Geometry and Energy of Non-abelian Vortices

22 Pages·2013·0.22 MB·English

DAMTP-2010-117
Geometry and Energy of Non-abelian Vortices
0
1
0
2 Nicholas S. Manton∗ and Norman A. Rink†

c
e
Department of Applied Mathematics and Theoretical Physics,
D
University of Cambridge,

4
Wilberforce Road, Cambridge CB3 0WA, England.
1

] December 2010
h
t
-
p
e
h Abstract
[ We study pure Yang–Mills theory on Σ×S2, where Σ is a compact
2
1 Riemann surface, and invariance is assumed under rotations of S . It is
v well known that the self-duality equations in this set-up reduce to vor-
4 tex equations on Σ. If the Yang–Mills gauge group is SU(2), the Bogo-
1 molny vortex equations of the abelian Higgs model are obtained. For
0
larger gauge groups one generally finds vortex equations involving sev-
3
eralmatrix-valuedHiggsfields. HerewefocusonYang–Millstheorywith
.
2 gauge group SU(N)/ZN and a special reduction which yields only one
1 non-abelian Higgs field.
0 One of the new features of this reduction is the fact that while the
1
instanton numberof the theory in four dimensions is generally fractional
:
v with denominator N, we still obtain an integral vortex number in the
i reduced theory. We clarify the relation between these two topological
X
chargesatabundlegeometriclevel. Anotherstrikingfeatureistheemer-
r genceofnon-triviallowerandupperboundsfortheenergyofthereduced
a
theory on Σ. These boundsare proportional to thearea of Σ.
We give special solutions of the theory on Σ by embedding solutions
oftheabelian Higgsmodelintothenon-abeliantheory,andwerelateour
worktothelanguageofquiverbundles,whichhasrecentlyprovedfruitful
in thestudy of dimensional reduction of Yang–Mills theory.
∗[email protected]
†[email protected]
1
1 Introduction
It was first noted in [30] that rotationally invariant instantons in Yang–Mills
theory can be interpreted as vortices in lower dimensions. This reduction was
originallycarriedoutforYang–MillstheorywithgaugegroupSU(2)andyielded
theabelianHiggsmodel. Inrecentyears,moregeneralreductionsofYang–Mills
theory on spaces of the formΣ S2 have been studied, where invarianceunder
×
rotationsofthesphereS2wasassumed,see[25,24,9,22]andreferencestherein.
This invariantset-upgenerallyleadsto severalmatrix-valuedHiggs fields onΣ,
and the precise number and shape of the Higgs fields is determined by the
Yang–Mills gaugegroupandthe specific way in which the rotationalsymmetry
is implemented in the theory. For example, it was shown in [22] that if the
Yang–Mills gauge group is SU(N), with N =2m, and one chooses a symmetry
reductionwhichbreaksthisgrouptoS(U(m) U(m)),thenasinglenon-abelian
×
Higgs field, which is a square (m m)-matrix, is obtained. Here we show how
×
a non-square Higgs field arises in the reduced theory on Σ when the Yang–
Mills gauge group, assumed to be locally the same as SU(N), is broken by the
rotational symmetry to a group which is locally the same as S(U(m) U(n)),
×
where N =m+n. The Higgs field is now an (n m)-matrix. We focus on the
×
case where Σ is a closed, compact Riemann surface, in order that the reduced
theory on Σ can have vortex solutions of finite energy.
This symmetry breaking pattern is desirable if one wants to construct the
Standard Model on Σ from a unified theory in higher dimensions. For the
electroweak sector, for example, one should take N =3, m=2, n=1. From a
purelytwo-dimensionalpointofview,theelectroweakmodelonΣwasstudiedby
BimonteandLozanoin[5],whereΣwastakentobeaflattorus,witheuclidean
signature. Inthissetting,aBogomolny-typeargumentcanbecarriedoutonthe
energy of the electroweak model, and the resulting Bogomolny equations were
derived in [5]. Vortex solutions to these equations were also obtained, related
to vortex solutions studied earlier in [17, 28]. Another result of [5] was a lower
boundontheenergyoftheelectroweakmodelonΣ. Thisboundisproportional
to the areaof Σ. Here we generalizethe results of [5]to arbitraryN, m, andn,
by viewing the theory as dimensionally reduced Yang–Mills theory on Σ S2.
