THE CONCENTRATION PRINCIPLE FOR DIRAC OPERATORS
By
Manousos Maridakis

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Mathematics - Doctor of Philosophy
2014

ABSTRACT
THE CONCENTRATION PRINCIPLE FOR DIRAC OPERATORS
By
Manousos Maridakis
The symbol map σ of an elliptic operator carries essential topological and geometrical
information about the underlying manifold. We investigate this connection by studying
Dirac operators with a perturbation term. These operators have the form Ds = D + sA :
Γ(E) → Γ(F ) over a Riemannian manifold (X, g) for special bundle maps A : E → F
and their behavior as s → ∞ is interesting. We start with a simple algebraic criterion
on the pair (σ, A) that insures that solutions of Ds ψ = 0 localize as s → ∞ around the
singular set ZA of A. Under certain assumptions of A, ZA is a union of submanifolds,
and this gives a new localization formula for the index of D as a sum over contributions
from the components of ZA . We give numerous examples.

Copyright by
MANOUSOS MARIDAKIS
2014

To my parents Petros and Pavlina

iv

ACKNOWLEDGMENTS

This work could not have been carried out without the help and support of a number
of people. Most of all I would like to thank my supervisor Thomas Parker for initiating
me to the mathematical aspects of gauge theory and for sharing with me the geometrical
significance of many developments on the subject. I was profited by his endless supply of
alternative points of view, his crucial suggestions on presentational issues and his constant
encouragement. Next I would like to thank Akos Nagy for interesting discussions on
mathematical physics. In my graduate years I had the company and the support of many
other friends. I am also grateful to Niko, Mixalh, Antigonh, Giwrgo, Nelh, Elenh, Zaxaria,
Gewrgia, Paata and the rest of HSA members to whom I owe a lot of joyful moments.
Last but not least I would like to thank my school teacher and friend Basiasdh Gewrgio
for many encouraging discussions throughout my whole mathematical quest.

v

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
Introduction

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Chapter 1 The Concentration Principle . . . . . . . . . . . . . . . . . . . . .

6

Chapter 2 Examples . . . . . . . . . . . . . .
2.1 First Examples . . . . . . . . . . . . . . .
2.2 Clifford Pairs . . . . . . . . . . . . . . . .
2.3 Self-dual spinor-form pairs in dimension 4

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

17
17
27
36

Chapter 3 Transverse Concentration . . . . . . . . . . . . . . . . . . . . . . . 56
3.1 Structure of A near the singular set . . . . . . . . . . . . . . . . . . . . . . 56
3.2 Structure of D + sA along the normal fibers . . . . . . . . . . . . . . . . . 63
Chapter 4 Constructing approximate solutions . . . . . . . . . . . . . . . . 68
4.1 The bundle of vertical solutions . . . . . . . . . . . . . . . . . . . . . . . . 69
ˆ Z . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.2 The operators DZ and D
Chapter 5 Approximate eigenvectors . . . . . . . . . . . . . . . . . . . . . . 82
5.1 Low-High separation of the spectrum . . . . . . . . . . . . . . . . . . . . . 82
5.2 A Poincar´e-type inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Chapter 6 Nonlinear concentration . . . . . . . . . . . . . . . . . . . . . . . 99
6.1 The nonlinear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
BIBLIOGRAPHY

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

vi

LIST OF TABLES

Table 2.1

Dimension count . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

vii

LIST OF FIGURES

Figure 2.1

The zero set of ψ . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

viii

Introduction

Many problems in geometric analysis involve studying solutions of first-order elliptic systems Dψ = 0 for operators
D : Γ(E) → Γ(F )
between bundles over a Riemannian manifold (X, g) . Given such an operator D, one can
look for zeroth-order perturbation A such that all finite-energy solutions of the equation

Ds ψ = (D + sA)ψ = 0

(0.0.1)

concentrate along submanifolds Z as s → ∞ in a precise sense. There are several
examples of this in the literature, the most well-known occurring in Witten’s approach to
Morse Theory. The aim of this thesis is to find a general setting for such concentration
phenomenon.
We start with a simple criterion that insures concentration : D + sA localizes if
the composition A∗ ◦ σD (α) of the adjoint of A with the symbol of D is self-adjoint
(Definition (1.0.1)). It is natural to regard D as a Dirac operator; its symbol is then

1

Clifford multiplication c and the concentration condition becomes

A∗ ◦ c(u) = c(u)∗ ◦ A

for every u ∈ T ∗ X.

This simple algebraic condition implies the analytic fact that solutions concentrate(in the
precise sense of Proposition 1.0.4).
This thesis has two main results. The first is a Spectral Decomposition Theorem for
operators that satisfy the concentration condition and three transversality conditions. It
shows that:
• The eigenvectors corresponding to the low eigenvalues of Ds∗ Ds concentrate near
the singular set of the perturbation bundle map A as s → ∞.
• The eigenvalues of Ds∗ Ds corresponding to the eigensections that do not concentrate
grow as at least O(s) as s → ∞.
• The perturbation A determines submanifolds Z and each determines an associated
decomposition of the normal bundle to the Z giving a specific asymptotic formula
for the solutions of (0.0.1) for large s.
Our second main result is an Index Localization Theorem (Theorem 1), which follows
from the Spectral Decomposition Theorem. It describes how the Atiyah-Singer index
formula for the index of D decomposes as a sum of local indices associated with specific
operators on bundles over the submanifolds Z .
I. Prokhorenkov and K. Richardson [PR] previously found the concentration condition
(1.0.1), but found few examples because they assumed A to be complex-linear and found
2

concentration only at points. Their list of examples does not include most of the’ examples
given in Chapter 2. The reason is that they assumed that A is self-adjoint and complexlinear, while in many of our examples A is conjugate-linear or has no complex structure
and in some examples concentration occurs along submanifolds i.e. the zero set of a
spinor. Thus it is essential to study (0.0.1) as a real operator.
This thesis has six chapters. Chapter 1 introduces the Concentration Condition, describes some elementary consequences, and states the main results, which are proved in
ˆ over the
later sections. An important part of the story is the vector bundles K and K
sub manifolds Z that are introduced in Chapter 1 and described in detail later.
Chapter 2 presents many examples, some already known, and some new. The point
here is to search for real linear perturbations on reducible Clifford bundles. The first
example is probably classical, but it already illustrates the idea that concentration occurs
when a complex operator is perturbed by a conjugate-linear operator. Example 2 stems
from an observation of Taubes. Taubes used a concentrating family of operators to give an
interesting new proof of the classical Riemann-Roch Theorem for line bundles on complex
curves; our Example 2 extends this to higher-rank bundles over curves. Example 3 is
Witten’s Morse Theory operator, which we show fits into the general setup of Chapter 1.
The next set of examples is more interesting because the singular set zA arises as the
zero set of a spinor on X. In these examples, D acts on “spinor-form pairs ”, meaning
sections of a subbundle of W ⊕ Λ∗ (T ∗ X) where W is a bundle of spinors, a Spinc bundle
and Λ∗ T ∗ X the bundle of forms. These examples are most natural in low dimensions, especially in dimension four. In particular, Examples 5 and 6 are linearizations of operators

3

that occur in Seiberg-Witten theory.
In Chapter 3 we examine the geometric structure of D + sA near the singular set ZA
of A. We first examine the perturbation term A, regarding it as a section of a bundle
of linear maps, and requiring it to be transverse to certain subvarieties, where the linear
maps jump in rank. This transversality condition, allows us to write down a Taylor series
expansion of A in the normal directions along each connected component Z of ZA , and
this gives a similar expansion of the coefficients of the operator D.
The technical analysis needed to prove the Spectral Separation Theorem is done in
Chapters 4 and 5. We use a maximum principle argument to obtain decay estimates
on the concentrating eigensections, and we use the Taylor expansions from Chapter 3 to
decompose D into the sum of a “vertical” operator DV that acts on sections along each
fiber of the normal bundle, and a “horizontal” operator DH that acts on sections of the
bundles K on each component Z of ZA . We then define a space of approximate eigensections, and work through estimates needed to show that these approximate eigensections
are uniquely corrected to true eigensections.
Chapter 6 heads in a new direction. It describes a simple “non-linearization” procedure
that associates to each operator (0.0.1) a non-linear elliptic system whose linearization
is (0.0.1). We show how this non-linearization procedure produces, quite automatically,
two famous non-linear elliptic systems: the Vortex equations in dimension two and the
Seiberg-Witten equations in dimension four. In these cases, concentration phenomena
are well-known as seen, for example, in the work Bradlow, Garcia-Prada, and Taubes.
One of the interesting consequences of this thesis is that the concentration that occurs in

4

these examples, which many believe to be a reflection of nonlinearity, appears instead to
be a result of an algebraic interplay between first and zeroth order terms that is already
present in the linearized equations.
Finally, we provide an appendix with the structure of the subvarieties that produce
the singular set of A in Chapter 3. A good reference is [K]. The extensive study of their
topology is crucial in the study of the localizing Dirac type operators.

5

Chapter 1
The Concentration Principle
The section describes some very general conditions in which one has a family Ds of first
order elliptic operators whose low eigenvectors concentrate around submanifolds Z as
s → ∞. At the end of the section, we state the main theorem of this thesis. We also
state one important corollary, which shows how the index of D decomposes as a sum of
indices of operators on the submanifolds Z .

Let (X, g) be a closed Riemannian manifold and E, F be real vector bundles over X.
Suppose that
• D : Γ(E) → Γ(F ) is a first order elliptic differential operator with its symbol σ.
• A : E → F is a real bundle map.
From this data we can form the family of operators

Ds = D + sA

where s ∈ R. Furthermore, assuming that the bundle E and F have metrics, we can form
the adjoint A∗ , and the formal L2 adjoint Ds∗ = D∗ + sA∗ of Ds . The symbol of D∗ is
−σ ∗ . The main point of this thesis is that such a family Ds is especially interesting when
6

A and the symbol σ are related in the following way.
Definition 1.0.1 (Concentrating pairs). In the above context, we say that (σ, A) is a
concentrating pair if it satisfies the algebraic condition

A∗ ◦ σ(u) = σ(u)∗ ◦ A

for every u ∈ T ∗ X.

(1.0.1)

Lemma 1.0.2. A pair (σ, A) is concentrating if and only if the operator

BA = D∗ ◦ A + A∗ ◦ D

has order 0, that is, is a bundle map. If so, then for each ξ ∈ C ∞ (E),

Ds ξ 22 = Dξ 22 + s2 Aξ 22 + s ξ, BA ξ

(1.0.2)

where these are L2 norms and inner products.
Proof. Given a tangent vector u ∈ Tp∗ X, choose a smooth function f with df |p = u. Then
for any smooth section ξ of E,

BA (f ξ) = D∗ (f A(ξ)) + A∗ (D(f ξ))

= −σ ∗ (df )Aξ + f D∗ Aξ + A∗ σ(df )ξ + f A∗ Dξ
− σ ∗ (u)A + A∗ σ(u) ξ + f BA (ξ).

=

Thus (1.0.1) holds if and only if BA (f ξ) = f BA (ξ), which means that BA is a zeroth

7

order operator. To obtain (1.0.2), expand |D + sA|2 and integrate; this gives

Ds ξ 22 =

Dξ 22 + s2 A(ξ) 22 + s Dξ, Aξ + s Aξ, Dξ

where, after integrating by parts, the last two terms are equal to s ξ, BA ξ .

✷

Remark 1.0.3. Given D and A as above, one can always form the self-adjoint operators

0
D = 
D

D∗
0






and

0
A = 
A

A∗
0



.

Then (σD , A) is a concentrating pair when σD (u) ◦ A = −A ◦ σD (u), u ∈ T ∗ X. This
implies that both (σ, A) and (−σ ∗ , A∗ ) are concentrating pairs.

The assumption that D : Γ(E) → Γ(F ) is elliptic mans that the bundles E and F
have the same rank. Thus a generic bundle map A : E → F is an isomorphism at almost
every point. In the anyalsis of the family D + sA, a key role will be played by the singular
set of A, defined as
ZA :=

x ∈ X | ker A(x) = 0 ,

that is, the set where A fails to be injective.

The following theorem shows the importance of the concentrating condition 1.0.1.
It shows that, under Condition 1.0.1, all solutions of Ds ξ = 0 concentrative along the
singular set ZA . More generally, it shows that all solutions of the eigenvalue problem
Ds ξ = λ(s)ξ with λ(s) = O(s) also concentrate along ZA .
8

Fix δ > 0, set Z(δ) be the δ-neighborhood of ZA and let Ω(δ) = X \ Z(δ) be its
complement.
Theorem 1.0.4 (Concentration Principle). There exist C = C (δ, A, C) > 0, such that
whenever ξ ∈ C ∞ (E) is a section with L2 norm 1 satisfying Ds ξ 22 ≤ C|s|, one has the
concentration estimate
|ξ|2 dvg <
Ω(δ)

C
.
|s|

(1.0.3)

Proof. By equation (1.0.2) and ξ as in assumption we get

C|s| ≥ Ds ξ 22 ≥ s ξ, BA ξ + s2 A(ξ) 22 .

By Lemma 1.0.2, BA is a tensor and compactness implies that M1 = supX |BA | is finite
hence by Cauchy-Schwartz

| ξ, BA (ξ) | ≤ M1

X

|ξ|2 dvg = M1 .

Also for each x ∈ X \ ZA , A is injective on fibers, so there is a positive constant κ(x) with
|A(ξ)| ≥ κ(x)|ξ|. By compactness, those inequalities hold with a constant κ > 0 uniform
on the closure of Ω(δ) and therefore

s2 A(ξ) 22 ≥ κ2 s2

9

|ξ|2 dvg .
Ω(δ)

Consequently

|ξ|2 dvg ≤
Ω(δ)

M1 + C
.
κ2 |s|

✷
Remark 1.0.5. The above proof can be refined if one has an estimate of the form
|A(ξ)|2 ≥ ra |ξ|2 on a tubular neighborhood of ZA , where r is the distance from ZA
and a > 0; this gives a bound on how the constant C in (1.0.3) depends on δ. Assumption (1.0.9) below gives such an estimate with a = 2. Later, in Chapters 4 and 5, we will
develop better estimates that show that the eigensections of Ds with low eigenvalues are
well-approximated by Gaussians as s → ∞.
It is convenient to work in the context of Dirac operators. Recall that a vector space V
is a representation of the Clifford algebra C(Rn ) if there is a linear map c : Rn → End(V )
that satisfies the Clifford relations

c(u)c(v) + c(v)c(u) = −2 u, v Id.

