# Radiation of Optical Angular Momentum by a Dipole Source inside a Magneto-
optical Environment

B.A. van Tiggelen<EMAIL_ADDRESS>Univ. Grenoble Alpes, CNRS,
LPMMC, 38000 Grenoble, France

(August 28, 2024)

###### Abstract

Radiation of electromagnetic energy by electric or magnetic multipole sources
can be modified by their local environment. In this work we demonstrate that a
magneto-optical environment of an unpolarized dipole source induces the
radiation of angular momentum into space. This radiation benefits from
Purcell-enhancement.

##  I Introduction

The power emitted by a light source depends on its local environment. This
notion, strongly related to the well-know Purcell effect [1], can be
quantified by the local density of electromagnetic modes available to the
photons leaving the source. It has important applications, e.g. to enhance
weak magnetic dipole transitions [2] or to suppress or enhance spontaneous
emission [3, 4, 5]. The statement holds for classical light sources and for
quantum sources only if they have low quantum efficiency [6]. Landau and
Lifshitz [7] calculate the torque on a classical emitting dipole in vacuum as
an exercise and find zero. A magneto-birefringent environment changes this
outcome and induces the radiation of angular momentum into space [8]. In this
work we calculate how much angular momentum is radiated into space by a
classical dipole inside a magneto-active Mie sphere that acts as a (resonant)
nano-antenna [9] for angular momentum . By conservation of angular momentum a
torque is exerted on the source, proportional to the power of the source.

##  II Radiation of energy and optical angular momentum

The radiation of electromagnetic waves by a classical source is described by
the vector Helmholtz equation for the electric field (in Gaussian units),

$-\frac{\varepsilon(\bf{r})}{c_{0}^{2}}\cdot\partial_{t}^{2}{\bf
E}({\bf{r}},t)-{\bf\nabla}\times{\bf\nabla}\times{\bf E}({\bf
r},t)=\frac{4\pi}{c_{0}^{2}}\partial_{t}{\bf J}_{S}({\bf r},t)$ (1)

featuring the current density $\bf{J}_{S}$ of the classical source. Its direct
environment is characterized by (here only) the dielectric tensor
$\varepsilon_{ij}(\bf{r})$. We shall ignore its frequency dependence, neglect
absorption and give up explicit reference to frequency dependence. For a
monochromatic source at frequency $\omega$ , the solution can be formally
written in Dirac notation as ${\bf E}({\bf r})=\langle{\bf r}|{\bf E}\rangle$
with $|{\bf E}\rangle={\bf G}\cdot(-4\pi i\omega/c_{0}^{2})|{\bf
J}_{S}\rangle$. The Helmholtz Green’s operator given by

${\bf G}({\bf x},{\bf p})=\frac{1}{\varepsilon({\bf
x})(k_{0}+i0)^{2}-p^{2}+{\bf pp}}$ (2)

where $k_{0}=\omega/c_{0}$ and ${\bf x},{\bf p}$ the usual canonical
operators, obeying $\left[{x}_{n},{p}_{m}\right]=i\delta_{nm}$. The small
positive imaginary part ensures the outwardly propagating solution relevant
for a source. According to Maxwell’s equations the cycle-averaged dissipation
density is $\frac{1}{2}\mathrm{Re}\left({\bf J}_{S}^{*}\cdot{\bf E}\right)$.
The emitted the power is therfore,

$P_{S}=-\frac{2\pi k_{0}}{c_{0}}\mathrm{Im}\int d^{3}{\bf r}\int d^{3}{\bf
r}^{\prime}\,{\bf J}_{S}({\bf r})^{*}\cdot{\bf G}({\bf r},{\bf
r}^{\prime})\cdot{\bf J}_{S}({\bf r}^{\prime})$ (3)

For a source with electric dipole (ED) moment ${\bf d}$ is ${\bf
J}_{\mathrm{ED}}({\bf r})=-i\omega{\bf d}\delta({\bf r})$, for a magnetic
dipole (MD) moment ${\bf m}$ is ${\bf J}_{\mathrm{MD}}({\bf
r})=c_{0}{\bf\nabla}\times\left[{\bf m}\delta({\bf r})\right]$. It follows [2]

$\displaystyle P_{\mathrm{ED}}$ $\displaystyle=$
$\displaystyle-\frac{2\pi}{3}k_{0}^{3}c_{0}|{\bf
d}|^{2}\mathrm{Im}\,G_{ii}(0,0)$ $\displaystyle P_{\mathrm{MD}}$
$\displaystyle=$ $\displaystyle-\frac{2\pi}{3}k_{0}c_{0}|{\bf
m}|^{2}\mathrm{Im}\,\hat{G}_{ii}(0,0)$ (4)

with
${\hat{G}}_{kk^{\prime}}=(i\epsilon_{knl}p_{l})G_{nn^{\prime}}(i\epsilon_{n^{\prime}k^{\prime}l^{\prime}}p_{l^{\prime}})$
the Helmholtz vector Green’s function associated with the magnetic field. The
same can be done for the radiation of angular momentum by the source at
$r\approx 0$. The electromagnetic angular momentum is defined as ${\bf J}_{\rm
rad}=\mathrm{Re}\int d^{3}{\bf r}\,{\bf r}\times({\bf E}^{*}\times{\bf
B})/8\pi c_{0}$ that can be split up into an orbital, a spin and a
longitudinal component [10, 11]. Without external torques the sum of ${\bf
J}_{\rm rad}$ and mechanical angular momentum ${\bf J}_{\rm mec}$ of the
source and its environment must be conserved. Let us consider a radius $R$
enclosing the environment. For a source that has been stationary during a time
$t\gg R/c_{0}$, total electromagnetic angular momentum inside the sphere $r<R$
should be steady. The balance becomes [7]

$\displaystyle 0$ $\displaystyle=$
$\displaystyle-{M}_{i}+\frac{R^{3}}{8\pi}\epsilon_{ijk}\mathrm{Re}\int_{4\pi}d\hat{{\bf
r}}\,\hat{r}_{l}\hat{r}_{j}(E_{l}^{*}E_{k}+B_{l}^{*}B_{k})(R\hat{{\bf r}})$
$\displaystyle\frac{d}{dt}{J}_{i,\mathrm{mec}}$ $\displaystyle=$
$\displaystyle+{M}_{i}$ (5)

This expresses that the total torque ${\bf M}$ on the matter is radiated away
as angular momentum to infinity ($r>R)$. The torque acts on both source and
its environment, ${\bf M}={\bf M}_{S}+{\bf M}_{E}$. The latter is

${\bf M}_{E}=\frac{1}{2}\mathrm{Re}\,\int d^{3}{\bf r}\,\left[{\bf
P}^{*}\times{\bf E}+P_{m}^{*}({\bf r}\times{\bf\nabla})E_{m}\right]$ (6)

but will vanish for a rotationally-invariant environment. The torque on the
source does not, and has an electric and magnetic part,

${\bf M}_{S}=\frac{1}{2}\mathrm{Re}\,\int d^{3}{\bf
r}\,\left[{\rho}_{S}^{*}({\bf r}\times{\bf E})+{\bf
r}\times\left(\frac{1}{c_{0}}{\bf J}_{S}^{*}\times{\bf B}\right)\right]$ (7)

For an ED or MD this simplifies to expressions similar to Eqs. (II),

$\displaystyle{M}_{i,\mathrm{ED}}$ $\displaystyle=$
$\displaystyle-\frac{2\pi}{3}k_{0}^{2}|{\bf
d}|^{2}\mathrm{Re}\,\epsilon_{ijk}{G}_{kj}(0,0)$ $\displaystyle
M_{i,\mathrm{MD}}$ $\displaystyle=$ $\displaystyle-\frac{2\pi}{3}|{\bf
m}|^{2}\mathrm{Re}\,\epsilon_{ijk}{\hat{G}}_{kj}(0,0)$ (8)

These torques vanish for an environment with isotropic optical response [7].
In the presence of magneto-birefringence, the torque is directed along the
external magnetic field ${\bf B}_{0}$. Both vectors have same parity but
behave differently under time-reversal, yet the relation ${\bf M}_{S}\sim{\bf
B}_{0}$ is allowed because the time-reversed process corresponds to the
_absorption_ of an incident spherical wave by a sink, implying
$i0\rightarrow-i0$ in Eq. (2). This changes the sign of $P_{S}$, whereas ${\bf
M}_{S}$ is left unchanged because ${\bf B}_{0}$ changes sign as well upon
time-reversal. The power $P_{S}$ is emitted as a radial Poynting vector that
decays as $S_{r}=P_{S}/4\pi r^{2}$ in the far field. The radiated angular
momentum emerges as a vortex with an azimuthal Poynting vector $S_{\phi}$
around ${\bf B}_{0}$ that decays with distance as $-M_{S}c_{0}\sin\theta/4\pi
r^{3}$. Hence the radiation of angular momentum is associated with a small
“photon Hall angle” $S_{\phi}/S_{r}=-(M_{S}c_{0}/P_{S})\sin\theta/r$ that
slowly decays in the far field [8].

##  III Application to a Mie sphere

We model the local environment of the source by a homogeneous dielectric
sphere subject to magneto-optical birefringence surrounding it symmetrically.
For an external magnetic field ${\bf B}_{0}$ in the $z$-direction, the
dielectric tensor of the sphere is given by $\varepsilon=m^{2}{\bf 1}-W(r){\bf
S}_{z}$ with ${S}_{z,ij}=-i\epsilon_{ijz}$ the electromagnetic spin operator
along the magnetic field[10]. The dimensionless material parameter
$W(r)=2mV(r)B_{0}/k_{0}$ involves the Verdet constant $V(r)$ that may vary
inside the sphere $r<a$ or even be outside. It is small, and we treat the
birefringence as a linear perturbation to the Mie sphere. If ${\bf G}^{M}$ is
the Helmholtz vector Green’s function for a Mie sphere with index of
refraction $m$, we can write Eq. (2) as

${\bf G}={\bf G}^{M}+{\bf G}^{M}\cdot k_{0}^{2}W(r){\bf S}_{z}\cdot{\bf
G}^{M}+\mathcal{O}(W^{2})$ (9)

We expand ${\bf G}^{M}$ into the orthonormal set of Mie eigenfunctions,
involving the spherical vector harmonics $\hat{{\bf Z}}_{JM},\hat{{\bf
N}}_{JM},\hat{{\bf X}}_{JM}$ that are also eigenfunctions of total
electromagnetic angular momentum ${\bf J}^{2}$ and its component along the
$z$-axis (with eigenvalue $M$)[10]. To this end we write

${\bf
G}^{M}=\varepsilon^{-1/2}\cdot\frac{1}{(k_{0}+i0)^{2}-\mathcal{H}}\cdot\varepsilon^{-1/2}$
(10)

with $\mathcal{H}=\varepsilon^{-1/2}\cdot(p^{2}-{\bf
pp})\cdot\varepsilon^{-1/2}$ a hermitian and semi-positive-definite operator.
It has positive eigenvalues $k^{2}$ associated with transverse magnetic ($e$)
and transverse electric ($m$) eigenfunctions, and an infinitely degenerated
subspace of eigenfunctions with eigenvalue $0$ associated with longitudinal
electric fields. We can decompose it as

${\bf G}^{M}({\bf r},{\bf
r}^{\prime})=\int_{0}^{\infty}dk\sum_{i=e,m,\ell}\sum_{J=1}^{\infty}\sum_{M=-J}^{J}\frac{{\bf
E}^{i}_{JMk}({\bf r}){\bf E}^{i}_{JMk}({\bf
r}^{\prime})^{*}}{(k_{0}+i0)^{2}-k_{i}^{2}}$ (11)

with $k_{e,m}=k$ and $k_{\ell}=0$ for all $k$. The longitudinal mode with
$J=0$ is ignored since it carries no angular momentum. For magneto-
birefringence inside the sphere we need the fields modes inside, properly
normalized in the far field [12]. With $y\equiv mkr$ and $\tilde{y}\equiv
kr/m$,

$\displaystyle{\bf E}^{e}_{JMk}({\bf r})=\frac{4\pi
kA^{e}_{kJ}}{(2\pi)^{3/2}}\left(\sqrt{J(J+1)}\frac{j_{J}(y)}{y}\hat{{\bf
N}}_{JM}(\hat{{\bf r}})\right.$ $\displaystyle\ \ \ \ \ \ \
+\left.\frac{(yj_{J}(y))^{\prime}}{y}\hat{{\bf Z}}_{JM}(\hat{{\bf r}})\right)$
$\displaystyle{\bf E}^{m}_{JMk}({\bf r})=\frac{4\pi
kA^{m}_{kJ}}{(2\pi)^{3/2}}j_{J}(y)\hat{{\bf X}}_{JM}(\hat{{\bf r}})$
$\displaystyle{\bf E}^{\ell}_{JMk}({\bf r})=\frac{4\pi
kA^{\ell}_{kJ}}{(2\pi)^{3/2}}\left(j_{J}^{\prime}(\tilde{y})\hat{{\bf
N}}_{JM}(\hat{{\bf
r}})+\sqrt{J(J+1)}\frac{j_{J}(\tilde{y})}{\tilde{y}}\hat{{\bf
Z}}_{JM}(\hat{{\bf r}})\right)$

