Manifold Ways to Darboux-Halphen System
Many distinct problems give birth to the 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 an infinite number of double orthogonal surfaces in R³. The second is a p...
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| description | Many distinct problems give birth to the 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 an infinite number of double orthogonal surfaces in R³. The second is a problem in general relativity about a gravitational instanton in the Bianchi IX metric space. The third problem stems from the new take on the moduli of enhanced elliptic curves called the 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.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 14 (2018), 003, 14 pages
Manifold Ways to Darboux–Halphen System
John Alexander Cruz MORALES †
1
, Hossein MOVASATI †
2
, Younes NIKDELAN †3,
Raju ROYCHOWDHURY †4 and Marcus A.C. TORRES †
2
†1 Departamento de Matemáticas, Universidad Nacional de Colombia, Bogotá, Colombia
E-mail: jacruzmo@unal.edu.co
†2 Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Brazil
E-mail: hossein@impa.br, mtorres@impa.br
URL: http://w3.impa.br/~hossein/
†3 Instituto de Matemática e Estat́ıstica (IME),
Universidade do Estado do Rio de Janeiro (UERJ), Rio de Janeiro, Brazil
E-mail: younes.nikdelan@ime.uerj.br
URL: https://sites.google.com/site/younesnikdelan/
†4 Instituto de F́ısica, Universidade de São Paulo (IF-USP), São Paulo, Brazil
E-mail: raju@if.usp.br
Received September 29, 2017, in final form January 03, 2018; Published online January 08, 2018
https://doi.org/10.3842/SIGMA.2018.003
Abstract. 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 or-
thogonal surfaces in R3. 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.
Key words: Darboux–Halphen system; Ramanujan system; Gauss–Manin connection; rela-
tivity and gravitational theory; Bianchi IX metric; Frobenius manifold; Chazy equation
2010 Mathematics Subject Classification: 34M55; 53D45; 83C05
1 Introduction
The Darboux–Halphen system of differential equations
ṫ1 = t1(t2 + t3)− t2t3,
ṫ2 = t2(t1 + t3)− t1t3,
ṫ3 = t3(t1 + t2)− t1t2, ˙ = ∂/∂τ, (1.1)
where τ 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 f ixed surfaces in the 3-dimensional Euclidean space R3 and
suppose that Σ 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 Σ′ is the family of surfaces
which are the locus of the points so that the difference of their distances from the surfaces A
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
mailto:jacruzmo@unal.edu.co
mailto:hossein@impa.br
mailto:mtorres@impa.br
http://w3.impa.br/~hossein/
mailto:younes.nikdelan@ime.uerj.br
https://sites.google.com/site/younesnikdelan/
mailto:raju@if.usp.br
https://doi.org/10.3842/SIGMA.2018.003
http://www.emis.de/journals/SIGMA/modular-forms.html
2 J.A.C. Morales, H. Movasati, Y. Nikdelan, R. Roychowdhury and M.A.C. Torres
and B are constant. Is there a third family of surfaces intersecting Σ and Σ′ 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 gra-
vitational 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 Σ is a family of surfaces parallel to A which is parameterized
by v, and Σ′ is a family of surfaces parallel to B that is parameterized by w. Is there a third
family of surfaces parameterized by τ such that it intersects Σ and Σ′ orthogonally? Note that,
two surfaces A1 and A2 are said to be parallel, if there exist a constant c ∈ R 6=0 and a continuous
one to one map between points a1 ∈ A1 and points a2 ∈ A2, such that the tangent planes at these
points are parallel and the position vector a2 = a1 + cN̂, where N̂ is the unitary vector normal
to the surface A1 at a1. We say that a family of surfaces is parameterized by ϕ = ϕ(x, y, z),
if any surface belonging to this family is given by ϕ(x, y, z) = const, in which x, y, z are the
standard coordinates of R3. If for a function ϕ = ϕ(x, y, z), we define
ϕx =
∂ϕ
∂x
, ϕy =
∂ϕ
∂y
, ϕz =
∂ϕ
∂z
,
Manifold Ways to Darboux–Halphen System 3
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
v2x + v2y + v2z = 1, w2
x + w2
y + w2
z = 1,
and the condition of orthogonality at points in the intersection of v and τ and w and τ , respec-
tively, is given by
τxvx + τyvy + τzvz = 0, τxwx + τywy + τzwz = 0.
