r_d} \right\}.
\end{eqnarray}
Here, $d_{(6)ABC}$ is proportional to the symmetric cubic invariant for the
$\bbox{15}
\in SU(6)$, and $d^{\Lambda \Sigma \Gamma \Pi}$ is proportional to
the $E_{7(7)}$
cubic invariant.

Two final comments are required regarding the decomposition $E_{7(7)}
\supset SU(8) \supset SU(2) \times SU(6)$.
First, it appears that we have assumed that the central charge
matrices for the black holes can be simultaneously diagonalized (in
the sense of equation~(\ref{ccharge})).
However, all we really
need to assume is that they can be simultaneously block-diagonalized
into $SU(2)$ and $SU(6)$ subgroups; our final expression, equation~%
(\ref{uresult}) is $SU(6)$ invariant and so does not require that the
matrices be diagonal.  That the block-diagonalization is possible is simply
the statement that the black holes preserve a common supersymmetry.
Incidentally, the block-diagonalization implies that the charges that
transform in the $(\bbox{2},\bbox{6})$ representation of $SU(2)\times SU(6)$
vanish; this is why there is no $(\bbox{1},\bbox{1})\otimes(\bbox{2},
\bbox{6})\otimes (\bbox{2},\bbox{\overline{6}})$ term in equation~%
(\ref{cubicu}) or~(\ref{uresult}).

Second, it
was implicitly assumed in the discussion
that the solution preserves exactly
$\frac{1}{8}$ of the supersymmetry.  If the solution preserves more
supersymmetry---i.e. if more than one $z_{\cdot a} = l_p^2 m_a$---then
there is no longer a natural $SU(2) \subset SU(8)$ but rather a larger
subgroup that is selected.  Nevertheless, it is easy from equation~%
(\ref{cubicu}) to see that no matter
how one chooses the $SU(2) \subset G$ (where, for a solution
preserving $\frac{1}{4}$ of the supersymmetry, $G=SU(4)$, for example)
one obtains the same answer for the cubic, namely zero, so there is no
ambiguity when there is more supersymmetry.

We now claim that equation~(\ref{uresult}), which was really only
derived for the special case of black holes with four charges, holds
for general (e.g.\ five charge) supersymmetric black hole configurations.
The forms of the two-point, three-point and four-point functions
have already been fixed uniquely by equation~(\ref{result}) and
the duality symmetries%
, so the only possible modification with the
additional charges, is the appearance of higher-point functions.  Since
these higher-point functions vanish when only four charges are
non-zero, they must be proportional to all five separate charges, and hence
cannot be made out of invariants of order less than five.  But the group
theory that led us to the ``U-duality'' invariant form for the cubic term
shows us that there are no such candidate invariants: all we can
work with
is the $(\bbox{1},\bbox{15})$
and its complex conjugate, and since
cubing one gives a singlet, and multiplying one by the other gives a singlet
we can never get a higher-order invariant.  This leaves
the $E_{7(7)}$ invariants, and the only symmetric one is the quartic.
Thus, equation~(\ref{uresult}) is the general result.