The factors $A^{m/e}_{kJ}$ can be found in standard literature [13, 14], and
$A^{\ell}_{kJ}=m/[mj_{J}^{\prime}(ka/m)h_{J}(ka)-j_{J}(ka/m)h^{\prime}_{J}(ka)]$
is less well known, because not excited in the scattering of a transverse
plane wave. Upon inserting Eq. (9) and Eq. (11) into expression (II) for an ED
source, we find that only modes with $i=e,\ell$ and $J=1$ survive,

$\displaystyle M_{\mathrm{ED},i}=-\frac{2\pi^{2}k_{0}^{3}|{\bf
d}|^{2}}{3}\mathrm{Re}\int_{0}^{\infty}dk\sum_{i=e,\ell}^{M=\pm
1}\frac{M\langle{\bf E}^{e}_{k_{0}1M}|W(r){\bf S}_{z}|{\bf
E}^{i}_{k1M}\rangle}{k_{0}^{2}-k_{i}^{2}+i0}$ (13) $\displaystyle\ \ \ \ \ \
\times\ {\bf E}^{e}_{k_{0}1M}(0)\cdot{\bf E}^{i}_{k1M}(0)^{*}$ $\displaystyle
P_{\mathrm{ED}}=|A^{e}_{k_{0}1}|^{2}\times\frac{k_{0}^{4}|{\bf
d}|^{2}c_{0}}{3}$

The matrix element is proportional to $M$ [15] and reduces to a radial
integral over de sphere. For the radiating MD we find similarly,

$\displaystyle M_{\mathrm{MD},i}=\frac{2\pi^{2}k_{0}|{\bf m}|^{2}}{3}$
$\displaystyle\ \ \ \ \ \ \times\ \mathrm{Re}\int_{0}^{\infty}dk\sum_{M=\pm
1}\frac{M\langle{\bf E}^{m}_{k_{0}1M}|W(r){\bf S}_{z}|{\bf
E}^{m}_{k1M}\rangle}{k_{0}^{2}-k^{2}+i0}$ $\displaystyle\ \ \ \ \ \ \times\
\frac{k_{0}^{2}k^{2}}{3\pi^{2}}m^{2}A^{m}_{1k}A^{m*}_{1k_{0}}$ $\displaystyle
P_{\mathrm{MD}}=m^{2}|A^{m}_{k_{0}1}|^{2}\times\frac{k_{0}^{4}|{\bf
m}|^{2}c_{0}}{3}$ (14)

where we have evaluated the electric field modes at $r=0$ explicitly. In both
cases, the $k$-integral suffers from oscillatory behavior and slow convergence
at large $k$. Some singularities result from the singular dipole field that
can be extracted using the asymptotic expressions for $|A^{i}_{kJ}|^{2}\sim
2/\left[m^{2}+1\pm(m^{2}-1)\cos(2k_{\mathrm{in}}a)\right]$ and, for $r<2a$,

$\displaystyle\frac{1}{{2\pi^{2}}r}\int_{0}^{\infty}dk\frac{2kj_{1}(kr)}{m^{2}+1\pm(m^{2}-1)\cos(2ka)}$
(15) $\displaystyle\ \ \ \ \ \ \ =\frac{1}{4\pi mr^{3}}+\frac{2\delta({\bf
r})}{3(m^{2}+1)}$ (16)

The singular dipole field has no effect on either torque or power.

Figure 1: The function $\Phi_{S}(r)$ as a function of distance inside the
dielectric sphere with index of refraction $m=2$ and size parameter
$k_{0}a=1.2$, for an electric dipole source (top), and a magnetic dipole
source (bottom). The blue dashed lines represent the analytical result for
$m=1$.

Figure 2: The function $\Phi_{S}(r)$ as a function of distance inside the
dielectric sphere with index of refraction $m=5.1$ and size parameter
$k_{0}a=0.599$ corresponding to a magnetic dipole resonance. Top: electric
dipole source; Bottom: magnetic dipole source. The blue dashed lines represent
the analytical result for $m=1$.

##  IV Numerical results

We have numerically evaluated the torques in Eqs. (13) and (III) and show the
results in Figs. (1,2) for $m=2$ and $m=5.1$. We calculate the dimensionless
ratio $M_{S}\omega/P_{S}$ that measures the amount of emitted angular momentum
in units of $\hbar$ per emitted photon. The ratio $M_{S}/P_{S}$ also
determines the photon Hall angle in the far field as mentioned above. The
Purcell factors $|A^{e}_{1k_{0}}|^{2}$ and $m^{2}|A^{m}_{1k_{0}}|^{2}$ for ED
and MD respectively cancel in this ratio. Because we focus here on magneto-
birefringence inside de sphere we can write,

$\frac{M_{S}\omega}{P_{S}}=k_{0}\int_{0}^{a}drW(r)\Phi_{S}(r)=2mVB_{0}a\times\eta_{S}$
(17)

The second equality holds for constant $W(r)$ inside the sphere. The entire
local environment $r<a$ contributes but not always with constant sign, and
singular dipole fields cancel at $r=0$. For $m=1$ we can derive analytically
that $\Phi_{\mathrm{ED}}(r)=(\sin x+4j_{1}(x)-16j_{2}(x)/x)/3$ and
$\Phi_{\mathrm{MD}}(r)=(\sin x-4j_{1}(x))/3$ with $x=2k_{0}a$. The linear
slopes of these curves in the vicinity of the source seem to survive for all
cases $m>1$ studied. For $r/a>0.25$ the radial profile changes significantly
and can have different sign. From Table 1 we see that the ratio, including its
sign, still depends on index of refraction $m$ and size parameter $k_{0}a$.
The Mie parameters are chosen for illustration and do not correspond to known
magneto-optical materials. Large Purcell factors
$\left|A^{e/m}_{1k_{0}}\right|^{2}$ indicate the presence of a ED or MD
resonance. For not too large $m$ there is no significant further enhancement
in the ratio (17) near resonances, because the $k$-integral in Eq. (13) is
less sensitive to what happens exactly on-shell ($k=k_{0}$). For large $m$
this is different. The extreme values $m=9.1$ and $k_{0}a=0.341$ in Table 1
correspond to a sharp MD resonance, studied for water droplets in the
microwave region [16]. The ratio ${M_{S}\omega}/{P_{S}}$ is much larger than
for the similar MD resonance at $m=5.1$ and $m=7.1$, and roughly follows the
Purcell factor $\left|mA^{m}_{1k_{0}}\right|^{2}$. Also at the first ED
resonance the ratio $M_{\mathrm{ED}}\omega/P_{\mathrm{ED}}$ is a factor $20$
larger than at the ED resonance for $m=5$, roughly equal to the ratio of ED
Purcell factors. In these extreme cases, the torque $M_{S}$ scales almost
quadratically with the Purcell factor. We mention also that the longitudinal
Mie eigenfunction is never negligible to calculate the ED torque.

$k_{0}a$ | $m$ | $\left|A^{e}_{1k_{0}}\right|^{2}$ | $\left|mA^{m}_{1k_{0}}\right|^{2}$ | $\eta_{ED}$ | $\eta_{MD}$
---|---|---|---|---|---
1.0 | 1.05 | 0.987 | 1.096 | 0.25 | -0.13
1.2 | 2.0 | 1.008 | 8.712 | -0.020 | -0.31
1 | 1.582 | 0.849 | 1.851 | 0.046 | -0.15
1.5 | 1.582 | 1.093 | 4.867 | 0.014 | -0.40
2 | 1.582 | 1.243 | 6.128 | -0.0013 | -0.051
3 | 1.582 | 2.157 | 2.428 | -0.13 | -0.22
3.5 | 1.582 | 1.233 | 4.302 | -0.25 | -0.57
2.5 | 2.73 | 1.764 | 14.66 | 0.17 | 0.24
4.5 | 2.73 | 1.077 | 44.84 | -0.12 | -0.68
0.855 | 5.0 | 41.00 | 12.28 | 0.020 | 0.0066
0.599 | 5.1 | 0.171 | 2343 | 0.0096 | 0.0089
0.617 | 7.1 | 282.1 | 15.32 | -0.023 | 0.0014
0.435 | 7.1 | 0.0427 | 15462 | 0.011 | -0.049
0.487 | 9.1 | 1039 | 17.10 | -0.41 | 0.00043
0.341 | 9.1 | 1.508 | 55805 | 0.0024 | -0.37
Table 1:  Numerical values for the ratio of torque to power
$\eta_{S}\equiv(M_{S}\omega/P_{S})/2mVB_{0}a$ for different Mie parameters,
off and on resonance, normalized to the dimensionless material parameter
$2mVB_{0}a$ in Eq. (17) associated with Faraday rotation, and assumed constant
inside the sphere. For large index of refraction $m$ the precision is only
$5\%$ due to slowly converging and oscillating integrals. The value $m=1.582$
corresponds to the paramagnetic compound EuF2 that has large Verdet constant.

##  V Discussion and Conclusion

The approach above reveals that the magneto-active sphere acts as a nano-
antenna for angular momentum. The emitted angular momentum feeds back to the
source as a torque and emerges as a vortex field of the Poynting vector around
the environment. This is a new effect and shows once more that the emitted
radiation couples back to the source after having interacted with its
environment. The rate of radiated angular momentum is proportional to the
product of a material parameter $2mVBa$, the power of the source $P_{S}$ and
the small value $\eta_{S}$ calculated in the two columns on the right of Table
1 that quantifies the local environment. Because magneto-optical effects are
small, the emitted angular momentum is clearly small. As an example we mention
EuF2 that has a huge paramagnetic Verdet constant $V\approx-70.000$ rad/T/m at
$4.5$ K and $\lambda=466$ nm [17] and previously used for optical magneto-
resistance measurements [18]. For $x=3.5$, $\lambda=488$nm implies $a=271$ nm,
about the EuF2 particle size in Ref. [18], so that $2mVBa=-6.0\cdot 10^{-2}$
at $B_{0}=1$ Tesla. The calculated value $\eta_{\mathrm{ED}}=-0.25$ implies
that each emitted photon sends an amount of angular momentum
$+0.015\times\hbar$ along the magnetic field into space. The radiated angular
momentum cumulates in time and is subject to resonant enhancement. Phase-
sensitive detection may be used to enhance the effect even further [8].

The nature of the emitted angular momentum is either spin or orbital but this
has not been investigated here. A magneto-optical coating outside the
dielectric sphere might be easier to design experimentally. Equations (13) and
(III) remain valid and without absorption the effect does not fall with the
distance of the coating to the source. Because the environment is supposed to
be rotationally-invariant, no torque is exerted on the environment. For any
disorder in the environment, this will probably change. A related intriguing
question is if the Poynting vortex will exert a radiation force on a gas
bounded outside the environment of the source.

##  VI Acknowledgments

The author thanks Romuald Le Fournis for drawing attention to Ref. [7].

Numerical data underlying the results presented in this paper are not publicly
available at this time but may be obtained from the author upon reasonable
request.

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# Discrete fluctuations in memory erasure without energy cost

Toshio Croucher  Centre for Quantum Dynamics,
Griffith University,
Brisbane, QLD 4111 Australia  Salil Bedkihal  Centre for Quantum Dynamics,
Griffith University,
Brisbane, QLD 4111 Australia  Joan A. Vaccaro  Centre for Quantum Dynamics,
Griffith University,
Brisbane, QLD 4111 Australia

###### Abstract

According to Landauer’s principle, erasing one bit of information incurs a
minimum energy cost. Recently, Vaccaro and Barnett (VB) explored information
erasure within the context of generalized Gibbs ensembles and demonstrated
that for energy-degenerate spin reservoirs, the cost of erasure can be solely
in terms of a minimum amount of spin angular momentum and no energy. As
opposed to the Landauer case, the cost of erasure in this case is associated
with the discrete variable. Here we study the discrete fluctuations in this
cost and the probability of violation of the VB bound. We also obtain a
Jarzynski-like equality for the VB erasure protocol. We find that the
fluctuations below the VB bound are exponentially suppressed at a far greater
rate and more tightly than for an equivalent Jarzynski expression for VB
erasure. We expose a trade-off between the size of the fluctuations and the
cost of erasure. We find that the discrete nature of the fluctuations is
pronounced in the regime where reservoir spins are maximally polarized. We
also state the first laws of thermodynamics corresponding to the conservation
of spin angular momentum for this particular erasure protocol. Our work will
be important for novel heat engines based on information erasure schemes that
do not incur an energy cost.

Understanding the thermodynamical costs of information erasure Szilard (1929);
Landauer (1961) is of fundamental importance for the design of nanoscale heat
engines Dechant  _et al._ (2015) and reversible computing Levitin and Toffoli
(2007); Vos (2011). Landauer’s erasure principle expresses a fundamental bound
for any erasure process that uses a thermal reservoir to store the erased
information. Plenio and Vitelli (2001). It states that the work cost $W$ to
erase one bit of information Landauer (1961) is given by

$\displaystyle W\geq\beta^{-1}\ln 2$ (1)

where $\beta=1/k_{B}T$, $T$ is the temperature of the reservoir and $k_{B}$ is
Boltzmann’s constant. Recently, Dillenschneider and Lutz Dillenschneider and
Lutz (2009) generalized Landauer’s principle to accomodate fluctuations and
obtained the probability of violating Eq. (1) as

$\displaystyle P(W\leq\beta^{-1}\ln 2-\epsilon)\leq e^{-\beta\epsilon}.$ (2)

In other words, small fluctuations $\epsilon$ below the Landauer bound are
possible and the probability of violation is exponentially suppressed in
accordance with the Jarzynski equality Jarzynski (1997a); Sagawa and Ueda
(2010).