×
Wealsoobtainlowerandupperboundsontheenergyinthisgeneralizedsetting.
Aninterpretationofthelowerboundisgivenintermsofthevacuumstructureof
the Yang–Mills theory in four dimensions. It should be noted that dimensional
reduction of Yang–Mills theory on R1,3 S2 to the electroweak model on R1,3
×
wasalreadycarriedout in[20], andwe use verycloselyrelatedmethods hereto
facilitate the reduction.
Thecrucialpointatanearlystageinouranalysisistheobservationthatthe
geometry ofS2 forces us to startwith Yang–Mills theory on Σ S2 with gauge
groupSU(N)/Z ,i.e.thequotientofSU(N)byitscentreZ .×Thisimpactson
N N
the bundle structures associated with the Yang–Mills theory and the reduced
theory on Σ. Most notably, it is no longer natural to think of Yang–Mills
theoryasbeing definedonavectorbundle overΣ S2 sincethere isno rankN
vectorbundlewithstructuregroupSU(N)/Z . Ins×teadweintroduceaprincipal
N
bundle with structure group SU(N)/Z , and we regard the gauge potential of
N
Yang–Mills theory as a connection on this principal bundle. As a consequence,
the instanton number (as conventionally normalized for a gauge potential on a
rank N vector bundle) need no longer be an integer but is generally a fraction
2
withdenominatornotbiggerthanN. TheHiggsfieldinthereducedtheorycan
still be regarded as a section of a vector bundle over Σ, and so the associated
vortexnumberisintegral. ThisvectorbundleoverΣ,however,isnotnecessarily
the bundle of homomorphisms between two distinct vector bundles as in the
literature [12, 1, 25, 9].
Inmuchofthe recentliteratureonnon-abelianvortices,see[14,3,27,10,4]
and references therein, Higgs fields are taken to be matrices whose columns are
charged under the gauge group, and different columns represent different fla-
vours. Then, in addition to gauge symmetry, there is also flavour symmetry
andthe correspondingsymmetry groupactsonthe Higgsfield onthe right. By
contrast, although the non-abelian Higgs field in the theory we study is gener-
allymatrix-valuedandactedonbysymmetrygroupsfromtheleftandtheright,
both group actions are gauged and neither is a flavour symmetry. Models con-
taining severalflavoursareusually obtainedfromsupersymmetric fieldtheories
by truncating these to their bosonic parts. Here we will not consider super-
symmetricmodels; neverthelessfermionscanconsistentlybe addedtoinvariant
Yang–Mills theory on Σ S2, as was done in [21, 18, 9].
×
This paper is organized as follows. In section 2 we review the most general
ansatz for the Yang–Mills gauge potential on Σ S2 that is invariant under
×
rotations of S2. Alongside of this we clarify which bundle structures are rel-
evant in the invariant Yang–Mills theory and in the reduced theory on Σ. In
section3 we specialize to the Yang–Mills gaugegroupSU(N)/Z and choose a
N
particular class of symmetry reductions which lead to a single Higgs field on Σ
withanassociatedvortexnumber. Section4isdedicatedtoreducingtheYang–
Mills actionandthe self-duality equationsinfourdimensions to the energyand
Bogomolny-typeequationsintwodimensions. Wealsofindtherelationbetween
the topological charges in four and two dimensions, the instanton and vortex
numbers. A first lower bound on the energy of the reduced theory on Σ is ob-
tained, and we comment on the implications of this bound for the existence of
invariantvacuainthe Yang–Millstheory. Asharperlowerboundonthe energy
of solutions to the Bogomolny equations as well as an upper bound are derived
in section 5, andwe presenta specialclass of solutions to the Bogomolnyequa-
tions in section 6. In section 7 we explicitly connect our work with [5], and we
comment on the allowed energy ranges for N = 3 and N = 5. In section 8 we
formulate the bundle theoretic features of our dimensional reduction scheme in
the language of quivers of vector bundles. Section 9 sums up our conclusions.