\section{Discussion} \label{conc}
We have given the effective action to ${\cal O}(\vec{v}^2)$
for scattering of an arbitrary number of charged supersymmetric
four dimensional black holes.  The U-duality invariant form required a
technical discussion of the
natural $SU(2)\times SU(6)$ decomposition of the four-dimensional
U-duality group $E_{7(7)}$.  We would now like to give a slightly
more detailed discussion of the asymptotically flat
moduli space for two black holes.  From equation~%
(\ref{uresult}), the moduli space is
\begin{mathletters} \label{mod2bh}
\begin{eqnarray}
\label{mod2bhmetric}
ds^2 &=& \frac{1}{2} f(\vec{r}) (dr^2 + r^2 d\Omega^2),
\end{eqnarray}
where
\begin{eqnarray}
\label{mod2bhf}
f(\vec{r}) &=& \mu + \frac{\Gamma_{II}}{r} + \frac{\Gamma_{III}}{r^2} +
   \frac{\Gamma_{IV}}{r^3
, \\
\label{mod2bhg2}
\Gamma_{II} &=& l_p^2 M \mu - q_{\Lambda 1} ({\cal M}_\infty^{-1})^%
        {\Lambda \Sigma} q_{\Sigma 2}, \\
\label{mod2bhg3}
\Gamma_{III} &=& \frac{3}{16} \left\{l_p^4 M^2 \mu
     + l_p d_{(6)ABC} q^A_1 q^B_2 (q^C_1 + q^C_2)
     + l_p d^*_{(6)\bar{A}\bar{B}\bar{C}} q^{\bar{A}}_1 q^{\bar{B}}_2
           (q^{\bar{C}}_1 + q^{\bar{C}}_2)
\right. \nonumber \\ & & \left.
     - \frac{l_p^2}{3} \left( M q^A_1 \delta_{A\bar{A}} q^{\bar{A}}_2
          + M q^A_2 \delta_{A\bar{A}} q^{\bar{A}}_1
          + m_1 q^A_2 \delta_{A\bar{A}} q^{\bar{A}}_2
          + m_2 q^A_1 \delta_{A\bar{A}} q^{\bar{A}}_1 \right) \right\}, \\
\label{mod2bhg4}
\Gamma_{IV} &=& \frac{l_p^2}{6} d^{\Lambda \Sigma \Gamma \Pi}
     q_{\Lambda 1} q_{\Sigma 2} \left(q_{\Gamma 1} q_{\Pi 1} + q_{\Gamma 2}
     q_{\Pi 2} \right)
\end{eqnarray}
and the centre of mass (relative mass) is $M=m_1+m_2$
($\mu=\frac{m_1 m_2}{m_1+m_2}$); also the relative coordinate is $\vec{r}=
\vec{r_2}-\vec{r_1}$.  In equation~(\ref{mod2bhmetric}), we have subtracted
away the centre of mass motion.  We have also omitted the term
\begin{equation} \label{omitmod2bhf}
\frac{\pi^3 l_p^4}{4} d^{\Lambda \Sigma \Gamma \Pi}
     q_{\Lambda 1} q_{\Sigma 1} q_{\Gamma 2} q_{\Pi 2} \delta^{(3)}(\vec{r}),,
\end{equation}
\end{mathletters}
from equation~(\ref{mod2bhf}) because this contact interaction
only occurs at $r=0$ by which point
the moduli space approximation has presumably broken down.
We recall that the moduli space approximation is only valid for small
velocities
and neglects radiation; in particular, in~\cite{fe} it was argued that the
moduli
space approximation breaks down for $r \lesssim v_\infty^2 M$.

The coordinate singularity at $r=0$ of equation~%
(\ref{mod2bhmetric}) is removed, when $\Gamma_{IV} \neq 0$,
by performing the coordinate
transformation $\xi=-2 \frac{\alpha'^{\frac{3}{4}}}{\sqrt{r}}$.
One then finds that as $r \rightarrow 0$,  ($\xi \rightarrow \infty$),
there is a second asymptotic region that is conical with deficit angle~%
$\pi$~\cite{fe}.  For $\Gamma_{IV} = 0$ but $\Gamma_{III} \neq 0$,
the coordinate singularity at
$r=0$ is removed by the coordinate transformation $\xi = \ln \frac{r}{\sqrt{%
\alpha'}}$
to find that the
asymptotic region has topology $\relax{\rm I\kern-.18em R} \times S^2$.
If both $\Gamma_{IV} = 0$  and $\Gamma_{III} = 0$, then one removes the
$r=0$ coordinate singularity via $\xi = \sqrt{\sqrt{\alpha'} r}$ to again find
a second asymptotic region that is conical with deficit
angle~$\pi$~\cite{shir}.
Thus, geodesics which extend to $r=0$ enter this second
asymptotic region; i.e.\ the black holes coalesce~\cite{fe,shir,dps}.
(It might seem strange that, having just rejected the contact interaction
for being at $r=0$, we are exploring the $r \rightarrow 0$ behaviour.
The point is that we can use equations~(\ref{mod2bh}), even as
$r \rightarrow 0$ (but $r \neq 0$), by taking
$v_\infty \rightarrow 0$.)

By examining
the geodesic equation as in~\cite{dps}, we find that the
turning point $r_c$ in the black hole motion is real and positive when
the impact parameter $b$ obeys
\begin{eqnarray} \label{bc}
b^2 &>& b_c^2 \equiv \mbox{\small $- \frac{\Gamma_{II}^2}{12 \mu^2}  +
     \frac{\Gamma_{III}}{\mu}
$}\nonumber \\ && \mbox{\small $
     + \frac{{\Gamma_{II}^4 + 18\,\Gamma_{II}\,\Gamma_{IV}\,{{\mu}^2}}}{
          {{12 \left( -\left( {\Gamma_{II}^6}\,{{\mu}^6}
                \right)  +
             540\,{\Gamma_{II}^3}\,\Gamma_{IV}\,
              {{\mu}^8} +
             24\,\left( 243\,{\Gamma_{IV}^2}\,
                 {{\mu}^{10}} +
                {\sqrt{3}}\,
                 {\sqrt{\Gamma_{IV}\,{{\mu}^{14}}\,
                     {{\left( -{\Gamma_{II}^3} +
                      27\,\Gamma_{IV}\,{{\mu}^2} \right) }^3
                      }}} \right)  \right) }^{{\frac{1}{3}}}}}
$}  \\ && \mbox{\small $
      + \frac{1}{12}
        {{\left( -\frac{ {\Gamma_{II}^6}}{\mu^6}  + 540\,
           \frac{{\Gamma_{II}^3}\,
           \Gamma_{IV}}{{\mu}^4} +
          24\,\left( 243\,\frac{\Gamma_{IV}^2}{\mu^2} +
             {\sqrt{3}}\,{\sqrt{\frac{\Gamma_{IV}}{{\mu}^{10}}\,
                  {{\left( -{\Gamma_{II}^3} +
                      27\,\Gamma_{IV}\,{{\mu}^2} \right) }^3
                    }}} \right)  \right) }^{{\frac{1}{3}}}}$}\nonumber.
\end{eqnarray}
In particular there is coalescence for $b \leq b_c$.  Na\"{\i}vely,
$b_c$ is only well-defined if either $\Gamma_{IV}=0$ or $\Gamma_{II}^3
\leq 27 \Gamma_{IV}  \mu^2$;
in fact, one must merely be careful
about how one chooses the cube roots in equation~(\ref{bc}).