The association between information erasure and energy embodied in Eq. (1) has
been widely accepted as a natural one. This is perhaps due to the deep
connection between energy and entropy in traditional thermodynamics. However,
in two classic papers, Jaynes Jaynes (1957a, b) formulated a generalized
theory of statistical mechanics using a principle of maximum entropy where not
only energy but all other measurable conserved quantities can be treated on an
equal footing. In this framework the notion of heat can be generalized to
incorporate an exchange of arbitrary conserved quantities such as quantized
spin angular momentum. In other words if there are $k$ conserved quantities
associated with the generalized reservoir, then the corresponding heat is
called the “$k^{th}$” heat and the corresponding measurement probe is called
the “$k^{th}$” meter Jaynes (1957a, b). If the $k^{th}$ conserved quantity
corresponds to energy then the corresponding meter is the thermometer.

Recently, Vaccaro and Barnett (VB) Vaccaro and Barnett (2011); Barnett and
Vaccaro (2013) applied Jaynes’ framework to the problem of erasing information
when multiple quantities are conserved. In particular, they formulated a
protocol based on energy-degenerate spin-$\frac{1}{2}$ reservoirs under the
conservation of spin angular momentum that gives the cost of erasure solely in
terms of spin angular momentum as

$\displaystyle\mathcal{L}_{\rm s}\geq\gamma^{-1}\ln{2}.$ (3)

Here $\mathcal{L}_{\rm s}$ is the spin equivalent of work which, henceforth,
we refer to it as a spinlabor 111 We introduce the composite words spinlabor
and spintherm to describe the spin equivalent of work and heat based on the
Latin word labor for work and the Greek word thermosh for warm
$\theta\epsilon\rho\mu$ ó $\zeta$ and $\gamma$ is the Lagrange multiplier
associated with the conservation of spin angular momentum which can be
expressed as Vaccaro and Barnett (2011); Barnett and Vaccaro (2013)

$\displaystyle\gamma=\frac{1}{\hbar}\ln\left[\frac{N\hbar-2\langle\hat{J}^{(R)}_{z}\rangle}{N\hbar+2\langle\hat{J}^{(R)}_{z}\rangle}\right]=\frac{1}{\hbar}\ln\left[\frac{1-\alpha}{\alpha}\right]$
(4)

where $\langle\hat{J}^{(R)}_{z}\rangle=\left(\alpha-\frac{1}{2}\right)N\hbar$
is the $z$ component of the total spin angular momentum, $N$ is the number of
spins in the reservoir and $0\leq\alpha\leq 1$ represents a convenient spin
polarisation parameter. It is interesting to note that the Boltzmann constant
does not appear in Eq. (4) since there is no energy cost involved due to the
degeneracy of the internal (intrinsic spin) degree of freedom. In other words,
the spatial and spin degrees of freedom are decoupled from each other, a
situation that is realized in optical trapping of cold atomic gases Leanhardt
_et al._ (2003). Also note that $\gamma$ in Eq. (4) has been called the
inverse “spin temperature” in optical pumping where spin exchange collisions
lead to the redistribution of spin angular momentum Anderson  _et al._ (1961);
Appelt  _et al._ (1998). If the energy degeneracy is broken by a Zeeman field
then the coupling of spatial and spin degrees of freedom will result in an
erasure with multiple costs in terms of both energy and spin polarization
Vaccaro and Barnett (2011); Barnett and Vaccaro (2013). Very recently, Jaynes’
framework has been further extended to incorporate non-commuting degrees of
freedom, and this led to a novel classification of Gibbs states, namely
Abelian and non-Abelian thermal states, and the possibility of extracting
arbitrary conserved quantities from individual quantum systems Guryanova  _et
al._ (2015); Halpern  _et al._ (2015); Lostaglio  _et al._ (2015).

In this work we consider a collection of energy-degenerate spins as an
explicit physical example of a generalised reservoir in Jaynes’ formalism, and
study the discrete fluctuations in an energy-free erasure scheme Vaccaro and
Barnett (2011); Barnett and Vaccaro (2013), a scenario that remains
unexplored. Our main results are a Jarzynski-like equality for quantized spin
angular momentum exchange, and the probability of violation of the VB bound
given by Eq. (3). We find that fluctuations below the VB bound are suppressed
at a rate far greater than expected from the equivalent Jarzynski equality.

We further find that the effects of discreteness are more pronounced at higher
degrees of reservoir spin polarization $\alpha\to 0$ or equivalently, from Eq.
(4), at low spin temperatures $\gamma^{-1}\to 0$. This is the first time that
the fluctuations in the memory erasure has been studied for discrete system.
Our work also opens up the possibilities of novel fluctuation theorems for
other conserved quantities, especially in the context of generalized Gibbs
ensembles, e.g. integrable spin chains Pozsgay (2013).

##  I The Model

We briefly outline the VB erasure scheme here—full details of can be found in
Ref. Vaccaro and Barnett (2011); Barnett and Vaccaro (2013). Studies of
information erasure typically involve two-state memory systems in contact with
one or more reservoirs. In the VB scheme the memory logic states are
associated with the $z$ component of spin polarization with the eigenstate
$|{\downarrow}\rangle$ corresponding to eigenvalue $-\hbar/2$ representing
logical $0$ and $|{\uparrow}\rangle$ corresponding to $\hbar/2$ representing
logical $1$. These states are assumed to be energy degenerate so that the
erasure incurs no energy cost. The reservoir that acts as an entropy sink
consists of $N$ similar energy-degenerate spins. The only other information we
have about the reservoir is the expectation value of the $z$ component of spin
polarization $\langle\hat{J}^{(R)}_{z}\rangle$ and the average energy
associated with the motional degrees of freedom $\langle\hat{H}^{(R)}_{\rm
ext}\rangle$. According to the maximum entropy principle Jaynes (1957a), the
best description of the reservoir is then given by the density operator
Barnett and Vaccaro (2013)

$\displaystyle\hat{\rho}=\frac{\exp(-\beta\hat{H}^{(R)}_{\rm
ext}-\gamma\hat{J}^{(R)}_{z})}{Z}$ (5)

where $\beta$ and $\gamma$ are corresponding Lagrange multipliers which are
independent of each other, and $Z$ is the partition function. Such density
operators may be realized in dilute cold atomic gases confined in optical
traps Leanhardt  _et al._ (2003) and for certain classes of integrable spin
chain models Pozsgay (2013).

We assume the reservoir is sufficiently large (i.e. $N\gg 1$) so that erasing
one bit of information changes $\gamma$ by a negligible amount and so the spin
angular momentum of the reservoir will be described by an approximately-fixed
probability distribution; specifically

$\displaystyle P_{\uparrow}(n,\nu)=\frac{\exp(-\gamma
n\hbar)}{Z_{R}},\vspace{-2mm}$ (6)

is the probability that the reservoir has a $z$ component of spin polarization
of $(n-\frac{1}{2}N)\hbar$ and the arrangement $\nu$ where
$\nu=1,2,\ldots{}^{N}C_{n}$ indexes a unique arrangement of the $N$ individual
spin states and $Z_{R}$ is the associated partition function. We also assume
that the memory spin is in an arbitrary state initially. It should be noted
that due to the energy degeneracy of the spins in the reservoir and memory, we
only need to consider spin angular momentum exchange. We allow energy flow
between spatial degrees of freedom, but it does not contribute to the erasure
cost.

Following the protocol in Refs. Vaccaro and Barnett (2011); Barnett and
Vaccaro (2013), we make use of an energy degenerate ancillary
spin-$\frac{1}{2}$ particle that is initially in a state
$|{\downarrow}\rangle\langle{\downarrow}|$ corresponding to logical zero. A
controlled-not (CNOT) operation is then applied to the memory-ancilla system
with the memory spin acting as the control and the ancilla spin as the target.
The spinlabor cost for this initial step is $\frac{\hbar}{2}$ and it leaves
both memory and ancilla spins in an equal mixture of both spin up and spin
down. After the CNOT operation we allow the memory-ancilla system to reach
spin equilibrium with the reservoir by the particular exchange of spin angular
momentum according to the protocol devised by VB Vaccaro and Barnett (2011);
Barnett and Vaccaro (2013). This equilibration step is assumed to conserve the
total spin angular momentum. Cycles consisting of adding an additional ancilla
to the memory-ancilla system, the CNOT operation, and spin equilibration with
the reservoir are repeated until the desired degree of erasure is achieved. In
each cycle, the CNOT operation incurs a cost of $\hbar$ in spin angular
momentum in transforming the newly added ancilla from
$|{\downarrow}\rangle\langle{\downarrow}|$ to
$|{\uparrow}\rangle\langle{\uparrow}|$ if the memory spin is in the state
$|{\uparrow}\rangle\langle{\uparrow}|$. On average, the cost of the CNOT
operation for the $m^{th}$ cycle is given by $\hbar Q_{\uparrow}(m)$ where

$\displaystyle
Q_{\uparrow}(m)=\frac{e^{-(m+1)\gamma\hbar}}{1+e^{-(m+1)\gamma\hbar}}$ (7)

is the probability of the memory spin being in
$|{\uparrow}\rangle\langle{\uparrow}|$. The total cost $\mathcal{L}_{\rm
s}=\frac{\hbar}{2}+\sum_{m=1}^{\infty}\hbar Q_{\uparrow}(m)$ is bounded below
by Eq. (3).

Our objective is to study the fluctuations in this cost. The combined
reservoir (R) and memory-ancilla (M) system is isolated except for times when
the unitary CNOT operations take place. By conservation of spin angular
momentum, any spinlabor performed by the CNOT operation will result in the
change $\mathcal{L}_{\rm s}=\Delta J_{z}^{(T)}$ to the total spin angular
momentum $J_{z}^{(T)}=J_{z}^{(R)}+J_{z}^{(M)}$. Using Liouville’s theorem, we
first obtain 222See supplementary material the equivalent of Jarzynski’s
equality Jarzynski (1997a, b)

$\displaystyle\langle e^{(-\gamma\mathcal{L}_{\rm s}+\ln 2)}\rangle$
$\displaystyle=$
$\displaystyle\sum_{\mathbf{z}}f(\mathbf{z})e^{[-\gamma\hbar(\Delta
m^{(R)}_{j}+\Delta m^{(M)}_{j})+\ln 2]}$ $\displaystyle=$
$\displaystyle\frac{1+e^{-\gamma\hbar}}{1+e^{-2\gamma\hbar}}=A,$

where $f(\mathbf{z})$ is the probability distribution over phase space vectors
$\mathbf{z}$ which indexes internal spin angular momentum eigenvalues
$m_{j}^{(\cdot)}$ and external spatial coordinates. We define the probability
that the cost in spinlabor be $\mathcal{L}_{s}$ as $Pr(\mathcal{L}_{s})\equiv
f(\Delta J_{z}^{T}=\mathcal{L}_{s})$. Following Jarzynski’s analysis Jarzynski
(2011) we find the probability that the spinlabor cost $\mathcal{L}_{s}$
violates VB’s bound by $\epsilon$ satisfies

$\displaystyle{Pr^{(v)}(\epsilon)}\equiv{Pr}(\mathcal{L}_{\rm
s}\leq\gamma^{-1}\ln{2}-\epsilon)\leq Ae^{-\gamma\epsilon},$ (9)

where $A$ is given by Eq. (I). The above result is analogous to the
Jarzynski’s work on the probability of observing a violation of the Clausius-
Duhem inequality Jarzynski (1999, 2011). Judiciously restricting the sum on
the right side of Eq. (I) yields a tighter bound on $Pr^{(v)}(\epsilon)$ Note
(2)

$\displaystyle{Pr^{(v)}(\epsilon)}\leq Be^{-\gamma\epsilon},$ (10)

where $B=\sum_{\Delta J_{z}^{(T)}\leq\gamma^{-1}\ln
2}f(\mathbf{z})e^{[-\gamma\hbar(\Delta m^{(R)}_{j}+\Delta m^{(M)}_{j})+\ln
2]}$ and $B\leq A$. Using semi-analytic methods Note (2) we find an even
tighter bound for $\gamma\to 0$ (i.e. $\alpha\to 0.5$)

$\displaystyle{Pr^{(v)}(\epsilon)}\leq
Ce^{-\sqrt{\frac{\gamma}{\hbar}}\epsilon},$ (11)

where $C=Pr(\mathcal{L}_{\rm s}\leq\gamma^{-1}\ln{2})$. Eqs. (8)-(11) are the
central novel results of our work.