2 Invariant Yang–Mills theory and bundles
Throughoutthis paper Σ is assumedto be a closed, compactRiemann surface1
withlocalcomplexcoordinatez. OnthesphereS2wetakethecomplexcoordin-
ate y obtained by stereographic projection. We also introduce real coordinates
x1, x2 on Σ and x3, x4 on S2 by the relations
z =x1+ix2, y =x3+ix4. (1)
1Manyresults,especiallyinthepresentsectionandthenext,generalizetoratherarbitrary
manifoldsΣ.
3
For the metric on M =Σ S2 we adopt the conventions of [22], i.e.
×
8
ds2 =σ(z,z¯)dzdz¯+ dydy¯, (2)
(1+yy¯)2
where σ is the conformal factor on Σ and the second term renders S2 a sphere
of radius √2 with Gauss curvature 1. The corresponding volume forms on Σ
2
and S2 are
8
dvolΣ =σdx1∧dx2, dvolS2 = (1+yy¯)2 dx3∧dx4, (3)
and the area of Σ is denoted by A .
Σ
We considerpure Yang–Millstheoryonthe productspaceM =Σ S2, and
×
regarditas atheory ofa connectionω definedona principalbundle P overM.
Thegaugepotential isobtainedfromωbymeansofalocalsections: U P,
A →
U M open,
⊂
=s∗ω, (4)
A
where the righthand side denotes the pull-back of ω under s. For our purposes
it is best to regard the sphere as the coset space S2 = SU(2)/U(1). This
introducesanaturaltransitiveactionofSU(2)onthesphereS2,andthisaction
extends to M by acting trivially onΣ. We canthen considerSU(2)-equivariant
principalbundlesoverM andSU(2)-invariantconnectionsonthem. Phrasedin
a less technical fashion, we are interested in SU(2)-invariantYang–Mills theory
onΣ S2. Fromthe pointofviewofthesurfaceΣthis amountstodimensional
×
reductionof Yang–Millstheory onΣ S2, where the sphere S2 is treatedas an
×
internal space.
The goal of the present section is to identify the geometric structures on Σ
that arise from the reduction of SU(2)-invariant Yang–Mills theory. The tools
wearegoingtousearetheresultsoftheanalysisin[16],whichgeneralizeWang’s
theorem[29]. Similar treatments which by-passthe analysis of connections and
principalbundles by focusingonthe gaugepotentialonlyare[11,18], andtheir
approach is usually referred to as coset space dimensional reduction.
First recall (from [16] for example) that every SU(2)-equivariant principal
bundle over S2 with structure group is isomorphic to a quotient space P
λ
G
defined by
P =SU(2) , (5)
λ λ
× G
where elements in SU(2) are identified by
×G
(S,g) (SS ,λ(S )−1g), S U(1), (6)
0 0 0
∼ ∈
and λ is a homomorphism λ: U(1) . The projection map π: P S2 is
λ
→ G →
given by
(S,g) [S], (7)
7→
where g and [S] denotes the left-coset S U(1) in SU(2). Note that the
∈ G { · }
P are isomorphic for different λ: U(1) within the same conjugacy class.
λ
→G
4
Now let P be an SU(2)-equivariant principal bundle on Σ S2 and choose
×
an open covering U of Σ such that all U are topologically trivial. Then
i i∈I i
{ }
therestrictionsP|Ui×S2 areSU(2)-equivariantbundleswhicharetrivialoverUi.