As equation~(\ref{bc}) is rather obscure, we point out some special cases.
If we consider the black holes of section~\ref{construct}, with
$Q_{1a} = Q_{5a} = Q_{Ra} = Q_{Ea} = l_p^2 \frac{m_a}{4}$, then we have the
Reissner-Nordstr\"{o}m black holes of~\cite{fe}%
\footnote{Equation~(\ref{bc}) corrects the polynomial equation
for $b_c$ that was given in the reference.}
for which the right-hand side of equation~(\ref{bc}) is real and positive,
and so there can be coalescence.  Note that this is despite the
fact that for Reissner-Nordstr\"{o}m black holes,
$27 \Gamma_{IV} \mu^2 - \Gamma_{II}^3 < 0$.
For two Reissner-Nordstr\"{o}m black
holes of equal mass ($m_1 = m_2$), $b_c \approx 2.3660 \, l_p^2
\frac{M}{4}$, in agreement with~\cite{fe}.

If the black holes only carry three charges, then $\Gamma_{IV} = 0$;
we find $b_c = \sqrt{\frac{\Gamma_{III}}{\mu}}$.%
\footnote{To obtain this from equation~(\ref{bc}) requires
choosing $(-1)^\frac{1}{3} = \frac{1}{2} \pm i \frac{\sqrt{3}}{2}$.}
In fact, if the black holes carry fewer than three charges, so that also
$\Gamma_{III}=0$, then there is never coalescence (except for the obvious
case $b=0$); this is in agreement with the results of~\cite{shir}.
This is different from higher dimensions, where there is always a
critical, non-zero impact parameter below which there is coalescence~%
\cite{shir,us}.

\acknowledgments
I thank Vijay Balasubramanian, Eric Gimon, Gary Horowitz, John Pierre,
Joe Polchinski, Andy Strominger and Haisong Yang
for useful discussions.  I also thank David Kaplan for
collaboration on an earlier, related paper.
I am grateful to Harald H. Soleng for making~\cite{cartan}
available, which was very useful in the computation.
The hospitality of the Physics Department at Harvard University is appreciated.
Financial support from NSERC and NSF is gratefully acknowledged;
this work was also supported in part by DOE Grant No. DOE-91ER40618.


\section*{Introduction}
Microbial Fuel Cells (MFCs) are devices that produce electricity from waste-water by utilizing microbial metabolic oxidation processes \cite{ieropoulos2013energy}. A MFC consists of a proton exchange membrane (PEM), an anode (negative half-cell) and a cathode (positive half-cell). Electricity is generated as a by-product of microbial metabolism, which results in electrons flowing from the bacterial cells onto the anode electrode, and then from anode to the cathode electrode via an external electrical circuit; this produces a flow of electrical current. Positively charged ions that are products of the oxidation reactions, such as protons, also flow out of the bacterial cells, due to electron-neutrality and diffuse to the cathode through the PEM. Electrons, cations and the cathodic oxidizing agent of choice (e.g. oxygen) recombine to complete the reaction and close the circuit. Despite the fact that MFC technology has been the subject of research for at least three decades, real-world implementation or commercialization is still limited \cite{cheng2011electricity,santoro2015cathode,ortiz2016study,mardanpour2012single,ledezma2013mfc,chouler2016towards}.

In addition to the utilization of MFCs in waste-water treatment and power production, their usage in other fields was suggested, like sensory applications \cite{kumlanghan2007microbial,abrevaya2015analytical}. Nonetheless, an application that is rarely studied is using configurations based on MFCs that realize computing units \cite{greenman2006microbial,greenman2006perfusion}. More specifically, the first approach \cite{greenman2006microbial} was to reproduce conventional binary logic gates using MFCs in order to examine the abilities of such a system, with an ultimate goal of constructing non-silicon multi-valued logic processing units being envisaged by the same authors in \cite{greenman2006microbial}. In that study, designs of hydraulic and electrical interconnections are suggested for building three basic logic gates (AND, OR and NOT) that can be combined to assemble universal gates, hence circuits capable of universal computation.

