arXiv:hep-th/0407011v1 2 Jul 2004 The Kähler Potential of Abelian Higgs Vortices

25 Pages·2008·0.22 MB·English

DAMTP-2004-60
The K¨ahler Potential of Abelian
Higgs Vortices
4
0 Heng-Yu Chen
∗
0
2
and

l
u
J
N. S. Manton
†
2

1
Department of Applied Mathematics and Theoretical Physics
v
1
University of Cambridge
1
0
7 Wilberforce Road, Cambridge CB3 0WA, UK
0
4
0
h/ July 2004
t
-
p
e
h
:
v Abstract
i
X
We calculate the Ka¨hler potential for the Samols metric on the
ar moduli space of Abelian Higgs vortices on 2, in two different ways.
R
Thefirstusesascalingargument. ThesecondisrelatedtothePolyakov
conjecture in Liouville field theory. The Ka¨hler potential on the mod-
uli space of vortices on 2 is also derived, and we are led to a geo-
H
metrical reinterpretation of these vortices. Finally, we attempt to find
the Ka¨hler potential for vortices on 2 in a third way by relating the
vortices to SU(2) Yang-Mills instantRons on 2 S2. This approach
R ×
does not give the correct result, and we offer a possible explanation
for this.
∗email [email protected]
†email [email protected]
1
1 Introduction
Vortex solutions are known to exist in the (2+1)-dimensional Abelian Higgs
model. They are the static field configurations minimizing the energy func-
tional. The Lagrangian density for this model is
1 1 λ
= FµνF + D ΦDµΦ ( Φ 2 1)2. (1.1)
µν µ
L −4 2 − 8 | | −
When the coupling constant λ takes the critical value of 1, there are no net
forces between the vortices. There then exist static configurations satisfying
the first order Bogomolny equations (see (2.3), (2.4) below). The N-vortex
solutions in 2 or equivalently the complex plane can be uniquely char-
R C
acterized by where the Higgs field Φ vanishes [1]. The N unordered Higgs
zero locations in are therefore the natural coordinates parameterising the
C
space of static N-vortex solutions. This space is called the moduli space for
N vortices, and we shall denote it by . These coordinates on are
N N
M M
called “collective coordinates”. has a natural K¨ahler structure inherited
N
M
from the kinetic terms of the Lagrangian.
The so-called “moduli space approximation” is a powerful approach for
studying the low energy dynamics of solitonic objects in field theories [2].
The idea is that, in the low energy limit, most of the field degrees of freedom
are effectively frozen. The solitonic dynamics can thus be described by the
dynamics in a reduced, finite-dimensional space of collective coordinates,
which is the moduli space. For N vortices, the potential energy is at the
absolute minimum everywhere on the space equipped with its K¨ahler
N
M
metric. If vortices move at small velocity, they are trapped close to , and
N
M
the kinetic energy term of the reduced dynamics dominates. The trajectories
arethe geodesics on , and the scattering of the vortices can beaccurately
N
M
modelled by such geodesic motions.
Finding the K¨ahler metric of has been the central problem in un-
N
M
derstanding vortex dynamics within the geodesic approximation. A general,
but not explicit formula for the metric was first derived by Samols [3]. Re-
cently, an explicit formula in terms of modified Bessel functions was given
for the K¨ahler metric on the moduli space of N well separated vortices [4].
A formula for the K¨ahler potential was also given. The main purpose of
this paper therefore is to construct a K¨ahler potential for the more general
Samols metric.
The K¨ahler potential and the K¨ahler metric on a complex manifold are
2
related via
∂2
g = K , (1.2)
rs ∂Z ∂Z¯
r s
whereZ (Z¯ ),g and aretheholomorphic(anti-holomorphic)coordinates,
r s rs
K
K¨ahlermetrictensorandK¨ahlerpotential, respectively. Noticethatifweadd
a holomorphic or an anti-holomorphic function to , it still gives the same
K
K¨ahler metric, so the modified K¨ahler potential is geometrically equivalent
to the original one. This property becomes important if we want to remove
undesirable singularities from the K¨ahler potential.
In this paper, we present three different approaches to calculating the N-
vortex K¨ahler potential. The first approach is to explicitly construct it from
¯
the quantities in the Samols metric. This is an example of the ∂-problem.
We show that, for vortices on , it can be solved by a scaling argument.