So the problem is equivalent to the following system of equations,
v2x + v2y + v2z = 1, (2.1)
w2
x + w2
y + w2
z = 1, (2.2)
τxvx + τyvy + τzvz = 0, (2.3)
τxwx + τywy + τzwz = 0. (2.4)
If for a function ϕ of three variables (x, y, z) we define the operator
∂ϕ := ϕx
∂
∂x
+ ϕy
∂
∂y
+ ϕz
∂
∂z
,
then equations (2.1), (2.2), (2.3) and (2.4), respectively, are given by ∂vv = 1, ∂ww = 1,
∂τv = ∂vτ = 0 and ∂τw = ∂wτ = 0, respectively. These equations imply ∂τ∂vv = 0, ∂τ∂ww = 0,
∂v∂vτ = 0, ∂w∂wτ = 0. Hence we get
2∂v∂vτ − ∂τ∂vv = 0, 2∂w∂wτ − ∂τ∂ww = 0. (2.5)
The situation is more interesting when the family (τ) is of second degree. Hence let us
suppose that the family (τ) is given by
ax2 + by2 + cz2 = 1, (2.6)
where a, b, c are functions of the parameter τ . By this assumption, equations (2.3) and (2.4)
yield
axvx + byvy + czvz = 0, axwx + bywy + czwz = 0. (2.7)
As well, from equation (2.5) we get
av2x + bv2y + cv2z = 0, aw2
x + bw2
y + cw2
z = 0. (2.8)
Equations (2.1), (2.6), (2.7) and (2.8) imply(
a2b′ + b2a′
)
(xvy − yvx)2 +
(
b2c′ + c2b′
)
(yvz − zvy)2 +
(
c2a′ + a2c′
)
(zvx − xvz)2 = 0,
ab(xvy − yvx)2 + bc(yvz − zvy)2 + ca(zvx − xvz)2 = 0, (2.9)
in which ′ = d
dτ . Analogously for w we find(
a2b′ + b2a′
)
(xwy − ywx)2 +
(
b2c′ + c2b′
)
(ywz − zwy)2 +
(
c2a′ + a2c′
)
(zwx − xwz)2 = 0,
ab(xwy − ywx)2 + bc(ywz − zwy)2 + ca(zwx − xwz)2 = 0. (2.10)
The two equations in (2.9) and the two equations in (2.10) become equivalent if
a2b′ + b2a′
ab
=
b2c′ + c2b′
bc
=
c2a′ + a2c′
ac
. (2.11)
4 J.A.C. Morales, H. Movasati, Y. Nikdelan, R. Roychowdhury and M.A.C. Torres
If in (2.6) we substitute a, b, c respectively by 1
t1
, 1
t2
, 1
t3
, then from (2.11) we get that the family
x2
t1
+
y2
t2
+
z2
t3
= 1,
is orthogonal to both Σ and Σ′ if t1, t2, t3 satisfy the following
t3
(
dt1
dτ
+
dt2
dτ
)
= t2
(
dt1
dτ
+
dt3
dτ
)
= t1
(
dt2
dτ
+
dt3
dτ
)
. (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,
t1 = 2(ln θ2(τ))′, t2 = 2(ln θ3(τ))′, t3 = 2(ln θ4(τ))′. (2.13)
with
θ2(τ) :=
∞∑
n=−∞
q
1
2
(n+ 1
2
)2 , θ3(τ) :=
∞∑
n=−∞
q
1
2
n2
,
θ4(τ) :=
∞∑
n=−∞
(−1)nq
1
2
n2
, q = e2πiτ , τ ∈ H.