Apart from basic logic gates, more complicated computational abilities were exhibited with the appropriate interconnection of a small number of MFCs, namely a simplified Pavlovian learning model \cite{greenman2006perfusion}. In this study, the symbiotic mix of natural biological cells, such as the anodophiles, and artificial systems, like electrodes, actuators, pumps and chemical solutions, were used to simulate a learning cycle. In detail, two signals representing the smell of food --- that activates production of saliva --- and hearing the sound of a bell --- that does not activate independently the production of saliva --- were associated, in order for the sound of the bell to trigger the production of saliva by itself, i.e. automatically.

Despite the seemingly simplistic, binary computational tasks performed in both of the aforementioned studies \cite{greenman2006microbial,greenman2006perfusion}, the authors make notice of the high number of states that could be realized by MFC devices, in order to enable complex computing. In this respect, a configuration of MFCs is proposed here to mimic the computation dynamics of Cellular Automata (CAs). A novel development of MFCs, which includes the introduction of additional electrodes, acting as poise/bias points --- or `pins' --- has been recently proposed [Patent filing number: GB1501570.4]. The invention introduces the principle of electrochemical redox (reduction--oxidation) bias via a third and/or fourth electrode, when an external power supply, or another MFC is connected to this third or fourth pin and to the working anode or cathode electrode, respectively. This is, by default, an unconventional means of connection, since the potential difference (voltage) of the external MFC (also known as `driver') can bias the redox potential difference (voltage) of the anode or cathode half-cell, depending of course on how the connection is made. Such a system is naturally subject to polarization and is therefore limited to a time constant ($t$), based on materials, voltage levels and oxidation/reduction states. Consequently, the system is an ideal platform for pulse-width-modulation techniques. For the purposes of the current study, a third electrode is used to achieve poise and, thus, have two MFCs behaving like a CA cell.

CAs can be considered an idealization of a physical system in which space and time are discrete, and the physical quantities take only a finite set of values \cite{chopard2009cellular,mizas2008reconstruction}. A CA comprises identical cells in a regular grid that are characterized by their state. The state of each cell is updated by a uniform local rule depending on the states of the cells in its vicinity. CAs can conceptually be identified as \textit{general} and \textit{simple} \cite{sipper1995quasi}. The term \textit{general} is referring to the fact that CAs can promote universal computation and that the state and local updating rules of the cells are not limited to specific regulations. Moreover, the term \textit{simple} is justified by the plain outline of CAs --- cells are characterized by basic states with local interactions --- compared with other computing machines. Finally, CAs can be considered as one of the most favorite candidates of the future computational architectures tackling the bottleneck of the von Neumann architecture when referring to the co-existence of computing and memory units in the same simple unit, or the CA cell.

Specifically, a well known CA model, namely Conway's Game of Life (GoL) is studied, which encapsulates the ability of universal computation and construction \cite{adamatzky2010game}. However, the realization of GoL here is not limiting the capabilities of possible configurations consisted of hydraulically and electrically linked MFCs. In fact, based on the local activity \cite{chua1998cnn} exhibited by MFCs, any CA local rule can be implemented in similar configurations. In addition to the advantage of the energy independence, along with water purification of the proposed computational scheme, when compared with other transducers of renewable energy sources, MFCs integrate both energy extracting mechanisms and computational units. Moreover, the ongoing miniaturization of MFCs \cite{chouler2016towards} will enable the production of smaller biological processing units. Finally, the amount of physicochemical parameters that can be externally manipulated and affect the performance and, thus, the outputs of the biofilms in MFCs is enormous \cite{greenman2006perfusion}. This fact can justify the utilization of MFCs for more complex computational schemes than the ones suggested up to this date.

\section*{Game of Life}

A Cellular Automaton (CA) consists of a regular grid of cells. Each cell takes $k$ different states, where $k > 2$, but not at once. The grid can be $n$-dimensional ($n \geq 1$). The evolution of the cells takes place at discrete points in time. That means that the state of each cell in the grid changes only at discrete moments of time, namely at time steps $t$. The time step $t = 0$ is usually considered as the initial step and therefore no changes at the state of the cells occur.

For each cell, a set of cells called its neighborhood (usually including the cell itself) is defined relative to the specified cell. Regarding the two dimensional CA, the two most common types of neighborhood that are mainly considered are:

\begin{itemize}
\item \textit{von Neumann} neighborhood, that consists of the central cell, whose condition is to be updated, and the four cells located to the north, south, east and west of the central cell

\item \textit{Moore} neighborhood, that consists of the same cells with the von Neumann neighborhood together with the four other adjacent cells of the central cell (the northwester, northeaster, south-east and south west cells).
\end{itemize}

The evolution of the cells demands the definition of a cell state, the neighboring cells as well as the local transition function:

\begin{itemize}
\item The local internal state of each cell of the CA:
\begin{equation}
C ( \vec{r},t) = \{ C_{1}( \vec{r},t), C_{2}( \vec{r},t),..., C_{m}( \vec{r},t)\}
\label{theory_state}
\end{equation}
\noindent at time step $t=0,1,2,...$ is described by a set of variables associated with each position $\vec{r}$ of the grid.