Figure 1: (a) Spinlabor statistics in the regime $\alpha\leq 0.4$. The
vertical black line represents the bound $\gamma^{-1}\ln{2}$. (b) Comparison
of bounds in Eq. (9)-(11) and the probability of violation. Figure 2: (a)
Spinlabor statistics in the limit $\alpha>0.4$. Here the vertical black line
represents the bound $\gamma^{-1}\ln{2}$. (b) Eq. (9), Eq. (10) and Eq. (11)
show an exponential suppression. Blue histogram shows probabilities below the
Vaccaro-Barnett bound.

We illustrate the spinlabor statistics and the probability of violation for
two distinct regimes of the reservoir spin polarization: (1) $\alpha\leq 0.4$
and (2) $\alpha>0.4$ in Fig. (1) and Fig. (2) respectively. Fig. 1(a) compares
the probability of the spinlabor cost $Pr(\mathcal{L}_{\rm s})$ for two values
of $\alpha\leq 0.4$ with the corresponding VB bounds $\gamma^{-1}\ln{2}$
(vertical black lines) and shows that there is a significant probability of
violating the bound (bars left of the bounds). This is analogous to the
fluctuation in the work cost below Landauer’s bound, as shown in Ref.
Dillenschneider and Lutz (2009). The probability of violation $Pr^{(v)}$ is
represented in Fig. 1(b) which shows that the fluctuations are exponentially
suppressed even faster than our tighter bounds in Eq. (10). and Eq. (11).

Fig. 2 shows the spinlabor statistics and the probability of violation for
three different values of $\alpha$ in the regime $\alpha>0.4$. As $\alpha$
increases, Fig. 2(a) shows the probability distribution $Pr(\mathcal{L}_{\rm
s})$ tends to become symmetric. Correspondingly, $\gamma\to 0$, the reservoir
approaches an equal mixture of spin up and spin down and its entropy
approaches its maximum value. This leads to increasing fluctuations in the
erasure which is reflected in $Pr(\mathcal{L}_{\rm s})$ becoming broader in
absolute terms. Nevertheless, the fluctuations relative to the erasure cost
become narrower and the cost diverges, as can be seen from Eq. (3).

Figures 1(a) and 2(a) also show a tradeoff between the average cost and the
relative size of the fluctuations compared to the cost, as follows. For
$\alpha\leq 0.4$ the average cost is lower and the fluctuations are relatively
pronounced, whereas for $\alpha>0.4$ the average cost is higher and the
fluctuations relative to it are lower.

It is interesting to recast the task of erasing information in terms of
spatial orientating. A reservoir with polarised spins (i.e. $\alpha<0.5$)
breaks rotational symmetry and becomes a resource for orientating the memory
spin. As $\alpha\to 0$, the polarisation increases and the reservoir becomes
more asymmetric; correspondingly, the orientating task becomes easier and so
the associated cost, $\mathcal{L}_{\rm s}$, lowers. For the opposite trend, as
$\alpha\to 0.5$, the polarisation reduces and the reservoir becomes more
rotationally symmetric. This makes the orientating task increasingly difficult
and so it incurs a rising cost.

##  II Discussion

Landauer’s erasure principle has been shown to be equivalent to the second law
of thermodynamics Plenio and Vitelli (2001). Similarly, Vaccaro and Barnett’s
erasure scheme gives an illustrative example of a generalized form of the
second law of thermodynamics for systems that exchange arbitrary conserved
quantities. Landauer’s erasure principle has been shown to require
modification for small systems where fluctuations are important
Dillenschneider and Lutz (2009). In this work we explored the corresponding
fluctuations in Vaccaro and Barnett’s erasure scheme. We showed that the
discrete nature of spin angular momentum exchange is reflected in the
spinlabor statistics and the probability of violation. Although we analyzed an
energy degenerate spin-$\frac{1}{2}$ systems in this work, our results can be
extended to energy-degenerate arbitrary-spin angular momentum reservoirs.

In addition to having an impact for a generalized second law, this work has
also implications for the first law as follows. In general, the average spin
angular momentum can be written as

$\displaystyle{J}_{z}=\sum_{j,m_{j}}\hbar p(j,m_{j})g(m_{j})$ (12)

where $p(j,m_{j})$ is the probability associated with the spin state
$(j,m_{j})$ and $g(m_{j})=m_{j}$ initially. The total change in ${J}_{z}$ is
given by Jaynes (1957a)

$\displaystyle\Delta{J_{z}}=\mathcal{L}_{s}+\mathcal{Q}_{s},$ (13)

where $\mathcal{L}_{s}=\sum_{j,m_{j}}\hbar p(j,m_{j})\Delta g(m_{j})$ is the
part of the change that is deterministic in origin, e.g. due to unitary
evolution by an external device, and $\mathcal{Q}_{s}=\sum_{j,m_{j}}\hbar
g(m_{j})\Delta p(j,m_{j})$ is the part that is non-deterministic, e.g. due to
the equilibration with another spin system. Eq. (13) is the spin equivalent of
the first law

$\displaystyle\Delta U=W+Q.$ (14)

Indeed, we refer to $\mathcal{Q}_{s}$ as spintherm (i.e. as the spin
equivalent of heat) in analogy with standard thermodynamics where a change in
the density matrix is associated with heat exchange. For the particular case
studied here, where one bit of information is erased from a spin-$\frac{1}{2}$
memory, the change in the spin angular momentum of the memory is
$\Delta{J_{z}^{(M)}}=-\frac{1}{2}\hbar$, and so according to Eq. (13)

$\displaystyle-\mathcal{Q}_{s}=\mathcal{L}_{s}+\frac{1}{2}\hbar.$ (15)

The total spintherm is greater than the spinlabor cost because the memory
initially contains inherent spintherm of $\frac{1}{2}\hbar$; this contrasts
with the usual case in Landauer’s erasure where heat and work costs are
equivalent. The negative sign in the expression $-\mathcal{Q}_{s}$ implies
that the spintherm is removed from the system. In this sense the VB erasure
protocol is an example of a _spintherm pump_.

Moreover, the number of first laws of generalized thermodynamics are equal to
number of conserved quantities if all the Lagrange multipliers are independent
of each other. In our case we have two independent laws: Eq. (14) which
corresponds to the conservation of energy in the energy exchange between the
spatial degrees of freedom (that does not contribute to the cost of the
erasure), and Eq. (13) which corresponds to the conservation of intrinsic spin
angular momentum.

Finally, we should mention that the experimental realisation and manipulation
of spin reservoirs is not new. For example, optical pumping is a well
established method to create spin polarized gases Anderson  _et al._ (1961);
Appelt  _et al._ (1998). In particular, in spin-exchange optical pumping of
${}^{3}{\rm He}$, the spin polarization is transferred between alkali atoms of
certain polarization and the ${}^{3}{\rm He}$ nuclei Appelt  _et al._ (1998).
The relaxation processes here include spin exchange collisions between alkali
atoms and the ${}^{3}{\rm He}$ nuclei. This is an example of entropy erasure
by spin exchange that does not appear to have been appreciated, especially in
the context of information erasure and thermodynamics.

To conclude, in contrast to recent work on abstract conserved quantities in
resource theories using a generalised Gibbs state formalism Guryanova  _et
al._ (2015); Halpern  _et al._ (2015); Lostaglio  _et al._ (2015), we have
considered Vaccaro and Barnett’s (VB) erasure scheme Vaccaro and Barnett
(2011); Barnett and Vaccaro (2013) in which the conserved quantity is the
physically-important spin angular momentum. This observable has the
distinction of having eigenvalues of integer multiples of $\hbar$. By studying
the discrete fluctuations in VB memory erasure scheme, we exposed several
interesting features; (a) a cross-over from the discrete to quasi-continuous
spinlabor probability distribution in the limit when $\alpha\to 0.5$, (b) the
trade-off between the cost of erasure and the relative fluctuations, (c) a
faster suppression (faster than Jarzynski’s analysis) of the probability of
violating VB bound (at a rate $\sqrt{\gamma/\hbar}$ found from the semi-
analytic calculation). Such a faster suppression than Jarzynski’s analysis is
the novel result of our work and has not been reported before in the context
of the fluctuations in memory erasure. Our work opens up an interesting
possibility of flucutation relations within the context of generalized-Gibbs
ensembles.

##  III Acknowledgements

This research was supported by the ARC Centre of Excellence Grant No.
CE110001027, ARC Linkage Grant No. LP140100797 and the Lockheed Martin
Corporation. JAV thanks S.M. Barnett, J. Jeffers, J.D. Cresser and B.
Schumacher for helpful discussions.

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Manifold Ways to Darboux–Halphen System

Manifold Ways to Darboux–Halphen System††This paper is a contribution to the
Special Issue on Modular Forms and String Theory in honor of Noriko Yui. The
full collection is available at http://www.emis.de/journals/SIGMA/modular-
forms.html

John Alexander Cruz MORALES ${}^{{\dagger}^{1}}$, Hossein MOVASATI
${}^{{\dagger}^{2}}$, Younes NIKDELAN ${}^{{\dagger}^{3}}$,
Raju ROYCHOWDHURY ${}^{{\dagger}^{4}}$ and Marcus A.C. TORRES
${}^{{\dagger}^{2}}$

J.A.C. Morales, H. Movasati, Y. Nikdelan, R. Roychowdhury and M.A.C. Torres

${}^{{\dagger}^{1}}$ Departamento de Matemáticas, Universidad Nacional de
Colombia, Bogotá, Colombia<EMAIL_ADDRESS>
${}^{{\dagger}^{2}}$ Instituto Nacional de Matemática Pura e Aplicada (IMPA),
Rio de Janeiro, Brazil<EMAIL_ADDRESS><EMAIL_ADDRESS>http://w3.impa.br/~hossein/

${}^{{\dagger}^{3}}$ Instituto de Matemática e Estatística (IME),
${}^{{\dagger}^{3}}$ Universidade do Estado do Rio de Janeiro (UERJ), Rio de
Janeiro, Brazil<EMAIL_ADDRESS>https://sites.google.com/site/younesnikdelan/

${}^{{\dagger}^{4}}$ Instituto de Física, Universidade de São Paulo (IF-USP),
São Paulo, Brazil<EMAIL_ADDRESS>
Received September 29, 2017, in final form January 03, 2018; Published online
January 08, 2018

Many distinct problems give birth to Darboux–Halphen system of differential
equations and here we review some of them. The first is the classical problem
presented by Darboux and later solved by Halphen concerning finding infinite
number of double orthogonal surfaces in $\mathbb{R}^{3}$. The second is a
problem in general relativity about gravitational instanton in Bianchi IX
metric space. The third problem stems from the new take on the moduli of
enhanced elliptic curves called Gauss–Manin connection in disguise developed
by one of the authors and finally in the last problem Darboux–Halphen system
emerges from the associative algebra on the tangent space of a Frobenius
manifold.

Darboux–Halphen system; Ramanujan system; Gauss–Manin connection; relativity
and gravitational theory; Bianchi IX metric; Frobenius manifold; Chazy
equation

34M55; 53D45; 83C05

##  1 Introduction

The Darboux–Halphen system of differential equations

$\displaystyle\dot{t}_{1}=t_{1}(t_{2}+t_{3})-t_{2}t_{3},$
$\displaystyle\dot{t}_{2}=t_{2}(t_{1}+t_{3})-t_{1}t_{3},$
$\displaystyle\dot{t}_{3}=t_{3}(t_{1}+t_{2})-t_{1}t_{2},\qquad\dot{}=\partial/\partial\tau,$
(1.1)

where $\tau$ is a free parameter, first came to existence when Darboux [9] was
studying the existence of an infinite number of double orthogonal system of
coordinates. He formulated the problem as follows: Let $A$ and $B$ be two
fixed surfaces in the $3$-dimensional Euclidean space $\mathbb{R}^{3}$ and
suppose that $\Sigma$ is the family of surfaces which are the locus of the
points such that the sum of their distances from the surfaces $A$ and $B$ are
constant; and $\Sigma^{\prime}$ is the family of surfaces which are the locus
of the points so that the difference of their distances from the surfaces $A$
and $B$ are constant. Is there a third family of surfaces intersecting
$\Sigma$ and $\Sigma^{\prime}$ orthogonally? When we restrict the third family
to the surfaces given by second degree equations, we find the Darboux–Halphen
system. In Section 2 we present Halphen’s solution to this problem.

The Darboux–Halphen system also emerge from a direct map from Ramanujan
relations (Section 3).

In 1979, Gibbons and Pope [14] found the Darboux–Halphen equations while
studying gravitational instanton solutions in Bianchi IX spaces without having
noticed it. Couple of decades later, Ablowitz et al. [1] pointed it out and
recently one of the authors [8] explored its integrability aspects. A
gravitational instanton is simply the (anti-)self-duality condition imposed on
the curvature of a Einstein manifold with asymptotic locally Euclidian
boundary conditions.

Hitchin [19] and Tod [28] realized that (anti-)self-duality in Bianchi IX
metric has a more general solution envolving to a Darboux–Halphen system
coupled to another system of linear differential equation similar to
Darboux–Halphen. A revised and simplified proof of the results of Tod and
Hitchin can be found in [4]. See [21] for a physical application in cosmology.
We review these works in Section 4.

