Singular Monopoles on Closed 3-Manifolds Saman Habibi Esfahani October 26 2022

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Singular Monopoles on Closed 3-Manifolds

Saman Habibi Esfahani

October 26, 2022

Abstract

We prove the existence of non-trivial irreducible SU(2)-monopoles with Dirac singu-

larities on any rational homology 3-sphere, equipped with any Riemannian metric, using

a gluing construction.

1 Introduction

The theory of Yang-Mills connections and, in particular, instantons revolutionized the study

of 4-manifolds [7,8,28]. The Bogomolny monopoles appear as the dimensional reduction of

instantons to 3-manifolds.

Deﬁnition 1 (The Bogomolny Monopole).Let (M, g)be an oriented Riemannian 3-manifold.

Let Gbe a compact Lie group. Let P→Mbe a principal G-bundle and gPthe associated

adjoint bundle. Let Abe a connection on Pand Φa section of gP. A pair (A, Φ)is called a

monopole if it satisﬁes the Bogomolny equation, which is

∗FA=dAΦ, (1)

where ∗is the Hodge star operator on the gP-valued diﬀerential forms on M, deﬁned using

the Riemannian metric gand the orientation on M.

The theory of monopoles on non-compact 3-manifolds is very rich. Jaﬀe and Taubes

proved the existence of non-trivial SU(2)-monopoles on R3, using a gluing construction [19].

The gluing constructions, originating from the works of Taubes, have been used to construct

solutions to various diﬀerential equations [29,11,20,30,25,12]. From the gluing construction

of monopoles, one can read the dimension of the moduli spaces of monopoles on R3. This can

also be proven using a variation of the Atiyah-Singer index theorem, called the Callias index

theorem, which is an index theorem for Dirac operators on non-compact odd-dimensional

manifolds [3,21]. The moduli spaces of monopoles on R3are ALF hyperk¨

ahler manifolds,

which have been extensively studied, originating from the works of Atiyah and Hitchin [3].

Furthermore, there exists an explicit parametrization of the moduli spaces of monopoles on

R3in terms of rational maps, due to Donaldson [9,18].

Floer studied monopoles on asymptotically Euclidean 3-manifolds [11] and, more recently,

Oliveira studied monopoles on asymptotically conical 3-manifolds and stated that there exists

a (4k−1)-dimensional family of non-trivial irreducible smooth SU(2)-monopoles on any

asymptotically conical 3-manifold (M, g) with b2(M) = 0 [25]. It is proven by Kottke that

arXiv:2210.13754v1 [math.DG] 25 Oct 2022

the expected dimension of the moduli space of monopoles on an asymptotically conical 3-

manifold, whose ends are asymptotic to a cone on Σ, is 4k+1

2b1(Σ) −b0(Σ) [21].

The theory of monopoles on compact 3-manifolds is quite diﬀerent from the ones on non-

compact manifolds. When the structure group Gis compact, every smooth monopole on a

closed oriented Riemannian 3-manifold satisﬁes a stronger condition,

∗FA=dAΦ= 0,

and therefore, Ais a ﬂat connection and Φis a covariantly constant section. These monopoles

are sometimes referred to as trivial monopoles.

There is another class of monopoles on compact 3-manifolds which have non-ﬂat curva-

ture. These monopoles are smooth on the complement of ﬁnitely many points with prescribed

Dirac singularities at these points.

Deﬁnition 2 (Dirac Singularity).Let P→M\ {p1, . . . , pn}be a principal SU(2)-bundle. A

monopole (A, Φ)on this bundle is called a monopole with Dirac singularities if close to the

singular point pi, we have

|Φ|=k

2r+m+O(r),(2)

where the norm is deﬁned with respect to the adjoint-invariant inner product on the adjoint

bundle gP,ris the geodesic distance from pi,k∈Nis a positive integer, called the charge of

the monopole at the singular point, and mis a constant, called the mass of the monopole at

the singular point. The pair (A, Φ)is called a monopole with a Dirac singularity.

Pauly studied the deformation of these singular monopoles with the structure group SU(2)

[26], and using the Atiyah-Singer index theorem and exploiting a theorem of Kronheimer [22]

— which states that close to the points with Dirac singularities, monopoles up to gauge, can

be understood as smooth S1-invariant instantons on a 4-dimensional space — he proved that

the expected dimension of the moduli space of singular monopoles with charge k∈Non

a compact Riemannian 3-manifold (M, g) is equal to 4k. However, this argument does not

imply that the moduli spaces are non-empty.

