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The moduli space of BPS vortex-antivortex pairs
René García
Work supervised by prof. Martin Speight
1
School of Mathematics
University of Leeds
Encuentro Iberoamericano de Geometría en Granada
Granada, Spain
June 10th – 12th, 2019
EIG-GR 2019 June 10th - 12th, Granada, Spain. 1 / 38

Overview
The O(3) Sigma model on euclidean space
Geometry on compact domains
Conclusions

O(3) Sigma Model on the Euclidean Plane
We consider space-time R1,2
with signature (+, −, −). Let
ϕ : R1,2
→ S2
, a ∈ Ω1
(
R1,2
)
and let us fix n ∈ S2
such that we can
define the height ϕ3 = ϕ · n. The abelian O(3) sigma model is given by
the functional
∫
R2
1
2
Dµϕ · Dµ
ϕ −
1
4
fµνfµν
−
1
2
(τ − ϕ3)
2
d2
x, (1)
where,
Dµϕ = ∂µϕ − aµn × ϕ, (2)
is the covariant derivative on the trivial fibered bundle R1,2
× S2
associated with the connection form and the real constant τ is chosen in
the interval [−1, 1].

Remarks
1. The abelian O(3) sigma model was proposed by Schroers as a
simplification of Skyrmion theory in two dimensions.
2. As we will see in the next slides, the O(3) sigma model shares several
analogies with Ginzburg-Landau’s functional of superconductivity.

A simplification with complex variables
We let ψ be the south pole projection of the field ϕ. Outside the set of
poles of ψ, the functional (1) is equivalent to
L =
∫
R2
1
2
ΩDµψ Dµψ −
1
4
fµνfµν
−
1
2
(
τ −
1 − |ψ|2
1 + |ψ|2
)2
d2
x, (3)
Ω is the conformal factor of the sphere,stereographically projected onto
the extended complex plane and the covariant derivatives are also
adjusted accordingly,
Dµψ = ∂µψ − iaµψ. (4)

Some simplifications on the model
In order to describe the set of minimizers of the O(3) Sigma model
Lagrangian, we look for stationary solutions.

The static Lagrangian
For a stationary configuration of fields, we can take a particular gauge,
such that a0 = 0. In this gauge, the potential energy becomes
V =
∫
R2
1
2
− ΩDjψ Djψ +
1
2
f2
12 +
1
2
(
τ −
1 − |ψ|2
1 + |ψ|2
)2
d2
x. (5)
From now on, we will use the convention that roman indices run on space
coordinates whereas greek indices run on space-time coordinates.

Bogomol’nyi’s trick
Claim. Let B = f12, the potential energy can be refactored into the form
V =
∫
R2
1
2
|D1ψ ± i D2ψ|2
+
1
2
(
B ±
(
τ −
1 − |ψ|2
1 + |ψ|2
))2
d2
x + Φ, (6)
where
Φ =
∫
R2
B d2
x, (7)
is the total magnetic flux. With the right boundary conditions for finite
energy, the flux is 2π n for some integer n.

Remarks
Let µ = log
(
(1 − τ)(1 + τ)−1
)
, and let (ρ, θ) be polar coordinates on
target space. The conditions for finite energy are
lim
ρ→∞
ψ = µeiχ(θ)
, lim
ρ→∞
aθ = aθ(θ). (8)
For suitable functions χ and a defined on the circle at infinity.

The Bogomol’nyi’s equations
Therefore, solutions to the set of equations,
D1ψ ± i D2ψ = 0, B = ±
(
1 − |ψ|2
1 + |ψ|2
− τ
)
, (9)
with the boundary conditions (8) are stationary extremals of the fields
Lagrangian preserving the total magnetic flux Φ.

Remarks
1. We will use the plus signs in Bogomol’nyi’s equations from now
onwards.
2. A solution is a vortex if its magnetic flux is 2π. If the flux is −2π,
the solution is an antivortex.
3. We can think of more general solutions as combinations of vortices
and antivortices.

Taubes’ Equation
Let h = log|ψ|2
and let P = ψ−1
(0) be the set of zeros of the field ψ
and let Q = ψ−1
(∞) be the set of points at infinity, then h is a solution
to the elliptic problem,
∆h = 2
(
eh
− 1
eh + 1
− τ
)
+
1
4π
∑
p∈P
δp −
1
4π
∑
q∈Q
δq, lim
|x|→∞
= µ. (10)
We call equation (10) Taubes’ equation, in analogy to the equation
studied by Taubes’ for the Ginzburg-Landau energy functional.

Regularity of the elliptic problem
Theorem. Given finite subsets P, Q of the euclidean plane, such that
P ∩ Q = ∅, there exists a unique solution h of equation (10). Such
solution is smooth in R2
P ∪ Q and converges logarithmically fast to µ
as |x| → ∞.