Another author [23, 24] among us met the Darboux–Halphen system while
exploring the Gauss–Manin connection of a universal family of elliptic curves.
This method is called Gauss–Manin connection in disguise, which also name the
vector field in this method that gives rise to the Darboux–Halphen equations
and we present it in Section 5.

The last interesting problem where Darboux–Halphen system appears is in the
context of a 3-dimensional Frobenius manifold with a certain potential
function $F(t)$. Frobenius manifold arose as a geometrization of
Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations [10, 30], an
overdetermined system of differential equations that appear in the physics of
topological field theories in 2 dimensions. In this particular case of
dimension 3, the WDVV equation is known as Chazy equation, which has a close
tie with the solutions of Darboux system. We present it in Section 6, where we
follow Dubrovin’s notes [11, 12].

We conclude this article, crossing information between problems displayed
here, which led us to interesting remarks and an evidence that leads to a new
way on how to examine spectral curves from monopoles using Gauss–Manin
connection in disguise, further explored in [29] by one of the authors.

Throughout this text we make extensive use of Einstein summation convention
where the sum over identical upper and lower indices is implicit.

##  2 The Darboux problem

The above Darboux problem given in Section 1 is equivalent to the following
problem: _Let $A$ and $B$ be as before and suppose that $\Sigma$ is a family
of surfaces parallel to $A$ which is parameterized by $v$, and
$\Sigma^{\prime}$ is a family of surfaces parallel to $B$ that is
parameterized by $w$. Is there a third family of surfaces parameterized by
$\tau$ such that it intersects $\Sigma$ and $\Sigma^{\prime}$ orthogonally?_
Note that, two surfaces $A_{1}$ and $A_{2}$ are said to be _parallel_ , if
there exist a constant $c\in\mathbb{R}_{\neq 0}$ and a continuous one to one
map between points $a_{1}\in A_{1}$ and points $a_{2}\in A_{2}$, such that the
tangent planes at these points are parallel and the position vector $\bf
a_{2}=a_{1}+{\rm c}\hat{N}$, where $\bf\hat{N}$ is the unitary vector normal
to the surface $A_{1}$ at $a_{1}$. We say that a family of surfaces is
parameterized by $\varphi=\varphi(x,y,z)$, if any surface belonging to this
family is given by $\varphi(x,y,z)={\rm const}$, in which $x$, $y$, $z$ are
the standard coordinates of $\mathbb{R}^{3}$. If for a function
$\varphi=\varphi(x,y,z)$, we define

$\displaystyle\varphi_{x}=\frac{\partial\varphi}{\partial
x},\qquad\varphi_{y}=\frac{\partial\varphi}{\partial
y},\qquad\varphi_{z}=\frac{\partial\varphi}{\partial z},$

then in the latter problem, Darboux chose a case of parametrization of
parallel surfaces that gives the Gauss map for points on the parallel surfaces

$\displaystyle v_{x}^{2}+v_{y}^{2}+v_{z}^{2}=1,\qquad
w_{x}^{2}+w_{y}^{2}+w_{z}^{2}=1,$

and the condition of orthogonality at points in the intersection of $v$ and
$\tau$ and $w$ and $\tau$, respectively, is given by

$\displaystyle\tau_{x}v_{x}+\tau_{y}v_{y}+\tau_{z}v_{z}=0,\qquad\tau_{x}w_{x}+\tau_{y}w_{y}+\tau_{z}w_{z}=0.$

So the problem is equivalent to the following system of equations,

$\displaystyle v_{x}^{2}+v_{y}^{2}+v_{z}^{2}=1,$ (2.1) $\displaystyle
w_{x}^{2}+w_{y}^{2}+w_{z}^{2}=1,$ (2.2)
$\displaystyle\tau_{x}v_{x}+\tau_{y}v_{y}+\tau_{z}v_{z}=0,$ (2.3)
$\displaystyle\tau_{x}w_{x}+\tau_{y}w_{y}+\tau_{z}w_{z}=0.$ (2.4)

If for a function $\varphi$ of three variables $(x,y,z)$ we define the
operator

$\displaystyle\partial_{\varphi}:=\varphi_{x}\frac{\partial}{\partial
x}+\varphi_{y}\frac{\partial}{\partial y}+\varphi_{z}\frac{\partial}{\partial
z},$

then equations (2.1), (2.2), (2.3) and (2.4), respectively, are given by
$\partial_{v}v=1$, $\partial_{w}w=1$, $\partial_{\tau}v=\partial_{v}\tau=0$
and $\partial_{\tau}w=\partial_{w}\tau=0$, respectively. These equations imply
$\partial_{\tau}\partial_{v}v=0$, $\partial_{\tau}\partial_{w}w=0$,
$\partial_{v}\partial_{v}\tau=0$, $\partial_{w}\partial_{w}\tau=0$. Hence we
get

$\displaystyle
2\partial_{v}\partial_{v}\tau-\partial_{\tau}\partial_{v}v=0,\qquad
2\partial_{w}\partial_{w}\tau-\partial_{\tau}\partial_{w}w=0.$ (2.5)

The situation is more interesting when the family $(\tau)$ is of second
degree. Hence let us suppose that the family $(\tau)$ is given by

$\displaystyle ax^{2}+by^{2}+cz^{2}=1,$ (2.6)

where $a$, $b$, $c$ are functions of the parameter $\tau$. By this assumption,
equations (2.3) and (2.4) yield

$\displaystyle axv_{x}+byv_{y}+czv_{z}=0,\qquad axw_{x}+byw_{y}+czw_{z}=0.$
(2.7)

As well, from equation (2.5) we get

$\displaystyle av_{x}^{2}+bv_{y}^{2}+cv_{z}^{2}=0,\qquad
aw_{x}^{2}+bw_{y}^{2}+cw_{z}^{2}=0.$ (2.8)

Equations (2.1), (2.6), (2.7) and (2.8) imply

$\displaystyle\big{(}a^{2}b^{\prime}+b^{2}a^{\prime}\big{)}(xv_{y}-yv_{x})^{2}+\big{(}b^{2}c^{\prime}+c^{2}b^{\prime}\big{)}(yv_{z}-zv_{y})^{2}+\big{(}c^{2}a^{\prime}+a^{2}c^{\prime}\big{)}(zv_{x}-xv_{z})^{2}=0,$
$\displaystyle
ab(xv_{y}-yv_{x})^{2}+bc(yv_{z}-zv_{y})^{2}+ca(zv_{x}-xv_{z})^{2}=0,$ (2.9)

in which ${}^{\prime}=\frac{d}{d\tau}$. Analogously for $w$ we find

$\displaystyle\big{(}a^{2}b^{\prime}+b^{2}a^{\prime}\big{)}(xw_{y}-yw_{x})^{2}+\big{(}b^{2}c^{\prime}+c^{2}b^{\prime}\big{)}(yw_{z}-zw_{y})^{2}+\big{(}c^{2}a^{\prime}+a^{2}c^{\prime}\big{)}(zw_{x}-xw_{z})^{2}=0,$
$\displaystyle
ab(xw_{y}-yw_{x})^{2}+bc(yw_{z}-zw_{y})^{2}+ca(zw_{x}-xw_{z})^{2}=0.$ (2.10)

The two equations in (2.9) and the two equations in (2.10) become equivalent
if

$\displaystyle\frac{a^{2}b^{\prime}+b^{2}a^{\prime}}{ab}=\frac{b^{2}c^{\prime}+c^{2}b^{\prime}}{bc}=\frac{c^{2}a^{\prime}+a^{2}c^{\prime}}{ac}.$
(2.11)

If in (2.6) we substitute $a$, $b$, $c$ respectively by $\frac{1}{t_{1}}$,
$\frac{1}{t_{2}}$, $\frac{1}{t_{3}}$, then from (2.11) we get that the family

$\displaystyle\frac{x^{2}}{t_{1}}+\frac{y^{2}}{t_{2}}+\frac{z^{2}}{t_{3}}=1,$

is orthogonal to both $\Sigma$ and $\Sigma^{\prime}$ if $t_{1}$, $t_{2}$,
$t_{3}$ satisfy the following

$\displaystyle t_{3}\left(\frac{{\rm d}t_{1}}{{\rm d}\tau}+\frac{{\rm
d}t_{2}}{d\tau}\right)=t_{2}\left(\frac{{\rm d}t_{1}}{{\rm d}\tau}+\frac{{\rm
d}t_{3}}{{\rm d}\tau}\right)=t_{1}\left(\frac{{\rm d}t_{2}}{{\rm
d}\tau}+\frac{{\rm d}t_{3}}{{\rm d}\tau}\right).$ (2.12)

A particular case of the equation (2.12), which is known as Darboux–Halphen
system, is given in (1.1). In 1881, G. Halphen [15] studied this system of
differential equations and expressed a solution of it in terms of the
logarithmic derivatives of the theta functions; namely,

$\displaystyle t_{1}=2(\ln\theta_{2}(\tau))^{\prime},\qquad
t_{2}=2(\ln\theta_{3}(\tau))^{\prime},\qquad
t_{3}=2(\ln\theta_{4}(\tau))^{\prime}.$ (2.13)

with

$\displaystyle\theta_{2}(\tau):=\sum_{n=-\infty}^{\infty}q^{\frac{1}{2}(n+\frac{1}{2})^{2}},\qquad\theta_{3}(\tau):=\sum_{n=-\infty}^{\infty}q^{\frac{1}{2}n^{2}},$
$\displaystyle\theta_{4}(\tau):=\sum_{n=-\infty}^{\infty}(-1)^{n}q^{\frac{1}{2}n^{2}},\qquad
q=e^{2\pi i\tau},\qquad\tau\in{\mathbb{H}}.$

These theta functions can be written in terms of the more general theta
functions with characteristics $r$ and $s$ and arguments $z$ and $\sigma$:

$\displaystyle\vartheta[r,s](z,\sigma)=\sum_{m\in\mathbb{Z}}\exp\big{\\{}\pi
i(m+r)^{2}\sigma+2\pi i(m+r)(z+s)\big{\\}},\qquad
z,r,s\in\mathbb{C},\qquad\sigma\in\mathbb{H},$

such that

$\displaystyle\theta_{2}(\tau)=\vartheta[1/2,0](0,\tau),\qquad\theta_{3}(\tau)=\vartheta[0,0](0,\tau),\qquad\theta_{4}(\tau)=\vartheta[0,1/2](0,\tau).$

##  3 Ramanujan relations between Eisenstein series

The following differential equation

$\displaystyle q\frac{\partial E_{2}}{\partial
q}=\frac{1}{12}\big{(}E_{2}^{2}-E_{4}\big{)},\qquad q\frac{\partial
E_{4}}{\partial q}=\frac{1}{3}(E_{2}E_{4}-E_{6}),\qquad q\frac{\partial
E_{6}}{\partial q}=\frac{1}{2}\big{(}E_{2}E_{6}-E_{4}^{2}\big{)},$ (3.1)

where $E_{i}$’s are the Eisenstein series

$\displaystyle E_{2i}(q):=1+b_{i}\sum_{n=1}^{\infty}\left(\sum_{d\mid
n}d^{2i-1}\right)q^{n},\qquad i=1,2,3,$

and $(b_{1},b_{2},b_{3})=(-24,240,-504)$, was discovered by Ramanujan in [27]
and it is mainly known as Ramanujan’s relations between Eisenstein series.
Ramanujan was a master of formal power series and had a very limited access to
the modern mathematics of his time. In particular, he and many people in
number theory didn’t know that the differential equation (3.1) had already
been studied by Halphen in his book [16, p. 331], thirty years before S.
Ramanujan. The equalities of the coefficients of $x^{i}$ in

$\displaystyle
4(x-t_{1})(x-t_{2})(x-t_{3})=4(x-a_{1}E_{2})^{3}-a_{2}E_{4}(x-a_{1}E_{2})-a_{3}E_{6},$
(3.2)

where

$\displaystyle(a_{1},a_{2},a_{3}):=\left(\frac{2\pi i}{12},12\left(\frac{2\pi
i}{12}\right)^{2},8\left(\frac{2\pi i}{12}\right)^{3}\right),$

gives us a map from ${\mathbb{C}}^{3}$ into itself which transforms
Darboux–Halphen into Ramanujan differential equation, see [24, pp. 330, 335].

##  4 Self-duality in Bianchi IX metrics

An instanton is a field configuration that vanishes at spacetime infinity. It
is the quantum effect that leads metastable states to decay into vacuum. It is
a phenomenon that takes place in usual spacetime with signature $(-,+,+,+)$
but in order to perform physical calculation we use its equivalence with a
soliton solution (static and energetically stable field configuration) in
Euclidean spacetime. In Yang–Mills theory, self-duality of the field strength
$F_{\mu\nu}=\epsilon_{\mu\nu\rho\sigma}F^{\rho\sigma}$ in four spacetime
dimensions is a widely known instanton configuration [5]. Similarly, self-
duality constraint on the curvature two-form (and connection $1$-form) in
Cartan’s formalism of general relativity characterizes a gravitational
instanton. An important feature of self-duality of the curvature is that the
Ricci-tensor vanishes and it is a solution of the vacuum Einstein equations.
Also, self-dual curvature leads to solving a linear differential equation, a
task much easier than solving the full non-linear Einstein equations.
Gravitational instantons were found in Bianchi IX metrics, by Gibbons and Pope
[14]. Without realizing it, they arrived at Darboux–Halphen system from self-
duality constraints.