(1.0.4)

for all u, v ∈ V . We will often denote u· = c(u).
Lemma 1.0.6. The concentration condition (1.0.1) with σ = c is equivalent to

c(u) ◦ A∗ = A ◦ c(u)∗

∀u ∈ T ∗ X.

Hence D + sA concentrates if and only if the adjoint operator D∗ + sA∗ does.
10

(1.0.5)

Proof. Multiplying (1.0.1) on the left by c(u) and on the right by c(u)∗ we get

|u|2 A ◦ c(u)∗ = c(u) ◦ A∗ |u|2

which gives (1.0.5). The proof in the opposite direction is similar.

✷

Dirac Operator Assumptions:
1. We assume that E, F are of equal rank, E ⊕ F admits a Z2 graded Spinc structure
induced from the symbol c and a Spinc connection preserving the grading so that
∇c = 0.
2. (c, A) is a concentrating pair: A∗ c(u) = c(u)∗ A for every u ∈ T ∗ X.
Then the composition D = c ◦ ∇ is a Dirac operator

D : Γ(E) → Γ(F ).

(1.0.6)

We also impose two further conditions on A that will guarantee that the components
Z of the singular set ZA are submanifolds and that the rank of A is constant on each
Z . For this, we regard A as a section of a subbundle L of Hom(E, F ) as in the following
diagram:

BL
A



/ Hom(E, F )



(X, g)
11

⊇ Fl

(1.0.7)

Here L is a bundle that parameterizes some family of linear maps A : E → F that satisfy
the concentration condition (1.0.1) for the operator (1.0.6), that is, each A ∈ L satisfies
A∗ ◦ c(u) = c(u)∗ ◦ A for every u ∈ T ∗ X. Inside the total space of the bundle Hom(E, F ),
the set of linear maps with l-dimensional kernel is a submanifold F l ; because E and
F have the same rank, this submanifold has codimension l2 . Assume that L ∩ F l is a
manifold for every l (see the Appendix).
Transversality Assumptions:
3. As a section of L, A is transverse to L ∩ F l for every l, and these intersections occur
at points where L ∩ F l is a manifold.
4. Z is closed for all .

As a consequence of the Implicit Function Theorem A−1 (L ∩ F l ) ⊆ X will be a submanifold of X for every l. The singular set decomposes as a union of these submanifolds,
and even further as a union of connected components Z :

A−1 (L ∩ F l ) =

ZA =

Z.

(1.0.8)

l

By Assumption 3, A has constant rank along each Z , so ker A and ker A∗ are bundles
over Z .
Our final assumption is a statement about the Taylor expansion of A.
Non-degeneracy Assumption:

12

5. Let K be the bundle obtained by parallel translating ker A → Z along geodesics
normal to Z in a tubular neighborhood of the singular set ZA . We require

A∗ A|K = r2 M + O(r3 )

(1.0.9)

where r is the distance function from ZA , and M is a positive-definite symmetric
endomorphism of the bundle K.
Now fix a point p ∈ Z and choose an orthonormal frame {eα } of the normal bundle
N → Z at p with dual frame {eα }. In Chapter 4 Lemma 4.1.1 we prove that the matrices

Mα = −c(eα )∇eα Ap : ker Ap → ker Ap

(1.0.10)

are a collection of commuting isomorphisms, and that each is self-adjoint (by Condition
1.0.1), and its spectrum is real, symmetric, and does not contain 0 (by Assumption 5).
Hence there exist a common decomposition into eigenspaces ker Ap =

i Ki

that simul-

taneously diagonalizes the family {Mα }. In this decomposition, it is the eigenspaces with
positive eigenvalue that are important — this positivity ultimately translates into the fact
that there are L2 concentrating sections in these directions.
Definition 1.0.7. For each component Z of ZA , let K → Z be the bundle whose fiber
at p ∈ Z is

K |p = span







ϕ ∈ ker Ap

ϕ



a common eigenvector of {Mα } with every eigenvalue 
.


positive

13

ˆ → Z defined in the same way with the matrices Mα replaced
There is a similar bundle K
ˆ α = −c(eα )∇α A∗ |p : ker A∗ → ker A∗ .
by M
p
p

ˆ are bundles
In Chapter 4 Proposition 4.1.5 we use assumption 5 to prove that K and K
over each component of ZA . Also in Proposition 4.1.6 of the same chapter, we show that,
ˆ is associated with a canonical Spinc structure
for each component Z , the bundle K ⊕ K
c on Z i.e the restriction of the Spinc structure on (X, g).
Finally we look at the pullback connection of the bundles
¯
((E ⊕ F )|Z , ∇)

✲

(E ⊕ F, ∇)

❄

❄

Z

✲

X

ˆ (see PropoIn Chapter 3, we show that this connection preserves the sub-bundle K ⊕ K
sition 4.1.5 and Proposition 4.1.6). We can compose this connection with Clifford multiˆ over Z .
plication to construct Dirac operators for the bundles K and K
Definition 1.0.8. On each component Z , we define

D

¯ : Γ(Z , K ) → Γ(Z , K
ˆ ).
= c ◦∇

ˆ .
and we denote its adjoint by D
In this definition, the Clifford multiplication c is compatible with the Levi-Civita
connection on X, not with the Levi-Civita connection of the induced metric on Z (the
two differ by a term involving the second fundamental form of Z ).

14

The main result of this thesis is a converse of Theorem 1.0.4. Recall that Theorem 1.0.4
shows that, for each C, the eigensections ξ satisfying Ds∗ Ds ξ = λ(s)ξ with |λ(s)| ≤ C,
concentrate around

Z for large s. The following Spectral Separation Theorem shows

that these localized solutions can be reconstructed using local data obtained from ZA .
Spectral Separation Theorem. Suppose that Ds = D + sA satisfies Assumptions 1-5
above. Let Eλ be the λ-eigenspace of the operator Ds∗ Ds and Fλ the corresponding space
for Ds Ds∗ . Then the low eigenspaces split according the decomposition (1.0.8): there exist
λ0 > 0 and s0 > 0 so that for every s > s0 , there exist vector space isomorphisms
∼
=

Eλ −→

ker D

λ≤λ0

and
∼
=

Fλ −→

ˆ .
ker D

λ≤λ0

In particular, if a component Z of ZA is a point p, then D = 0.

As a corollary we get the following localization for the index :
Index Localization Theorem. Suppose that Ds = D + sA satisfies Assumptions 1-5
above. Then the index of D can be written as a sum of local indices as

index D =

index D .

Proof. By the Spectral Separation Theorem there exist λ0 > 0 and s0 > 0 so that for
15

every s > s0

index Ds = dim ker Ds − dim ker Ds∗ = dim ker Ds∗ Ds − dim ker Ds Ds∗
Eλ − dim

= dim
λ≤λ0

=

Fλ
λ≤λ0

index D

where the third equality holds because Ds∗ Ds and Ds Ds∗ have the same spectrum and
their eigenspaces corresponding to a common non zero eigenvalue are isomorphic. Since
Ds and D differ by a compact perturbation they have the same index. This finishes the
proof.

✷

16

Chapter 2
Examples

The concentration condition (1.0.1) is clearly an algebraic condition on the symbol c of
the Dirac operator D. The existence of the perturbation term A and the construction
of interesting examples of concentrating pairs (c, A) is an algebraic problem about representations of Clifford algebras and their connection with geometry. In the next several
sections, we start with basic examples and progressively built more elaborate ones.

2.1

First Examples

Our first two examples are in dimension two. These have the form Ds = D + sA where
D is a ∂ operator and A is a conjugate-linear zeroth-order operator. Thus Ds is a real
operator, although in the examples it is convenient to write it using complex notation.

Example 1: For functions f, g : C → C, consider the operators

Ds f = ∂f + sz f¯ and Ds g = −∂g + sz¯
g.

These have the form D + sA where A is the self-adjoint real linear map Af = z f¯. Using

17

Lemma 1.0.2, the calculations

∗

BA = (∂ A + A∗ ∂)f = −∂(z f¯) − z∂f = −f¯

and
BA = (∂A∗ − A∂)f = ∂(z f¯) − z∂f = 0

(2.1.1)

show that both (σD , A) and (σD , A∗ ) are concentrating pairs. Theorem 1.0.4 then shows
that as s → ∞, all solutions of Df = 0 and D g = 0 concentrate around the zero set of
A, which is the origin. For these equations, we can find the solutions explicitly:
• Equations (1.0.2) and (2.1.1) show that any solution of Ds g = 0 satisfies

∗
0 = ∂ g + sA∗ g 22 = ∂g 22 + s2 A∗ g 22 .

This means that g is an anti-holomorphic function that vanishes for z = 0, so g = 0.
Thus ker Ds = 0 for all s = 0.
• If f satisfies Ds f = 0, we can apply the operator −∂ to get −∂∂f −sf¯+s2 |z|2 f = 0.
Writing f = f1 + if2 , the imaginary part f2 satisfies

−∂∂f2 + sf2 + s2 |z|2 f2 = 0.

Taking L2 inner product with f2 and integrating by parts, we see that f2 ≡ 0, so

18

f = f1 is a real-valued function. Finally, by completing the differential, we obtain

2

∂(es|z| f ) = 0.

2

But the a real-valued holomorphic function is constant, so f (z) = Ce−s|z| for some
C ∈ R. Thus ker Ds is real and one-dimensional, and the non-zero solutions of
Ds f = 0 clearly concentrate at the origin as s → ∞.
Similarly the problem ∂f + s¯
z f¯ = 0 has trivial solutions and its adjoint has a real
one-dimensional kernel.

It is more interesting to consider real Dirac operators on Riemann surfaces. In Section 7
of [T1], C. H. Taubes showed a concentration property for perturbed ∂-operators on
complex line bundles over Riemann surfaces. The following example generalizes Taubes’
observation to higher rank bundles.
¯ and
Example 2: Let (Σ, g) be a closed Riemann surface with anticanonical bundle K,
let E be a holomorphic bundle of rank r with a Hermitian metric ·, · conjugate linear
¯
in the second argument. The direct sum of the ∂-operator ∂ : Γ(E) → Γ(KE)
and its
adjoint is a self-adjoint Dirac operator




¯∗
0 ∂ 
¯
¯
D=
→ Γ(E ⊕ KE).
 : Γ(E ⊕ KE)
∂¯ 0

19

The symbol of D, applied to a (0, 1)-form u is

c(u)(ξ) = u ∧ ξ − ιu ξ,

¯
ξ ∈ E ⊕ KE.

(2.1.2)

One checks that this satisfies the Clifford relations (1.0.4), so defines a Clifford bundle
¯
structure on E ⊕ KE.
Now choose

¯ ⊗C Sym2 E).
µ ∈ Γ(Σ, K
C

Combined with the conjugate linear isomorphism E ∼
= E ∗ defined by the hermitian metric,
¯
Set
overlineµ becomes a conjugate linear map µ : E → KE.

0
A=
µ

µ∗
0



¯
 ∈ EndR (E ⊕ KE).

Lemma 2.1.1. (c, A) is a concentrating pair.
Proof. It suffices to fix a point p ∈ Σ and verify that c(u) ◦ A = −A ◦ c(u) for all u ∈ Tp∗ Σ.
This is equivalent to proving that µ and it’s adjoint µ∗ satisfy the two identities

ιu (µ(ξ)) = µ∗ (u ∧ ξ)

and

u ∧ µ∗ (η) = µ(ιu (η))

¯ ⊗ E)p . Choose orthonormal bases {ei } of Ep and
for all ξ in the fiber Ep and η in (K
¯ ij ei ⊗ ej ∈ K
¯ Then µ = kµ
¯ ⊗C Sym2 (E) corresponds to the map µ : E → KE
¯
k¯ of K.
C

20

defined by
µ(ξ) = k¯ ei , ξ µij ej .
¯ we have
Thus for u = λk,

¯ ij ei , ξ )ei = λµij ei , ξ ej
ιu µ(ξ) = λ ιk¯ (kµ

and
¯ j , kξ
¯ ei .
µ∗ (u ∧ ξ) = µ∗ (u ∧ ξ), ei ei = u ∧ ξ, µ(ej ) ei = λµij ke
These are equal since µij = µji . The second identity is proved from the first one using
Lemma 1.0.6.

✷

Lemma 2.1.1 shows that Theorem 1.0.4 applies. Thus as s → ∞ the low eigensections
of the operator
¯
¯
Ds = D + sA : Γ(E ⊕ KE)
→ Γ(E ⊕ KE)
concentrate on the singular set ZA . The following lemma describes the structure of ZA .

Lemma 2.1.2. For generic µ, ZA is a finite set of oriented points {p }. Furthermore,
ˆ = 0, and
• At each positive p , K ∼
= R and K
ˆ ∼
• At each negative p , K = 0 and K
= R.
¯ fails to be an
Proof. The singular set of A is the set of points in Σ where µ : E → KE
¯
isomorphism. Thus ZA is the zero set of det µ : Λr E → Λr (KE).
Using the isomorphism
21

Λr E ∼
= Λr E ∗ of the induced hermitian metric on Λr E, this becomes a complex map
¯
Λr E ∗ → Λr (KE),
or equivalently a section

det µ ∈ Γ(L)

of the complex line bundle
¯ r ⊗C Λr E ⊗C Λr E.
L=K

(2.1.3)

Note that while L is a holomorphic bundle, this section is only assumed to be smooth. For
a generic choice of µ, the section det µ will have only transverse zeros, which are therefore
isolated points. By compactness the set {p } of zeros is finite. At each p , the derivative
(∇ det µ) is an isomorphism from Tp Σ to the fiber of L at p. Both of these spaces are
oriented; p is called positive if this isomorphism is orientation-preserving and is called
negative if orientations are reversed.
Let z be a local holomorphic coordinate on Σ centered at p ∈ {p }. Because det µ has
a zero at p, there is a non-vanishing section e1 of E so that µ(e1 ) vanishes at z = 0. Since
µ is conjugate-linear, the section e2 = ie1 also satisfies µ(e2 ) = 0 at z = 0. Hence we can
¯ in which µ has the local expansion
choose real local framings of E and KE




H 0 
2
µ = 
 + O(|z| )
0 ∗
where ∗ denotes an invertible (n − 2) × (n − 2) real matrix and

H : ker µ0 → ker µ∗0 .
22

is the real 2 × 2 matrix that corresponds to multiplication by f → (αz + β z¯)f¯ under the
identification C = R2 .
For a generic section we have |α| = |β|. It follows that det µ has a positive zero at p
if |α| > |β|, and a negative zero if |α| < |β|.
Suppose |α| < |β|. By changing coordinates if necessary, we may assume that α = 0
and β = 1. One then sees that A∗ A has the expansion (1.0.9), so all of the assumptions of
the Spectral Separation Theorem hold. Write z = x + iy, and use the basis {e1 , e2 = ie1 }
of ker µ0 and {d¯
z e1 , d¯
z e2 } of ker µ∗0 to write f = (f1 , f2 ) ∈ ker µ0 . Then




 f1 
H(x, y) 
 =
−f2

 
f 1 
xA1 + yA2   ,
f2

where




1 0 
A1 = 

0 −1




 0 −1
and A2 = 
.
−1 0

In this basis, one can calculate that the Clifford multiplication (2.1.2) is given by




√
1 0 
c(dx) = − 22 

0 1




√
0 −1
and c(dy) = 22 
,
1 0

where these are maps ker µ∗0 → ker µ0 . The corresponding matrices (1.0.10) are therefore




√
1 0 
M1 = −c(dx)A1 = 22 

0 −1





√
−1 0
and M2 = −c(dy)A2 = 22 
.
0 1

23

Applying Definition 1.0.7, one then sees that Kp = 0 in this case. Analoguous calculations
show that




ˆ1 = 2 
M
2 

−1 0

0 1

√

√



ˆ2 = 2 
and M
2 


−1 0
,
0 1

ˆ p is one dimensional.
and hence K
The case |α| > |β| is similar.
✷
Corollary 2.1.3. (Riemann-Roch) If E is a rank r holomorphic bundle over a complex
curve C, then
index ∂¯E = 2c1 (E)[Σ] − rχ(Σ).