In this article, we prove the existence of SU(2)-monopoles with Dirac singularities on

rational homology 3-spheres equipped with any Riemannian metric. The proof is based

on a gluing construction. Furthermore, this construction gives a geometric interpretation

to Pauly’s dimensional formula for the moduli spaces of singular monopoles on rational

homology 3-spheres. This gluing construction is also motivated by the study of monopoles

in higher dimensions [15].

Theorem 1. Let Mbe an oriented rational homology 3-sphere equipped with a Riemannian

metric g. Let Sp={p1, . . . , pn}and Sq={q1, . . . , qk}be two sets of points in Mwhere all

n+kpoints are disjoint. Let k1, . . . , knbe nnegative integers, where k+Pn

i=1 ki= 0. Then

there exists an irreducible SU(2)-monopole (A, Φ)with Dirac singularities with charge |ki|at

pifor all i∈ {1, . . . , n}on a principal SU(2)-bundle P→M\Spsuch that

(A, Φ) = (A0, Φ0)+(a, ϕ),

where (A0, Φ0)is equal to a scaled BPS-monopole on a small neighbourhood Bj(qj)of each

point qjfor j∈ {1, . . . , k}and is equal to the lift of a U(1)-Dirac monopole with charge ki

at pifor i∈ {1, . . . , n}on M\∪k

j=1B3j(qj). Moreover, the pair (a, ϕ)∈W1,2

α1,α2for suitable

values of α1and α2, where W1,2

α1,α2is a weighted Sobolev space, deﬁned in Deﬁnition 6, such

that k(a, ϕ)kW1,2

α1,α2→0as the masses at the Dirac singularities go to inﬁnity.

The proof of Theorem 1is based on a gluing construction.

•The ﬁrst step is to produce an Abelian Dirac monopole on (M, g) with some singular

points piwith negative charges and some singular points qjwith charge +1 such that

the total charge of the monopole is zero. We construct the Dirac monopole in Section

•The second step is to smooth out the singularities with charge +1 and construct an

approximate monopole. The smoothing process is carried over by gluing model SU(2)-

monopoles — called the scaled BPS-monopoles — to the singular points with charge

+1 and leaving out the rest of the singular points not smoothed-out. This has been

done in Section 3.

•The third step is the deformation. The resulting conﬁguration from the previous step is

an approximate monopole and it does not necessarily satisfy the Bogomolny equation.

However in a suitable norm, it is close to a solution and can be deformed into a genuine

monopole. We solve the linearized Bogomolny equation in Section 4, and then consider

the quadratic terms, and solve the full Bogomolny equation in Section 5.

Remark 1. The gluing construction for SU(2)-monopoles with Dirac singularities at the points

piwith charges |ki|depends on 4k-parameters, where k=−Pn

i=1 ki. This is equal to the

expected dimension of the moduli space of singular SU(2)-monopoles with charge k, as com-

puted by Pauly. 3kof this number is accounted by the position of the highly concentrated

BPS-monopoles, k−1of this number by choices of the framings at these points. There are k

points which we can ﬁx the frames there; however, 1parameter vanishes after taking the ac-

tion of the gauge group into account. The remaining 1 degree of freedom comes from changing

the average mass of the monopole.

Acknowledgments. This article is part of the PhD thesis of the author at Stony Brook

University. I am grateful to my advisor Simon Donaldson for his guidance, encouragement,

and support. Moreover, I would like to thank Aliakbar Daemi, Lorenzo Foscolo, Jason Lotay,

Gon¸calo Oliveira, and Yao Xiao for helpful conversations. This work was completed while the

author was in residence at the Simons Laufer Mathematical Sciences Institute (previously

known as MSRI) Berkeley, California, during the Fall 2022 semester, supported by NSF

Grant DMS-1928930.

2 Dirac Monopoles on Closed 3-Manifolds

In this section, we study and later construct Dirac monopoles on rational homology 3-spheres.

2.1 Local Model of Dirac Monopoles

In this subsection, we study Dirac monopoles close to the points with Dirac singularities.

Let (AD, ΦD) be a U(1)-monopole with a Dirac singularity at p∈Mwith signed charge

k∈Z\ {0}, deﬁned on a small neighbourhood of pin M. A Dirac monopole is a monopole

with Dirac singularities on a bundle with structure group U(1). Close to a singular point p,

the Higgs ﬁeld ΦDhas the following form,

ΦD=−k

2r+m+O(r),(3)

where rdenotes the geodesic distance from pand kis the signed charge at p. Note that

unlike 2, the left-hand-side is the section itself and not its norm.

The Bianchi identity shows that the curvature 2-form of a U(1)-connection is closed.

Furthermore, from the Chern-Weil theory we know that the 2-form

FAD

2π=∗dΦD

2π,

presents c1(L), the ﬁrst Chern class of a line bundle Lwhere the monopole is deﬁned on.