In [6], L. Bianchi studied continuous isometries of 3-dimensional spaces. He
noticed that the continuous isometries (continuous motion that preserve ${\rm
d}s^{2}$) of a space form a finite-dimensional Lie group and he classified
such spaces according to the corresponding group of isometries. Bianchi IX
corresponds to a 3-dimensional space with ${\rm SO}(3)$ or ${\rm SU}(2)$ as
Lie group of isometries. When we consider it in the context of 4-dimensional
cosmology, the isometries lie in the 3 spacial directions [21], but since we
are working in Euclidean signature we consider the isometry group ${\rm
SO}(3)$ as a subgroup of ${\rm SO}(4)$. In this configuration, as the
instanton vanishes at infinity, Lorentz symmetry is recovered and the space is
called asymptotically locally Euclidean (ALE). This same manifold describes
the reduced111Moduli of charge 2 monopoles reduced by quotient by
$\mathbb{R}^{3}$ action. moduli $M^{0}_{2}$ of charge 2 monopoles in a ${\rm
SU}(2)$ Yang–Mills–Higgs theory.

A magnetic 2-monopole is a soliton solution of charge 2 of Bogomolny equations
in the Yang–Mills–Higgs theory in $\mathbb{R}^{3}$, where ${\rm SU}(2)$
Yang–Mills is a gauge theory of 1-form connections $\bf{A}$ on a principal
${\rm SU}(2)$-bundle while the Higgs field $\Phi$ correspond to a section of
an associated $\mathfrak{su}(2)$-bundle [3, 20]. In [3], Atiyah and Hitchin
showed that the reduced moduli $M^{0}_{2}$ of 2-monopoles is a 4-dimensional
hyperkähler manifold and an anti-self-dual (curvature-wise) Einstein manifold.
Since $M^{0}_{2}$ admits ${\rm SO}(3)$ isometry, the metric is a Bianch
IX222Note that here the four coordinates of the moduli are not spacetime
directions, but internal parameters of a 2-monopole solution. (4.2). This is a
consequence of the hyperkähler structure of $M^{0}_{2}$ which has an
$S^{2}$-parameter family of complex structures, i.e., if $I$, $J$, $K$ are
covariantly constant complex structures in $M^{0}_{2}$ then $aI+bJ+cK$ is also
a covariantly constant complex structure in $M^{0}_{2}$ given that
$a^{2}+b^{2}+c^{2}=1$.

Here we present a detailed derivation of the Darboux–Halphen system starting
from the Euclidean Bianchi IX metric with ${\rm SO}(3)$ symmetry with an
imposition of the constraints of self-duality at the level of Riemann
curvature. The constraint of anti-self-duality yields an anti-instanton, a
solution with negative instanton number and we present this solution together
by using $\pm\operatorname{sign}$. We follow the steps of [8] and [14], see
also [26].

###  4.1 Geometric analysis

A metric for a 4-dimensional spacetime with coordinates
$(x^{1},x^{2},x^{3},x^{4})$, Euclidean time $x^{4}$ and ${\rm
SO}(3)\subset{\rm SO}(4)$ isometry is written in terms of invariant 1-forms
$\sigma^{i}$ on ${\rm SO}(3)$, dual to the standard basis $X_{1}$, $X_{2}$,
$X_{3}$ of its Lie algebra

$\displaystyle\sigma^{i}=-\frac{1}{r^{2}}\eta^{i}_{\mu\nu}x^{\mu}{\rm
d}x^{\nu},$

where $\mu,\nu=1,2,3,4$ and $i=1,2,3$ and $\eta^{i}_{\mu\nu}$ is a ’t Hooft
symbol given by

$\displaystyle\eta^{i}_{\mu\nu}=\begin{cases}\varepsilon_{i\mu\nu}+\delta_{i\mu}\delta_{\nu
4}-\delta_{\mu 4}\delta_{i\nu}\qquad\text{or},\\\
\varepsilon_{i\mu\nu}-\delta_{i\mu}\delta_{\nu 4}+\delta_{\mu
4}\delta_{i\nu}.\end{cases}$

according to two different choices of $\mathfrak{so}(3)$ generators in the Lie
algebra of the group ${\rm SO}(4)={\rm SU}(2)\times{\rm SU}(2)$. Among the
symbols presented above, $\varepsilon_{i\mu\nu}$ is the Levi-Civita symbol

$\displaystyle\varepsilon_{123}=\varepsilon_{231}=\varepsilon_{312}=-\varepsilon_{213}=-\varepsilon_{321}=-\varepsilon_{132}=1,\qquad\text{and
zero elsewhere,}$

and $\delta_{i\mu}$ refers to the Kronecker delta. The $\sigma^{i}$’s obey the
structure equation:

$\displaystyle{\rm
d}\sigma^{i}=-{\varepsilon}_{ijk}\sigma^{j}\wedge\sigma^{k},$ (4.1)

where we use Einstein summation in the repeated upper and lower indices here
and what follows below. This choice of ${\rm SO}(3)$ isometry leads to a 4D
spherically symmetric Bianchi IX metric

$\displaystyle{\rm d}s^{2}=c_{0}(r)^{2}{\rm
d}r^{2}+c_{1}^{2}(r)\big{(}\sigma^{1}\big{)}^{2}+c_{2}^{2}(r)\big{(}\sigma^{2}\big{)}^{2}+c_{3}^{2}(r)\big{(}\sigma^{3}\big{)}^{2},$
(4.2)

with $r=\sqrt[]{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}}$,
$c_{0}(r)=c_{1}(r)c_{2}(r)c_{3}(r)$ and $c_{1}$, $c_{2}$, $c_{3}$ being
functions of $r$.

We can impose self-duality in Bianchi IX metric in two ways:

  1. 1)

connection wise self-duality,

  2. 2)

curvature wise self-duality.

The connection wise self-duality is a stronger form of self-duality that leads
to self-dual curvature tensor [13]. This form of self-duality does not present
Darboux–Halphen system, but the Lagrange or Euler-top system [8]. It is not in
our goal to describe it here.

One can perform a standard analysis using vierbeins, leading to Cartan’s
structure equation. The vierbeins could be chosen as

$\displaystyle e^{0}=c_{0}{\rm d}r,\qquad
e^{i}=c_{i}\sigma^{i}\quad(\textnormal{no sum in}\ i),\qquad i=1,2,3,$

and the connection $1$-form can be obtained from the structure equation

$\displaystyle{\rm d}e^{a}=e^{b}\wedge{\omega^{a}}_{b},$

where $a,b=0,1,2,3$. Obviously, $e^{0}$ produces no connections while other
three does

$\displaystyle{\rm d}e^{0}=0,\qquad{\rm d}e^{i}=\partial_{r}c_{i}{\rm
d}r\wedge\sigma^{i}-c_{i}{\varepsilon}_{ijk}\sigma^{j}\wedge\sigma^{k}.$ (4.3)

The first term on the r.h.s. above gives ${\omega^{i}}_{0}$ while the second
term needs to be rewritten in order to produce a antisymmetric connection
$1$-form

$\displaystyle\varepsilon_{ijk}\frac{c_{i}^{2}}{c_{i}}\sigma^{j}\wedge\sigma^{k}=\varepsilon_{ijk}\frac{2c_{i}^{2}+\big{(}c_{j}^{2}-c_{k}^{2}\big{)}-\big{(}c_{j}^{2}-c_{k}^{2}\big{)}}{2c_{i}}\sigma^{j}\wedge\sigma^{k}$
$\displaystyle\hphantom{\varepsilon_{ijk}\frac{c_{i}^{2}}{c_{i}}\sigma^{j}\wedge\sigma^{k}}{}=\varepsilon_{ijk}\frac{c_{i}^{2}+c_{j}^{2}-c_{k}^{2}}{2c_{i}c_{j}}e^{j}\wedge\sigma^{k}+\varepsilon_{ikj}\frac{c_{i}^{2}+c_{k}^{2}-c_{j}^{2}}{2c_{i}c_{k}}e^{k}\wedge\sigma^{j}$
$\displaystyle\hphantom{\varepsilon_{ijk}\frac{c_{i}^{2}}{c_{i}}\sigma^{j}\wedge\sigma^{k}}{}=\varepsilon_{ijk}\frac{c_{i}^{2}+c_{j}^{2}-c_{k}^{2}}{c_{i}c_{j}}e^{j}\wedge\sigma^{k}.$

Rewriting (4.3),

$\displaystyle{\rm d}e^{i}=-\frac{\partial_{r}c_{i}}{c_{0}}\sigma^{i}\wedge
e^{0}-\varepsilon_{ijk}\frac{c_{i}^{2}+c_{j}^{2}-c_{k}^{2}}{c_{i}c_{j}}e^{j}\wedge\sigma^{k}.$

Hence,

$\displaystyle{\omega^{i}}_{0}=\frac{\partial_{r}c_{i}}{c_{0}}\sigma^{i}\quad(\text{no
sum in}\
i),\qquad{\omega^{i}}_{j}=-\varepsilon_{ijk}\frac{c_{i}^{2}+c_{j}^{2}-c_{k}^{2}}{c_{i}c_{j}}\sigma^{k}.$
(4.4)

Here the connection 1-form components are anti-symmetric under permutation of
its indices.

###  4.2 Curvature wise self-duality and Darboux–Halphen system

Curvature-wise self-duality was first studied in search of gravitational
instantons. It is a more general solution than imposing self-duality on
connection 1-forms. The Cartan-structure equation for Ricci tensor is

$\displaystyle R_{ij}={\rm d}\omega_{ij}+\omega_{im}\wedge{\omega^{m}}_{j}.$

The (anti-)self-duality of curvature demands that

$\displaystyle R_{0i}=\pm\frac{1}{2}{\varepsilon_{0ilm}}R^{lm}=\pm R_{jk},$
(4.5)

where $\\{i,j,k\\}$, in this order, are a cyclic permutation of $\\{1,2,3\\}$
and we used the fact that Euclidean vierbein indices are raised and lowered
with Kronecker deltas ${\delta^{i}}_{j}$. Comparing the l.h.s. and r.h.s. of
(4.5), we have

$\displaystyle{\rm
d}(\omega_{0i}\mp\omega_{jk})=\pm(\omega_{0k}\mp\omega_{ij})\wedge(\omega_{0j}\mp\omega_{ki}),$
$\displaystyle{\rm
d}\big{(}\lambda_{1}(r)\sigma^{i}\big{)}=\pm\big{(}\lambda_{3}(r)\sigma^{k}\big{)}\wedge\big{(}\lambda_{2}(r)\sigma^{j}\big{)}=\lambda_{3}(r)\lambda_{2}(r)\sigma^{k}\wedge\sigma^{j},$
$\displaystyle\partial_{r}\lambda_{1}dr\wedge\sigma^{i}+\lambda_{1}d\sigma^{i}=\mp\lambda_{2}\lambda_{3}\sigma^{j}\wedge\sigma^{k},$
(4.6)

where the second line comes from equation (4.4) with $\lambda_{1}$,
$\lambda_{2}$, $\lambda_{3}$ being functions of $r$. But the third line and
(4.1) show that $\lambda_{i}$’s are constants and
$\lambda_{1}=\pm\tfrac{1}{2}\lambda_{2}\lambda_{3}$. From cyclicity of $i$,
$j$, $k$ we obtain two more copies of (4.6). Therefore,

  1. 1)

$\lambda_{1}=\lambda_{2}=\lambda_{3}=0$ or

  2. 2)

$(\lambda_{1})^{2}=(\lambda_{2})^{2}=(\lambda_{3})^{2}=4$ with
$\lambda_{1}\lambda_{2}\lambda_{3}=\pm 8$.