Vortex-Antivortex on the plane
Figure: A vortex (right) - antivortex (left) system on euclidean plane

The moduli space of stationary solutions
1. The moduli space M of stationary solutions modulo gauge
equivalence is a stratified space with countably infinitely many
strata.
2. Each stratus Mn+,n−
is topologically a subspace of the quotient
Cn+
× Cn−
/sym, where n+ = |P|, n− = |Q|, and the symmetric
group acts permuting vortices and antivortices independently.
3. We aim to approximate dynamics of the full vortex equations on
moduli space.

Approximated dynamics on moduli space
1. If a system of vortices and antivortices is moving slowly, it can be
approximated to some degree by parametric motion on moduli space.
2. We make the assumption that stationary solutions dependent on a
real parameter are approximations to truly dynamical solutions to
the field equations.
3. This assumption has been thoroughly tested for Ginzburg-Landau
vortices [2].

Localization formula
Theorem. Let z = x1 + ix2, and let Z1, . . . , Zn+
be the position of n+
vortices. Likewise, let Zn++1, . . . , Zn++n−
be the position of n−
antivortices. Let h be the solution to Taubes’ equation for the given
distribution of vortices and antivortices. If ϵs denotes the symbol
ϵs =
{
1, if 1 ≤ s ≤ n+,
−1, if n+ + 1 ≤ s ≤ n+ + n−,
(11)
and we define,
bs = 2 ∂z|Zs
(
ϵsh − log|z − Zs|2
)
, gjk = π
(
(1 − ϵjτ)δjk +
∂bj
∂Zk
)
, (12)
then the complex bilinear form
ds2
= gjkdZj dZk
, (13)
defines an hermitian product in moduli space.

Localization formula
1. The metric is not only hermitian, but Kahler with the obvious
pseudocomplex structure in moduli space.
2. The vortices and antivortices behave as localized lumps of energy
whose effective mass is roughly the area of a spherical sector of area
π(1 ∓ τ).
3. The geometrical properties of moduli space are encoded in the
coefficients bs.
4. Unlike Ginzburg-Landau vortices which at frontal collision will
scatter off at 90 degrees, numerical evidence suggests
vortex-antivortex pairs colliding frontally will bubble.

Vortex-Antivortex systems
If there are only one vortex and one antivortex at positions z1, z2, the
metric can be separated in a product in centre of mass coordinates. If we
let,
Z =
z1 + z2
2
, z1 − z2 = reiθ
, (14)
then symmetry considerations show that the metric is the product,
g = |dZ|2
+ η(r)
(
dr2
+ r2
dθ2
)
. (15)

Vortex-antivortex systems
It can be shown the coefficient b1 has the functional form,
b1 = −eiθ
(
1
r
+ b(r)
)
, (16)
and b1 + b2 = 0 as a consequence of Taube’s equation invariance under
the action of the Euclidean group. Moreover,
η =
π
2
(
1 − τ2
−
1
r
d
dr
(rb(r))
)
. (17)

Vortex-Antivortex collisions
Figure: A vortex approaches an antivortex on euclidean plane. The coordinate
frame was chosen such that the antivortex remains at origin. Instead of solving
the full field equations, we solved the low energy approximation in moduli
space.

The centre of mass metric on a vortex-antivortex system
Figure: A tipical conformal factor in the centre of mass frame. As
vortex-antivortex separation decreases, the conformal factor grows indefinitely.

Geometry on compact domains
Let Σ be a closed surface with a Riemannian metric g and compatible
pseudocomplex structure. A construction by Sibner et al. [3] extends the
fields to take values on a fiber bundle with base Σ and fibers S2
. This
construction is further generalized in the work of Romao and Speight [1].
We can extend the previous discussion about the O(3) Sigma model to
R × Σ with the metric dt2
− g.

Governing elliptic problem
Assume ˆC → L → Σ is a sphere line bundle in which there is a
representation of a U(1) → P → Σ principal bundle and ψ : Σ → L is a
section of this line bundle. Let h = log|ψ|2
, P = ψ−1
(0), Q = ψ−1
(∞),
then h is a solution to the governing elliptic problem,
∆gh = 2
(
eh
− 1
eh + 1
+ τ
)
+
∑
p∈P
δp −
∑
q∈Q
δq. (18)

Bradlow bound
Unlike the euclidean plane, there is a constraint on the number of
coexisting vortices and antivortices.
− (1 + τ)
|Σ|
2π
< |P| − |Q| < (1 − τ)
|Σ|
2π
. (19)
This kind of constraint is called a bradlow bound.

Existence of solution to the governing elliptic problem
Theorem. If P, Q are finite sets such that Bradlow’s bound is satisfied,
there exists a unique solution h to the governing elliptic problem such
that h is of class C∞
(Σ P ∪ Q).
The proof of the theorem can be deduced along the lines of the original
proof by Sibner, Sibner and Yang for τ = 0.

The moduli space of a compact surface
1. In this case, the moduli space is a subspace of Σn+
× Σn−
/sym.
2. On a holomorphic chart U ⊂ Σ, the localization formula (11)
remains unchanged.
3. In the compact case, we can talk about the volume of moduli space.

