Non-abelian vortices on CP^ 1 and Grassmannians

43 Pages·2013·0.32 MB·English

DAMTP-2012-69
Non-abelian vortices on CP1 and Grassmannians
3
1
0
Norman A. Rink
2
[email protected]
r
p
A
Department of Applied Mathematics and Theoretical Physics,

8 University of Cambridge,

Wilberforce Road, Cambridge CB3 0WA, England.
]
h
t 8 April 2013
-
p
e
h
Abstract
[

Manypropertiesofthemodulispaceofabelian vorticesonacompact
2
RiemannsurfaceΣareknown. Fornon-abelianvorticesthemodulispace
v
2 is less well understood. Here we consider non-abelian vortices on the
6 RiemannsphereCP1,andwestudytheirmodulispacesneartheBradlow
6 limit. We give an explicit description of the moduli space as a K¨ahler
1 quotient of a finite-dimensional linear space. The dimensions of some
. of these moduli spaces are derived. Strikingly, there exist non-abelian
1
1 vortex configurations on CP1, with non-trivial vortex number, for which
2 themodulispaceisapoint. This isinstark contrasttothemodulispace
1 of abelian vortices.
: Foraspecialclassofnon-abelianvorticesthemodulispaceisaGrass-
v
mannian,andthemetricneartheBradlowlimitisanaturalgeneralization
i
X oftheFubini–Studymetriconcomplexprojectivespace. Weusethismet-
r ric to investigate the statistical mechanics of non-abelian vortices. The
a partitionfunctionisfoundtobeanalogoustotheoneforabelianvortices.
1
Contents
1 Introduction 3
2 A non-abelian Yang–Mills–Higgs model 4
3 Near the Bradlow limit 6
3.1 The dissolved limit . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 Dissolving vortices . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.3 The case k =0 and the small sphere . . . . . . . . . . . . . . . . 13
3.4 Holomorphic gauge transformations. . . . . . . . . . . . . . . . . 14
4 K¨ahler quotient construction of the moduli space 15
4.1 Moment maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.2 The moduli space metric . . . . . . . . . . . . . . . . . . . . . . . 18
5 Moduli spaces and their dimensions 20
5.1 The Bogomolny equations and linear maps. . . . . . . . . . . . . 20
5.2 The case n=1 and Grassmannians . . . . . . . . . . . . . . . . . 21
5.3 The moduli space for m=(k+1)n . . . . . . . . . . . . . . . . . 22
5.4 Dimension of the moduli space for m=n . . . . . . . . . . . . . 23
5.5 The case k =0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
6 Grassmannians and applications 25
6.1 The moduli space metric revisited . . . . . . . . . . . . . . . . . 25
6.2 Statistical mechanics on the moduli space . . . . . . . . . . . . . 27
6.3 The Plu¨cker embedding and abelian vortices . . . . . . . . . . . . 30
7 Summary and outlook 31
8 Acknowledgements 33
Appendix 33
A Geometry and topology of Grassmannians 33
A.1 Grassmannians as K¨ahler quotients . . . . . . . . . . . . . . . . . 33
A.2 The tautological sequence and the Fubini–Study form . . . . . . 35
A.3 Metric aspects of the Plu¨cker embedding . . . . . . . . . . . . . . 36
A.4 Cohomology and the Fubini–Study form . . . . . . . . . . . . . . 38
References 41
2
1 Introduction
Gauged vortices with abelian gauge group are a time-honoured subject (see
e.g. [3, 4, 5, 6, 7, 8, 9, 10]), and various models with non-abelian gauge groups
have been studied (e.g. [11, 12, 13, 14, 15, 16, 17, 18]). The term non-abelian
vortex refers to a solution of the first order Bogomolny equations in a model
with non-abelian gauge group. Quite naturally the intuition we have for the
behaviour of non-abelian vortices is much less developed than in the abelian
case: The only degrees of freedomof abelian vortices in two dimensions are the
coordinates of their centres; they have no internal degrees of freedom. Based
on this, one would expect that the degrees of freedom associated with a non-
abelian vortex are comprised of its spatial coordinates, as in the abelian case,
and additional internal degrees of freedom. The latter are expected to capture
the fields’ orientations within suitable representations of the gauge algebra. In
this paper we study a model on CP1 which accommodates non-abelian vortices
for which this separation into spatial and internal degrees of freedom breaks
down. Insuchasituationvorticescannotbelocalized,andthereforeitisperhaps
unjustified to speak of individual vortices. Nonetheless, we continue to refer to
solutions of the relevant Bogomolny equations as vortices, by analogy with the
abelian Higgs model.