These theta functions can be written in terms of the more general theta functions with charac-
teristics r and s and arguments z and σ:
ϑ[r, s](z, σ) =
∑
m∈Z
exp
{
πi(m+ r)2σ + 2πi(m+ r)(z + s)
}
, z, r, s ∈ C, σ ∈ H,
such that
θ2(τ) = ϑ[1/2, 0](0, τ), θ3(τ) = ϑ[0, 0](0, τ), θ4(τ) = ϑ[0, 1/2](0, τ).
3 Ramanujan relations between Eisenstein series
The following differential equation
q
∂E2
∂q
=
1
12
(
E2
2 − E4
)
, q
∂E4
∂q
=
1
3
(E2E4 − E6), q
∂E6
∂q
=
1
2
(
E2E6 − E2
4
)
, (3.1)
where Ei’s are the Eisenstein series
E2i(q) := 1 + bi
∞∑
n=1
∑
d|n
d2i−1
qn, i = 1, 2, 3,
and (b1, b2, b3) = (−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 xi in
4(x− t1)(x− t2)(x− t3) = 4(x− a1E2)
3 − a2E4(x− a1E2)− a3E6, (3.2)
Manifold Ways to Darboux–Halphen System 5
where
(a1, a2, a3) :=
(
2πi
12
, 12
(
2πi
12
)2
, 8
(
2πi
12
)3
)
,
gives us a map from C3 into itself which transforms Darboux–Halphen into Ramanujan diffe-
rential 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 Fµν = εµνρσF
ρσ 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 ds2) 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 SO(3) or 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
SO(3) as a subgroup of 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 reduced1 moduli M0
2 of charge 2 monopoles in a 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 R3, where SU(2) Yang–Mills is a gauge theory of 1-form connections A on
a principal SU(2)-bundle while the Higgs field Φ correspond to a section of an associated su(2)-
bundle [3, 20]. In [3], Atiyah and Hitchin showed that the reduced moduli M0
2 of 2-monopoles is
a 4-dimensional hyperkähler manifold and an anti-self-dual (curvature-wise) Einstein manifold.
Since M0
2 admits SO(3) isometry, the metric is a Bianch IX2 (4.2). This is a consequence of the
hyperkähler structure of M0
2 which has an S2-parameter family of complex structures, i.e., if I,
J , K are covariantly constant complex structures in M0
2 then aI + bJ + cK is also a covariantly
constant complex structure in M0
2 given that a2 + b2 + c2 = 1.
Here we present a detailed derivation of the Darboux–Halphen system starting from the
Euclidean Bianchi IX metric with 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 ± sign. We follow the steps of [8] and [14], see also [26].
1Moduli of charge 2 monopoles reduced by quotient by R3 action.
2Note that here the four coordinates of the moduli are not spacetime directions, but internal parameters of
a 2-monopole solution.
6 J.A.C. Morales, H. Movasati, Y. Nikdelan, R. Roychowdhury and M.A.C. Torres
4.1 Geometric analysis
A metric for a 4-dimensional spacetime with coordinates (x1, x2, x3, x4), Euclidean time x4 and
SO(3) ⊂ SO(4) isometry is written in terms of invariant 1-forms σi on SO(3), dual to the
standard basis X1, X2, X3 of its Lie algebra
σi = − 1
r2
ηiµνx
µdxν ,
where µ, ν = 1, 2, 3, 4 and i = 1, 2, 3 and ηiµν is a ’t Hooft symbol given by
ηiµν =
{
εiµν + δiµδν4 − δµ4δiν or,
εiµν − δiµδν4 + δµ4δiν .