\item The local transition function is defined as:
\begin{equation}
R=\{R_{1},R_{2},...,R_{m}\}
\label{theory_function}
\end{equation}
\noindent and determines the evolution during time of the internal state  of each cell according to the following equation:
\begin{equation}
C_{j}( \vec{r},t+1) =R_{j} \left( C( \vec{r},t), C( \vec{r}+\vec{\delta}_{q},t),..., C( \vec{r}+\vec{\delta}_{m},t) \right)
\label{theory_evolution}
\end{equation}
\noindent where $\vec{r}+\vec{\delta}_{k}$ designate the cells which belong to a given neighborhood of cell $\vec{r}$.
\end{itemize}

The state of cell $\vec{r}$, at time step ($t+1$), is computed according to $R$. $R$ is a function of the state of this cell at time step ($t$) and the states of the cells in its neighborhood at time step ($t$). In the above definition, the function $R$ is identical for all sites and it is applied simultaneously to each of them, leading to synchronous dynamics. It is important to notice that the rule is homogeneous, i.e. it does not depend explicitly on the cell position $\vec{r}$. However, spatial inhomogeneities can be introduced by having some cells' states $C_{j}(\vec{r})$ systematically at a fixed value, i.e. 1, in some given locations of the lattice, to mark particular cells for which a different rule applies. Furthermore, the new state at time $t+1$ is only a function of the previous state at time $t$. It is sometimes necessary to have a longer memory and introduce a dependence on the states at times $t-1$, $t-2$, …, $t-k$. Such a situation is already included in the definition, if one keeps a copy of the previous state in the current state.

Conway's Game of Life (GoL) is a two-dimensional CA with binary states \cite{conway1970game} that had significantly contributed to the extensive attention CA theory has gained. The neighborhood considered is \textit{Moore} neighborhood and the two states that each cell can adopt is \textit{alive} and \textit{dead} (or `1' and `0', respectively). The local transition rule uses the states of all nine cells in the neighborhood, during the directly preceding time step, to determine the new state of the central cell in the neighborhood. More specifically, the following transitions between states can occur:

\begin{enumerate}

\item When a cell is \textit{dead} at time \textit{t} and precisely three of the eight neighbors are \textit{alive}, the cell adopts the state \textit{alive} at time \textit{t}+1.

\item When a cell is \textit{alive} at time \textit{t} and none, one or more than three of the eight neighbors are \textit{alive}, the cell adopts the state \textit{dead} at time \textit{t}+1.

\end{enumerate}

Note that if none of the two aforementioned cases are true, the local rule dictates that the cell retains its previous state. Assuming $i$ and $j$ the dimension indexes that establish the location of each cell in the grid and $t$ the current time step, the transition rule can be expressed as:

\begin{equation}
C_{i,j}^{t+1}= \begin{cases}
        0, & \text{if }  (\sum_{k=-1,l=-1}^{k=1,l=1} C_{i+k,j+l})-C_{i,j} \leq 1 \text{ or} \\
        & (\sum_{k=-1,l=-1}^{k=1,l=1} C_{i+k,j+l})-C_{i,j} \geq 4 \\
        1, & \text{if } (\sum_{k=-1,l=-1}^{k=1,l=1} C_{i+k,j+l})-C_{i,j} = 3\\
        C_{i,j}^{t} & \text{else }
        \end{cases}
\label{golrule}
\end{equation}

Note that \textit{Moore} neighborhood of cell $C_{i,j}$ is consisted of cells $C_{i+1,j}$, $C_{i,j+1}$, $C_{i-1,j}$, $C_{i,j-1}$, $C_{i+1,j+1}$, $C_{i+1,j-1}$, $C_{i-1,j+1}$, $C_{i-1,j-1}$ and the cell itself.

It is suggested that the inherent complexity of GoL is due to the fact that its transition rule is non-monotonic and nonlinear \cite{adamatzky2010game,chua1998cnn,Rendell2002}. Moreover, the aforementioned rule is characterized as an outer totalistic rule, given that it only accounts for the value of the central cell during the last time step and on the sum of values of cells in the outer \textit{Moore} neighborhood.