(2.1.4)

Proof. Lemma 2.1.1 and the proof of Lemma 2.1.2 show that so all of the assumptions
of the Index Localization Theorem 1 hold. In this case, Z = {p } is the set of zeros of a
generic section det µ of the complex line bundle L defined by (2.1.3). By Lemma 2.1.2,
ˆ = 1, and similarly each negative
each positive zero has local index D = dim K − dim K
zero has index D = −1. The Index Localization Theorem therefore says that index D is
given by the Euler number

index ∂¯E = χ(L)[Σ] = c1 (L)[Σ].

This Riemann-Roch formula (2.1.4) follows because

¯ ⊗C E) = 2c1 (Λr E) − c1 (K r ) = 2c1 (E) − rc1 (K)
c1 (Λr E ⊗C Λr (K

24

and c1 (K)[Σ] = χ(Σ).

✷

Example 3: On a closed Riemmanian manifold (X, g) the bundle E ⊕ F = Λev T ∗ X ⊕
Λodd T ∗ X is a Clifford algebra bundle in two ways:

c(v) = v ∧ −ιv#

and

cˆ(w) = w ∧ +ιw#

(2.1.5)

for v, w ∈ T ∗ X. One checks that these anti-commute:

c(v)ˆ
c(w) = −ˆ
c(w)c(v).

(2.1.6)

Note that D = d + d∗ is a first-order operator whose symbol is c. Fix a 1-form γ with
transverse zeros and set Aγ = cˆ(γ). Then Theorem 1.0.4 shows that the low eigenvectors
of
Ds = D + sAγ = (d + d∗ ) + sˆ
c(γ) : Ωev (X) → Ωodd (X)
concentrate around the zeros of γ.
This is the localization in E. Witten well-known paper on Morse Theory [W1]. After
fixing a Morse function f ∈ C ∞ (X) setting Ds = d + d∗ + sˆ
c(df ), Witten considered the
corresponding Laplacian ∆s = Ds∗ Ds + Ds Ds∗ on the space Ω∗ (X) of differential forms
on X. He showed that the q-forms are low eigenvectors of ∆s concentrate at the index q
critical points of f as s → ∞. In fact, using the natural Z-grading on Ω∗ (X), Witten is
able to prove a refined localization of the low eigenvectors: the low eigenvectors of ∆s on
Ωq (X) localize around the critical points of f with index q.

25

Let {p } be the set of critical points of f and choose one of them p 0 = p with index q.
In Morse coordinates around p, df has the form

ηα xα dxα where ηα = 1 for α = 1, . . . q

and ηα = 1 for α > q. Then

Mα = −ηα c(dxα )ˆ
c(dxα ) : Λev Tp∗ X → Λev Tp∗ X∀α

are invertible self-adjoint matrices with symmetric spectrum of eigenvalues ±1 that commute with each other. In particular if ϕ belongs in the +1- eigenspace of Mα then

Kα+ = {c(dxI )ϕ : |I| = even α ∈
/ I} and Kα− = {c(dxI )ϕ : |I| = even α ∈ I}.

Lemma 2.1.4.

ˆ = 0, and
• At each positive p , K ∼
= R and K

ˆ ∼
• At each negative p , K = 0 and K
= R.
ˆ is a q -form.
• If p has index q then a basis element for K ⊕ K
Proof. The proof of the first two bullets is a verbatim of the proof of Lemma 2.3.7.
For the last bullet, suppose p is a positive zero of index q and ϕ ∈ K is a basis vector.
ϕ is an even form and we use the notation α ∈ ϕ to denote that when ϕ = ϕI dxI then
α ∈ I. Using (2.1.5)

ϕ = Mα ϕ = −ηα c(dxα )ˆ
c(dxα )ϕ = −ηα (dxα ∧ (ι∂α ϕ) − ι∂α (dxα ∧ ϕ)
= ηα (ϕ − 2dxα ∧ (ι∂α ϕ)) =

−ηα ϕ, if α ∈ ϕ
ηα ϕ,

26

if α ∈
/ϕ

where in the fourth equality we used the Cartan’s identity. Hence we must have that
α ∈ {1, . . . , q} if and only if α ∈ ϕ i.e. ϕ has to be a q- form. A similar calculation shows
✷

ϕ to be a q-form when p is a negative zero.

The Index Localization Theorem then shows the well-known fact that the index of
d + d∗ : Ωev (X) → Ωodd (X) is the Euler characteristic χ(X). Witten further showed how
the Morse flow gives rise to “tunneling” maps between the spaces of low eigenvectors, and
how this data enables one to compute the total homology of the manifold.

2.2

Clifford Pairs

Examples 1-3 can be extended and placed in a general context by working with Clifford
algebra bundles. A bundle W → X is called a Clifford algebra bundle if it is equipped
with a a bundle map c : Cl(T ∗ X) → End(W ) that is an algebra homomorphism, meaning
that it satisfies the Clifford relation (1.0.4). For each connection ∇ on W , there is an
associated Dirac operator D = c ◦ ∇ on Γ(W ) whose symbol is c. This section shows
how interesting examples arise by taking W to be the direct sum of two Clifford bundles
associated with different representations of the groups Spin(n) or Spinc (n).
To describe the general context, let (E, c) and (E , c ) be two Clifford algebra bundles
on (X, g) with connection and with corresponding Dirac operators D and D . Suppose

27

there is a bundle map P : E → E; one can then consider the diagram
E
P

c(v) ✲ E
P

✻

E

✻

c (v)✲

E

(2.2.1)

for each v ∈ T ∗ X. Then the perturbed operator

Ds = D + s A : Γ(E ⊕ E ) → Γ(E ⊕ E )

with




D 0 
D = 

0 D


and



P
 0
A = 

∗
−P
0

satisfies the concentration principle if and only if Diagram (2.2.1) is commutes for every
v ∈ T ∗ X. Furthermore, if E and E are reducible Clifford bundles, then one can restrict
Ds to sub-bundles to produce additional examples of concentrating pairs.
The examples in this section are special cases in which we take E and E to be of
the form W ⊗ Λ∗ (T ∗ X) where W is a bundle of spinors. We next describe this setup,
beginning with some linear algebra.
Let ∆ be the fundamental Spinc representation of the group Spinc (n); ∆ is irreducible
for n odd and the sum ∆+ ⊕ ∆− of two irreducible representations for n even. Clifford
multiplication is a linear map c : Rn → EndC (∆); we will often use Hitchin’s “lower dot”

28

notation
v.ϕ := c(v)(ϕ).
There is also a Clifford algebra map cˆ : Rn → End (Λ∗ Rn ) given by

cˆ(v) := σd+d∗ (v) = (v ∧ ·) − ιv (·).

Lemma 2.2.1. Clifford multiplication extends to a Spinc (n)-equivariant linear map c :
Λ∗ Rn → EndC (∆) that satisfies

v.b.ψ = (ˆ
c(v)b).ψ

for all v ∈ Rn , b ∈ Λ∗ Rn and ψ ∈ ∆.

(2.2.2)

Proof. Define the extension c : Λ∗ Rn → EndC (∆) using the standard basis {ej } of Rn by

p

c(e1 ∧ · · · ∧ ep ) = e1· . . . e·

(2.2.3)

for each p-tuple (i1 , . . . , ip ) with i1 < · · · < ip . This map is Spin(n)-equivariant because
for every g ∈ Spin(n), η ∈ Λ∗ X we have that

c(Ad(g)∗ η) = g· c(η)g·−1

Indeed if η = e1 ∧ · · · ∧ ep then according to (2.2.3) and since {ad(g)∗ ei } is also an

29

orthonormal coframe with the same orientation

c(Ad(g)∗ η)(g· ψ)

= c(Ad(g)∗ e1 )c(Ad(g)∗ e2 ) . . . c(Ad(g)∗ ep )(g· ψ)
= c(Ad(g)∗ e1 )c(Ad(g)∗ e2 ) . . . (g· (c(ep )ψ)
= . . . = g· (c(e1 )c(e2 ) . . . c(ep )ψ
= g· (c(η)ψ).

To verify (2.2.2) note that for all k, l

el· (e1 ∧ · · · ∧ ek )· ψ

= el· e1· . . . ek· ψ
(el ∧ e1 ∧ · · · ∧ ek )· ψ,

if l > k

(−1)l (e1 ∧ · · · ∧ eˆl ∧ · · · ∧ ek )· ψ

if 1 ≤ l ≤ k

=

(el ∧ e1 ∧ · · · ∧ ek )· ψ,

if l > k

=
−(ιel (e1 ∧ · · · ∧ ek ))· ψ, if 1 ≤ l ≤ k
= cˆ(el )(e1 ∧ · · · ∧ ek )· ψ

✷

Because of Spinc (n)-equivariance, the map of Lemma 2.2.1 globalizes. Let (X, g) be
an oriented Riemannian n-manifold with a Spinc bundle W and Hermitian metric ·, ·
conjugate linear in the second factor and determinant bundle L = detC (W ). Clifford
multiplication defines bundle maps

30

c : Λ∗ T ∗ X → EndC (W )

and

cˆ : T ∗ X → End (Λ∗ T ∗ X)

(2.2.4)

that satisfy (2.2.2). Given a Hermitian connection A on L with curvature FA we get an
induced spin covariant derivative ∇A on W compatible with the Levi-Civita connection
∇ on T ∗ X and a Dirac operator DA on W .
Example 4: Spinor-form pairs
In the above context, consider the map

P : W → HomC Λ∗ TC∗ X, W

: ψ → c(·)ψ.

(2.2.5)

For ψ a spinor on W we consider the operator

Ds = D + s Aψ : Γ(W ⊕ Λ∗C T ∗ X) → Γ(W ⊕ Λ∗C T ∗ X)

with




0 
DA
D = 

∗
0 d+d


and

Pψ 
 0
Aψ = 

∗
−Pψ
0

where Pψ ∗ denotes the complex adjoint of Pψ .

Lemma 2.2.2. (σD , Aψ ) is s concentrating pair.

31



Proof. The symbol of D, applied to a covector v, and his adjoint are given by


c(v) 0 
σD (v) = 

0 cˆ(v)



0 
−c(v)(·)
σD (v)∗ = 

0
−ˆ
c(v)

and

Formula (2.2.2) expresses the fact that the diagram
W
Pψ

c(v) ✲ W

✻

Λ∗ X

✻

Pψ

cˆ(v) ✲

Λ∗ X

commutes for every v ∈ T ∗ X, which means that (σD , Aψ ) is s concentrating pair.

✷

Unfortunately, Lemma 2.2.2 does not automatically mean that the theorems in the
introduction apply to general spinor-form pairs. The difficulty is seen when one examines
the singular set
ZA = {x ∈ X | ker Pψ = 0}.
The dimension of the exterior algebra Λ∗ (Rn ) is 2n , and the fundamental representation
n
of Spin(n) has complex dimension 2[ 2 ] (see the chart). Thus if whenever dim X > 2,

every map Pψ : Λ∗ (T ∗ X) → W has a non-trivial kernel at each point, so ZA is all of X.
n

2

3

4

5

6

7

dimR Λ∗ (Rn )

4

8

16

32

64

128

dimR W

4

4

8

8

16

16

Table 2.1: Dimension count
32

To avoid this difficulty, we look for sub-bundles L of Hom(Λ∗ (T ∗ X), W ) as in diagram (1.0.7). One way to obtain such sub-bundles is via bundle involutions.


τ 0
Suppose that T = 
 is a metric invariant bundle involution on E ⊕ F so that
0 τˆ
σD (v)τ = ± τˆσD (v) and ∇T = 0

(2.2.6)

for every covector v. Let E = E + ⊕ E − and F = F + ⊕ F − be the decompositions into
±1 eigenspaces of τ and τˆ with p± = 21 (1E ± τ ) : E → E ± and pˆ± = 12 (1F ± τˆ) : F → F ±
the corresponding projections. Set

D+ = σD ◦ p+ ∇|E +

and

A+ = pˆ± A|E +

the restrictions of D and A to E + with values in F + or F − depending on the sign of
(2.2.6).
Lemma 2.2.3. If Ds satisfies the concentration condition (1.0.1), then so does

Ds+ = D+ + sA+ : Γ(E + ) → Γ(F ± ).

(2.2.7)

Proof. The operator p+ ∇|E + defines a metric compatible connection on sections of E + →
X. Also

∗ (v)A)|
(A+ )∗ σD (v)|E + + (σD (v)|E + )∗ A+ = p+ (A∗ σD (v) + σD
E+ = 0

33

for every v ∈ T ∗ X.

✷

In the examples below, we will build involutions by combining three bundle maps. All
three are defined when X is an oriented Riemannian n-manifold.
• The parity operator (−1)p that is (−1)p Id on p-forms.
• The Hodge star operator, which satisfies ∗2 = (−1)p(n−p) .
• Clifford multiplication by the volume form dvol, which satisfies (dvol).2 = (−1)[n/2] .
The parity involution. When dim X = 2n is even, the endomorphism

τ = τˆ = in dvol· ⊕ (−1)p+1 ∈ End(W ⊕ Λ∗C X)

is an involution; its ±1 eigenbundles are

E + = W + ⊕ Λodd
C X

F + = W − ⊕ Λev
C X,

and

and σD∗ (v)τ = −τ σD (v). Furthermore, the restriction of (2.2.5) decomposes as

W + → HomC Λev TC∗ X, W + ⊕ HomC Λodd TC∗ X, W − .