according to two different choices of so(3) generators in the Lie algebra of the group SO(4) =
SU(2)× SU(2). Among the symbols presented above, εiµν is the Levi-Civita symbol
ε123 = ε231 = ε312 = −ε213 = −ε321 = −ε132 = 1, and zero elsewhere,
and δiµ refers to the Kronecker delta. The σi’s obey the structure equation:
dσi = −εijkσj ∧ σk, (4.1)
where we use Einstein summation in the repeated upper and lower indices here and what follows
below. This choice of SO(3) isometry leads to a 4D spherically symmetric Bianchi IX metric
ds2 = c0(r)
2dr2 + c21(r)
(
σ1
)2
+ c22(r)
(
σ2
)2
+ c23(r)
(
σ3
)2
, (4.2)
with r =
√
x21 + x22 + x23 + x24, c0(r) = c1(r)c2(r)c3(r) and c1, c2, c3 being functions of r.
We can impose self-duality in Bianchi IX metric in two ways:
1) connection wise self-duality,
2) curvature wise self-duality.
The connection wise self-duality is a stronger form of self-duality that leads to self-dual curva-
ture 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
e0 = c0dr, ei = ciσ
i (no sum in i), i = 1, 2, 3,
and the connection 1-form can be obtained from the structure equation
dea = eb ∧ ωab,
where a, b = 0, 1, 2, 3. Obviously, e0 produces no connections while other three does
de0 = 0, dei = ∂rcidr ∧ σi − ciεijkσj ∧ σk. (4.3)
The first term on the r.h.s. above gives ωi0 while the second term needs to be rewritten in order
to produce a antisymmetric connection 1-form
εijk
c2i
ci
σj ∧ σk = εijk
2c2i +
(
c2j − c2k
)
−
(
c2j − c2k
)
2ci
σj ∧ σk
Manifold Ways to Darboux–Halphen System 7
= εijk
c2i + c2j − c2k
2cicj
ej ∧ σk + εikj
c2i + c2k − c2j
2cick
ek ∧ σj
= εijk
c2i + c2j − c2k
cicj
ej ∧ σk.
Rewriting (4.3),
dei = −∂rci
c0
σi ∧ e0 − εijk
c2i + c2j − c2k
cicj
ej ∧ σk.
Hence,
ωi0 =
∂rci
c0
σi (no sum in i), ωij = −εijk
c2i + c2j − c2k
cicj
σk. (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 equa-
tion for Ricci tensor is
Rij = dωij + ωim ∧ ωmj .
The (anti-)self-duality of curvature demands that
R0i = ±1
2
ε0ilmR
lm = ±Rjk, (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 δij . Comparing the
l.h.s. and r.h.s. of (4.5), we have
d(ω0i ∓ ωjk) = ±(ω0k ∓ ωij) ∧ (ω0j ∓ ωki),
d
(
λ1(r)σ
i
)
= ±
(
λ3(r)σ
k
)
∧
(
λ2(r)σ
j
)
= λ3(r)λ2(r)σ
k ∧ σj ,
∂rλ1dr ∧ σi + λ1dσ
i = ∓λ2λ3σj ∧ σk, (4.6)
where the second line comes from equation (4.4) with λ1, λ2, λ3 being functions of r. But the
third line and (4.1) show that λi’s are constants and λ1 = ±1
2λ2λ3. From cyclicity of i, j, k we
obtain two more copies of (4.6). Therefore,
1) λ1 = λ2 = λ3 = 0 or
2) (λ1)
2 = (λ2)
2 = (λ3)
2 = 4 with λ1λ2λ3 = ±8.
The first case leads to self-dual connection 1-forms and Euler-top system, while the second case
can be resumed to λ1 = λ2 = λ3 = ±2 by an appropriate change of sign in ci [14]. Therefore,
from equations (4.4) and (4.6) we get(
∂rci
c0
)
= ∓
(
c2j + c2k − c2i
cjck
− 2
)
, ∂r
(
ln c2i
)
= ∓2
(
c2j + c2k − c2i − 2cjck
)
.