A continuous version of GoL, namely a discrete-time, continuous spatial automaton with the same behavior as GoL has been also proposed \cite{maclennan1990continuous}. In continuous spatial automata, the cells and their states form a continuum. A similar behavior to GoL in a continuous field can be realized with a local rule implemented by a non-monotonic function of the population density in the neighborhood. That is assuming the function will be graphically represented in two dimensions, namely the input or the population density in $x$-axis and the output or the next state of the cell in $y$-axis. Keeping in mind the rules of the original GoL, the continuous local rule function has to be continuously increasing in the space $(0,m)$ and continuously decreasing in the space $(m,1)$, for instance an inverted parabola; where $m$ is a value in space $(0,1)$, the minimum value of population density is $0$, while the maximum value is $1$. Note that value $m=3/8$ provides the closest analogy to the standard, binary GoL.

Another study \cite{adachi2004game} has proposed a CA model with a local rule based on a continuously valued expression with three parameters. That model matches GoL when one of these parameters, namely one defined as temperature $T$, is approximately zero and the other two have appropriate values. The upper limits of the temperature parameter were investigated, where formations of GoL, like gliders, start to decay. Nonetheless, it is suggested that for higher values of $T$ parameter, the model's behavior is increasingly biased towards chaos \cite{adamatzky2010game}.

Furthermore, the realization of GoL by Cellular Neural Networks (CNNs) have been suggested \cite{274337}. In that study, a two layer single-step CNN template, a multi-step three layer discrete time CNN template with threshold sigmoid and a multi-step and a multi-step piecewise-linear discrete time CNN template have been presented.

\section*{Proposed implementation}

Drawing inspiration from an implementation using MFCs to build logic gates \cite{greenman2006microbial}, we propose a new implementation to execute a popular example of CA, namely the aforementioned GoL. The instance of GoL is selected here as it is an excellent example of the fact that complex behaviors emerge from a trivial local interaction of simple agents. Nonetheless, the application of the GoL rules to realize functions that can be translated as global computations \cite{adamatzky2010game}, can define the limitations of the proposed configuration. The functionality of a MFC is affected by the voltage applied on the third electrode (see Fig. \ref{fig1}A) of the device, which introduces an electrochemical redox bias.

\begin{figure}[tb]
				\centering
				\begin{subfigure}[b]{0.45\textwidth}
					\includegraphics[width=\textwidth]{fig1A.pdf}
					\caption{}
					\label{fig1a}
        \end{subfigure}
        \begin{subfigure}[b]{0.4\textwidth}
					\includegraphics[width=\textwidth]{fig1B.pdf}
					\caption{}
					\label{fig1b}
        \end{subfigure}
\caption{{\bf Schematic of the configuration of MFCs implementing a GoL CA cell.} A: Inputs and outputs of one MFC. B: The configuration of a MFC duet.}
\label{fig1}
\end{figure}

The scheme realizing one cell of GoL CA consists of two MFCs, one primary and one secondary. The two MFCs are both hydraulically and electrically connected as shown in Fig. \ref{fig1}B. The secondary MFC is fed by a main/initial source of fuel, while the primary is fed by the effluent of the second. Both are operating under continuous flow conditions. MFCs stacked in cascades have been proven to produce higher power and current densities when their position is higher up the cascade, thus fed directly by the fuel source, than the ones placed lower downstream \cite{ledezma2013mfc,walter2015microbial}. Consequently, setting a fuel source that provides a balanced but limited substrate concentration, will enable only one of the MFCs to function.

The selection of which of the two MFCs will be functional is a result of their electrical interconnection. Both MFCs are equipped with a third electrode, independently. These electrodes are connected, via different resistances, with a single pin representing the electrical input of the CA cell implementation. Note that the different values of resistance separate the operation of the MFC GoL cell into three regions depending on the input voltage applied as explained below. The output power of the primary MFC is used to describe the state of the CA cell, i.e. its output.

The secondary MFC will act as a control unit, through the hydraulic link, for the primary one. The main fuel source is considered to provide a solution with limited carbon-energy, so that only one of the two MFCs will be able to fully process this by its anodophilic biofilm in order to produce electricity. When an appropriate bias is applied to the input port of the CA cell, thus, on the third electrode of the secondary MFC, the processes in the biofilm of the anode of the secondary MFC will be activated. Consequently, the biofilm will be utilizing the nutrients from the source, resulting in an effluent depleted from carbon-energy, which is used as the influent for the primary MFC. This means that the primary MFC will be unable to produce electrical energy and its power output will be low, hence the state of the proposed CA cell will be `0' (see Fig. \ref{fig2}C).

\begin{figure}[!tb]
				\centering
				\begin{subfigure}[b]{0.4\textwidth}
					\includegraphics[width=\textwidth]{fig2A.pdf}
					\caption{}
					\label{fig2a}
        \end{subfigure}
        \begin{subfigure}[b]{0.4\textwidth}
					\includegraphics[width=\textwidth]{fig2B.pdf}
					\caption{}
					\label{fig2b}
        \end{subfigure} \\
				\begin{subfigure}[b]{0.4\textwidth}
					\includegraphics[width=\textwidth]{fig2C.pdf}
					\caption{}
					\label{fig2c}
        \end{subfigure}
\caption{{\bf The three operation regions of the MFC-based scheme.} A: Low input bias resulting state `0'. B: Intermediate input bias resulting state `1'. C: High input bias resulting state `0'.}
\label{fig2}
\end{figure}

On the other hand, when the bias applied to the third electrode of both MFCs is low enough (or zero), neither of them will be activated. Consequently, the power produced by the primary MFC will be low and the state of the proposed CA cell will be `0' (see Fig. \ref{fig2}A).