Therefore, by the previous paragraph,
P|Ui×S2 ∼=Ui×Pλi, (8)
where the homomorphisms λ : U(1) may be different for different open
i
→ G
sets U . However,by looking at non-empty overlaps U =U U , one finds,
i ij i j
∩
Uij ×Pλi ∼=P|Uij×S2 ∼=Uij ×Pλj. (9)
This showsthat P =P and hence λ and λ must lie in the same conjugacy
λi ∼ λj i j
class. If Σ is connected, which we have assumed as part of the definition of a
Riemann surface, we can therefore choose a single λ: U(1) such that
→G
P|Ui×S2 ∼=Ui×Pλ. (10)
The isomorphisms in (9) therefore give rise to anautomorphismof U P
ij λ
×
which is determined by a transition function h : U such that
ij ij
→G
λ=h−1λh , (11)
ij ij
i.e.h takesvaluesin (λ(U(1))),thecentralizerofλ(U(1))in . Furthermore,
ij G
on triple overlaps U CU U = the Cˇech cocycle conditionGholds,
i j k
∩ ∩ 6 ∅
h =h h . (12)
ik ij jk
Thus the h define a principal bundle P over Σ with structure group
ij Σ
= (λ(U(1))). (13)
G
H C
This centralizer, , is the residual gauge group after dimensional reduction.
H
Before giving the general form of an SU(2)-invariant connection on P, we
notethatthehomomorphismλisdeterminedbyauniqueΛ g,theLiealgebra
∈
of , which is defined as follows: Introduce the Pauli matrices
G
0 1 0 i 1 0
σ1 = 1 0 , σ2 = i −0 , σ3 = 0 1 . (14)
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) − (cid:19)
Then E = iσ , for i = 1,2,3, form a basis of the Lie algebra su(2) and E
a −2 a 3
generates the subgroup U(1) which is the isotropy group at [id] SU(2)/U(1).
∈
Then, for a matrix Λ g,
∈
λ eE3t =eΛt, (15)
(cid:0) (cid:1)
i.e.ΛgeneratesaU(1)subgroupin whichistheimageofλ. Fromexp(4πE )=
3
G
1 it follows that Λ must satisfy
2
e4πΛ =id . (16)
G
Now let ω be an SU(2)-invariant connection on the equivariant bundle P
over M. Over the open set U Σ, we can pull ω back to an SU(2)-invariant
i
⊂
5
connectiononU P , andthis connectioncorrespondstoa gaugepotential
i λ i
× A
on U S2 which is given by
i
×
=A (z,z¯), (17)
i,z i,z
A
=A (z,z¯), (18)
i,z¯ i,z¯
A
1
= ( iΛy¯ Φ (z,z¯)), (19)
i,y i
A 1+yy¯ − −
1
= iΛy+Φ (z,z¯)† , (20)
i,y¯ i
A 1+yy¯
(cid:0) (cid:1)
subject to the constraints
[Λ,A ]=[Λ,A ]=0, (21)
i,z i,z¯
[Λ,Φ ]= iΦ , [Λ,Φ†]=iΦ†. (22)
i − i i i
The above formulae are a special case of the results derived in [11, 16, 18], but
also compare [12, 25, 9, 22]. Note that A , A and Φ take values in g∗,
i,z i,z¯ i
the complexification of the Lie algebra g, which is merely a consequence of our
choice to express in terms of complex coordinates.