The modelwe considerwasderivedin[18], where itwasobtainedfrompure
Yang–Mills theory by a symmetry reduction: Starting with Yang–Mills theory
on the euclidean background Σ S2, with Σ a Riemann surface, and imposing
×
invarianceunderrotationsofS2reducestheYang–MillstheorytoaYang–Mills–
Higgs model on Σ. The self-duality equation of Yang–Mills theory reduces to
Bogomolny equations in the Yang–Mills–Higgs model on Σ. Similar symmetry
reductions of Yang–Mills theory have been looked at in many places both in
the mathematics [11, 19, 20] and the physics literature [3, 14, 15, 17, 21]. The
novelty of the derivation in [18] lies in the choice of Yang–Mills gauge group,
which was taken to be PSU(N), i.e. SU(N) modulo its centre.
TheBogomolnyequationsin[18]areexpectedtobeintegrable,providedΣis
equippedwithahyperbolicmetric,i.e.hasconstantnegativecurvature. Thisisa
consequenceoftheintegrabilityoftheself-dualYang–Millsequations,see[8,15].
FortheBogomolnyequationsoftheabelianHiggsmodelsomeexplicitsolutions
have been found in special cases [3, 9, 10, 22], and also in a model with gauge
group U(1) U(1) [21]. However, there is a fundamental mathematical
×···×
problem that impedes finding explicit solutions if Σ has non-trivial topology:
Part of such a solution is an explicit expression for the Higgs field, which can
be regardedasaholomorphicsectionofavectorbundle overΣ. The sectionsof
holomorphiclinebundlesoveranarbitraryRiemannsurfaceΣarenotgenerally
known in closed form. Even the number of linearly independent sections may
be unknown. Needless to say that the situation is even more complicated for
vector bundles.
If, however, one is prepared to give up integrability, one can consider the
Bogomolny equations on the Riemann sphere CP1. The advantage of this is
that the Grothendieck Lemma (see e.g. chapter 5 in [23]) serves to classify
holomorphic vector bundles over CP1. Moreover, holomorphic sections can be
regarded as vectors whose entries are homogeneous polynomials in the homo-
geneouscoordinatesofCP1. ThisgivesanexplicitdescriptionoftheHiggsfield,
andthe coefficients ofthe polynomialsarethe moduli. For abelianvorticesthis
3
idea was used in [24] to derive the moduli space metric near the Bradlow limit
[5]. In this paper we generalize the work in [24] to non-abelian vortices. Near
theBradlowlimittheBogomolnyequationsreducetoalgebraicconstraints. To-
gether with a gauge fixing procedure these algebraic constraints give a descrip-
tionofthevortexmodulispaceasaK¨ahlerquotient(cf.[6,11,25]andappendix
B of[26]). The K¨ahlerquotientdescriptionalsoequips the moduli spacewith a
naturalmetric,andthismetricagreeswiththeonewhosegeodesicsdescribethe
slow motion of vortices [27]. In an interesting special case the moduli space is
a Grassmannian,and the moduli space metric is a naturalgeneralizationof the
Fubini–Study metric on complex projective space. This allows us to calculate
the volume of the moduli space and hence to study its statistical mechanics,
generalizing [28]. The volume of the moduli space of non-abelian vortices in a
different model was recently calculated in [29].