Finally, an intermediate value of bias applied to the CA cell input, or both MFCs' third electrodes, will allow the primary MFC to produce electricity. Provided a design with appropriately chosen values of the resistances connected to the third electrodes, the voltage drop over the resistance connected to the secondary MFC will be sufficiently large not to enable its activation and, thus, the depletion of carbon-energy in its effluent. Note that the values of the resistances should satisfy $R_1 > R_2$, in order to set voltage drops that will activate only the primary MFC. The voltage drop over the resistance connected to the primary MFC will not be large enough, but given that the influent will be carbon-energy replete (i.e. rich in metabolites), it will result in the production of electrical current (state `1') (see Fig. \ref{fig2}B).

The power output ($P$) of each CA cell for the three regions of operation is expressed in relation to the input bias applied ($V$) as in Eq. \ref{eq1}. Note that the time variant is included in the equation, to conform to CA terminology. The time variant has an actual real analogy with the transition response for the establishment of a new steady-state, resulted in changes in the conditions applied to the biofilm component. That transition response is identified as approximately four minutes \cite{greenman2006microbial}.

\begin{equation} \label{eq1}
P_{out}^{t+1} =  \begin{cases}
        P_{low}, & \text{for }  V_{in}^{t} \leq V_{thr\_low}\\
        P_{high}, & \text{for } V_{thr\_low}\leq V_{in}^{t}\leq V_{thr\_high}\\
        P_{low}, & \text{for } V_{thr\_high}\leq V_{in}^{t}
        \end{cases}
\end{equation}

Each CA cell, realized by a duet of MFCs, is connected with its eight neighbors, as illustrated in Fig. \ref{fig3}, to form \textit{Moore} neighborhood which is used in GoL. The current produced by each primary MFC is used to convey the information about central cell's state to all of its neighbors.

\begin{figure}[!tb]
\centering
\includegraphics[width=0.5\textwidth]{fig3.pdf}
\caption{{\bf Electrical interconnection of neighboring CA cells.}}
\label{fig3}
\end{figure}

\section*{Electrical Circuit Equivalent}
An electrical circuit that has an equivalent behavior to the MFC configuration presented in the previous Section and, thus, implements the GoL cell state transition rule, is presented in Fig \ref{fig4}A. However, there are significant differences. Firstly, the MFC logic can combine hydraulic and electrical links, thus, it can accommodate more complex computations with the same amount of basic building units. As mentioned previously, there is an enormous amount of physicochemical parameters that affect biofilms in MFCs. As a result, a MFC --- that is utilized as a computing unit --- except from the electrical pins, has also hydraulic inputs that affect its functionality. Note that there is no possible connection of two transistors that will act as an electrical counterpart of the proposed MFC duet configuration and, thus, in Fig. \ref{fig4}A three transistors are used. Moreover, MFC schemes do not require an external power supply, but a fuel source; a fuel that is inexpensive and abundant. Consequently, the proposed computing configurations instead of consuming energy, they are able to produce electrical energy, the level of which will depend on the requirements of the desired task; in other words, a more complicated computational task, requiring a higher number of MFCs, will naturally generate more power. In contrast, the transition response is faster in the electrical circuit.

\begin{figure}[!tb]
\centering
				\begin{subfigure}[b]{0.45\textwidth}
					\includegraphics[width=\textwidth]{fig4A.pdf}
					\caption{}
					\label{fig4a}
        \end{subfigure}
        \begin{subfigure}[b]{0.4\textwidth}
					\includegraphics[width=\textwidth]{fig4B.pdf}
					\caption{}
					\label{fig4b}
        \end{subfigure}
\caption{{\bf Equivalent electrical circuit to the proposed MFC configuration.} A: Schematic of the equivalent circuit. B: Output of equivalent circuit.}
\label{fig4}
\end{figure}

The circuit consists of three transistors in total. Transistors \texttt{Q1}-\texttt{Q2} ensure a high output when the input voltage is higher than a lower threshold (i.e. 2V as depicted in Fig. \ref{fig4}B), while \texttt{Q3} results in a high output when the input voltage is lower than a high threshold (i.e. 7V as depicted in Fig. \ref{fig4}B). Also, the interconnection of collectors of transistors \texttt{Q2} and \texttt{Q3} forms a hard-wired {\sc and} gate. The voltage output (blue line) of the circuit correlated with a sinusoid voltage input (black line) is illustrated in Fig. \ref{fig4}B. The behavior of the circuit imitates the behavior of a continuous GoL cell \cite{maclennan1990continuous}.