i
A
On non-empty overlaps U one finds the relations,
ij
A =h−1A h +h−1dh , (23)
j ij i ij ij ij
Φ =h−1Φ h , (24)
j ij i ij
where the h : U denote the transition functions of P defined above,
ij ij Σ
→ H
andA =A dz+A dz¯andanalogouslyforA . Thereforethecollectionofthe
i i,z i,z¯ j
localgaugepotentials A defines a connectiononP . Note that the constraints
i Σ
(21) imply that the A take their values in h, the Lie algebra of , which is
i
H
consistent with P having structure group . In the same vein, the Φ define
Σ i
H
a section of the vector bundle E which is associated to P by the adjoint
Σ Σ
representation of on g∗. In symbols,
H
E =P g∗. (25)
Σ Σ ad
×
Toconcludethissection,weremarkthattheinverseoperationsofrestriction
and induction work for principal bundles over product spaces in precisely the
same way as they do in the vector bundle case, which has been looked at in
[1, 2, 25, 9]. Starting with the SU(2)-equivariant bundle P over M, we can
define its restriction to Σ [id] which we denote as P . This is a U(1)-
Σ×[id]
× |
equivariant bundle with structure group , where U(1) acts trivially on the
G
base and its action on the fibre is defined by the homomorphism λ: U(1)
→ G
associated with P in the same way as above. It can be shown that
P =P . (26)
|Σ×[id] ∼ Σ
The inverse operation is given by the formula
P =SU(2) P . (27)
λ Σ×[id]
× |
However, our construction of the bundle P by analyzing the restrictions of P
Σ
to patches U Σ rather than using restriction and induction has clarified that
i
×
6
1. thestructuregroupofP canbereducedto (λ(U(1)))andtheequivari-
Σ G
C
ant connection on P naturally leads to a connection on P ,
Σ
2. there is an associated vector bundle E of which the Higgs field Φ is a
Σ
section.
Furthermore, the analysis carried out in this section should generalize to the
situation studied in [8], where M =Σ S2 is replaced with a flat fibration
×
S2 ֒ M Σ. (28)
→ →
3 Two-block reduction with Yang–Mills gauge
group SU(N)/Z
N
To make further progress,we need to solve explicitly the constraints(21), (22),
and in order to do so we have to make choices for Λ and the Yang–Mills gauge
group . For the rest of the paper let = SU(N)/Z , which has the Lie
N
G G
algebra g = su(N). Note that at the level of pure Yang–Mills theory this is
locally indistinguishable from the case where the gauge group is SU(N) since
thecentreZ actstriviallyonthe gaugepotential . However,wewillseethat
N
we are forced to take =SU(N)/Z by the geomAetry of S2.
N
G
Now, since Λ su(N), it must be an anti-hermitian and traceless (N
∈ ×
N)-matrix. By conjugating Λ with a suitable SU(N)-matrix, we can make Λ
diagonal and hence we choose
α1 0
Λ=i m , (29)
0 β1
n
(cid:18) (cid:19)
whereN =m+nandα, β arerealconstants. Toallowfornon-trivialsolutions
oftheconstraint(22),itisnecessarytorequireα β = 1. Restrictingattention
− ±
to α β =1 and using the tracelessness of Λ, we find
−
n m
α= , β = . (30)
N −N
We check that this is consistent with (16),
e4πΛ = e4πiN0n 1m e−4π0imN 1n =e4πiNn 10m 10n =id∈SU(N)/ZN,
(cid:18) (cid:19) (cid:18) (cid:19)
(31)
which clarifies our choice of the Yang–Mills gauge group. Note that for special
values of N, m, n one may be able to choose a bigger gauge group, i.e. SU(N)
moduloonlya subgroupofZ . Forexample,ifN iseven,itisclearlysufficient
N
to mod out by Z , and in the case m = n, N = 2m, it is consistent to work
N/2
with the gauge group SU(N), as was done in [22]. The special case N = 2,
m = n = 1, gives the traditional reduction of SU(2)-instantons to abelian
vortices, which was first discussed in [30].
The constraints (21), (22) are now solved by
0 0 0 φ†
Φ= , Φ† = , (32)
φ 0 0 0
(cid:18) (cid:19) (cid:18) (cid:19)
7
and
a 0
A= , (33)
0 b
(cid:18) (cid:19)
whereφisan(n m)-matrix-valuedfieldonΣandwehaveintroducedthegauge
×
potentials a = a dz+a dz¯ and b = b dz +b dz¯ on Σ. The index i I on A
z z¯ z z¯
∈
and Φ has been omitted since, as a consequence of (23), (24), the constraints
(21), (22) are globally meaningful.