Thus for any ψ ∈ Γ(W + ), we can write Pψ = Pψev + Pψodd under this decomposition, and
set


A+
ψ = 



Pψodd 

0
−Pψev∗
34

0

.

(2.2.8)

Then by Lemma 2.2.3 the operator

−
ev
+
odd
Ds+ = D+ + s A+
ψ : Γ(W ⊕ ΛC X) → Γ(W ⊕ ΛC X)

satisfies the concentration condition (1.0.1).
The self-duality involution. In even dimensions there is a second self-duality invoodd/ev

lution on the bundles W ± ⊕ ΛC

X namely τ = Id ⊕ ∗. The self-duality involution

preserves the eigenspaces of the parity involution and the two involutions commute. Then
∗ +
+
σD∗ (v)τ = τ σD (v) and A+∗
ψ τ = τ Aψ for ψ ∈ W in this case. Hence

D++ + sA++
ψ : Γ(E) → Γ(F )

(2.2.9)

satisfies the concentration condition (1.0.1), where




k
2p−1

E = W+ ⊕ 
p=1

ΛC



X

2k,+

F = W − ⊕ ΛC

and



k−1
2p

X⊕
p=0

ΛC X  .

when dim X = 4k, and




k
2k+1,+

E = W + ⊕ ΛC

2p−1

X⊕
p=1

ΛC

X


and



k
2p

F = W− ⊕ 
p=0

ΛC X  .

when dim X = 4k + 2. In the next section, we will use these involutions to construct
spinor-form pairs that display the concentrating property with a non-trivial singular set
ZA .

35

2.3

Self-dual spinor-form pairs in dimension 4

When X is an oriented Riemannian 4-manifold, the self-duality involution produces a
Dirac operator (2.2.9) with the concentration property and with a singular set ZA that,
we will show next, is not all of X.
In dimension four, the self-dual spinor-form spaces from section 2.2 are

and F = W − ⊕ (Λ0 ⊕ Λ2,+ X)

E = W + ⊕ Λ1 X

and the concentrating pair is given by






c(v) 0 
σD (v) = 

0 cˆ(v)

 0
and Aψ = 
−Pψev ∗

where cˆ(v) is the symbol map of the Dirac operator
W+
Pψev

0



√ +
2d + d∗ . In order for the diagram

c(v) ✲ W −
✻ odd
Pψ

✻

Λ0 ⊕ Λ2,+



Pψodd 

cˆ(v) ✲

Λ1

to commute and the concentration condition to hold, we have to slightly modify Aψ from
(2.2.8) by defining

Pψodd : Λ1 → W − ,

b → b· ψ

and

2,+

Pψev : Λ0 ⊕ΛC

→ W+

(ρ, θ) → (ρ+ √1 θ)· ψ.
2

Lemma 2.3.1. W + is a Clifford bundle for the bundle of Clifford algebras Cl(Λ2,+ (X)).
36

Proof. It suffices to show this at a point p ∈ X. Let now {ei } be an orthonormal coframe
and define Λ2,+ (X) = span{η0 , η1 , η2 } where

η0 = √1 (e1 ∧e2 + e3 ∧e4 ),
2

η1 = √1 (e1 ∧e3 + e4 ∧e2 ),
2

η2 = √1 (e1 ∧e4 + e2 ∧e3 )
2

are orthonormal. Note that

ηi· ηj· + ηj· ηi· = 2 ηi , ηj (dvol· − IdW )

(2.3.1)

for every i, j and so the same identity holds for every other two forms in Λ2,+ (X). Restricted to W + we get

η· θ· + θ· η· = −4 η, θ IdW +

(2.3.2)

for every η, θ ∈ Λ2,+ (X) which is an analog of the Clifford relation for the self dual
2-forms acting on W + . This finishes the proof.

✷

Remark 2.3.2. Choose ψ ∈ W + \{0}. Then from the above proof it follows that the set
{ψ, ηk· ψ} ⊂ W + is orthogonal and ηk· ηk· ψ = −2ψ which implies |ηk· ψ|2 = 2|ψ|2 . Both
 
ϕ
E and F are 8-dimensional real vector bundles. The volume form acts on ξ =   ∈ Ep
b
 
−ϕ
by dvol· ξ =  . Hence choosing|ϕ| = |b| = √1 , the Clifford action produces an
2
b

37

orthonormal basis for Ep and Fp so that

Ep = span{eI· ξ : I even string } and Fp = span{eJ· ξ : J odd string }.

Regard now W + as a real vector bundle of rank 4 with the induced metric. By
considering the negative definite quadratic form produced by that metric we can form the
algebra bundle Cl0,4 (W + ). The perturbation Aψ enjoys the following property:




 0
Lemma 2.3.3. The map W + → End(E⊕F ) : ψ → 
Aψ

A∗ψ 
0

 defines a representation

of the real Clifford algebra bundle Cl0,4 (W + ) on E.
Proof. Fix ψ ∈ Wp+ . By the Clifford relations, the sets {ek· ψ} ⊂ W − and {ψ, ηk· ψ} ⊂ W +
are orthogonal. Therefore for b ∈ Λ1C ,
∗

∗

Pψodd ◦ Pψodd (b) = bl Pψodd ◦ Pψodd (el ) =

el· ψ, ek· ψ bl ek = |ψ|2 b

2,+

and similarly for (ρ, θ) ∈ Λ0C ⊕ ΛC
Pψev ∗ ◦ Pψev (ρ, θ)

1
1
1
= ρ|ψ|2 + √ θl ηl· ψ, ψ + √ ρ ψ, ηk· ψ ηk + θl ηl· ψ, ηk· ψ ηk
2
2
2
1
= ρ|ψ|2 + ηk· ψ, ηk· ψ θk ηk
2
= |ψ|2 (ρ, θ).

38

This proves that

A∗ψ ◦ Aψ = |ψ|2 IdE

Aψ ◦ A∗ψ = |ψ|2 IdF .

and

Finally, polarization gives the relations

A∗ψ ◦ Aψ2 + A∗ψ ◦ Aψ1 = 2

ψ1 , ψ2 IdE

Aψ1 ◦ A∗ψ + Aψ2 ◦ A∗ψ = 2
2
1

ψ1 , ψ2 IdF

1

2

and

for every ψ1 , ψ2 ∈ W + . Hence get a well defined algebra map

 0
Cl0,4 (W + ) → End(E ⊕ F ) : ψ → 
Aψ
The result follows.



A∗ψ 
0

.

✷

Corollary 2.3.4. The mapping ψ → Aψ defines an injection W + → Isom(E, F ).
Proof. Let ξ ∈ E. Then by Lemma 2.3.3

|Aψ ξ|2 = |ξ|2 |ψ|2 .

implying that if ξ ∈ ker Aψ is nontrivial then ψ = 0. Therefore ψ = 0 if and only if Aψ
✷

is non singular which implies the corollary.
39

Example 5: We would like to the study the operator D + sAψ . Choosing a transverse
section ψ : X → W + the singular set of Aψ will be a finite set of oriented points.
Let p ∈ X be such a point and let a coordinate chart (U, {xα }) with xα (p) = 0 and
tangent frame and coframe {eα } and {eα } respectively. Expanding we get sections

ψ(x) = xα ψα + O(|x|2 )

for some elements ψα ∈ Wp+ . Extend these smoothly to sections, still called ψα , of W +
near p. By transversality at p we have

xα ψα = 0 for all x = 0. Setting

Aα := ∇eα Aψ = Aψα ,

we see that

ˆp
xα Aα : Kp = ker(Aψ(p) ) → coker (Aψ(p) ) = K

(2.3.3)

is an isomorphism for every x ∈ Tp X − {0}.
The following technical lemma assures that Aψ can be perturbed to satisfy the nondegeneracy assumption (1.0.9).
Lemma 2.3.5. We can modify ψ without changing its zero set Z(ψ) to insure that {ψα }
are orthonormal.
Proof. Let H : Wp+ → Wp+ be a real orientation preserving linear isomorphism with
eigenvalues {µα } such that {Hψα } is an orthonormal basis for W + . We may assume that
40

H has no real eigenvalues otherwise replace H by λH where λ ∈ S 1 such that none of the
numbers {λµα } is real. As a consequence there exist constant C > 0 such that

inf

(t,x)∈R×S 3

tH(xα ψα ) + (1 − t)xα ψα > C.

Let B(0, 2R) ⊂ U and ρ be a smooth cutoff function with supp(ρ) ⊂ B(0, 2R) and
ρ|B(0,R) ≡ 1. We redefine ψ in (U, x) as

Ψ(x) := ψ(x) + ρ(x)(H − Id)(xα ψα ) = ρ(x)(H − Id)(xα ψα ) + xα ψα + O(|x|2 ).

where |O(|x|2 )| ≤ C1 |x|2 for x ∈ U . Clearly Ψ has a transverse zero at p and satisfies the
C we get that for x ∈ B(0, 2R)\{0}
conclusion of the lemma at p. Chossing R < 2C
1

xα
xα
|O(|x|2 )|
≤ 2C1 R < C < ρ(x)H( ψα ) + (1 − ρ(x)) ψα
|x|
|x|
|x|

therefore there are no other zeros of Ψ in U except at p. Repeating this proccess for each
✷

of the finitely many zeros of the original ψ we are done.

α ∗
ˆ
ˆ
Recall the matrices Mα = −eα
· Aα ∈ End(Kp ) and Mα = −e· Aα ∈ End(Kp ). Let

ˆ α± be the positive/negative eigenspaces of Mα and M
ˆ α respectively. We are
Kα± and K
interested in describing their common positive eigenspaces

Kα+

Kp =

ˆ α+ .
K

ˆp =
and K

α

α

Lemma 2.3.6. The eigenvalues of Mα are λα = ±1 and the corresponding eigenspaces
41

can be described as






α
e· b· ψα 

Kα+ = span 

b

1
: b∈Λ X

and



α
−e· b· ψα 

Kα− = span 

b

1
: b∈Λ X

for every α.
Proof. By relation (1.0.1) and Corollary 2.3.4

α
∗
Mα2 = eα
· Aα e· Aα = Aα Aα = Id

 
ξ 
i.e. Mα has eigenvalues ±1. Let now ϕ =   ∈ Kp is a λα - eigenvector of Mα . Then
b



Mα ϕ = −eα
· Aα ϕ = − 

−c(eα )
0

 
0   b· ψα 
ξ 
=
λ


k  .
α
ev∗
b
−ˆ
c(e )
−Pψ ξ




(2.3.4)

α

By comparing the first rows of (2.3.4) we see that

ξ =

1 α
e b· ψα = λα eα · b· ψα
λα ·

since λ2α = 1. It remains to show that given b ∈

42

1

(2.3.5)

X, the above choice of ξ gives equality

of second rows of (2.3.4). Using (2.2.4)

−ˆ
c(eα )Pψev∗
ξ
α

=

α ∗
α ev
(Pψevα cˆ(eα ))∗ λα eα
· b· ψα = λα (b· e· Pψα cˆ(e )) ψα

= −λα (Pψevα cˆ(b))∗ ψα
= λα cˆ(b)Pψev∗
ψ .
α α

Also for every η ∈ Λ0 X ⊕ Λ2,+ X

η,

Pψev∗
ψ
α α

=

η· ψα , ψα =





0

if η ∈ Λ2,+ X




η

if η ∈ Λ0 X

showing that Pψev∗ ψα = 1. Hence
α

λα cˆ(b)Pψev∗
ψ = λα cˆ(b)1 = λα b
α α

proving equality of the second rows of (2.3.4).

✷

ˆ α } and {Mα } are related as
In Chapter 4 we prove that the families {M

ˆ α eI·
eI· Mα = −M

if α ∈ I

and

ˆ α eI·
eI· Mα = M

if α ∈ I

for every string I of odd length. It follows

ˆ ∓ ) if α ∈ I
eI· ∈ Hom(Kα± , K
α

and

43

ˆ ± ) if α ∈ I.
eI· ∈ Hom(Kα± , K
α

(2.3.6)

Lemma 2.3.7. The spaces

+
α Kα

ˆ+
α Kα

and

Tp X → Wp+ preserves orientation then Kp =
{0}. If ∇ψ reverses orientation then
Proof. Let ϕ ∈

+
α Kα .

+
α Kα

are at most one dimensional. If ∇ψ :

+
α Kα

ˆp =
is non trivial and K

= {0} and

ˆ+
α Kα

ˆ+
α Kα

=

is nontrivial.

Then for every even string I and α ∈ I

Mα eI· ϕ = −eI· ϕ

which implies that eI· ϕ ∈ Kα− . By Remark 2.3.2 Ep = span{eI ϕ : I = even} therefore
+
α Kα

= ϕ is at most one dimensional. The case with

Suppose now that J is a string and ϕ ∈

α∈J

Kα− ∩

ˆ+
α Kα

is analogous.

+
α∈J c Kα

vector.
• J is an even string if and only if




eJ· Mα ϕ = eJ· ϕ
if α ∈
/J
J
Mα (e· ϕ) =



−eJ· Mα ϕ = eJ· ϕ if α ∈ J

for every α so that

+
α Kα

= eJ· ϕ .

• J is an odd if and only if




eJ· Mα ϕ = eJ· ϕ
if α ∈
/J
J
ˆ α (e ϕ) =
M
·



−eJ· Mα ϕ = eJ· ϕ if α ∈ J

for every α so that

ˆ+
α Kα

= eJ· ϕ .
44

is a nontrivial

This dichotomy shows also that either

+
α Kα

or

ˆ+
α Kα

should be nontrivial at each

zero of ψ. Say that α ∼ β iff α, β ∈ J or α, β ∈ J c . By Lemma (2.3.6) if α ∼ β we




β
α
±e· b· ψα 
±e· b· ψβ 
can write ϕ = 
=


 for some common b ∈ Λ1 X and if α ∼ β
b
b




β
α
e· b· ψα 
−e· b· ψβ 
then ϕ = 
 = 
 for the same b ∈ Λ1 X. In particular we have a
b
b
description of the orthonormal basis {ψα } in terms of ψ1 and b as

ψα =





1
b· eα
· e· b· ψ1

if α ∼ 1




1
−b· eα
· e· b· ψ1

if α ∼ 1

.