This work is structured as follows: In section 2 we review the model that
was introduced in [18] and set up notation. In sections 3 and 4 we identify
the moduli space of non-abelian vortices near the Bradlow limit as a K¨ahler
quotient, and we give a semi-explicit expression for its metric. We derive the
dimensions of possible moduli spaces in section 5. In a special case the moduli
space is shownto be a Grassmannian. In this case the moduli space metric can
be givenexplicitly,whichwedoinsection6. Thevolumeofthe modulispaceis
also derived, and this is used to study the statistical mechanics of non-abelian
vortices. Section 7 summarizes our results and suggests directions for future
work.
2 A non-abelian Yang–Mills–Higgs model
InthissectionwereviewtheYang–Mills–Higgsmodelfrom[18],withparticular
emphasisonthegeometricstructuresinvolved. Ournotationislargelythesame
as in [18].
LetP be a principalbundle onthe RiemannsurfaceΣ with structuregroup
G = S(U(m) U(n))/ζ , where N = m+n. The leading S means that the
N
×
overalldeterminant is one, andζ denotes the centre of S(U(m) U(n)), which
N
×
is the cyclic group of order N. By g we denote the Lie algebra of G, i.e. g =
s(u(m) u(n)). AunitaryconnectiononP islocallygivenbythegaugepotential
×
A,
a 0
A= . (1)
0 b
(cid:18) (cid:19)
Hereaandbarelocallydefined1-formswithvaluesintheLiealgebrasu(m)and
u(n) respectively, and suchthat tr(a)+tr(b)=0. The correspondingcurvature
FA decomposes accordingly,
da+a a 0 fa 0
FA =dA+A A= ∧ = , (2)
∧ 0 db+b b 0 fb
(cid:18) ∧ (cid:19) (cid:18) (cid:19)
where the last equality defines the curvatures fa, fb.
The gauge group G acts on M (C), the space of N N matrices with
N×N
×
complex entries, by the adjoint representation of U(N),
M Ad(g)M =gMg−1, M M (C), g U(N). (3)
N×N
7→ ∈ ∈
4
This action of U(N) does not depend on elements in the centre ζ , and hence
N
yields an action of G. Therefore we can introduce on Σ the vector bundle
associated to P via the representation Ad,
E =P M (C). (4)
Ad N×N
×
The bundle E inherits a covariant derivative from the connection A,
DE =d+[A, ]. (5)
·
For g G choose the representative
∈
g = g0a g0b e2πiNk, ga ∈U(m), gb ∈U(n), k ∈N, (6)
(cid:18) (cid:19)
and write M M (C) as
N×N
∈
M M
M = 11 12 , (7)
M M
21 22
(cid:18) (cid:19)
with M M (C), M M (C), M M (C), M M (C).
11 m×m 12 m×n 21 n×m 22 n×n
∈ ∈ ∈ ∈
It follows from
g M g−1 g M g−1
Ad(g)M = a 11 a a 12 b , (8)
g M g−1 g M g−1
(cid:18) b 21 a b 22 b (cid:19)
that the bundle E decomposes as
E =E E E E . (9)
11 12 21 22
⊕ ⊕ ⊕
The symmetry reduction in [18] leads to a Higgs field φ which is a section of
E . Restricting the covariantderivative DE to E yields
21 21
Dφ=dφ+bφ φa. (10)
−
Consequently φ† is a section of E , with covariant derivative
12
Dφ† =dφ†+aφ† φ†b. (11)
−
TowritedownanenergyfunctionalfortheYang–Mills–Higgsmodelandthe
corresponding Bogomolny equations, we need a metric on Σ. We take
ds2 =Ω(x1,x2)((dx1)2+(dx1)2)=Ω(z,z¯)dzdz¯, (12)
where z is a local complex coordinate on Σ, and the realcoordinates x1, x2 are
defined by z = x1 +ix2, z¯ = x1 ix2. Since Σ has real dimension two, the
−
K¨ahler form ω and the volume form agree,
Σ
i
ω = Ω(z,z¯)dz dz¯=dvol , (13)
Σ Σ
2 ∧
For future reference we also give explicitly the Hodge operator on functions,
∗
1-forms, and 2-forms,
f =fω , (14)
Σ
∗
(α dz+α dz¯)= iα dz+iα dz¯, (15)
z z¯ z z¯
∗ −
η dz dz¯
(η dz dz¯)= 2iΩ−1η = zz¯ ∧ . (16)
zz¯ zz¯
∗ ∧ − ω
Σ
5
Note in particular that on 1-forms 2 = 1. From this it follows that is a
∗ − ∗
complexstructureonthespaceofconnectionsonP. Wewillusethisobservation
at the end of subsection 4.2.