C
The second approach is inspired by a conjecture of Polyakov [5] relat-
ing the so-called “accessory parameters” in the context of uniformization of
Riemann surfaces to the regularized Liouville action. An accessible mathe-
matical proof was given by Takhtajan and Zograf based on their earlier work
in [6]. Proofs for the Polyakov conjecture on Riemann surfaces with a range
of singularities can be found in [7], [8]. It turns out that in the vortex situa-
tion, there are analogous quantities to the accessory parameters, and we can
construct a modified regularized Liouville action as the generating function
forthese quantities, which acts as theinteracting part of the K¨ahler potential
for the moduli space .
N
M
Our third approach was motivated by considering SU(2) Yang-Mills in-
stantons on a K¨ahler 4-manifold, the K¨ahler potential of whose moduli space
was given by Maciocia [9]. The appropriate 4-manifold here is 2 S2.
R ×
Dimensional reduction of the instantons over 2 S2, using the SO(3) sym-
R ×
metry of S2, results in Abelian Higgs vortices over 2. Maciocia’s formula
R
suggests a way of obtaining the K¨ahler potential for the vortices over 2 by
R
relating it to the K¨ahler potential of the instantons. A promising result is
derived, but it appears to be incorrect. We discuss this difficulty and its
possible resolution in section 6.
An interesting variation of the vortices on 2 is the Abelian Higgs model
R
forvortices defined onthehyperbolic plane 2 with constant Ricci scalar 1.
H −
Such vortices were shown to be integrable by Witten [10], as the Bogomolny
equations in this case can be reduced to the Liouville equation. The metric
on the moduli space of vortices on 2 was first derived by Strachan [11]. In
H
3
this paper we construct a K¨ahler potential for the metric and discuss the
geometrical interpretation of the Higgs field and K¨ahler metric.
This paper is organized as follows: In section 2, we shall briefly review
Abelian Higgs vortices, and Samols’ metric on the moduli space of N-vortex
¯
solutions. In section 3, the relevant ∂ problem is described, and a general for-
mula for the K¨ahler potential based on a scaling argument will be presented.
It is tested for the case of two well separated vortices. In section 4, we shall
brieflyreviewtheregularizedactionofLiouvillefieldtheoryandthePolyakov
conjecture, and show how the interacting part of the K¨ahler potential can be
constructed for vortices on from an analogous regularized action. In sec-
C
tion 5, we shall discuss vortices in the hyperbolic plane, present a geometric
interpretation for the Higgs field, and show that in this case the regularized
action is the entire K¨ahler potential. In section 6, dimensional reduction of
the instantons over 2 S2 is presented. Maciocia’s formula and its relation
R ×
to the vortex K¨ahler potential are discussed. A possible explanation for the
discrepancy in applying Maciocia’s formula to the two-dimensional system
will also be given.
2 Abelian Higgs Vortices
We shall be working in the A = 0 gauge and at critical coupling λ = 1. The
0
total energy in the static situation is
1 1
E = d2x F2 +D ΦD Φ+ ( Φ 2 1)2 , (2.1)
2 12 i i 4 | | −
Z (cid:26) (cid:27)
where F = ∂ A ∂ A = B is the magnetic field in the plane and D Φ =
12 1 2 2 1 j
−
∂ Φ iA Φ, j = 1,2, is the covariant derivative of the complex Higgs field
j j
−
Φ. The boundary condition for Φ is that Φ 1 as x , so Φ becomes
| | → | | → ∞
pure phase and finite energy implies that the gauge field becomes pure gauge
at spatial infinity, such that D Φ vanishes. The winding number of Φ at
j
infinity is denoted by N, and is assumed to be a positive integer.
We can rearrange E into the Bogomolny form using the standard trick of
completing the square
2
1 1
E = d2x F + ( Φ 2 1) +(D Φ iD Φ)(D Φ+iD Φ)+F .
12 1 2 1 2 12
2 2 | | − −
( )
Z (cid:18) (cid:19)
(2.2)
4
Inderiving (2.2), we have discarded theboundary terms which give vanishing
contributions at spatial infinity. As the first two terms in (2.2) are both non-
negative, the minimal E is obtained when A and Φ satisfy the Bogomolny
i
equations
1
F + ( Φ 2 1) = 0, (2.3)
12
2 | | −
D Φ+iD Φ = 0. (2.4)
1 2
The minimal value of E, which is related to the winding number through
Stokes’ theorem, is
1
E = d2xF = Nπ, (2.5)
12
2
Z
and it can be interpreted as the energy of N non-interacting vortices.