But |J| + |J c | = 4 hence J, J c are both even or both odd. Therefore {ψα } is positively
oriented in Wp+ for J even and negatively oriented for J odd.

✷

Corollary 2.3.8. As a consequence of Spectral Separation Theorem the index of D :
Γ(E) → Γ(F ) is the signed count of the zeros of ψ i.e.

index D = c2 (W + )[X]

the second Chern class of the bundle W + evaluated on the fundamental class of X.

Example 6: J-holomorphic curves in symplectic four-manifolds.
Recall the philosophy of Diagram 1.0.7: if we can find a sub-bundle L of Hom(E, F )
whose sections satisfy the concentration condition, then we obtain concentrating operators
Ds with singular sets ZA of possibly different dimensions. This example illustrates this
45

phenomenon in dimension four, by showing how a sub-bundle L can be constructed from
a symplectic structure.
Let (X 4 , ω) a closed symplectic manifold with a complex hermitian line bundle L and
a section ψ ∈ Γ(L) whose zero set is a transverse disjoint union Zψ = ∪ Z of symplectic
submanifolds of X. Let N be the symplectic normal bundle of Z . Choose an almost
complex structure J and a Riemannian metric g on X so that (ω, J, g) is a compatible
triple and each Z is J-holomorphic; the symplectic and metric normal bundles of Z are
the same.
As usual, write T X ⊗ C = T 1,0 X ⊕ T 0,1 X and define the canonical bundle to be the
complex line bundle K = Λ2,0 X. In this context, the direct sum W = W + ⊕ W − of the
complex rank 2 bundles

W − = L ⊗ Λ0,1 X

¯
W + = L ⊕ LK,

has a Spinc structure and a Clifford multiplication T ∗ M ⊗ W → W from wedging and
contracting (0, 1) forms as in formula (2.1.2).
Let ∇L be a hermitan connection on L and ∇X the Levi-Civita connection on X.
¯

These can be used to build a Spinc connection ∇ = ∇L ⊕ ∇LK on W + . There is also the
projection to the (0, 1) part of T ∗ X of ∇L namely ∂ L ψ := 12 (∇L ψ + i∇L ψ ◦ J). Then a
Dirac operator is defined by

D=

√
∗
¯ → Λ0,1 X ⊗ L.
2(∂ L + ∂ L ) : L ⊕ KL

46

We would like to study the perturbed operator D + sAψ . Fix one component Z = Z
with normal bundle N = N . By the transversality of ψ, the map ∇L ψ : N → L is an R
linear isomorphism.
In order for the nondegeneracy condition (1.0.9) to hold we need the following:
Lemma 2.3.9. We can change ψ without changing its zero set so that ∇L ψ : N → L
becomes orthogonal.
Proof. We consider the bundles N with the induced metric and L|Z as a real vector
bundle with the induced metric h from the hermitian metric. Let O(N, L|Z ) = {H ∈
Hom(N, L|Z )|H ∗ h = g}, a deformation retraction of Hom(N, L|Z ). Therefore there is a
smooth path of bundle maps [0, 1]

t → Ht ∈ Hom(N, L|Z ) so that H0 = ∇L ψ and

H1 ∈ O(N, L|Z ). This path can be chosen so that Ht is invertible for every t ∈ [0, 1]. As
a consequence there exist constant C > 0 such that

inf
(t,v)∈[0,1]×S

Ht (v) > C

where S is the unit sphere bundle of the normal bundle N .
Now use the exponential map on the normal bundle N of Z to define a tubular neigbroorhood N , a parallel transport map τ : L|Z → L|N along normal geodesics and set
x = exp(v). Let B(Z, R) ⊂ N and ρ be a smooth cutoff function with supp(ρ) ⊂ B(Z, 2R)
and ρ|B(Z,R) ≡ 1. We redefine ψ in B(Z, 2R) as

Ψ(v) := ψ(v) + τ (Hρ(v) (v) − ∇L
v ψ|Z ).

47

Note that
2
2
|ψ(v) − τ ∇L
v ψ|Z | = O(|v| ) ≤ C1 |v|

for v ∈ N . Clearly Ψ has also transverse intersection with the zero section at Z and
C we get that for v ∈
satisfies the conclusion of the lemma at Z. Choosing R < 2C
1

B(Z, 2R)\{0}
|O(|v|2 )|
v
≤ 2C1 R < C < τ Hρ(v) ( )
|v|
|v|
therefore there are no other zeros of Ψ in U except at p. Repeating this proccess for each
component of the singular set of the original ψ we are done.

✷

Fix now p ∈ Z and local coordinates {xi } in X so that Z = {x1 = x2 = 0} and
orthonormal frames {e1 , e2 = J(e1 )} and {e3 , e4 = J(e3 )} trivializing N and T Z respectively around p. By the Lemma {ψi = ∇L
ei ψ} is an orthonormal frame trivializing L|Z
around p and we extend it to the normal directions to a frame trivializing L. Then ψ
expands in the normal directions of Z as

ψ = x1 ψ1 + x2 ψ2 + O(|x|2 ).

Denote ∇ei Aψ = A L = Ai .
∇i ψ
We now have to consider the matrices Mα = −eα
· Aα , α = 1, 2 and their common
positive spectrum. By Lemma 2.3.6 the positive eigenspaces are given by




α
e· b· ψα 

Kα+ = span 

b

1
 : b ∈ Λ X , α = 1, 2.

48

There are two cases:
• The map ∇L ψ : N → L|Z preserves the natural orientations as an R linear map. Then

L
e2 = J(e1 ) and ∇L
J(e1 ) ψ = ψ2 = iψ1 = i∇e1 ψ

so that e1· ψ1 + e2· ψ2 = Dψ|Z = ∂ L ψ|Z = 0. Then
   
ψ1  ψ2 
K = K1+ ∩ K2+ = span   ,  
e1
e2


=

L
∇ v


v


ψ
∗
: v∈N

N ∗ . (2.3.7)

Also

ˆ = span{e3 K, e4 K}
K
·
·

orthogonal complement of ω in Λ2,+ X

K X |Z

T ∗Z ⊗ N ∗.

Hence the local operator is

DZ = ∂¯N ∗ : Γ(N ∗ ) → Γ(T ∗ Zi ⊗ N ∗ )

and by Riemann-Roch

index DZ = 2N 2 − 2(g − 1) = (L|Z )2 − 2(g − 1) where g = genus of Z.

Also if since ∇L ψ|Z : N → L|Z preserves orientation then the adjunction formula applies
to give
2(g − 1) = (L|Z )2 + L|Z K.
49

Z

3

X

Z

2

Z

1

Figure 2.1: The zero set of ψ
Hence index DZ = (L|Z )2 − K|Z · L|Z in this case.
• If ∇L ψ|Z reverses orientation then adjunction formula gives

¯ Z )2 + K L
¯ Z = (L|Z )2 − K · L|Z
2(g − 1) = (L|

and a similar calculation computes the operator in this case DZ = ∂¯∗ : T ∗ Z → C. By
Riemann-Roch we then have

index DZ = 2(g − 1) = (L|Z)2 − K · L|Z .

Applying the Index Localization Theorem and using the contributions of the local indices
from all the components Z with L = L|Z we get

c2 (W + )[X] = index D =

(L2 − K · L ) = L2 − KL.

This is a familiar formula is SW theory. It describes the dimension of the SW moduli
50

space in terms of the bundled K and L.

Example 7: Spinor/form pairs twisted by SU (2)-bundles.
Let (X, g, W ± , c(·)) be a Riemmannian manifold with a Spinc - structure and (E, h) →
X be a Hermitian SU (2) - bundle. Set also su(E) := {A ∈ EndC (E) : A+A∗ = 0, trC A =
0} where A∗ is the Hermitian adjoint of A. Differences of Hermitian connections on E
are sections of Λ1 X ⊗ su(E). Equip E with a Hermitian connection ∇E and W with a
Spinc connection. We get induced connections ∇W ⊗E on W ⊗ E and ∇ on Λ∗ X ⊗ su(E).
The symbol maps c and cˆ extend as

c(v) ⊗ idE : W + ⊗ E → W − ⊗ E

and cˆ(v) ⊗ idsu(E) : Λodd X ⊗ su(E) → Λev X ⊗ su(E).

Finally we get operators

DE = (c ⊗ idE ) ◦ ∇W ⊗E

and dE + d∗E = (ˆ
c ⊗ idsu(E) ) ◦ ∇.

We define a Clifford multiplication cE to include End(E)- valued forms by

cE : Λ∗ (X) ⊗ End(E) → End(W ⊗ E)
η ⊗ A → η· ⊗ A

The restriction of cE to the subspace Λ∗ (X) ⊗ su(E) defines maps

P ev : W + ⊗ E → HomC Λev (X) ⊗ su(E), W + ⊗ E
51

(2.3.8)

and
P odd : W + ⊗ E → HomC Λodd (X) ⊗ su(E), W − ⊗ E
both given by ψ ⊗ e → cE (·)ψ ⊗ e.
Proposition 2.3.10. For fixed Ψ ∈ W + ⊗ E the perturbed operator

Ds = D + s AΨ : Γ(W + ⊗ E) ⊕ Ωodd (X, su(E)) → Γ(W − ⊗ E) ⊕ Ωev (X, su(E))

with






0
DE

D = 

∗
0 dE + dE


Aψ⊗e = 

and



odd
PΨ


0
ev∗
−PΨ

0



ev∗ denotes the adjoint of P ev .
satisfies the concentration relation 1.0.1. Here PΨ
Ψ

Proof. It suffices to show the proposition for Ψ = ψ ⊗ e. The symbol of D, applied to a
covector v, and his adjoint are given by


0
c(v) ⊗ idE

σD (v) = 

0
cˆ(v) ⊗ idsu(E)
and




0
−c(v) ⊗ idE

σD (v)∗ = 
.
0
−ˆ
c(v) ⊗ idsu(E)

52

Checking the concentration relation is just proving the identity

ev ◦ (ˆ
odd .
Pψ⊗e
c(v) ⊗ idsu(E) ) = (c(v) ⊗ idE ) ◦ Pψ⊗e

By linearity it is enough to check the identity for b ⊗ B ∈ Λodd X ⊗ su(E). Then

odd (b ⊗ B) = (c(v) ⊗ id )(c(b)ψ ⊗ B(e)) = (c(v)c(b)ψ) ⊗ B(e)
(c(v) ⊗ idE ) ◦ Pψ⊗e
E

and

ev ◦ (ˆ
ev (ˆ
Pψ⊗e
c(v) ⊗ idsu(E) )(b ⊗ B) = Pψ⊗e
c(v)b ⊗ B) = (c(ˆ
c(v)b)ψ) ⊗ B(e)

= (c(v)c(b)ψ) ⊗ B(e)

where in the third equality we used relation (2.2.2).

✷

Finally the map cE has an interesting property:
Lemma 2.3.11. On Λ∗ (X) ⊗ End(E) the bracket

[η1 ⊗ A1 , η2 ⊗ A2 ] = cˆ(η1 )η2 ⊗ A1 A2 − cˆ(η2 )η1 ⊗ A2 A1

(2.3.9)

defines a Lie algebra structure. The map cE becomes then a Lie algebra homomorphism.
Proof. On Λ∗ X the bracket

(η1 , η2 ) → cˆ(η1 )η2 − cˆ(η2 )η1

53

defines a Lie algebra structure. Then [·, ·] is an extended Lie algebra from Λ∗ (X) and
End(E). Using (2.3.9) we see that

cE ([η1 ⊗ A1 , η2 ⊗ A2 ]) = cE (η1 ⊗ A1 ) ◦ cE (η2 ⊗ A2 ) − cE (η2 ⊗ A2 ) ◦ cE (η1 ⊗ A1 ).

✷

Example 8: Witten’s deformations twisted by SU (2)-bundles.
Let (X, g) closed Riemannian and, as in the previous example, (E, h) → X be a
Hermitian SU (2) - bundle with Hermitian connection ∇E and set

Sym(E) := {A ∈ EndC (E) : A = A∗ }.

a subspace of End(E) with the trace product. Recall the two Clifford representations c
and cˆ of Λ∗ (X) described in (2.1.5) and extend them to maps

σ(v) = c(v) ⊗ idEnd(E) : Λodd (X) ⊗ End(E) → Λev (X) ⊗ End(E)

and, fixing α × A ∈ Λ1 (X) ⊗ Sym(E)

Aα⊗A : Λodd (X) ⊗ End(E) → Λev (X) ⊗ End(E)
β⊗B

→

cˆ(v)β ⊗ A ◦ B

for every v ∈ T ∗ X and A ∈ Sym(E). Finally the induced connection ∇ on Λ∗ (X) ⊗

54

End(E) gives operator

DE = σ ◦ ∇ : Γ(Λodd (X) ⊗ End(E)) → Γ(Λev (X) ⊗ End(E)).

Proposition 2.3.12. The perturbed operator

Ds = DE + s Aα⊗A

satisfies the concentration condition (1.0.1).
Proof. By linearity it is enough to check the identity for b ⊗ B ∈ Λodd X ⊗ End(E). Then

(c(v)∗ ⊗ idEnd(E) ) ◦ Aα⊗A (b ⊗ B) = (c(v)∗ ⊗ idEnd(E) (ˆ
c(α)β ⊗ AB)
= c(v)∗ cˆ(α)β ⊗ AB

and

A∗α⊗A ◦ (c(v) ⊗ idEnd(E) )(b ⊗ B) = Aα⊗A (c(v)b ⊗ B)
= cˆ(α)∗ c(v)b ⊗ A∗ B.

Since A∗ = A and by relation (2.1.6) the two lines are equal.