8 J.A.C. Morales, H. Movasati, Y. Nikdelan, R. Roychowdhury and M.A.C. Torres
One may suppose that we must parametrize the l.h.s. to match the linear form in c2i , c
2
j and c2k
of the r.h.s. in the equation above. Essentially, the derivative operator aside, c2i must be
parametrized such that
ln c2i = ln Ωj + ln Ωk − ln Ωi + const = ln
(
ΩjΩk
Ωi
)
+ const.
We choose new parametrization
(ci)
2 =
ΩjΩk
2Ωi
⇒ Ωi = 2cjck.
which enable us to decouple the individual parameters into their own equations turning into
simpler expressions. This allows us to continue our analysis
∂r
[
ln
(
ΩjΩk
2Ωi
)]
=
Ω̇j
Ωj
+
Ω̇k
Ωk
− Ω̇i
Ωi
= ∓
(
ΩkΩi
Ωj
+
ΩiΩj
Ωk
− ΩjΩk
Ωi
− 2Ωi
)
.
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
Ω̇j
Ωj
+
Ω̇k
Ωk
− Ω̇i
Ωi
= ∓
(
ΩkΩi
Ωj
+
ΩiΩj
Ωk
− ΩjΩk
Ωi
− 2Ωi
)
+
Ω̇k
Ωk
+
Ω̇i
Ωi
− Ω̇j
Ωj
= ∓
(
ΩiΩj
Ωk
+
ΩjΩk
Ωi
− ΩkΩi
Ωj
− 2Ωj
)
y
⇒ Ω̇k
Ωk
= ∓2
(
ΩiΩj
Ωk
− Ωi − Ωj
)
⇒ Ω̇k = ∓(ΩiΩj − ΩkΩi − ΩkΩj),
where throughout derivative (denoted by dot) is taken with respect to r. Self-duality proceeds
to give us the classical Darboux–Halphen system
Ω̇i + Ω̇j = 2ΩiΩj .
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
ds2 = F
(
dt2 +
σ21
Ω2
1
+
σ22
Ω2
2
+
σ23
Ω2
3
)
.
where t is the cosmological time and different from before, here the isometry is SU(2) and (σi)
are the corresponding SU(2) invariant forms along the spacial directions with structure constant
dσ1 = σ2 ∧ σ3, dσ2 = σ3 ∧ σ1, dσ3 = σ1 ∧ σ2.
We define the new variables Ai(t) by the equations
∂tΩi = −ΩjΩk + Ωi(Aj +Ak), (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 Ai in the form of the Darboux–Halphen system
∂tAi = −AjAk +Ai(Aj +Ak). (4.8)
Manifold Ways to Darboux–Halphen System 9
Therefore we find Ωi’s by first solving system (4.8) and applying its solution in (4.7). A non-
trivial solution is given by (2.13)
A1 = 2
∂
∂t
(ln θ2(it)), A2 = 2
∂
∂t
(ln θ3(it)), A3 = 2
∂
∂t
(ln θ4(it)).