Thefieldsa, b, φarethe contentofthetheoryonΣarisingasthe symmetry
reduction of Yang–Mills theory on M. The gauge group of the theory on Σ
coincides with the structure groupof P from (13), and with our choice of Λ is
Σ
=S(U(m) U(n))/Z , (34)
N
H ×
where the leading S indicates that the overall determinant is one. Modding
out by Z is meaningful since Z is contained in S(U(m) U(n)) as a normal
N N
×
subgroup. Because of the structure of , the transitionfunctions h of P can
ij Σ
H
be written as
hij(z,z¯)= hmij(0z,z¯) hn(0z,z¯) e2πNik, (35)
(cid:18) ij (cid:19)
where hm U(m) and hn U(n) such that det(hm)det(hn) = 1, and k is an
ij ∈ ij ∈ ij ij
integer between 0 and N 1. The transformation law for the section φ on U
ij
−
can then be read off from (24),
φ = hn −1φ hm, (36)
j ij i ij
(cid:0) (cid:1)
where φ and φ are local expressions for φ on the open sets U and U re-
i j i j
spectively. We thus obtain a more refined picture of Φ and E : There exists
Σ
a vector bundle E over Σ of rank mn. The structure group of E is also
mn mn
S(U(m) U(n))/Z , and its fibres transform according to the law (36). The
N
×
collectionofthe φ then comprisea sectionφof E , andwe shallreferto φ as
i mn
thenon-abelianHiggsfield. ThegaugepotentialAfrom(33)definesacovariant
derivative on E by virtue of
mn
0 0
DΦ=dΦ+[A,Φ]= , (37)
Dφ 0
(cid:18) (cid:19)
withDφ=dφ+bφ φa. Fromthiswecancalculatethecurvatureofthebundle
−
E ,whichwedenoteasf. Thecurvatureactsonsectionsasanendomorphism
mn
in the following way,
fφ=fbφ φfa, (38)
−
where fa =da+a a and fb =db+b b. Then a straightforwardcalculation
∧ ∧
shows that
tr(f)=mtr fb ntr(fa), (39)
−
(cid:0) (cid:1)
8
where on the left hand side the trace is takenin the space of endomorphismsof
(n m)-matrices. Thus, the first Chern number of E is
mn
×
i
c (E )= tr(f )dz dz¯ (40)
1 mn zz¯
2π ∧
ZΣ
i
= mtr fb ntr(fa ) dz dz¯. (41)
2π zz¯ − zz¯ ∧
ZΣ
(cid:0) (cid:0) (cid:1) (cid:1)
Since the non-abelianHiggs fieldφ is a sectionofE , we willtake c (E ) as
mn 1 mn
a generalization of the vortex number in the abelian Higgs model. This will be
motivated more in the next section, where we relate c (E ) to the instanton
1 mn
number of the Yang–Mills theory on Σ S2.
×
The above expression for c (E ) can be simplified: Note that the gauge
1 mn
potential A is traceless, and therefore also tr(fa)+tr fb =0. Hence,
iN iN (cid:0) (cid:1)
c (E )= tr fb dz dz¯= tr(fa )dz dz¯. (42)
1 mn 2π zz¯ ∧ −2π zz¯ ∧
ZΣ ZΣ
(cid:0) (cid:1)
Itfollowsfromthegeneraltheoryofcomplexvectorbundles thatc (E )is an
1 mn
integer. Therefore,
i 1 i 1
tr(fa )dz dz¯ Z, tr fb dz dz¯ Z. (43)
2π zz¯ ∧ ∈ N 2π zz¯ ∧ ∈ N
ZΣ ZΣ
(cid:0) (cid:1)
Sincetheseexpressionsneednotbeintegral,thisshowsthatingeneralthereare
no vector bundles with gauge potentials a and b. This is in agreementwith the
general form of the transition function (35): The entries hm and hn need not
ij ij
satisfy the Cˇech cocycle conditions in U(m) or U(n) respectively, but only up
to an element of Z . As a consequence, unlike in [12, 1, 25, 9], E cannot be
N mn
thoughtofasthebundleofhomomorphismsbetweentwodistinctvectorbundles
over Σ. This can be traced back to the fact that since we consider Yang–Mills
theorywithgaugegroupSU(N)/Z ,thereisnovectorbundleofrankN which
N
is naturally associated with this Yang–Mills theory on Σ S2.