55

✷

Chapter 3
Transverse Concentration

3.1

Structure of A near the singular set

In proving the Spectral Separation Theorem we will have to analyze the geometry of the
operator
Ds = c ◦ ∇ + sA : Γ(E) → Γ(F )
near the singular set ZA . The idea is to expand into Taylor series along the normal
directions of each component Z of the singular set ZA .
Fix a Z =: Z an m-dimensional submanifold of the n dimensional manifold X. Let
π : N → Z be the normal bundle of Z in X with pZ : T X|Z → T Z and pN : T X|Z → N
the orthogonal projections along Z. The Levi-Civita connection ∇X , when restricted
on sections of T X|Z decomposes to pZ ∇X and pN ∇X i.e. the Levi Civita connection
∇Z of Z and a connection ∇N of the normal bundle respectively. Our first task is to
understand the perturbation term A on a tubular neighborhood N of Z. For that purpose
we introduce the following coordinates:
Fix normal coordinates (U, {zi }) centered at p ∈ Z and choose orthonormal moving
frames {eα } parallel at p with respect to ∇N on N |U . The frame {eα } at z identifies an
open subset Nz ⊂ Nz with an open subset of Np ⊂ Rn−m with coordinates {yα }. We get
56

the chart

U × Np → NU ⊂ X, (z, y) → expz (yα eα )

(3.1.1)

with tangent frame and coframe {∂i } and {dxi } and on the normal fibers {∂α } and {dyα }
so that ∂a |U ×{0} = eα and ∂i |U ×{0} =: ei . These are normal coordinates, the distance
function r from Z writes in this chart r(z, y) = (

2 1/2
α za )

and g = exp∗ gX has the form

gαβ = δαβ + O(z, r2 ) in normal directions.
The Levi Civita connection from X pullback to U × Np and writes

j

β

∇X
∂i ∂α = Γiα ∂j + Γiα ∂β .
β

where Γiα = 0 at p.
Next introduce the rank l - subbundles

Kz := ker Az ≤ Ez

and

ˆ z := ker A∗z ≤ Fz
K

of E|Z and F |Z as z runs in Z. By ∇E - parallel transporting along the normal fibers
ˆ ⊆ F |N . Recall now the non-degeneracy
of N we create subbundles K ⊆ E|N and K
assumption
A∗ A|K = r2 M + O(r3 )
where M is positive definite and symmetric. We choose orthonormal frame {σk } that
diagonalize M at Ep and extend locally in U ⊂ Σ to a parallel frame trivializing E|U =

57

(K ⊕ K ⊥ )|U . Extend over NU by parallel transporting along the normal radial geodesics.
Since index A = 0 a consequence of the concentration condition (1.0.1) is

ˆ Z
u· K|Z = K|

and

ˆ ⊥
u· KZ⊥ = K|
Z

(3.1.2)

ˆ Z over Z has a natural Z2 - graded
for every u ∈ T ∗ X|Z . In particular the bundle (K ⊕ K)|
Spinc structure. Derivating relations (1.0.1) and (1.0.5) along Z we also get

u· ∇α A = −∇α A∗ u·

and ∇α A u· = −u· ∇α A∗

(3.1.3)

for every u ∈ T ∗ X|Z . Transversality condition (see Appendix) of A along the normal
directions of U ⊂ Z gives

ˆ U
∇α A(K|U ) ⊆ K|

and

ˆ U ) ⊆ K|U .
∇α A∗ (K|

Here the second relation is obtained by the first one using (3.1.3). Hence




0 
Aα
⊥
ˆ ⊕K
ˆ ⊥ )U .
∇α A = 
 : (K ⊕ K )|U → (K
0 ∗(z)
Hence the Taylor expansion with respect to the decompositions E|N = K ⊕ K ⊥ and
ˆ ⊕K
ˆ ⊥ of the perturbation term A along the normal directions of NU write
F |N = K




0 
1
Aα
3
A = A0 + yα 
 + yα yβ ∇α ∇β A + O(r )(z)
2
0 ∗(z)
58

(3.1.4)





0 0 
where A0 = 
 is the evaluation of A at z ∈ Z. We have the very useful technical
0 ∗(z)
lemma:
Lemma 3.1.1. The restriction of ∇E and ∇F to Z ⊂ X preserve the splittings E =
ˆ K
ˆ ⊥ . Since ∇A preserves those splittings all the covariant derivatives
K⊕K ⊥ and F = K⊕
∇k A preserve these splittings.
Proof. Let ξ ∈ Γ(N , K) with ξ|Z ∈ K|Z . By (3.1.4) at p

F
A(∇E
α ξ) = ∇α (Aξ) − (∇α A)ξ = Aα ξ − Aα ξ = 0

and since Aξ|Z ≡ 0
A(∇E
i ξ) = −(∇i A)ξ = 0
i.e. ∇E
A ξ|p ∈ Kp for A = i, α. But the last conclusion is an independent statement of the
frame hence it holds in Z. Also the Riemannian metric on E is parallel with respect to
∇E therefore ∇E satisfies the same property with the bundle K ⊥ . The case where E is
replaced by F is the same.

✷

Proposition 3.1.2. The 2 -jets of A∗ A and A along Z satisfy

∇2v,v (A∗ A)|K > 0,

∇2v,w (A∗ A)|K = 0

and

∇2u,v A|K = 0

(3.1.5)

for every u ∈ T Z and v, w ∈ N \{0} with v ⊥ w. The 1- jet of the perturbation A along

59

Z ∩ NU satisfies

ˆ
Aα ∈ Isom(K, K),

A∗α Aα = A∗β Aβ ,

and

A∗α Aβ + A∗β Aα = 0

(3.1.6)

for every i, α, β with α = β.
Proof. The first couple of relations are a direct consequence of the assumption of (1.0.9).
Now if R denotes the curvature on End(E, F )

RF (∂i , ∂α )(Aξ) = (R(∂i , ∂α )A)ξ + A(RE (∂i , ∂α )ξ).

By Lemma 3.1.1 RE (ei , eα )ξ|p ∈ K|p and Aξp = 0 hence (R(ei , eα )A)ξ|p = 0. Also
(∇i A)ξ|p = 0 and [ei , eα ]|p = [∂i , ∂α ]|p = 0 therefore

0 = (R(ei , eα )A)ξ|p = (∇i ∇α A)ξ|p = (∇2i,α A)ξp .

This proves the last relation of the Hessian of A.
For relations on the 1 -jet we just notice by using (3.1.4) to expand A∗ A

A∗α Aα = ∇2α,α (A∗ A)|K > 0

so Aα is invertible for every α. Also eα ± eβ are orthogonal and the relations (3.1.5) give

0 = ∇eα +eβ ∇eα −eβ (A∗ A)|K = ∇2α,α (A∗ A)|K − ∇2β,β (A∗ A)|K = A∗α Aα − A∗β Aβ .

60

Similarly by (3.1.4)
0 = ∇2α,β A|K = A∗α Aβ + A∗β Aα
i.e the last relation.

✷

Finally we can change A so that the following happens :
Lemma 3.1.3. We can choose our perturbation A so that ∇2u,v A|Z ≡ 0 for every u, v ∈
N.
Proof. According to properties (3.1.6) of the family {Aα } we see that

(A∗α Aβ + A∗β Aα ) +

(yα Aα )∗ (yβ Aβ ) =

α

α=β

yα2 A∗α Aα

yα2 A∗α Aα =
α

and since A∗α Aα is positive definite for every α there exist C > 0 so that

|yα Aα ξ| ≥ C|y||ξ|

ˆ Z is an isomorphism for every y = 0
for every ξ ∈ K|Z . In particular yα Aα : K|Z → K|
and
|(yα Aα )−1 | ≤

C
.
|y|

Hence there exist 1 > 0 so that for every 0 < |y| < 1

A0 + yα ∇α A : E|Z → F |Z

61

is invertible with
|(A0 + yα ∇α A)−1 | ≤

C
.
|y|

Introduce now a cut off function supported on N , a tubular neighborhood around Z
of radius

to be chosen later. Let ρˆ : [0, ∞) → [0, 1] smooth cut off with ρˆ−1 ({0}) =

r(q)
) on N
[1, ∞), ρˆ−1 ({1}) = [0, 1/2] and strictly decreasing in [1/2, 1], define ρ(q) = ρˆ(

and extend as 0 on X − N . We can form the bundle map

B : E → F,

B(q) =

ρ(q) 2
∇v,v A|Z ,
2

q = expz (v).

Derivating relation (3.1.3) we get that

u· ∇α ∇β A = −∇α ∇β A∗ u·

hence
u· B = −B ∗ u·
for every u ∈ T ∗ X|Z . Hence A − B satisfies (1.0.1) and using (3.1.7) on NU

A − B = A0 + yα ∇α A +

1 − ρ(y)
yα yβ ∇α ∇β A + O(|y|3 )(z).
2

Choose 0 < < 1 so that for every 0 < |y| <
1 − ρ(y)
|y|
1
−1
yα yβ ∇α ∇β A + O(|y|3 ) <
< (A0 + yα ∇α A)−1 .
2
2C
2

62

Then A − B is invertible on N − Z and agrees with A outside N therefore ZA−B = ZA .
Also by construction
∇2u,v (A − B) ≡ 0 : E|Z → F |Z
for every u, v ∈ N . Finally we have only changed the 2-jet of A around Z to produce
A − B. Hence condition (1.0.9) still holds for A − B since it relates only to the 1-jet of A
on Z. Replacing A with A − B on every component Z of ZA we are done.

✷

According to Lemma 3.1.3 on a sufficiently small tubular neighborhood N around Z




0 
Aα
3
A = A0 + yα 
 + O(r )(z)
0 ∗(z)

3.2

(3.1.7)

Structure of D + sA along the normal fibers

Our next task is understanding Ds near the set Z. Denote by g Z and g N the metric
g X restricted on T Z and on N respectively. Recall the chart (3.1.1) centered at p ∈ Z
with ({xi }, {yα }) being the horizontal and vertical coordinates respectively and the frame
{σk } on E|N . Denote by ∂ V /H the local directional derivatives of those vertical and
U

horizontal tangent frames in this chart.
Except for the tangent frame on T N we can parallel transport the local frames {eα }
and {ei } from U to N |U along the normal radial geodesics using ∇X to construct new
63

frames {τa } and {τi } with dual frames τ α and τ i respectively.
Comparing the parallel frame with the tangent frame at p we see

τα − ∂α = O(r2 )V + O(r2 )H

and

τi − ∂i = O(r2 )V + O(r)H

hence, action on the local differentials will write

∂τα = ∂α + O(r2 )∂ V + O(r2 )∂ H

and

∂τi = ∂i + O(r2 )∂ V + O(r)∂ H (3.2.1)

where the symbols O(r)V and O(r)H are used to denote decomposition in the vertical
and horizontal frames respectively of order r. Also

l
∇E
τA (f σk ) = (∂τA f )σk + f ωAk σl

for A = i, α and note that the Taylor expansions of the connection components is

ωαk = O(r2 )

and

0 + O(r).
ωik = ωik

Definition 3.2.1. The dilation operators from Γ(N , E|N ) → Γ(N , F |N ) are defined as

DN = τ·α ∇E
τα

and

DH = τ·i ∇E
τi .

Those are globally defined operators on the tubular neighborhood N of Z independent of
the choice of the above frames and D = DN + DH . Also for fixed z ∈ Z, the Euclidean

64

Clifford action eα
· : Ez → Fz induce a Euclidean Dirac operator

/ 0 = eα
D
· ∂α : Γ(Nz , Ez ) → Γ(Nz , Fz ).

Finally observe that the coframe {ei } acts on (E ⊕ F )|Z making it a Z2 - graded Spinc
bundle over Z. The restriction of the connection components {ωik } on U give rise to
¯ i σk = ω 0 regarded otherwise as the pullback connections
connections denoted both as ∇
ik
of the bundles
¯
(π ∗ (E|Z ), ∇)

❄

N

¯
✲ (E|Z , ∇)
π

✲

❄
✲

Z

(E|N , ∇E )

❄
✲

N

Definition 3.2.2.

¯ i ξ.
DZ : Γ(N, π ∗ (E|Z )) → Γ(N, π ∗ (F |Z )) : ξ → ei· ∇

ˆ Z is analogous.
The definitions for D
/ 0 and DZ
Notice that DN and DH are only defined on sections ξ : N → E|N while D
/ 0 and DH to DZ
are defined on sections ξ : N → π ∗ (E|Z ). In order to relate DN with D
we introduce the parallel transport map along the radial geodesics of Nz

τ : Γ(Nz , (E ⊕ F )z ) → Γ(Nz , (E ⊕ F )|Nz )
f (z, w)σk (z, 0)

(3.2.2)

→ f (z, w)σk (z, w)

for every z ∈ Z. Hence τ operates from sections of π ∗ (E ⊕ F )|Z to give sections of
65

(E ⊕ F )|N . Also since the symbol map is ∇X - parallel

(τ·A σk )(z, v) = τ (eA
· σk (z, 0)), A = α, i

for every (z, v) ∈ N where τ is the parallel transport map defined by (3.2.2).
Proposition 3.2.3. Let ξ : N → π ∗ (E|K ). We have the relations

/ 0 ξ + O(r2 ∂ V + r2 ∂ H + r2 )τ ξ
DN (τ ξ) = τ D

and
DH (τ ξ) = τ DZ ξ + O(r2 ∂ V + r∂ H + r)τ ξ
Proof. By linearity and compactness it suffices to work in the chart NU centered at p ∈ Z
with τ ξ = f σk . Since the rates of the Taylor expansion do not depend on the choice of
the frame we can restrict our calculations at p and use (3.2.1).
For the vertical operator DN = τ·α ∇E
τα we estimate

DN (τ ξ) = τ·α ((∂α f )σk + O(r2 (∂ V f ) + r2 (∂ H f ) + r2 f )σk )
/ 0 ξ + O(r2 ∂ V + r2 ∂ H + r2 )τ ξ.
= τD

Also

i
0
2 V
H
DH (τ ξ) = τ·i ∇E
τi (τ ξ) = τ· ((∂i f )σk + f τ ωik + O(r (∂ f ) + r(∂ f ) + rf )σk )

= τ DZ ξ + O(r2 ∂ V + r∂ H + r)τ ξ.
66

✷
Combining Proposition 3.2.3 with expansion 3.1.7 we get:
Corollary 3.2.4. Ds expands along the normal directions of the singular set Z with
ˆ ⊕K
ˆ ⊥ as
respect to the decompositions E|N = K ⊕ K ⊥ and F |N = K

/ s + DZ )ξ + sA0 τ ξ + O(r2 ∂ V + r∂ H + r + sr3 )τ ξ.
(D + sA)τ ξ = τ (D

Here




0 
Aα
/ s := eα
D
.
· ∂α + syα 
0 ∗(z)
The same construction holds for the dual operator Ds∗ : Γ(F |N ) → Γ(E|N ) with the
ˆ Z respectively.
/ ∗s and D
normal and horizontal operators denoted as D
Proposition 3.2.4 shows the rates of each of the horizontal and vertical derivatives
along Z as s >> 0. In particular, for large s, sections (ξ1 , ξ2 ) : N → K ⊕ K ⊥ satisfying
 
ξ1 
Ds   = 0
ξ2
/ ∩ ker DZ . We
are well-approximated by sections ξ = (ξ1 , 0) of the bundle K in ker D
will construct approximate solutions of Ds ξ = 0 by finding solutions to the first order
approximation
/ s + DZ )ξ1 = 0
(D
where ξ1 : N → π ∗ (K|Z ). Solving the first order approximation is the next main topic.
67

Chapter 4
Constructing approximate solutions

In this chapter we explicitly describe solutions to

/ s + DZ )ξ = 0
(D

where ξ : N → π ∗ (K|Z ). By Proposition 4.2.1 this amounts to finding solutions of the
system
/ sξ = 0
D

and

DZ ξ = 0.