For simplicity, we rename ϑ2 ≡ θ2(it), ϑ3 ≡ θ3(it), ϑ4 ≡ θ4(it). The system (4.7) thus becomes
∂tΩ1 = −Ω2Ω3 + 2Ω1∂t ln(ϑ3ϑ4),
∂tΩ2 = −Ω3Ω1 + 2Ω2∂t ln(ϑ4ϑ2),
∂tΩ3 = −Ω1Ω2 + 2Ω3∂t ln(ϑ2ϑ3). (4.9)
There is a class of solutions of this system that satisfies vacuum Einstein equations
Rab −
1
2
Rgab + Λgab = 0,
once we choose the appropriate conformal factor F [28]. This class depend on the values of the
cosmological constant Λ and satisfy the constraint
ϑ42Ω
2
1 − ϑ43Ω2
2 + ϑ44Ω
2
3 =
π2
4
ϑ42ϑ
4
3ϑ
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
Ω1 = − i
2
ϑ3ϑ4
d
dqϑ
[
p, q + 1
2
]
eπipϑ[p, q]
, Ω2 =
i
2
ϑ2ϑ4
d
dqϑ
[
p+ 1
2 , q + 1
2
]
eπipϑ[p, q]
,
Ω3 =
i
2
ϑ2ϑ4
d
dqϑ
[
p+ 1
2 , q + 1
2
]
eπipϑ[p, q]
,
where ϑ[p, q] denotes the theta function ϑ[p, q](0, ir), p, q ∈ C. The corresponding metric is real
and satisfies the Einstein equations for negative cosmological constant Λ if p ∈ R and R{q} = 1
2
(real part of q) or for positive cosmological constant if q ∈ R and R{p} = 1
2 . In both the cases
the corresponding conformal factor is given by
F =
2
πΛ
Ω1Ω2Ω3(
d
dq lnϑ[p, q]
)2 .
There is another family of solutions
Ω1 =
1
t+ q0
+ 2
∂
∂t
lnϑ2, Ω2 =
1
t+ q0
+ 2
∂
∂t
lnϑ3, Ω3 =
1
t+ q0
+ 2
∂
∂t
lnϑ4,
with q0 ∈ R, that defines manifolds with vanishing cosmological constant if
F = C(t+ q0)
2Ω1Ω2Ω3.
10 J.A.C. Morales, H. Movasati, Y. Nikdelan, R. Roychowdhury and M.A.C. Torres
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
Et : y2 − 4(x− t1)(x− t2)(x− t3) = 0, t ∈ C3\ ∪i,j {ti = tj},
is the universal family for the moduli of 3-tuple (E, (P,Q), ω), where E is an elliptic curve and
ω ∈ H1
dR(E)\F 1. There is a unique regular differential 1-form in the Hodge filtration ω1 ∈ F 1,
such that 〈ω, ω1〉 = 1 and ω, ω1 together form a basis of H1
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 (t1, 0) and (t2, 0) and ω = xdx
y and ω1 = dx
y . The Gauss–Manin connection
of the family of elliptic curves Et written in the basis dx
y , xdx
y is given as bellow
∇
(
dx
y
xdx
y
)
= A
(
dx
y
xdx
y
)
,
where
A =
dt1
2(t1 − t2)(t1 − t3)
(
−t1 1
t2t3 − t1(t2 + t3) t1
)
+
dt2
2(t2 − t1)(t2 − t3)
(
−t2 1
t1t3 − t2(t1 + t3) t2
)
+
dt3
2(t3 − t1)(t3 − t2)
(
−t3 1
t1t2 − t3(t1 + t2) t3
)
.
The reader who is not familiar with the Gauss–Manin connection must replace ∇ with d
∫
δt
,
where ti’s are assumed to depend on some parameter τ , d = ∂
∂τ and δt is a 1-dimensional
homology class in Et. In the parameter space of the family of elliptic curves Et there is a unique
vector field R, such that
∇R
(
dx
y
)
= −xdx
y
, ∇R
(
xdx
y
)
= 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:
∂3F (t)
∂tα∂tβ∂tλ
ηλµ
∂3F (t)
∂tµ∂tγ∂tδ
=
∂3F (t)
∂tδ∂tβ∂tλ
ηλµ
∂3F (t)
∂tµ∂tγ∂tα
,
where F (t), with t = (t1, t2, . . . , tn), 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 (t1, t2, . . . , tn). That’s the
essence of a Frobenius manifold that we will detail below starting with the algebraic structure
in TM .