Restricting the bundle to a suﬃciently small punctured neighbourhood of a singular point p

with charge k, the line bundle L|B(p)\{p}→B(p)\{p}is isomorphic to Hk

p, where Hpis the

Hopf line bundle centered at p, with the ﬁrst Chern number c1

c1= lim

→0

2πZ∂B(p)∗dΦD= lim

→0

2πZ∂B(p)k

22+O(1)vol∂B(p)=k.

The model connection Aof a Dirac monopole on R3close to a singular point 0 ∈R3with

charge kis an SO(3)-invariant connection deﬁned on the line bundle Hk

0→R3\ {0}. Let

0(1) be the unit 2-sphere centred at the origin in R3. We can cover S2

0(1) by U+and

U−, where U+=S2

0(1) \ {(0,0,−1)}and U−=S2

0(1) \ {(0,0,1)}. In spherical coordinates

(ρ, θ, ϕ), the connection Aon U+and U−is given by the following 1-forms,

A|U−=k(1 −cos(ϕ))

2dθ, A|U+=k(−1−cos(ϕ))

2dθ,

with the transition function eikθ. Note that on U−∩U+

A|U−−A|U+=kdθ.

We extend the connection radially to Hk

0→R3\ {0}to get A.

Using geodesic normal coordinates, we can deﬁne a diﬀeomorphism

η:B(0) ⊂R3→B(p)⊂M,

between a small neighbourhood of the origin in R3and a small neighbourhood of a point

p∈M. Furthermore, by choosing a bundle isomorphism, covering η, we can identify the

bundles above these open neighbourhoods and pull back the connection ADto a punctured

neighbourhood of the origin in R3.

Lemma 1. The connection of the Dirac monopole with charge k, denoted by AD, close to a

singular point p∈M, up to a gauge transformation, can be written as the following,

η∗AD=A+a, with |a|=O(r),(4)

where the gauge transformation — which is just addition by an exact 1-form — corresponds

to tensoring Hk

pby a ﬂat line bundle.

Proof. The pair (η∗AD, η∗ΦD) is not necessarily a monopole with respect to the Euclidean

metric on B(0) ⊂R3; however, it is a monopole with respect to the pull-back metric η∗g,

and therefore, η∗ΦD=−k

2r+m+O(r), where ris the geodesic distance from the origin with

respect to η∗g.

Ais the connection of a monopole with a Higgs ﬁeld Φ=−k

2r0+m0+O(r0), where r0

is the distance to the origin with respect to the Euclidean metric, and therefore,

| ∗0d(η∗AD−A)|g0=|d(η∗ΦD−Φ)|g0=|d(k

2r−k

2r0

)|g0+O(1).

Moreover,

|r−r0|= max{Ri,j,k,l}O(r3

0) + O(r4

0),

where Ri,j,k,l is the Riemann curvature tensor of η∗g, and therefore,

|d(η∗AD−A)|η∗g=O(1),

which shows in a suitable gauge, |a|η∗g=O(r).

2.2 Construction of Dirac Monopoles

In this section, we construct a Dirac monopole (AD, ΦD) with prescribed charges and singu-

larities on a rational homology 3-sphere (M, g). One can construct Dirac monopoles on any

closed Riemannian 3-manifold; however, here we only consider the case where H2(M, Q) = 0.

Lemma 2. Let (M, g)be an oriented rational homology 3-sphere equipped with a Rieman-

nian metric g. Let p1, . . . , pnbe ndistinct points in Mwith non-zero integer-valued charges

k1, . . . , kn, respectively, where Pn

i=1 ki= 0. Then there exists a monopole (AD, ΦD)with

Dirac singularities with charge kiat pion a principal U(1)-bundle P→M\ {p1, . . . , pn}.

This monopole, up to gauge transformations and adding a constant to the Higgs ﬁeld, is

unique.

Proof. From the monopole equation it can be seen that on the complement of the singular

points we have ∆ΦD= 0, and therefore, ΦDis a harmonic section of the adjoint bundle on

M\ {p1, . . . , pn}. A Dirac monopole singular at the points p1, . . . , pnwith corresponding

signed charges k1, . . . , knis a solution to the equation

∆ΦD=

i=1

kiδpi,(5)

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SingularMonopolesonClosed3-ManifoldsSamanHabibiEsfahaniOctober26,2022AbstractWeprovetheexistenceofnon-trivialirreducibleSU(2)-monopoleswithDiracsingu-laritiesonanyrationalhomology3-sphere,equippedwithanyRiemannianmetric,usingagluingconstruction.1IntroductionThetheoryofYang-Millsconnectionsand,inpart...

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