Freezing z ∈ Z the first equation can be solved in sections ξz : Nz → Kz , since we have
only derivatives in the normal directions of Z. The family of solutions spaces over z ∈ Z
form the so called bundle of vertical solutions and can be constructed using the bundle
K → Z introduced in Definition 1.0.7. An analogue bundle is constructed in the dual case
ˆ → Z introduced in the same definition. Then DZ restricts to the sections
/ ∗s using K
of D
of those solution bundles to give Dirac type operator there. The second equation can be
interpreted as the kernel of that operator.

68

4.1

The bundle of vertical solutions

Recall the normal coframe {eα } on N |U introduced in Section 3.1and the terms Aα =
∇eα A|K in the Taylor expansion of A around Z. For this section we will be using w
coordinates moving in the whole fiber of the normal bundle N . One can think of those
coordinates as the blown up coordinates w =

√
sy when s = ∞ (compare with Remark

1.0.5).
ˆ
Fix z ∈ U . The Euclidean Clifford action eα
· : Kz → Kz induce Euclidean Dirac
operators

/ 0 ξ = (eα
by D
· ∂α )ξ

(4.1.1)

/ = (eα
by Dξ
· ∂α )ξ + wα Aα ξ.

(4.1.2)

ˆ z ),
/ 0 : Γ(Nz , Kz ) → Γ(Nz , K
D

and

ˆ z ),
/ : Γ(Nz , Kz ) → Γ(Nz , K
D

The purpose of this section is to count the dimension of decaying solutions ξ : Nz → Kz
of the equation

/ =0
Dξ

and show that they form a bundle as z varies in Z.

69

(4.1.3)

We start by noticing that {Aα } is a family of invertible matrices satisfying the relations

β

β

e· Aα + A∗α e· = 0,

A∗α Aα = A∗β Aβ , ∀a, b and A∗α Aβ + A∗β Aα = 0, a = b. (4.1.4)

The first of these is obtained by differentiating the concentration condition (1.0.1), while
the other two are due to Proposition 3.1.2.
Lemma 4.1.1. Under the relations (4.1.4) the set of matrices

{Mα = −eα
· ∇eα A|K } ⊆ End(Kz )

constitutes of commuting invertible self-adjoint endomorphisms satisfying

β
β α
eα
· e· Mβ = Mβ e· e·

and

Mα2 = Mβ2

(4.1.5)

for every α, β. Each Mα has symmetric spectrum and opposite eigenvalues have the same
multiplicity. Furthermore for every string I of even length eI· satisfies

eI· ∈ Hom(Kα± , Kα± )

if

α∈I

and

eI· ∈ Hom(Kα± , Kα∓ ) if

α ∈ I. (4.1.6)

where Kα± denote the eigenspaces of Mα corresponding to the ±µα - eigenvalues respectively.
Proof. The first and second of relations (4.1.4) directly imply the first and second relations
of (4.1.5) respectively and show that Mα is self-adjoint. Finally the third of relations

70

(4.1.4) implies that {Ma } is a commuting family. Also

β

γ β γ

γ

γ β

β

α
α
α
eα
· e· Mγ = e· e· e· e· Mγ = Mγ e· e· e· e· = Mγ e· e·

for α, β = γ. This imply the relations

Mα eI· = eI· Mα

showing (4.1.6).

if α ∈ I

and

eI· Mα = −Mα eI·

if α ∈ I

✷

Remark 4.1.2. 1) Relations (4.1.4) show that {eα
· ϕ, Ab ϕ}α,β is a family of orthogonal
vectors when ϕ ∈ Kz is non trivial.
2) Using (4.1.5), and the commutativity of {Mα }

wβ Aβ ξ, wα Aα ψ

β
wα wβ e· Mβ ξ, eα
· Mα ψ = −

=

α,β

α,β

β

wα wβ Mα eα
· e· Mβ ξ, ψ
α,β

β
wα wβ e· eα
· Mα Mβ ξ, ψ +

= −

2 M 2 ξ, ψ
wα
α
α

α=β
β
wα wβ eα
· e· [Mα , Mβ ]ξ, ψ +

=

2 M 2 ξ, ψ
wα
α
α

α<β

for every w ∈ Nz and every ξ, ψ ∈ K|Z . In particular, if the Mα commute then

2 M 2 ξ.
(wα Aα )∗ (wβ Aβ )ξ = wα
α

Let

(4.1.7)

Ki be the decomposition of Kz into the common eigenspaces of the family

{Mα }. Hence Ki = Kα+ when viewed as eigenspace where Mα has a positive eigenvalue
71

or Ki = Kα− otherwise.
Since Mα2 = Mβ2 the eigenvalues µαi of Mα on Ki are equal in absolute value to a
common number µi . Fixing now a summand Ki let tri be the normalized trace over
End(Ki ). We have the following:
Lemma 4.1.3 (Exponential decay estimates). Set

wα wβ eα
· (∇eβ A)|K ∈ End(Kz ),

Mw = −
α,β

and let ξ : Nz → Ki be a C 2 section, decaying at infinity and satisfying (4.1.3). Then
there exist a constant M0 > 0 such that
1

|ξ(w)| ≤ M0 e 2 tri (Mw ) .

Proof. Let ∆ denote the analyst’s Laplacian on Nz

Rn−m and ∇ the Euclidean gradi-

/ ∗ to equation 4.1.3 and using Remark 4.1.2 (2), we obtain
ent. Applying D

0 =

/ ∗ Dξ,
/ ξ = D
/ ∗0 D
/ 0 ξ, ξ + eα· Aα ξ, ξ + |wα Aα ξ|2
D

=

∇∗ ∇ξ, ξ −

=

∇∗ ∇ξ,

2 |M ξ|2
wα
α

Mα ξ, ξ +
α

α
2 µ2
(wα
αi

ξ +

− µαi )|ξ|2 .

α

Therefore

∆|ξ|2 = 2|∇ξ|2 − 2

∇∗ ∇ξ, ξ = 2|∇ξ|2 + 2

2 µ2 − µ )|ξ|2 .
(wα
αi
αi
α

72

(4.1.8)

Combining with the identity

∆|ξ|2 =

∂j (2|ξ|∂j |ξ|) = 2|ξ|∆|ξ| + 2|∇|ξ||2
j

and Kato’s inequality |∇|ξ|| ≤ |∇ξ| gives

2 µ2 − µ )|ξ|2 .
(wα
αi
αi

|ξ|∆|ξ| ≥
α

Fix now > 0 and define

1

F : Nz → R, F (w) = M e 2 tri Mw ,

where M is to be defined. Assuming {σl } is an orthonormal base for Ki and using
Remark 4.1.2 (1) we calculate

tri Mw = −wα wβ

2
eα
· Aβ σl , σl = −wα

eα
· Aα σl , σl

(4.1.9)

2µ .
= −wα
αi

Therefore
2 µ2 − µ )F
(wα
αi
αi

∆F =
α

and the difference satisfies:

|ξ|∆(F − |ξ|) ≤

α

2 µ2 − µ )|ξ|(F − |ξ|).
(wα
αi
αi

73

(4.1.10)

Now set

|µαi |)1/2

R = µ1 (
i

1

and M = ( + sup |ξ|)e 2 µi R

2

|w|=R

α

When |w| = R
2 (|µ | − µ ) ≥ 0
tri (Mw ) + µi R2 ≥ wα
αi
αi

so (F −|ξ|) |w|=R ≥ > 0 and when |w| > R the term

2 2
α (wα µαi −µαi ) is strictly positive.

Set V := {w : |w| > R, |ξ(w)| > F (w)}, an open set that satisfies the properties:
(i) V¯ ∩ {|w| = R} = ∅
(ii) |ξ| > 0 on V¯ . Enlarging slightly V we may assume that ∂V is smooth, properties (i)
and (ii) are satisfied, and furthermore
(iii) (F − |ξ|) ∂V > 0.
Suppose that V = ∅. If F − |ξ| had a negative minimum w0 ∈ V then, by (4.1.10) and the
maximum principle, F (w0 ) > |ξ(w0 )| which is a contradiction. By Property (iii) above
and the assumption that ξ decays at infinity, the function F −|ξ| cannot attend a negative
minimum in the boundary of V or at infinity, if V is unbounded, therefore V = ∅ and
1

|ξ(w)| ≤ M e 2 tri Mw for every > 0 and |w| > R and the result follows.

✷

Denote by Kz is the sum of those Ki ’s where Mα |Ki = µi IdKi for every α. This will
essentially be the space of L1,2 solutions of the equation (4.1.3):

74

Theorem 4.1.4. Suppose that X = Rn with coordinates {wα } and Aα is a collection of
matrices satisfying (4.1.4). All L1,2 solutions of the equation

(eα
· ∂α + wα Aα )ξ = 0

(4.1.11)

α

are linear combinations of sections of the form

1

2

ξi (w) = e− 2 µi |w| ϕ

where ϕ : Nz → Ki ⊂ Kz is a constant section.
Proof. Given ξ ∈ Γ(Nz , Kz ) using (4.1.7) we compute

∗
/ ∗ Dξ
/ = D
/ ∗0 D
/ 0 ξ + eα
D
· Aα ξ + wα wβ Aα Aβ ξ

/ ∗0 D
/ 0ξ −
= D

2 M 2 ξ.
wα
α

Mα ξ +
α

(4.1.12)

α

Now when ξ is a L1,2 solution of (4.1.11) then by regularity ξ is a C 2 decaying solution
and equivalently ξ evaluates the above presentation to 0. Since Mα are all simultaneously
diagonalizable we can decompose ξ =

i ξi

according to the decomposition Kz =

Ki .

Because of linear interdependency ξi : Nz → Ki will be an L1,2 solution of (4.1.11) for
each i and we can assume that ξ = ξi for some i.
1

Now recall F (w) = e 2 tri (Mw ) : Nz → R and note that by calculation (4.1.9) F satisfies

75

the system ∂α F + µαi wα F = 0 for every α. Replacing ξ by F ξ we calculate

α
eα
· ∂α (F ξ) + wα e· Mα (F ξ)

/ ξ) =
D(F

/ 0ξ
∂α F + wα µαi F eα
· ξ + FD

=
α

α

/ 0 ξ.
= FD

Hence if ξ : Nz → Ki obey (4.1.11)

/ = D(F
/ F −1 ξ) = F D
/ 0 (F −1 ξ)
0 = Dξ

/ 0 (F −1 ξ) = 0. But by Lemma 4.1.3 the section F −1 ξ is bounded and harmonic on
i.e. D
the entire Nz in the Euclidean sense hence F −1 ξ = ϕ ∈ Ki is a constant vector and so
ξ = F ϕ : Nz → Ki where ϕ now is viewed as a constant section. This section belongs in
L2 iff ϕ = 0 or if Ki ⊂ Kz . In the later case tri (Mw ) = −µi |w|2 .

✷

Proposition 4.1.5. The spaces {Kz : z ∈ Z} are independent of the choice of {eα }, and
they form a subbundle of K over Z.
Proof. Let {eα } be a second orthonormal frame of Nz with {Mα } the corresponding
family of commuting self-adjoint matrices:
Claim: When all {Ma } have same eigenvalue ±µi on Ki then all {Mα } have the same
eigenvalue on Ki .
Suppose eα = dαβ eβ for some orthogonal matrix d and let σ ∈ Ki so that Ma σ = µi σ

76

for every α. Then dαβ dγβ = δαγ and therefore

β γ

β γ

d2αβ σ − µi

Mα σ = −dαβ dαγ e· e· Mγ σ = µi
β

dαβ dαγ e· e· σ = µi σ
β=γ

for every Mα . This proves the claim.
Suppose now p ∈ Z and that Ki ⊂ Kp , so that the {Mα } have common eigenvalue µi
on Ki . To construct local bundles recall chart 3.1.1 with U ⊂ Z a geodesic ball of center p
with orthonormal frame {eα } of Np . This induce the family {Mα } ⊂ End(Kp ) and {σk }
of Kp so that the set partition to orthonormal frames of each Ki . We extend both frames
to moving frames over U centered at p and we extend the family {Mα } accordingly by
means of {eα }. Let γ(t) be a geodesic of U with γ(0) = p and γ(0)
˙
∈ Tp Z. Fix a parallel
local section σ so that σ(p) ∈ Ki . By the last relation of (3.1.5)

α
α
2
∇E
A)σ = 0
˙ Aα )σ = −e· (∇γ(t),α
γ(t)
˙ (Mα σ) = −e· (∇γ(t)
˙

i.e. Mα σ is also parallel hence a linear combination of {σk }. Since at the origin Mα σ =
µi σ this equality holds in U proving that Mα has constant spectrum over U with {σk }
being the eigenvectors. Hence each Ki ⊂ Kp extend over U providing a local bundle of
solutions. But by the previous claim in the intersections of the various U ’s each local
version Ki is independent of the frame {eα } used and the various local versions of the Ki
patch together to give a global bundle Ki over Z. Hence also K is a well defined bundle
of solutions along Z, and a subbundle of K|Z .

✷

ˆ = ker A∗ and we define similarly M
ˆ α = −eα A∗ = eα Mα eα ∈
In the dual case K
· α
·
·
77

ˆ Then the family {M
ˆ α } satisfy the same relations as the family {Mα } and give
End(K).
ˆ of solutions of D
/ ∗.
rise to a well defined bundle K
ˆ is a Z2 -graded Spinc bundle over Z.
Proposition 4.1.6. K ⊕ K
ˆ ± to be the ±µα -eigenspaces of M
ˆ α for every α and
Proof. To start we define similarly K
α
notice that the frame {eA }, A = i, α acts analogously to relations to (4.1.6)

ˆ α eI
eI· Mα = −M
·

if α ∈ I

and

ˆ α eI
eI· Mα = M
·

if α ∈ I

(4.1.13)

for every string I of odd length implying that Mα has the same spectrum and multiplicities
ˆ α and furthermore
as M

ˆ ± ) if α ∈ I
eI· ∈ Hom(Kα± , K
α

and

ˆ ∓ ) if α ∈ I.
eI· ∈ Hom(Kα± , K
α

ˆ ⊂ (K ⊕ K)|
ˆ Z is preserved by the Clifford action of the tangent coframe
In particular K⊕ K
{ei } to Z thus being a Z2 - graded Spinc -bundle over Z.