The first case leads to self-dual connection $1$-forms and Euler-top system,
while the second case can be resumed to
$\lambda_{1}=\lambda_{2}=\lambda_{3}=\pm 2$ by an appropriate change of sign
in $c_{i}$ [14]. Therefore, from equations (4.4) and (4.6) we get

$\displaystyle\left(\frac{\partial_{r}c_{i}}{c_{0}}\right)=\mp\left(\frac{c_{j}^{2}+c_{k}^{2}-c_{i}^{2}}{c_{j}c_{k}}-2\right),\qquad\partial_{r}\big{(}\ln
c_{i}^{2}\big{)}=\mp
2\big{(}c_{j}^{2}+c_{k}^{2}-c_{i}^{2}-2c_{j}c_{k}\big{)}.$

One may suppose that we must parametrize the l.h.s. to match the linear form
in $c^{2}_{i}$, $c^{2}_{j}$ and $c^{2}_{k}$ of the r.h.s. in the equation
above. Essentially, the derivative operator aside, $c_{i}^{2}$ must be
parametrized such that

$\displaystyle\ln
c_{i}^{2}=\ln\Omega_{j}+\ln\Omega_{k}-\ln\Omega_{i}+\text{const}=\ln\left(\frac{\Omega_{j}\Omega_{k}}{\Omega_{i}}\right)+\text{const}.$

We choose new parametrization

$\displaystyle(c_{i})^{2}=\frac{\Omega_{j}\Omega_{k}}{2\Omega_{i}}\
\Rightarrow\ \Omega_{i}=2c_{j}c_{k}.$

which enable us to decouple the individual parameters into their own equations
turning into simpler expressions. This allows us to continue our analysis

$\displaystyle\partial_{r}\left[\ln\left(\frac{\Omega_{j}\Omega_{k}}{2\Omega_{i}}\right)\right]=\frac{\dot{\Omega}_{j}}{\Omega_{j}}+\frac{\dot{\Omega}_{k}}{\Omega_{k}}-\frac{\dot{\Omega}_{i}}{\Omega_{i}}=\mp\left(\frac{\Omega_{k}\Omega_{i}}{\Omega_{j}}+\frac{\Omega_{i}\Omega_{j}}{\Omega_{k}}-\frac{\Omega_{j}\Omega_{k}}{\Omega_{i}}-2\Omega_{i}\right).$

Adding up the above equation with cyclic permutations of $i$, $j$, $k$ we will
find that (anti-)self-dual cases of the Bianchi IX metric gives us

$\displaystyle\frac{\dot{\Omega}_{j}}{\Omega_{j}}+\frac{\dot{\Omega}_{k}}{\Omega_{k}}-\frac{\dot{\Omega}_{i}}{\Omega_{i}}=\mp\bigg{(}\frac{\Omega_{k}\Omega_{i}}{\Omega_{j}}+\frac{\Omega_{i}\Omega_{j}}{\Omega_{k}}-\frac{\Omega_{j}\Omega_{k}}{\Omega_{i}}-2\Omega_{i}\bigg{)}$
$\displaystyle\hphantom{\frac{\dot{\Omega}_{j}}{\Omega_{j}}+\frac{\dot{\Omega}_{k}}{\Omega_{k}}-\frac{\dot{\Omega}_{i}}{\Omega_{i}}}{}+$
$\displaystyle\frac{\dot{\Omega}_{k}}{\Omega_{k}}+\frac{\dot{\Omega}_{i}}{\Omega_{i}}-\frac{\dot{\Omega}_{j}}{\Omega_{j}}=\mp\bigg{(}\frac{\Omega_{i}\Omega_{j}}{\Omega_{k}}+\frac{\Omega_{j}\Omega_{k}}{\Omega_{i}}-\frac{\Omega_{k}\Omega_{i}}{\Omega_{j}}-2\Omega_{j}\bigg{)}$
$\displaystyle\hphantom{\frac{\dot{\Omega}_{j}}{\Omega_{j}}+\frac{\dot{\Omega}_{k}}{\Omega_{k}}-\frac{\dot{\Omega}_{i}}{\Omega_{i}}}{}\big{\downarrow}$
$\displaystyle\Rightarrow\ \frac{\dot{\Omega}_{k}}{\Omega_{k}}=\mp
2\left(\frac{\Omega_{i}\Omega_{j}}{\Omega_{k}}-\Omega_{i}-\Omega_{j}\right)\
\Rightarrow\
\dot{\Omega}_{k}=\mp(\Omega_{i}\Omega_{j}-\Omega_{k}\Omega_{i}-\Omega_{k}\Omega_{j}),$

where throughout derivative (denoted by dot) is taken with respect to $r$.
Self-duality proceeds to give us the classical Darboux–Halphen system

$\displaystyle\dot{\Omega}_{i}+\dot{\Omega}_{j}=2\Omega_{i}\Omega_{j}.$

###  4.3 General Bianchi IX self-dual Einstein metric

Following [4], we rewrite the Bianchi IX by adding a conformal scaling term
$F$ in the metric

$\displaystyle{\rm d}s^{2}=F\left({\rm
d}t^{2}+\frac{\sigma^{2}_{1}}{\Omega^{2}_{1}}+\frac{\sigma^{2}_{2}}{\Omega^{2}_{2}}+\frac{\sigma^{2}_{3}}{\Omega^{2}_{3}}\right).$

where t is the cosmological time and different from before, here the isometry
is ${\rm SU}(2)$ and ($\sigma_{i}$) are the corresponding ${\rm SU}(2)$
invariant forms along the spacial directions with structure constant

$\displaystyle{\rm d}\sigma_{1}=\sigma_{2}\wedge\sigma_{3},\qquad{\rm
d}\sigma_{2}=\sigma_{3}\wedge\sigma_{1},\qquad{\rm
d}\sigma_{3}=\sigma_{1}\wedge\sigma_{2}.$

We define the new variables $A_{i}(t)$ by the equations

$\displaystyle\partial_{t}\Omega_{i}=-\Omega_{j}\Omega_{k}+\Omega_{i}(A_{j}+A_{k}),$
(4.7)

for distinct $i$, $j$ and $k$ taking values in the set $\\{1,2,3\\}$. The
curvature-wise self-duality condition is expressed in terms of the new
variables $A_{i}$ in the form of the Darboux–Halphen system

$\displaystyle\partial_{t}A_{i}=-A_{j}A_{k}+A_{i}(A_{j}+A_{k}).$ (4.8)

Therefore we find $\Omega_{i}$’s by first solving system (4.8) and applying
its solution in (4.7). A non-trivial solution is given by (2.13)

$\displaystyle A_{1}=2\frac{\partial}{\partial t}(\ln\theta_{2}(it)),\qquad
A_{2}=2\frac{\partial}{\partial t}(\ln\theta_{3}(it)),\qquad
A_{3}=2\frac{\partial}{\partial t}(\ln\theta_{4}(it)).$

For simplicity, we rename $\vartheta_{2}\equiv\theta_{2}(it)$,
$\vartheta_{3}\equiv\theta_{3}(it)$, $\vartheta_{4}\equiv\theta_{4}(it)$. The
system (4.7) thus becomes

$\displaystyle\partial_{t}{\Omega}_{1}=-\Omega_{2}\Omega_{3}+2\Omega_{1}\partial_{t}\ln(\vartheta_{3}\vartheta_{4}),$
$\displaystyle\partial_{t}{\Omega}_{2}=-\Omega_{3}\Omega_{1}+2\Omega_{2}\partial_{t}\ln(\vartheta_{4}\vartheta_{2}),$
$\displaystyle\partial_{t}{\Omega}_{3}=-\Omega_{1}\Omega_{2}+2\Omega_{3}\partial_{t}\ln(\vartheta_{2}\vartheta_{3}).$
(4.9)

There is a class of solutions of this system that satisfies vacuum Einstein
equations

$\displaystyle R_{ab}-\frac{1}{2}Rg_{ab}+\Lambda g_{ab}=0,$

once we choose the appropriate conformal factor $F$ [28]. This class depend on
the values of the cosmological constant $\Lambda$ and satisfy the constraint

$\displaystyle\vartheta^{4}_{2}\Omega_{1}^{2}-\vartheta^{4}_{3}\Omega_{2}^{2}+\vartheta^{4}_{4}\Omega_{3}^{2}=\frac{\pi^{2}}{4}\vartheta^{4}_{2}\vartheta^{4}_{3}\vartheta^{4}_{4},$
(4.10)

The general two-parametric family of solutions of the system (4.9) satisfying
condition (4.10), is given by the following formulas

$\displaystyle\Omega_{1}=-\frac{i}{2}\vartheta_{3}\vartheta_{4}\frac{\frac{{\rm
d}}{{\rm d}q}\vartheta\big{[}p,q+\tfrac{1}{2}\big{]}}{e^{\pi
ip}\vartheta[p,q]},\qquad\Omega_{2}=\frac{i}{2}\vartheta_{2}\vartheta_{4}\frac{\frac{{\rm
d}}{{\rm d}q}\vartheta\big{[}p+\tfrac{1}{2},q+\tfrac{1}{2}\big{]}}{e^{\pi
ip}\vartheta[p,q]},$
$\displaystyle\Omega_{3}=\frac{i}{2}\vartheta_{2}\vartheta_{4}\frac{\frac{{\rm
d}}{{\rm d}q}\vartheta\big{[}p+\tfrac{1}{2},q+\tfrac{1}{2}\big{]}}{e^{\pi
ip}\vartheta[p,q]},$

where $\vartheta[p,q]$ denotes the theta function $\vartheta[p,q](0,ir)$,
$p,q\in{\mathbb{C}}$. The corresponding metric is real and satisfies the
Einstein equations for negative cosmological constant $\Lambda$ if
$p\in\mathbb{R}$ and $\mathcal{R}\\{q\\}=\tfrac{1}{2}$ (real part of $q$) or
for positive cosmological constant if $q\in\mathbb{R}$ and
$\mathcal{R}\\{p\\}=\tfrac{1}{2}$. In both the cases the corresponding
conformal factor is given by

$\displaystyle
F=\frac{2}{\pi\Lambda}\frac{\Omega_{1}\Omega_{2}\Omega_{3}}{\big{(}\frac{{\rm
d}}{{\rm d}q}\ln\vartheta[p,q]\big{)}^{2}}.$

There is another family of solutions

$\displaystyle\Omega_{1}=\frac{1}{t+q_{0}}+2\frac{\partial}{\partial
t}\ln\vartheta_{2},\qquad\Omega_{2}=\frac{1}{t+q_{0}}+2\frac{\partial}{\partial
t}\ln\vartheta_{3},\qquad\Omega_{3}=\frac{1}{t+q_{0}}+2\frac{\partial}{\partial
t}\ln\vartheta_{4},$

with $q_{0}\in\mathbb{R}$, that defines manifolds with vanishing cosmological
constant if

$\displaystyle F=C(t+q_{0})^{2}\Omega_{1}\Omega_{2}\Omega_{3}.$

##  5 Gauss–Manin connection in disguise

In this section we explain how one can derive the Darboux–Halphen equations
from the Gauss–Manin connection of a universal family of elliptic curves. This
has been taken from the references [23, 24]. The family of elliptic curves

$\displaystyle E_{t}\colon\ y^{2}-4(x-t_{1})(x-t_{2})(x-t_{3})=0,\qquad
t\in{\mathbb{C}}^{3}\backslash\cup_{i,j}\\{t_{i}=t_{j}\\},$

is the universal family for the moduli of $3$-tuple $(E,(P,Q),\omega)$, where
$E$ is an elliptic curve and $\omega\in H^{1}_{\rm dR}(E)\backslash F^{1}$.
There is a unique regular differential 1-form in the Hodge filtration
$\omega_{1}\in F^{1}$, such that $\langle\omega,\omega_{1}\rangle=1$ and
$\omega$, $\omega_{1}$ together form a basis of $H^{1}_{\rm dR}(E)$. $P$ and
$Q$ are a pair of points of $E$ that generate the $2$-torsion subgroup with
the Weil pairing $e(P,Q)=-1$. The points $P$ and $Q$ are given by $(t_{1},0)$
and $(t_{2},0)$ and $\omega=\frac{x{\rm d}x}{y}$ and $\omega_{1}=\frac{{\rm
d}x}{y}$. The Gauss–Manin connection of the family of elliptic curves $E_{t}$
written in the basis $\frac{{\rm d}x}{y}$, $\frac{x{\rm d}x}{y}$ is given as
bellow

$\displaystyle\nabla\begin{pmatrix}\frac{{\rm d}x}{y}\vspace{1mm}\\\
\frac{x{\rm d}x}{y}\end{pmatrix}=A\begin{pmatrix}\frac{{\rm
d}x}{y}\vspace{1mm}\\\ \frac{x{\rm d}x}{y}\end{pmatrix},$

where

$\displaystyle A=\frac{{\rm
d}t_{1}}{2(t_{1}-t_{2})(t_{1}-t_{3})}\begin{pmatrix}-t_{1}&1\\\
t_{2}t_{3}-t_{1}(t_{2}+t_{3})&t_{1}\end{pmatrix}$
$\displaystyle\hphantom{A=}{}+\frac{{\rm
d}t_{2}}{2(t_{2}-t_{1})(t_{2}-t_{3})}\begin{pmatrix}-t_{2}&1\\\
t_{1}t_{3}-t_{2}(t_{1}+t_{3})&t_{2}\end{pmatrix}$
$\displaystyle\hphantom{A=}{}+\frac{{\rm
d}t_{3}}{2(t_{3}-t_{1})(t_{3}-t_{2})}\begin{pmatrix}-t_{3}&1\\\
t_{1}t_{2}-t_{3}(t_{1}+t_{2})&t_{3}\end{pmatrix}.$

The reader who is not familiar with the Gauss–Manin connection must replace
$\nabla$ with ${\rm d}\int_{\delta_{t}}$, where $t_{i}$’s are assumed to
depend on some parameter $\tau$, ${\rm d}=\frac{\partial}{\partial\tau}$ and
$\delta_{t}$ is a 1-dimensional homology class in $E_{t}$. In the parameter
space of the family of elliptic curves $E_{t}$ there is a unique vector field
$R$, such that

$\displaystyle\nabla_{R}\left(\frac{{\rm d}x}{y}\right)=-\frac{x{\rm
d}x}{y},\qquad\nabla_{R}\left(\frac{x{\rm d}x}{y}\right)=0.$

The vector field $R$ is given by the Darboux–Halphen system (1.1) and it is
called Gauss–Manin connection in disguise.