×
4 Actions, energy and vortex equations
We denote the field strength of Yang–Mills theory as = d + . The
F A A∧A
actionofYang–Mills theoryonΣ S2 in terms ofthe complexcoordinatesz, y
×
is given by
4 (1+yy¯)2
SYM =ZΣ×S2tr(cid:18)σ2Fz2z¯− σ (FzyFz¯y¯+Fzy¯Fz¯y)
(1+yy¯)4
+ 16 Fy2y¯ dvolΣdvolS2. (44)
(cid:19)
OneshouldbearinmindthatSYM isnon-negativesincethecomponentsof in
real directions are anti-hermitian for unitary gauge groups such as SU(N)/FZ .
N
Substitutinginthesymmetricansatzfor from(17)-(20)weobtainthereduced
A
9
action on Σ,
4 1
S =8π tr F2 + D ΦD Φ†+D ΦD Φ†
Σ σ2 zz¯ σ z z¯ z¯ z
ZΣ (cid:18)
+ 1(cid:0) 2iΛ [Φ,Φ†] 2 σdx1(cid:1) dx2, (45)
16 − ∧
(cid:19)
(cid:0) (cid:1)
where F =dA+A A and we have performed the integral over S2,
∧
8
dx3 dx4 =8π. (46)
ZS2 (1+yy¯)2 ∧
It is sensible to identify S with the potential energy E of a field theory on Σ
Σ
since it is a static action. We use the convention E = S /16π, where we have
Σ
divided by the area of S2 and introduced a factor of 1. Henceforth we shall
2
assumem n. ThenwecanexpresstheenergyE intermsoftheunconstrained
≥
fields a, b, φ on Σ,
1 4
E = tr(fa fa )+tr fb fb
2 σ2 zz¯ zz¯ zz¯ zz¯
ZΣ(cid:18)
1(cid:0) (cid:0) (cid:1)(cid:1)
+ tr D φD φ† +tr D φD φ†
z z¯ z¯ z
σ
+ 1n(cid:0)(m(cid:0)−n) + 1(cid:1)tr 1 (cid:0) φφ† 2 (cid:1)σ(cid:1)dx1 dx2, (47)
n
8 N 8 − ∧
(cid:19)
(cid:0) (cid:1)
where the constant term arises because
tr 2iΛ [Φ,Φ†] 2 =tr 2n1 +φ†φ 2+tr 2m1 φφ† 2 (48)
m n
− − N N −
(cid:0) (cid:1) =2n(cid:16)(m−n) +2tr (cid:17)1 φφ(cid:16)† 2. (cid:17) (49)
n
N −
(cid:0) (cid:1)
From the above expression for E we can immediately read off the lower bound
A
Σ
E n(m n), (50)
≥ 16N −
whichweshallcalltheBimonte–Lozano boundsinceasimilarboundwasderived
in [5] in a purely two-dimensional context. This lower bound for E provides
good motivation for putting the theory on compact Σ, as here and in [5]. If Σ
had infinite area, then for m = n the energy of the theory on Σ would always
6
be infinite. The action SYM of the corresponding Yang–Mills theory on Σ S2
×
wouldalsobeinfinite,andhenceinstantonswithSU(2)-symmetryandwiththis
choiceofΛdonotcontributetothepartitionfunction. Fromatwo-dimensional
pointofviewthisinfinity canofcoursebecuredbysubtractingaconstantfrom
E, but from a four-dimensional point of view this is unnatural.
Itisclearfrom(50)thatE =0cannotbeachievedform=n. Thisisnotin
6
disagreementwiththefactthatYang–MillstheoryonΣ S2 alwaysadmitsthe
×
vacuum solution =0 since, for m=n, this solutiondoes not lie in the sector
F 6
ofSU(2)-invariantsolutionswearestudying. Wecanmakethisstatementmore
precise: By looking at (45) we see that if the vacuum of Yang–Mills theory is
10