Regular part of h on the sphere
2 1 0 1 2
1.5
1.0
0.5
0.0
0.5
1.0
1.5
R=1, =0, =0.25
x
1.5 1.0 0.5 0.0 0.5 1.0 1.5
y
1.5
1.0
0.5
0.0
0.5
1.0
1.5
z
0.4
0.2
0.0
0.2
0.4
Figure: Regular part of the solution h to the governing elliptic problem on the
sphere for a vortex-antivortex system, stereographically projected on the
complex plane. The cores are located at ±ϵ in the projection.

The volume formula on a compact domain
Unlike in the Euclidean plane, on a compact manifold it is conjectured
the volume of moduli space is finite.
Conjecture. Let Σ be a compact surface of genus g with k+ vortices
and k− antivortices defined on it. Let us define,
J± = 2π (1 ∓ τ) Vol (Σ) − 2π2
(k± − k∓) , (20)
K± = ∓2π2
. (21)
Then it is conjectured that the volume of M is
Vol (M) =
g
∑
ℓ=0
g!(g − ℓ)!
(−1)ℓℓ!
∏
σ=±
g
∑
jσ=ℓ
(2π)2ℓ
Jkσ−jσ
σ Kjσ−ℓ
σ
(jσ − ℓ)!(g − jσ)!(kσ − jσ)!
. (22)

Volume conjecture
The conjecture comes from an elaborated idea from Romao and Speight.
Specialising to the case of the sphere, we could corroborate the validity
of the conjecture if there are only vortices or antivortices and if there is
exactly one vortex and antivortex.

The volume of moduli space
Theorem. If there are only vortices on the sphere, the volume of moduli
space is
Vol(M) =
(
4π2
R2
(
2(1 − τ) − k+
R2
))k+
k+!
, (23)
on the other hand, if there are exactly one vortex and one antivortex in
the sphere, the volume is,
Vol(M) = (8π2
R2
)2
(1 − τ2
). (24)

Volume formula for a vortex-antivortex pair on the sphere
Let
A = 2π
(
4R2
1 + ϵ2
− ϵB1 − 2R2
− 1
)
, c = 8πR2
τ (25)
the volume of the vortex-antivortex moduli can be calculated with the
formula
Vol = 4π2
lim
ϵ→0
A(ϵ)2
− c2
π2
, (26)
numerical evidence suggested limϵ→0 ϵB1 = −1, which can be probed
with elliptic estimates.

Numerical estimates of limϵ→0 ϵB1
0.2 0.4 0.6 0.8 1.0
1.00
0.95
0.90
0.85
0.80
0.75
b
=0.00, R=1.0
0.2 0.4 0.6 0.8 1.0
1.000
0.975
0.950
0.925
0.900
0.875
0.850
b
=0.71, R=1.0
Figure: ϵB1 for the symmetric case and an example antisymmetric pair.

The moduli is incomplete
Moduli space is ill-defined at ϵ = 0. The length of a geodesic segment
joining a vortex located at ϵ0 with the origin depends on the derivative of
the function ϵB1. Using a priory estimates, the following claim can be
proved.
Theorem. If there is only one vortex and one antivortex in the sphere,
there is a constant C such that the length ℓ of the trajectory joining both
in moduli space is bounded,
ℓ ≤
√
π
(
πR + 2
√
2 C
)
. (27)
Therefore, the moduli space of vortex-antivortex pairs on the sphere is
incomplete.

Behaviour of the family hϵ as ϵ → 0
0.0 0.2 0.4 0.6 0.8 1.0
/
1.8
2.0
2.2
2.4
2.6
2.8
3.0
h
= 0.71, R = 1.00
1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00
x
1.4
1.2
1.0
0.8
0.6
hlower
R=1.0, =0.4
=0.25
=0.20
=0.15
=0.10
=0.05
Figure: Left, unwrap of the real profile of a sample of the functions hϵ. As
ϵ → 0, the profile converges uniformly to a constant value. Right, Behaviour of
the real profile of the functions hϵ at south pole as ϵ → 0. The image was
stereographically projected from the north pole. The profile suggests that
outside a small neighbourhood of the south pole, the solutions converge
uniformly to a constant depending on τ and R, the radius of the target sphere.

Conclusions
1. The presence of an inner spin in the abelian O(3) Sigma model splits
the cores in two types: vortices and antivortices.
2. It is conjectured that moduli space is geodesically incomplete for any
compact surface. This claim was proved in general for a spherical
domain.
3. In the euclidean plane, numerical evidence suggests vortex-antivortex
scattering angle can grow arbitrarily large as the impact parameter
converges to 0.
4. At impact parameter 0, the vortex and antivortex collide in finite
time, hence the fields are expected to develop singularities.

References I
N. M. Romão and J. M. Speight.
The geometry of the space of BPS vortex-antivortex pairs.
jul 2018.
T. M. Samols.
Mathematical Physics Vortex Scattering.
Commun. Math. Phys, 145(1):149–179, 1992.
L. Sibner, R. Sibner, and Y. Yang.
Abelian gauge theory on Riemann surfaces and new topological
invariants.
Proceedings of the Royal Society of London A: Mathematical,
Physical and Engineering Sciences, 456(1995):593–613, 2000.

Thank you!
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