If we introduce the complex coordinate z = x1 +ix2, (2.3) and (2.4) can
be written as
1
iF = ( Φ 2 1), (2.6)
zz¯
4 | | −
D Φ = ∂ Φ iA Φ = 0. (2.7)
z¯ z¯ z¯
−
Equation(2.7)allows usto writeA = i∂ logΦ. Wecanexpress Φinterms
z¯ z¯
of a gauge invariant quantity h and a−phase factor χ as Φ = e21h+iχ, where
the boundary condition for Φ implies h 0 at spatial infinity. Substituting
→
these into (2.6), we obtain the gauge invariant governing equation for the
vortex solutions [1]
N
4∂ ∂ h eh +1 = 4π δ(z Z ), (2.8)
z z¯ r
− −
r=1
X
where Z ,...,Z are the vortex positions in . These positions are taken
1 N
{ } C
as distinct, simple zeros of Φ, although they can coalesce. There is a unique
solution for any choice of positions. Notice (2.8) has a form very similar to
the Liouville equation on a punctured Riemann surface. The Higgs vacuum
expectation value 1 in (2.8) sets the scale for the system; it thus breaks the
conformal invariance.
Close to the r-th vortex position Z , h has the following expansion
r
1 1
h = log z Z 2 +a + b¯(z Z )+ b (z¯ Z¯ )
r r r r r r
| − | 2 − 2 −
1
+c¯(z Z )2 z Z 2 +c (z¯ Z¯ )2 +... . (2.9)
r r r r r
− − 4| − | −
5
¯
The expansion coefficients a , b , b , c and c¯ are all functions of the sep-
r r r r r
arations between the r-th vortex position Z and all other vortex positions
r
Z , s = r. The coefficient a plays the role of a local scaling factor. b and
s r r
¯ 6
b measure the deviation from circular symmetry of h around Z due to in-
r r
¯
teractions with other vortices. Samols showed that b and b play a central
r r
role in the formula for the metric on the moduli space. He calculated that
the metric is [3]
N N
∂b
g dZ dZ¯ = δ +2 s dZ dZ¯ . (2.10)
rs r s rs r s
∂Z
r,s=1 r,s=1(cid:18) r(cid:19)
X X
Notice that while this formula is very general, it is not explicit, as we do not
have the exact analytic expression for b in general. However, we can deduce
s
from the hermiticity of the metric that
¯
∂b ∂b
s r
= , (2.11)
∂Z ∂Z¯
r s
and from this it follows easily that the metric is K¨ahler. The translational
N N ¯
invariance of the entire system implies that b = b = 0 and the
r=1 r r=1 r
N ¯ N ¯
rotational invariance gives b Z = b Z [12].
r=1 r r r=P1 r r P
In an analysis of the conservation laws of a model of first order vortex
P P
dynamics [13], it has been shown that some conserved quantities can be
expressed in terms of integrals involving h and its derivatives. The integral
of h itself can also be computed. For N non-coincident vortices
N
lim d2xh = π (b Z¯ +b¯Z +6). (2.12)
r r r r
ǫ→0 ˜ −
ZC Xr=1
Theintegrationregion ˜ istheentirecomplexplane withtheN smalldisks
C C
of radius ǫ centred at the vortex locations Z ,...,Z being punctured out,
1 N
{ }
and the limit ǫ 0 taken.
→
In this paper, we aim to find a suitable integral expression for the K¨ahler
potential involving functions of, and derivatives of, the gauge invariant
K
quantity h, such that the mixed double derivative of with respect to the
K
holomorphic and anti-holomorphic collective coordinates gives us the K¨ahler
metric in the form (2.10).