Manifold Ways to Darboux–Halphen System 11
6.1 Frobenius algebra
An algebra A over C is Frobenius if
• it is a commutative associative C-algebra with unity e,
• it has a C-bilinear symmetric non-degenerate inner product
〈 , 〉 : A×A −→ C,
(a, b) 7→ 〈a, b〉,
which is invariant, i.e., 〈a.b, c〉 = 〈a, b.c〉
Properties: Let eα, α = 1, . . . , N , be any basis in A, such that e1 = e is the unity. By
notation, we define ηαβ := 〈eα, eβ〉, which yields the matrix η := [ηαβ]1≤α,β≤N and its inverse
η−1 := [ηαβ]1≤α,β≤N , and it follows ηαβηβγ = δαγ . By writing eα · eβ in the given basis, we find
the structure constants cγαβ defined by eα · eβ = cγαβeγ . If we set cαβγ = cεαβηεγ , then we get
cγαβ = cαβεη
εγ . Note that in all above expressions, and in what follows, Einstein summation of
indices is implicit. Therefore, ηαβ and the structure constants cγαβ satisfy
commutativity ηαβ = ηβα, (6.1)
associativity (eα.eβ).eγ = eα.(eβ.eγ) ∴ cεαβc
δ
εγ = cδαεc
ε
δβ, (6.2)
normalization cα1β = δαβ , (6.3)
invariance&commutat. cαβγ = 〈eαeβ, eγ〉 = cβαγ = cαγβ. (6.4)
Now consider an n-parametric deformation of the Frobenius algebra At, t = (t1, t2, . . . , tn), with
structure constants cγαβ(t) preserving relations (6.1) to (6.4). Such deformed algebra At can be
seen as a fiber bundle with the space of parameters t ∈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 ∈ M the tangent space TtM contains the structure of a Frobenius algebra (At, 〈 , 〉t),
satisfying the following axioms:
A.1. The metric 〈 , 〉t on M is flat. The unit vector e must be flat, i.e., ∇e = 0, where ∇ is the
Levi-Civita connection for the metric.
A.2. Let c be the 3-tensor c(x, y, z) = 〈x.y, z〉, with x, y, z ∈ TtM . Then the 4-tensor (∇wc)(x,
y, z) must be symmetric in x, y, z, w ∈ TtM .
A.3. A linear vector field E must be fixed on M , i.e., ∇(∇E) = 0 such that the corresponding
one-parameter group of diffeomorphisms acts by conformal transformations of the metric
〈 , 〉 and by rescaling on the Frobenius algebras TtM .
The flatness of the metric 〈 , 〉 implies the existence of a system of flat coordinates t1, . . . , tn
on M . In these flat coordinates the structure constants of At are given by
∂
∂tα
.
∂
∂tβ
= cγαβ(t)
∂
∂tγ
.
12 J.A.C. Morales, H. Movasati, Y. Nikdelan, R. Roychowdhury and M.A.C. Torres
Potential deformation. If there is a function F (t), called potential, such that the structure
constants of At, t ∈M , can be locally represented as
cαβγ(t) =
∂3F (t)
∂tγ∂tα∂tβ
,
satisfying A.2 with unity vector e = ∂
∂t1
, and the metric given by
ηβγ = c1βγ =
∂3F (t)
∂tγ∂t1∂tβ
s.t.
∂4F (t)
∂tα∂tγ∂t1∂tβ
= 0,
satisfying A.1, and the associativity property (6.2) represented by the WDVV equations
∂3F (t)
∂tα∂tβ∂tλ
ηλµ
∂3F (t)
∂tµ∂tγ∂tδ
=
∂3F (t)
∂tδ∂tβ∂tλ
ηλµ
∂3F (t)
∂tµ∂tγ∂tα
,
then M is a Frobenius manifold and the Frobenius algebra At is called a potential deformation.
Note that the condition A.3 is satisfied by a quasihomogeneous function F (t).