The energy functional of the Yang–Mills–Higgs model derived in [18] is
1 1
E = tr(fa fa) tr fb fb + tr Dφ Dφ†
2d
2 − ∧∗ − ∧∗ 2 ∧∗
ZΣ(cid:18)
+1tr I φφ† 2ω(cid:0) + 1 (cid:1)n(m−(cid:0)n)Vol(Σ), (cid:1) (17)
n Σ
8 − 16 N
(cid:19)
where the minus signs occ(cid:0)ur in fron(cid:1)t of the Yang–Mills kinetic terms since fa
andfb areanti-hermitian. TheconstanttermproportionaltoVol(Σ)wasshown
in[18]tobe aconsequenceofthe factthatE isobtainedfromtheYang–Mills
2d
action in four dimensions by a symmetry reduction. A Bogomolny argument
can be carried out on the energy functional (17), leading to the Bogomolny
equations
D φ=0, (18)
z¯
Ω 2n
fa = I +φ†φ , (19)
zz¯ 8 −N m
(cid:18) (cid:19)
Ω 2m
fb = I φφ† . (20)
zz¯ 8 N n−
(cid:18) (cid:19)
Itwasshownin[18]thatwhentheBogomolnyequationsaresatisfied,theenergy
functional (17) reduces to
π
= c (E ), (21)
1 21
E N
where c denotes the first Chern number. We refer to solutions of (18)-(20) as
1
non-abelian vortices, and we call c (E ) the non-abelian vortex number. This
1 21
isjustifiedsinceform=n=1theequations(18)-(20)reducetotheBogomolny
equations of the abelian Higgs model [1].
Solutionsof (18)-(20)arephysicallyequivalentiftheyarerelatedbyagauge
transformation. We are interested in the moduli space of non-abelian vortices,
whichis the space ofsolutions of (18)-(20)modulo gaugetransformations. The
moduli space can be described as a K¨ahler quotient [6, 11] by identifying equa-
tions (19) and (20) as the level set equations of the moment map for the action
of gauge transformations. In the next section we show how equations (19) and
(20) reduce to algebraic constraints if Σ=CP1 and near the Bradlow limit. In
section4 we identify these constraintswith momentmaps onfinite dimensional
linear spaces.
3 Near the Bradlow limit
From now on we take Σ = CP1. This allows us to solve explicitly the first
Bogomolnyequation (18). Namely, the first Bogomolnyequation says that φ is
a holomorphic section of E . By the Grothendieck Lemma the vector bundle
21
E over CP1 decomposes into a sum of line bundles,
21
mn
E = (k ), (22)
21 i
O
i=1
M
6
where (k ) denotes the holomorphic line bundle over CP1 of degree k Z.
i i
O ∈
It follows that the entries of φ are sections of the (k ), which in turn can be
i
O
described as homogeneous polynomials of degree k .
i
The Bogomolny equations (18), (19) have solutions only if
c (E )
Vol(CP1) 4π 1 21 . (23)
≥ mn
This is the generalized Bradlow bound for non-abelian vortices, cf. [5, 18]. In
thestrictBradlowlimit,i.e.whenequalityholdsin(23),theHiggsfieldvanishes
identically, φ = 0, and the moduli space consists of a single point, as we shall
see. Near the Bradlow limit, when Vol(CP1) slightly exceeds the lower bound
(23),themagnitudeoftheHiggsfieldφissmall,andasaconsequencetheBogo-
molny equations simplify. We will take advantage of this to study properties of
the moduli space. Since increasing Vol(CP1) is a smooth process, statements
aboutthe topologicalpropertiesofthe modulispace nearthe Bradlowlimitare
expected to hold for general values of Vol(CP1).