6
3 The Vortex K¨ahler Potential and the Scal-
ing Integral
IntheK¨ahlermetriconthemodulispaceofvortices, theexpansioncoefficient
b plays the central role. The exact expression for b is only known in a few
r r
limitingcases, e.g. N overlapping vorticesonasphere[14]. However weknow
b obeys the hermiticity identity (2.11) and a recently derived symmetric
r
identity [15]
∂b ∂b
r s
= . (3.1)
∂Z¯ ∂Z¯
s r
Clearly (2.11) and (3.1) can both be satisfied if there exists a real function
of the collective coordinates Z ,...,Z ;Z¯ ,...,Z¯ such that
1 N 1 N
K { }
∂ ∂
e ¯
K = 2b , K = 2b . (3.2)
∂Z¯ r ∂Z r
r r
e e
Moreover, the existence of such , which plays the role of interacting part
K
for the K¨ahler potential, follows from these identities. Solving (3.2) for
¯ K
is analogous to the so-called ∂-preoblem in the mathematical literature. We
do not solve these equations separately, but consider the following lineaer
combination of the equations
N N
∂ ∂
Z +Z¯ = 2 Z b¯ +Z¯ b . (3.3)
r∂Z r∂Z¯ K r r r r
r=1 (cid:26) r r(cid:27) r=1
X X(cid:8) (cid:9)
e
This can be written as
N N
∂
λ = 2 Z b¯ +Z¯ b , (3.4)
r r r r r
∂λ K
r=1 (cid:18) r(cid:19) r=1
X X(cid:8) (cid:9)
e
where Z = λ eiφr, and λ and φ are the distance and the angle of the r-th
r r r r
vortex away fromthe origin. On one side we have the overall scaling operator
of the vortex moduli space acting on , and on the other side a real quantity
K
which is essentially the integral of h, using (2.12). We have now reduced a
problem depending on2N parameterse Z ,...,Z ;Z¯ ,...,Z¯ to a problem
1 N 1 N
{ }
that only depends on the N parameters λ ,...,λ . Effectively we have
1 N
{ }
eliminated the angular dependences and shall subsequently keep the angles
φ ,...,φ fixed.
1 N
{ }
7
With the scaling operator in mind, we make an ansatz that all λ are
r
parameterised by a single variable τ, i.e. λ λ (τ) and Z = Z (τ) =
r r r r
≡
λ (τ)eiφr, where τ is a dimensionless parameter measuring how changes
r
K
under overall scaling of the moduli space. So we can express the overall
scaling operator in terms of τ e
d N dZ (τ) ∂ dZ¯ (τ) ∂
r r
τ = τ +τ
dτK dτ ∂Z dτ ∂Z¯ K
r=1 (cid:26) (cid:18) (cid:19) r (cid:18) (cid:19) r(cid:27)
X
e N e
∂ ∂
= Z +Z¯ . (3.5)
r∂Z r∂Z¯ K
r=1 (cid:26) r r(cid:27)
X
e
The second line in (3.5) follows from the first if τdZr = Z r, i.e. Z itself
dτ r ∀ r
is also proportional to the scaling parameter τ and we have restricted all the
vortices to the ”scaling” motion. We can set the constant of proportionality
to be Z0 to obtain Z (τ) = Z0τ, so that Z0,...,Z0 are the vortex posi-
r r r { 1 N}
tions where we want to evaluate the K¨ahler potential. Moreover, with this
scaling argument, we can now combine (3.3) and (3.5) and write an integral
expression for by integrating with respect to τ. We take τ ranging from
K ∞
to 1. This corresponds to bringing the vortices from spatial infinity to their
desired positiones, so we obtain
1 dτ N
= 2 b¯(τ)Z (τ)+b (τ)Z¯ (τ)
r r r r
K τ
∞
Z r=1
X(cid:0) (cid:1)
e 1 dτ 1
= 2 d2xh(x;τ)+6N , (3.6)
− τ π
∞
Z (cid:26) Z (cid:27)
where the ǫ 0 limit is implied in the integral of h. We have written
→
b b (τ) to highlight that b only depends on τ, as we have restricted the
r r r
≡
vortices to the “scaling” motion, and h(x;τ) means the function h in the
plane, again with this scaling motion of the vortices. With a suitable change
of coordinates, this expression reduces to the expression given in [16] for
the case of N = 2. The entire K¨ahler potential by this scaling argument is
therefore given by
N 1 dτ 1
= Z Z¯ 2 d2xh(x;τ)+6N , (3.7)
r r
K − τ π
∞
r=1 Z (cid:26) Z (cid:27)
X
8
where the first term gives the δ in (2.10).