The functionality of the equivalent circuit for the three different states is presented in Fig. \ref{fig5} and can be compared with the functionality of the MFC configuration presented in Fig. \ref{fig2}.

\begin{figure}[!tbp]
\centering
				\begin{subfigure}[b]{0.4\textwidth}
					\includegraphics[width=\textwidth]{fig5A.pdf}
					\caption{}
					\label{fig5a}
        \end{subfigure}
        \begin{subfigure}[b]{0.4\textwidth}
					\includegraphics[width=\textwidth]{fig5B.pdf}
					\caption{}
					\label{fig5b}
        \end{subfigure} \\
				\begin{subfigure}[b]{0.4\textwidth}
					\includegraphics[width=\textwidth]{fig5C.pdf}
					\caption{}
					\label{fig5c}
        \end{subfigure}
\caption{{\bf The three operation regions of the equivalent circuit.} A: Low current input resulting state `0'. B: Intermediate current input resulting state `1'. C: High input current resulting state `0'.}
\label{fig5}
\end{figure}

In order to illustrate the functionality of the equivalent circuit under the rules of GoL, a grid of  $3 \times 3$ cells is designed using LTspice software as shown in Fig. \ref{fig6}. Despite the fact that the length of the grid can be characterized as small, this grid is used for demonstration reasons, thus its simplicity enhances readability and in detail comprehension of the proposed electronic circuits. Nevertheless, designing larger grids, i.e. of $n \times n$ cells, is a trivial and effortless procedure, due to the well known prominent and inherent characteristics of CA, like local interconnections, simplicity, uniformity and area utilization. The grid is initialized using transistors \texttt{Q1}, \texttt{Q2} and \texttt{Q3} to set the inputs of cells \texttt{X4}, \texttt{X5} and \texttt{X6} to the voltage needed to trigger the `1' state in the next time step. Note in the results depicted in Fig. \ref{fig7} that the outputs of the central cell (\texttt{X5}) and the west cell (\texttt{X4}) are high (state `1') at $t=2ms$. Also, note that the state of the central cell remains `1', while the states of the west (\texttt{X4}) and the north (\texttt{X2}) cells are oscillating between `1' and `0' states, whereas they are not both in the same state for any time step. Each cell in the grid presented in Fig. \ref{fig6} is comprised of the circuit depicted in Fig. \ref{fig4}A and a circuit adding some time delay between its input and its output. The time delay circuit for this example is adding $1ms$ from the moment that the input is changed to the cell's response, to realize the GoL rules in a synchronous matter and avoid the loss of signals. This procedure is inherent to the MFC configuration as the transition time between states is reported to be consistent around four minutes \cite{greenman2006microbial}.

\begin{figure}[!tbp]
\centering
\includegraphics[width=0.9\textwidth]{fig6.pdf}
\caption{{\bf A $3 \times 3$ grid of equivalent circuit cells.}}
\label{fig6}
\end{figure}

\begin{figure}[!tbp]
\centering
\includegraphics[width=0.7\textwidth]{fig7.pdf}
\caption{{\bf The results of the grid initialized to oscillate as a GoL blinker.}}
\label{fig7}
\end{figure}

\section*{Conclusions}
The capabilities of MFCs in water purification, extraction of useful elements, sensory applications and energy production have been thoroughly studied. Another proposed application for MFCs is the carrying out of computational functions, which has been expressed as the construction of conventional logic gates, a Pavlovian learning model and, in this study, an implementation of a CA paradigm, namely GoL.

The ability of interconnecting MFCs via hydraulic and electrical links and the multiple states that can be adopted by each MFC, makes the possible computing configurations more complex than conventional ones. Moreover, the MFC computing units are not limited by a power source; on the contrary they are powered by sustainable, diverse and abundant fuel sources. A disadvantage of these systems is the long transition times experienced between steady states that can reach up to four minutes; however, they have been reported to be consistent.

Here the design of a duet of MFCs interconnected hydraulically and electrically to form a unit that behaves like a cell of GoL was proposed. Namely, the effluent of one MFC is used as an influent of the other, the third electrodes of both are connected with the electrical input of the cell, while the anode of one of the MFCs is used as the output of the cell. Given the fact that the proposed configuration consisted of two MFCs has the same behavior as a GoL CA cell, the realization of universal computation is possible.

An aspect of future work is the implementation of different CA local rules with configurations of real interconnected MFCs. Furthermore, the possibility of using conventional computing machines for the initialization of the CA grid and the exploitation of the outputs will be investigated, to design and realize a hybrid computing system.

\section*{Acknowledgments}
This work was supported by the European Union's Horizon 2020 Research and Innovation Programme under Grant Agreement No. 686585.

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