Example 6.1. Let dimM = 3, and consider the basis e = e1 = ∂
∂t1
, e2 = ∂
∂t2
and e3 = ∂
∂t3
of
the 3-dimensional algebra At. Then the multiplication law is given by
e22 = fxxye1 + fxxxe2 + e3, e2e3 = fxyye1 + fxxye2, e23 = fyyye1 + fxyye2,
where the funtion F (t) has the form F (t) = 1
2(t1)2t3 + 1
2 t
1(t2)2 + f(t2, t3) and the notation
fx = ∂xf(x, y), fy = ∂yf(x, y). The associativity condition (e22)e3 = e2(e2e3) implies the
following PDE for f(x, y):
f2xxy = fyyy + fxxxfxyy. (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 mani-
fold. We follow Dubrovin’s notes [11, 12]. Let dim M = 3, and consider the potential function
F (t) =
1
2
(
t1
)2
t3 +
1
2
t1
(
t2
)2 − (t2)4
16
γ
(
t3
)
,
where γ(τ) is an unknown 2π-periodic function that is analytic at τ = i∞. Then the associativity
condition (6.5) leads to the Chazy equation
γ′′′ = 6γγ′′ − 9(γ′)2.
The solution, up to a shift in τ , is given by γ(τ) = πi
3 E2(τ), where E2 is the weight-2 Eisenstein
series. Notice that the Darboux–Halphen solution (2.13) leads to
t1 + t2 + t3 =
πi
2
E2(τ),
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 τ 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
y3 − 3
2
γ(τ)y2 +
3
2
γ′(τ)y − 1
4
γ′′(τ) = 0.
Manifold Ways to Darboux–Halphen System 13
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 ques-
tions. 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 gene-
rator of the modular group SL(2,Z) in t3 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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1 Introduction
2 The Darboux problem
3 Ramanujan relations between Eisenstein series
4 Self-duality in Bianchi IX metrics
4.1 Geometric analysis
4.2 Curvature wise self-duality and Darboux–Halphen system
4.3 General Bianchi IX self-dual Einstein metric
5 Gauss–Manin connection in disguise
6 Frobenius manifolds and Chazy equation
6.1 Frobenius algebra
6.2 Frobenius manifold
6.3 Chazy equation and Darboux–Halphen system
7 Conclusion
References
|
| id | nasplib_isofts_kiev_ua-123456789-209461 |
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| issn | 1815-0659 |
| language | English |
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| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Morales, J.A.C. Movasati, H. Nikdelan, Y. Roychowdhury, R. Torres, M.A.C. 2025-11-21T19:15:28Z 2018 Manifold Ways to Darboux-Halphen System / J.A.C. Morales, H. Movasati, Y. Nikdelan, R. Roychowdhury, M.A.C. Torres // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 30 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 34M55; 53D45; 83C05 arXiv: 1709.09682 https://nasplib.isofts.kiev.ua/handle/123456789/209461 https://doi.org/10.3842/SIGMA.2018.003 Many distinct problems give birth to the 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 an infinite number of double orthogonal surfaces in R³. The second is a problem in general relativity about a gravitational instanton in the Bianchi IX metric space. The third problem stems from the new take on the moduli of enhanced elliptic curves called the 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. During the manuscript preparation period, MACT was fully sponsored by CNpQ-Brasil. The research of RR was supported by FAPESP through the Instituto de Física, Universidade de São 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. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Manifold Ways to Darboux-Halphen System Article published earlier |
| spellingShingle | Manifold Ways to Darboux-Halphen System Morales, J.A.C. Movasati, H. Nikdelan, Y. Roychowdhury, R. Torres, M.A.C. |
| title | Manifold Ways to Darboux-Halphen System |
| title_full | Manifold Ways to Darboux-Halphen System |
| title_fullStr | Manifold Ways to Darboux-Halphen System |
| title_full_unstemmed | Manifold Ways to Darboux-Halphen System |
| title_short | Manifold Ways to Darboux-Halphen System |
| title_sort | manifold ways to darboux-halphen system |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209461 |
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