rs
As a simple test for this formula, we consider the case of two well sepa-
rated vortices. We work in the centre of mass frame, and use the asymptotic
b values as calculated in [4]. Using the notation Z = σeiθ = Z and
r 1 2
−
b = b(σ)eiθ = b where σ is the separation from the origin, we have
1 2
2 Z b¯ +Z¯ b− = 4σb(σ). b(σ) equals q2 K (2σ) for large σ, where
r=1 r r r r 2π2 1
the constant q2 was calculated to be 4π2√8 by Tong using string duality
P (cid:8) (cid:9)
[17]. Applying the integral expression (3.7), we find the asymptotic K¨ahler
potential for two vortices is
= 2σ2 8√8K (2σ). (3.8)
0
K −
The functions K ,K are the modified Bessel functions of the second kind,
0 1
and we have used Bessel function identities in deriving (3.8). Equation (3.8)
coincides with the formula for the K¨ahler potential given in [4] up to an
overall factor of π.
4 Regularized Liouville Action and Vortex
K¨ahler Potential
In this section we present a different approach to solving the equations (3.2).
It turns out that is in fact a suitably regularized modified Liouville action
K
which gives rise to the vortex equation (2.8). We shall consider the case of
vortices on heree and discuss vortices on the hyperbolic plane 2 in the
C H
next section. The motivation for this section was drawn from the striking
similarity of b to the so-called “accessory parameters” in the Liouville field
r
theory. We first review ideas concerning Liouville theory.
Let us consider a Liouville field φ defined over an n-punctured Riemann
sphere, Σ = ˆ/ z ,z ,...,z , ˆ = , where z = and n 3.
∼ 1 2 n n
C { } C C ∪ {∞} ∞ ≥
In the following, we shall adopt the conventions in [6], [18]. φ satisfies the
Liouville equation
1
∂ ∂ φ = eφ, (4.1)
z z¯
2
and we assume the punctures are parabolic singularities, with asymptotic
behaviour
2log z z 2log log z z +O(1) as z z , r = n ,
φ = − | − r|− | | − r|| → r 6
2log z 2loglog z +O(1) as z z = .
n
(cid:26)− | |− | | → ∞
(4.2)
9
φ determines the Poincar´e metric on Σ, ds2 = eφdzdz¯, with the Ricci scalar
given by R = 4e−φ∂ ∂ φ. The Liouville equation implies R = 2. (The
Σ z z¯ Σ
− −
Gaussian curvature is half this.)
The uniformization theorem of the Riemann surface states that Σ has a
universal covering space, the Poincar´e half plane 2 = z ,Im z > 0 ,
H { ∈ C }
and Σ = 2/Γ. Here Γ is a finitely generated Fuchsian group, which is a
∼
H
subgroup of PSL(2, ) acting discretely on . eφ can be expressed as
2
R H
4∂ w∂ w u
eφ = z z , w = 1 , (4.3)
(1 ww¯)2 u
2
−
where u andu area pair of suitably normalized, linearly independent, holo-
1 2
morphic solutions of the Fuchsian differential equation with the monodromy
group Γ,
1
∂ ∂ u+ T (z)u = 0. (4.4)
z z φ
2
T (z) is a meromorphic function over ˆ and plays the role of zz component
φ
C
of the energy-momentum tensor for the Liouville field φ,
1
T (z) = ∂ ∂ φ (∂ φ)2 , ∂ T (z) = 0. (4.5)
φ z z z z¯ φ
− 2
Here, T (z) is given by
φ
n−1
1 c
r
T (z) = + ,
φ
2(z z )2 (z z )
r=1 (cid:26) − r − r (cid:27)
X
1 c 1
n
= + +O , (4.6)
2z2 z3 z4
(cid:18) (cid:19)
where the second expression is the expansion around z = . The coefficient
∞
of each term 1/(z z )2, r = 1,...,n 1 and also of 1/z2 for z = , is half
r n
− − ∞
the conformal weight of each z , so in this case each parabolic singularity
r
z has conformal weight 1. The complex numbers c , r = 1,...,n, are the
r r
accessory parameters. They are uniquely determined by the positions of
the punctures z ,...,z . If we match the first expression with the second
1 n
expression for T (z) as z , we can derive the following constraints
φ
→ ∞
n−1 n−1 n−1
n
c = 0 , c z = 1 , (z +c z2) = c . (4.7)
r r r − 2 r r r n
r=1 r=1 r=1
X X X
10