We now briefly introduce our conventions regarding CP1. Thinking of CP1
as the sphere S2, it can be covered with two open sets, U and U , where U
0 1 0
consist of all points of S2 except the north pole, and U consists of all points
1
except the south pole. We denote the complex coordinate on U as z, and
0
the one on U as z′. The coordinate z is obtained by stereographic projection
1
from the north pole onto the equatorial plane, and z′ is obtained analogously
by projecting from the south pole. On U U the local coordinates satisfy
0 1
z′ =1/z. For k Z the holomorphic transiti∩on function
∈
g : U U C∗, (24)
01 0 1
∩ →
g (z)=zk, (25)
01
definestheholomorphiclinebundle (k)ofdegreekoverCP1. Unlessotherwise
stated, we always work on the openOset U . We equip CP1 with the standard
0
round metric given by the conformal factor
4R2
Ω(z,z¯)= , (26)
(1+zz¯)2
whereRistheradiusofCP1whenregardedasthesphereS2. Thecorresponding
K¨ahler form,
i 4R2
ωCP1 = dz dz¯, (27)
2(1+zz¯)2 ∧
is a multiple of the Fubini–Study form on CP1. The area of CP1 is
Vol(CP1)= ωCP1 =4πR2. (28)
CP1
Z
To solve the Bogomolny equations, it is convenient to work in holomorphic
gauge: The gauge potential A can be expanded in its 1-form components,
A=A dz+A dz¯. (29)
z z¯
7
Holomorphic gauge is defined by the condition A =0. In [10] it was explained
z¯
how to go to holomorphic gauge in the abelian Higgs model. The procedure is
completely analogousin the non-abelianmodel we are studying here: First one
introduces a hermitian structure h on the bundle P. In unitary gauge one has
h=I , and the gauge groupof P is S(U(m) U(n))/ζ . In a generalgauge h
N N
×
is locally given by positive definite hermitian matrices of the form
ha 0
h= , ha M (C), hb M (C), (30)
0 hb ∈ m×m ∈ n×n
(cid:18) (cid:19)
and the structure group of P is S(GL(m) GL(n))/ζ . In unitary gauge the
N
×
connection A on P satisfies
A† = A. (31)
−
In a general gauge this equation is replaced by the compatibility condition
dh=A†h+hA. (32)
Therefore, in holomorphic gauge the condition A = 0 fully determines A in
z¯
terms of the hermitian structure h,
A =h−1∂ h, (33)
z z
and this is known as the Chern connection, see e.g. [23, 30]. It follows that
a =(ha)−1∂ ha, a =0, (34)
z z z¯
b =(hb)−1∂ hb, b =0. (35)
z z z¯
On CP1 and in holomorphic gauge the Bogomolny equations (18)-(20) read
∂ φ=0, (36)
z¯
R2 2n
fa = I +(ha)−1φ†hbφ , (37)
zz¯ 2(1+zz¯)2 −N m
(cid:18) (cid:19)
R2 2m
fb = I φ(ha)−1φ†hb , (38)
zz¯ 2(1+zz¯)2 N n−
(cid:18) (cid:19)
with the field strengths
fa = ∂ ((ha)−1∂ ha), (39)
zz¯ − z¯ z
fb = ∂ ((hb)−1∂ hb). (40)
zz¯ − z¯ z
3.1 The dissolved limit
Usingtheterminologyof[31],wealsorefertotheBradlowlimitasthedissolved
limit. Thisisbecauseabelianvorticesarecentredatthezerosofφ,andvortices
are fully dissolved in the Bradlow limit since φ=0 identically.
We make the following ansatz for the hermitian structure h,
ha(z,z¯)=(1+zz¯)daI , (41)
m
hb(z,z¯)=(1+zz¯)dbI . (42)
n
8
Theexponentsd andd arerealconstants,andtheirvaluesmustbeconsistent
a b
with the topology of P. By this is meant that ha and hb must transformunder
gauge transformations as
ha (g−1)†hag−1, (43)
7→ a a
hb (g−1)†hbg−1, (44)
7→ b b
where g and g are as in (6). (Note that no index contraction is implied. The
a b
symbols a and b merely label the block entries of g in (6) and of h in (30).)