##  6 Frobenius manifolds and Chazy equation

Frobenius manifolds were developed in order to give a geometrical meaning to
WDVV equations:

$\displaystyle\frac{\partial^{3}F(t)}{\partial t^{\alpha}\partial
t^{\beta}\partial
t^{\lambda}}\eta^{\lambda\mu}\frac{\partial^{3}F(t)}{\partial t^{\mu}\partial
t^{\gamma}\partial t^{\delta}}=\frac{\partial^{3}F(t)}{\partial
t^{\delta}\partial t^{\beta}\partial
t^{\lambda}}\eta^{\lambda\mu}\frac{\partial^{3}F(t)}{\partial t^{\mu}\partial
t^{\gamma}\partial t^{\alpha}},$

where $F(t)$, with $t=(t^{1},t^{2},\dots,t^{n})$, is a quasi-homogeneous
function on its parameters. The above equations conceal properties of an
associative commutative algebra on the tangent space of a manifold $M$ of
dimension $n$ defined by the parameter space $(t^{1},t^{2},\dots,t^{n})$.
That’s the essence of a Frobenius manifold that we will detail below starting
with the algebraic structure in $TM$.

###  6.1 Frobenius algebra

An algebra $A$ over $\mathbb{C}$ is Frobenius if

  * •

it is a commutative associative $\mathbb{C}$-algebra with unity $e$,

  * •

it has a $\mathbb{C}$-bilinear symmetric non-degenerate inner product

$\displaystyle\langle\,,\,\rangle\colon\ $ $\displaystyle A\times
A\longrightarrow\mathbb{C},$ $\displaystyle(a,b)\mapsto\langle a,b\rangle,$

which is invariant, i.e., $\langle a.b,c\rangle=\langle a,b.c\rangle$

Properties: Let $e_{\alpha}$, $\alpha=1,\dots,N$, be any basis in $A$, such
that $e_{1}=e$ is the unity. By notation, we define
$\eta_{\alpha\beta}:=\langle e_{\alpha},e_{\beta}\rangle$, which yields the
matrix $\eta:=[\eta_{\alpha\beta}]_{1\leq\alpha,\beta\leq N}$ and its inverse
$\eta^{-1}:=[\eta^{\alpha\beta}]_{1\leq\alpha,\beta\leq N}$, and it follows
$\eta^{\alpha\beta}\eta_{\beta\gamma}=\delta^{\alpha}_{\gamma}$. By writing
$e_{\alpha}\cdot e_{\beta}$ in the given basis, we find the structure
constants $c_{\alpha\beta}^{\gamma}$ defined by $e_{\alpha}\cdot
e_{\beta}=c_{\alpha\beta}^{\gamma}e_{\gamma}$. If we set
$c_{\alpha\beta\gamma}=c_{\alpha\beta}^{\epsilon}\eta_{\epsilon\gamma}$, then
we get
$c_{\alpha\beta}^{\gamma}=c_{\alpha\beta\epsilon}\eta^{\epsilon\gamma}$. Note
that in all above expressions, and in what follows, Einstein summation of
indices is implicit. Therefore, $\eta_{\alpha\beta}$ and the structure
constants $c_{\alpha\beta}^{\gamma}$ satisfy

commutativity $\displaystyle\eta_{\alpha\beta}=\eta_{\beta\alpha},$ (6.1)
associativity
$\displaystyle(e_{\alpha}.e_{\beta}).e_{\gamma}=e_{\alpha}.(e_{\beta}.e_{\gamma})\therefore
c_{\alpha\beta}^{\epsilon}c_{\epsilon\gamma}^{\delta}=c_{\alpha\epsilon}^{\delta}c_{\delta\beta}^{\epsilon},$
(6.2) normalization $\displaystyle
c_{1\beta}^{\alpha}=\delta_{\beta}^{\alpha},$ (6.3) invariance&commutat.
$\displaystyle c_{\alpha\beta\gamma}=\langle
e_{\alpha}e_{\beta},e_{\gamma}\rangle=c_{\beta\alpha\gamma}=c_{\alpha\gamma\beta}.$
(6.4)

Now consider an $n$-parametric deformation of the Frobenius algebra $A_{t}$,
$t=(t^{1},t^{2},\dots,t^{n})$, with structure constants
$c_{\alpha\beta}^{\gamma}(t)$ preserving relations (6.1) to (6.4). Such
deformed algebra $A_{t}$ can be seen as a fiber bundle with the space of
parameters $t\in M$ as base space. We identify this fiber bundle with the
tangent bundle $TM$ to arrive at the definition of a Frobenius manifold. The
requirements for this to happen are presented in the definition below.

###  6.2 Frobenius manifold

A Frobenius manifold M of dimension $n$, is an $n$-dimensional Riemannian
manifold, such that for all $t\in M$ the tangent space $T_{t}M$ contains the
structure of a Frobenius algebra $(A_{t},\langle\,,\,\rangle_{t})$, satisfying
the following axioms:

  1. A.1.

The metric $\langle\,,\,\rangle_{t}$ on $M$ is flat. The unit vector $e$ must
be flat, i.e., $\nabla e=0$, where $\nabla$ is the Levi-Civita connection for
the metric.

  2. A.2.

Let $c$ be the 3-tensor $c(x,y,z)=\langle x.y,z\rangle$, with $x,y,z\in
T_{t}M$. Then the 4-tensor $(\nabla_{w}c)(x,$ $y,z)$ must be symmetric in
$x,y,z,w\in T_{t}M$.

  3. A.3.

A linear vector field $E$ must be fixed on $M$, i.e., $\nabla(\nabla E)=0$
such that the corresponding one-parameter group of diffeomorphisms acts by
conformal transformations of the metric $\langle\,,\,\rangle$ and by rescaling
on the Frobenius algebras $T_{t}M$.

The flatness of the metric $\langle\,,\,\rangle$ implies the existence of a
system of flat coordinates $t^{1},\dots,t^{n}$ on $M$. In these flat
coordinates the structure constants of $A_{t}$ are given by

$\displaystyle\frac{\partial}{\partial t^{\alpha}}.\frac{\partial}{\partial
t^{\beta}}=c_{\alpha\beta}^{\gamma}(t)\frac{\partial}{\partial t^{\gamma}}.$

Potential deformation. If there is a function $F(t)$, called potential, such
that the structure constants of $A_{t}$, $t\in M$, can be locally represented
as

$\displaystyle c_{\alpha\beta\gamma}(t)=\frac{\partial^{3}F(t)}{\partial
t^{\gamma}\partial t^{\alpha}\partial t^{\beta}},$

satisfying A.2 with unity vector $e=\tfrac{\partial}{\partial t^{1}}$, and the
metric given by

$\displaystyle\eta_{\beta\gamma}=c_{1\beta\gamma}=\frac{\partial^{3}F(t)}{\partial
t^{\gamma}\partial t^{1}\partial
t^{\beta}}\qquad\textnormal{s.t.}\qquad\frac{\partial^{4}F(t)}{\partial
t^{\alpha}\partial t^{\gamma}\partial t^{1}\partial t^{\beta}}=0,$

satisfying A.1, and the associativity property (6.2) represented by the WDVV
equations

$\displaystyle\frac{\partial^{3}F(t)}{\partial t^{\alpha}\partial
t^{\beta}\partial
t^{\lambda}}\eta^{\lambda\mu}\frac{\partial^{3}F(t)}{\partial t^{\mu}\partial
t^{\gamma}\partial t^{\delta}}=\frac{\partial^{3}F(t)}{\partial
t^{\delta}\partial t^{\beta}\partial
t^{\lambda}}\eta^{\lambda\mu}\frac{\partial^{3}F(t)}{\partial t^{\mu}\partial
t^{\gamma}\partial t^{\alpha}},$

then $M$ is a Frobenius manifold and the Frobenius algebra $A_{t}$ is called a
potential deformation. Note that the condition A.3 is satisfied by a
quasihomogeneous function $F(t)$.

######  Example 6.1.

Let $\dim M=3$, and consider the basis $e=e_{1}=\tfrac{\partial}{\partial
t^{1}}$, $e_{2}=\tfrac{\partial}{\partial t^{2}}$ and
$e_{3}=\tfrac{\partial}{\partial t^{3}}$ of the 3-dimensional algebra $A_{t}$.
Then the multiplication law is given by

$\displaystyle e_{2}^{2}=f_{xxy}e_{1}+f_{xxx}e_{2}+e_{3},\qquad
e_{2}e_{3}=f_{xyy}e_{1}+f_{xxy}e_{2},\qquad
e_{3}^{2}=f_{yyy}e_{1}+f_{xyy}e_{2},$

where the funtion $F(t)$ has the form
$F(t)=\frac{1}{2}(t^{1})^{2}t^{3}+\frac{1}{2}t^{1}(t^{2})^{2}+f(t^{2},t^{3})$
and the notation $f_{x}=\partial_{x}f(x,y)$, $f_{y}=\partial_{y}f(x,y)$. The
associativity condition $(e_{2}^{2})e_{3}=e_{2}(e_{2}e_{3})$ implies the
following PDE for $f(x,y)$:

$\displaystyle f^{2}_{xxy}=f_{yyy}+f_{xxx}f_{xyy}.$ (6.5)

###  6.3 Chazy equation and Darboux–Halphen system

In this section we explain how the Chazy equation arises from a 3-dimensional
Frobenius manifold. We follow Dubrovin’s notes [11, 12]. Let dim $M=3$, and
consider the potential function

$\displaystyle
F(t)=\frac{1}{2}\big{(}t^{1}\big{)}^{2}t^{3}+\frac{1}{2}t^{1}\big{(}t^{2}\big{)}^{2}-\frac{\big{(}t^{2}\big{)}^{4}}{16}\gamma\big{(}t^{3}\big{)},$

where $\gamma(\tau)$ is an unknown $2\pi$-periodic function that is analytic
at $\tau=i\infty$. Then the associativity condition (6.5) leads to the Chazy
equation

$\displaystyle\gamma^{\prime\prime\prime}=6\gamma\gamma^{\prime\prime}-9(\gamma^{\prime})^{2}.$

The solution, up to a shift in $\tau$, is given by $\gamma(\tau)=\frac{\pi
i}{3}E_{2}(\tau)$, where $E_{2}$ is the weight-2 Eisenstein series. Notice
that the Darboux–Halphen solution (2.13) leads to

$\displaystyle t_{1}+t_{2}+t_{3}=\frac{\pi i}{2}E_{2}(\tau),$

which can be easily checked from (3.2) or by writing the theta functions in
terms of Dedekind eta function, see [7, Chapter 3, p. 29]. Applying $\tau$
derivatives on both sides and using Darboux–Halphen equations, one can also
check that the solution to Darboux–Halphen system (2.13) are the roots of the
cubic equation

$\displaystyle
y^{3}-\frac{3}{2}\gamma(\tau)y^{2}+\frac{3}{2}\gamma^{\prime}(\tau)y-\frac{1}{4}\gamma^{\prime\prime}(\tau)=0.$

##  7 Conclusion

The study of Darboux–Halphen equations in several different problems in
theoretical physics and mathematics raised more and more questions that
eventually lead us to further studies.

The problem involving Gauss–Manin connection in disguise lies at the center of
some questions. It shows that the Darboux–Halphen system corresponds to a
vector field in the moduli of an enhanced elliptic curve. As mentioned in
Section 4, the Bianchi IX four-manifold (4.2) also describes the reduced
moduli of 2-monopoles and its self-dual curvature equations can be
reparametrized to the Darboux–Halphen equations. Furthermore, in the problem
of 2-monopoles it has been found that a 2-monopole solution relates to an
elliptic curve as its spectral curve [17, 18, 20]. Therefore, we believe that
Gauss–Manin connection in disguise is a new way to demonstrate the association
of spectral curves and the curvature equations of the moduli of monopole
solutions. Starting from these coincidences, in [29] one of the authors
started to find more evidences to support this idea.

Another interesting remark is the fact that potential functions and structure
constants in Frobenius manifolds correspond to prepotentials (or genus zero
topological partition function) and Yukawa couplings in topological string
theory and Gauss–Manin connection in disguise has been used in the moduli of
enhanced Calabi–Yau varieties to find polynomial expressions for Yukawa
couplings and higher genus topological partition functions [2, 25]. It would
be interesting to find cases where the moduli of enhanced Calabi–Yau varieties
are also Frobenius manifolds. In particular, the Frobenius manifold presented
in Section 6.3 is a case of modular Frobenius manifold where the prepotential
is preserved under a inverse symmetry that acts as an $S$ generator of the
modular group ${\rm SL}(2,\mathbb{Z})$ in $t^{3}$ direction [22]. Such
modularity is a desirable property that can establish a relation to
Gauss–Manin connection in disguise and may be extended to the group of
transformations of Calabi–Yau modular forms [2, 25].

### Acknowledgements

During the period of preparation of the manuscript MACT was fully sponsored by
CNpQ-Brasil. The research of RR was supported by FAPESP through Instituto de
Fisica, Universidade de Sao Paulo with grant number 2013/17765-0. The work was
initiated during the visit of RR to IMPA, he would like to thank IMPA for the
hospitality during the course of this project.

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