Setting φ = 0 and using the above ansa¨tze for ha, hb, equations (37), (38)
lead to
n
d = R2, (45)
a
N
m
d = R2. (46)
b
−N
This does not appear to constrain d and d in any way since R can take any
a b
value. However,intheBradlowlimittheconstantsd andd arefixedbyvirtue
a b
of (23). For the vortex number in the Bradlow limit we find
i
c (E )= tr mfb tr(nfa) (47)
1 21
2π CP1 −
Z
=mn(d (cid:0)d(cid:0)) (cid:1) (cid:1) (48)
a b
−
=mnR2. (49)
Since c (E ) is integral, this leads to the constraint
1 21
1
R2 Z, (50)
∈ mn
and we also have
c (E )
1 21
d = , (51)
a
mN
c (E )
1 21
d = . (52)
b
− nN
We givea specialnameto the value ofR forwhichthe bound (23)is saturated,
c (E )
R2 = 1 21 . (53)
B mn
Note that c (E ) = 0 is compatible with the Bradlow limit only if R = 0,
1 mn B
i.e. CP1 degenerates to a point. We usually assume R = 0, and we will deal
B
6
with the case R =0 separately.
B
The degrees k in (22) are uniquely determined in the Bradlow limit. This
i
canbeseenbyinspectingthehermitianstructureonE . Forarbitrarysmooth
21
sections ψ ,ψ Γ(CP1,E ),
1 2 21
∈
(ψ ,ψ ) =tr (ha)−1ψ†hbψ =(1+zz¯)db−datr ψ†ψ . (54)
1 2 h 1 2 1 2
(cid:16) (cid:17) (cid:16) (cid:17)
This implies that k = k = d d for all i = 1,...,mn. Alternatively, one
i a b
−
canemploy amoreabstractargumentto showthatallthe k areidentical: The
i
9
Bogomolny equations (37), (38) with φ = 0 imply that the vector bundle E
21
is Einstein–Hermitian. Therefore E decomposes into a direct sum of stable
21
bundles all of which have the same slope, see section V.2 in [32]. We also refer
to [32] for the definitions of Einstein–Hermitian vector bundles, stable bundles,
and slope. In the case where E is an Einstein–Hermitian bundle over CP1
21
it follows again from the Grothendieck Lemma that E decomposes into line
21
bundles of the same degree k. Since k is fully determined by the Bogomolny
equationswith φ=0,italsofollowsthatinthe Bradlowlimitthe modulispace
is a point.
Now, c (E )=mnk, and thus (51) and (52) read
1 21
nk
d = , (55)
a
N
mk
d = , (56)
b
− N
which is consistent with k = d d . Moreover, the fact that d ,d 1Z is
a − b a b ∈ N
consistentwiththestructuregroupofP,inholomorphicgauge,beingS(GL(m)
×
GL(n))/ζ (cf.[18]). The localformonU ofthe hermitianstructurehonP is
N 0
h(z,z¯)= (1+zz¯)nNkIm 0 . (57)
0 (1+zz¯)−mNkIn !
On U the hermitian structure is fully determined by h(z,z¯) and the transition
1
function
g01(z)= z−nN0kIm zmN0kIn !e2πiNl , l∈Z. (58)
Hence,
h(z′,z¯′)= (1+z′z¯′)nNkIm 0 . (59)
0 (1+z′z¯′)−mNkIn !
The expression for g may require clarification: Determining the value g (z)
01 01
requirestakingtheN-throotofz,whichisdefinedonlyuptoanelementofζ .
N
Since the structuregroupofP is S(GL(m) GL(n))/ζ , the transitionfunction
N
×
g is well-defined if the same representative is chosen for the N-th root of z in
01
all entries of g . This can be achieved by defining
01
zN1 =eN1 logz, (60)
and using a fixed branch of the logarithm.
The upshot of the discussion in this subsection is that in the Bradlow limit
d and d are determined by (55) and (56), and the radius R is fixed by the
a b
vortex number,
c (E )=mnk, (61)
1 21
R2 =R2 =k. (62)
B
10