✷

Remark 4.1.7. We end up this paragraph with a remark. The non - degeneracy assumption (1.0.9) can be weaken for the proof of the Spectral Separation Theorem. It is
included for making the construction of the bundle of solutions simpler. In general one has
to examine the various normal vanishing rates of the eigenvalues of A∗ A. In particular
the bundles of solutions examined here will have a layer structure corresponding to those
rates and possibly will have a jumping locus in their dimension.

78

4.2

ˆZ
The operators DZ and D

Recall the Spinc bundle
π ∗ (E ⊕ F )|Z → N
¯ introduced in Section 2.2. In view of Lemma 3.1.1, this connection
with connection ∇
ˆ Z → N . Accordingly, the restrictions
restricts to a connection of the bundle π ∗ (K ⊕ K)|

ˆ z ),
/ : Γ(Nz , Kz ) → Γ(Nz , K
D

∀z ∈ Z

and
ˆ Z ))
DZ : Γ(N, π ∗ (K|Z )) → Γ(N, π ∗ (K|
ˆ Z and satisfy the following:
/ ∗ and D
are well-defined with adjoints D
Proposition 4.2.1. For every section ξ : N → π ∗ (K|Z ), we have

ˆ Z D)ξ
/ ∗ DZ + D
/ = 0,
(D

and therefore
/ + DZ )ξ 22 = Dξ
/ 22 + DZ ξ 22 .
(D

Proof. Recall the chart (3.1.1) centered at p with tangent frame {∂A }, A = i, α so that
¯ i σk = ∂i + ω 0 . Then by
∂A |U = eA and the parallel frame {σk } on E|N so that ∇
ik
U
α
∗ α
∗
(2.2.2) ei· ∇X
i e· = (d + d )e· = ((d + d )dyα |U )· = 0. Moreover the last relation of (3.1.5)

79

evaluated at p gives ∇i Aα = 0. Hence if ξ = f σk at p

∗ i¯
/ ∗ DZ ξ = (eα
D
· ∂α + wα Aα )e· ∇i ξ
X α
i
¯ i (eα
= −ei· ∇
· (∂α f )σk + e· (∇i e )· (∂α f )σk

¯ i (Aα f σk ) + wα ei· (∇i Aα )f σk
− wα ei· ∇
ˆ Z Dξ
/
= −D

Since p was arbitrary this holds everywhere. The last identity follows since the cross terms
are zero by the first.

✷

¯ is really a restriction of ∇ to sections of the above bundles and that K
Note that ∇
ˆ can be viewed as subbundles. Recall the construction of the bundles Ki , Kˆi from
and K
¯ preserves them inducing in that way a connection
Proposition 4.1.5. It follows that ∇
ˆ i and hence a well defined connection on their direct sum bundle
on sections of Ki ⊕ K
ˆ → Z. This connection will not in general be compatible with the Spinc structure
K⊕K
ˆ unless the second fundamental form of the embedding Z → X is trivial.
on K ⊕ K
Definition 4.2.2. The restriction operator

¯ i : Γ(Z, K) → Γ(Z, K)
ˆ
DZ = ei· ∇

is well-defined Dirac operator.
Hence solutions ξ : N → π ∗ (K|Z ) of the equation

/ s + DZ )ξ = 0
(D
80

are explicitly given in terms of the distance r from Z by

1

2

e− 2 sµi r ϕi

ξ=

(4.2.1)

i

where µi > 0 and where ϕi ∈ Γ(Z, Ki ) satisfies DZ ϕi = 0. In Chapter 5 we use these
Gaussian sections (see Definition 5.1.2) to construct a space of approximate solutions of
the equation
Ds ξ = 0.

81

Chapter 5
Approximate eigenvectors

5.1

Low-High separation of the spectrum

Our main goal for this chapter is to prove the Spectrum Separation Theorem stated in
the introduction. For that purpose we will use the bundles K introduced in Chapter
1 and define a space of approximate solutions to the equation Ds ξ = 0. The space of
approximate solutions is linearly isomorphic to a certain “thickening” of ker Ds by “low”
eigenspaces of Ds∗ Ds for large s. The same result will apply to ker Ds∗ . The “thickening”
will occur by a phenomenon of separation of the spectrum of Ds∗ Ds into low and high
eigenvalues for large s. The following lemma makes this idea precise:
Lemma 5.1.1. Let L : H → H be a densely defined closed operator with between the
Hilbert spaces H, H so that L∗ L has descrete spectrum. Denote Eµ the µ- eigenspace of
L∗ L. Suppose V is an k- dimensional subspace of H so that

|Lv|2 ≤ C1 |v|2 , ∀v ∈ V

and

|Lw|2 ≥ C2 |w|2 , ∀w ∈ V ⊥ .

Then there exist consecutive eigenvalues µ1 , µ2 of L∗ L so that µ1 ≤ C1 , µ2 ≥ C2 and if

82

in addition 4C1 < C2 , the orthogonal projection

Eµ → V

P :
µ≤µ1

is an isomorphism.
Proof. Let µ1 be the k-th eigenvalue of the self-adjoint operator L∗ L with counted multiplicity and µ2 be the next eigenvalue. Denote by Gk (H) the set of k- dimensional
subspaces of H and set W = ⊕µ≤µ1 Eµ , also k-dimensional. By the Rayleigh quotients
we have
µ2 =

inf

max
S∈Gk (H)

v∈S ⊥ ,|v|=1

|Lv|2 ≥

inf

v∈V ⊥ ,|v|=1

|Lv|2 ≥ C2 .

and also
µ1 =

max

inf

v∈S ⊥ ,|v|=1

S∈Gk−1 (H)

|Lv|2 .

But for any k − 1-dimensional subspace S ⊂ H there exist a unit vS ∈ S ⊥ ∩ V so

µ1 ≤

max
S∈Gk−1 (H)

|LvS |2 ≤ C1 .

Finally, given w ∈ W write w = v0 + v1 with v0 = P (w) and v1 ∈ V ⊥ . Then

C2 |w − P (w)|2 = C2 |v1 |2 ≤ |Lv1 |2 ≤ 2(|Lw|2 + |Lv0 |2 ) ≤ 2(µ1 + C1 )|w|2 ≤ 4C1 |w|2

C

and so |idW − P |2 ≤ 4 C1 . If additionally 4C1 < C2 and P (w) = 0 for some w = 0 then
2
|w|2 = |w − P (w)|2 ≤ |idW − P |2 |w|2 < |w|2
83

a contradiction. Hence P is injective and by dimension count an isomorphism.

✷

We have to construct an appropriate space V that will be viewed as the space of
approximate solutions to the problem Lξ = Ds ξ = 0. It is enough to describe the
construction for a fixed m - dimensional component Z = Z . For this purpose we introduce
a cutoff function ρ near Z.
Let N = N2 be the tubular neighborhood of radius 2 around Z. The exponential map
N → exp(N ), is a radial isometry along the vertical directions and the distance function
r from Z. Let ρˆ : [0, ∞) → [0, 1] be a smooth cut off with ρˆ−1 ({0}) = [1, ∞), ρˆ−1 ({1}) =
[0, 21 ] and strictly decreasing in [ 12 , 1] with |ˆ
ρ | ≤ 3 and define

ρˆ(

r(p)

)

p = exp(z, v) ∈ exp(N )

ρ(p) =
p ∈ X\ exp(N ).

0

Let K be the bundle over Z as in Definition 1.0.7; K is the direct sum of common
eigenspaces Ki of the family {Mα = −eα
· Aα } of positive common eigenvalues µi . As in
Definition 4.2.2, the Dirac operator D acts on sections of K along each component Z
of ZA .
Definition 5.1.2. For each component Z of ZA , set
1

2

V s = span ρ · e− 2 sµi r τ ϕi

ϕi ∈ Γ(Z , Ki ), µi > 0, DZ ϕi = 0 ,

where τ is the parallel transport map defined by (3.2.2). Taking the direct sum over all

84

components of ZA we construct the space of approximate solutions along ZA

Vs =

V s ⊂ L1,2 (X, E).

and denote by Vs⊥ the closure of its L2 -perpendicular in the L1,2 -norm:
⊥ 2
Vs⊥ := Vs L ∩ L1,2 (X, E).

Notice that VZs is L2 -perpendicular to VZs for Z1 = Z2 since their corresponding
1
2
sections have disjoint supports. There are completely analogous constructions of the
spaces VˆZs , Vˆs for Ds∗ and L1,2 (X, F ) = Vˆs ⊕ Vˆs⊥ .
Theorem 5.1.3. There exist an s0 > 0 and constants Ci = Ci (s0 ) > 0, i = 1, 2 so that
when s > s0
(a) For every η ∈ Vs ,

C
Ds η 22 ≤ 1 η 22 .
s

(5.1.1)

Ds η 22 ≥ C2 η 22 .

(5.1.2)

(b) For every η ∈ Vs⊥ ,

The same estimates hold for the L2 -adjoint operator Ds∗ .
In proving estimate (5.1.1) we will use the following growth rates of Lemma 5.1.4. The
85

proof of estimate (5.1.2) will be given in the next section.
Lemma 5.1.4. For every k > 0 there exist a constant C > 0 depending on the eigenvalues
{µi } of the family {Mα } so that for every ρ · ξ approximate solution with ξ 2 = 1

|y|2k (|ξ|2 + |∂ H ξ|2 )dydz ≤ Cs−k

|y|2k |∂ V ξ|2 dydz ≤ Cs1−k

and

N

N

Furthermore there exist an s0 = s0 ( ) so that

|ρ(z, y) · ξ(z, y)|2 dydz ≥
N

1
2

for every s > s0 uniformly on ξ 2 = 1. Here r = |y| is the distance from the singular
set ZA .
Proof. We write
1

n−m

2

e− 2 sµi r ϕi

ξ(z, y) = s 4

i

with DZ ϕi = 0 and µi > 0 for all i. Denote by Vn−m the measure of the n − m − 1 dimensional unit sphere. Then there exist C > 0 with

n−m

2

|y|2k |ξ|2 dydz = s 2
N

i

N

|y|2k e−sµi |y| |ϕi |2 dydz
∞

= s−k Vn−m

2

rn−m+2k−1 e−µi r dr

0

i

Z
∞

≤ Cs−k Vn−m
i

= Cs−k .

86

0

2

rn−m−1 e−µi r dr
Z

|ϕi |2 dz

|ϕi |2 dz

and
∞

|y|2k |∂ V ξ|2 dydz = s1−k Vn−m
N

0

i

2

µ2i r2k+n−m+1 e−µi r dr

∞

≤ Cs1−k Vn−m

2

rn−m−1 e−µi r dr
Z

0

i

Z

|ϕi |2 dz

|ϕi |2 dz

= Cs1−k .

Elliptic regularity gives
|∂ H ϕi |2 dz ≤ C1

Z

Z

|ϕi |2 dz

for every i. Using 2ab ≤ a2 + b2

|y|2k |∂ H ξ|2 dydz

≤

s−k V

∞
n−m

N

r2k+n−m−1 e−

µi +µj
2
2 r dr

0

i,j

Z
∞

≤ Cs−k Vn−m

2

rn−m−1 e−µi r dr

0

i

Z

∂ϕi , ∂ϕj dz

|ϕi |2 dz

= Cs−k .

For the last part we change to w-variables and estimate

|ρ(z, y)ξ(z, y)|2 dy =
N

√ ρ
N ( s)

w
2( √

s

)|ξ|2 dw ≥

√

s

= Vn−m
i

0

2
s |ξ| dw
N( 2 )
√

2

rn−m−1 e−µi r dr
Z

|ϕi |2 dz

√
√
where N ( s) denote the tubular neighborhood in the normal bundle of radius
s. But
√

there exist s0 = s0 ( ) so that 0

s n−m−1 −µ r2
r
e i dr

and every s > s0 . The result follows.

✷
87

2
> 21 0∞ rn−m−1 e−µi r dr for every i

Proof of estimate (5.1.1) in Theorem 5.1.3. Recall the tubular neighborhood N of Z, the
chart (NU , {zi , yα }) described at (3.1.1), the frames {∂i |U = ei }, {∂α |U = eα }, the
ˆ = ker A∗ from Chapter 2.
parallel transport map τ and the bundles K = ker A and K
Choose η = ρ·ξ ∈ Vs with ξ 2 = 1. With respect the decompositions E|N = K ⊕K ⊥
ˆ ⊕K
ˆ⊥
and F |N = K


η =ρ·ξ =

n−m
1
2
ρ · s 4 e− 2 sµi r τ ϕi

0



,

DZ ϕi = 0.

The Taylor expansion from Corollary 3.2.4 gives

1

2

(D + sA)η = dρ· ξ + ρ · e− 2 sµi r τ DZ ϕi

 

− 12 sµi r2
0  e
ϕi 
Aα
+ ρ · τ eα
 

· ∂α + syα 
0 ∗(z)
0
+ ρ · O r2 ∂ V + r∂ H + r + sr3 ξ
= dρ· ξ + ρ · O r2 ∂ V + r∂ H + r + sr3 ξ.

(5.1.3)

√

Because dρ has support outside the

s
2 -neighborhood

of ZA , the L2 norm of the first

term on the right hand side is bounded as

N

|dρ· τ ξ|2 dydz ≤ C
i

∞
2
√ r n−m−1 e−µi r dr
s
2

≤

C
.
s

Also, by Lemma 5.1.4 the squared L2 - norm of the error term on the right hand side of
(5.1.3) is bounded by Cs . Furthermore, there is an s0 > 0, independent of that choice of

88

η, so that η 22 > 21 for every s > s0 . This proves the result.

✷

Applying Lemma 5.1.1 we get a proof of Spectrum Separation Theorem stated in the
Introduction:
Proof of Spectral Separation Theorem. Choose s0 > 0 so that the constants of Theorem
C

5.1.3 satisfy 4 s1 < C2 for every s > s0 . Then apply Lemma 5.1.1 for L = Ds with
H = L2 (X, E), H = L2 (X, F ) and Vs constructed above. But by construction

VZs

ˆ )}
ker{DZ : Γ(Z , K ) → Γ(Z , K

VˆZs

ˆ Z : Γ(Z , K
ˆ ) → Γ(Z , K )}
ker{D

for every . This completes the proof.

✷

Remark 5.1.5. Combining Theorem 5.1.3 with the proof of Lemma 5.1.1 we actually
get the stronger statement that if ξ 2 = 1 and Ds ξ = 0 then
4C1
→ 0 as s → ∞.
ξ − P (ξ) 2 ≤
sC2

5.2

A Poincar´
e-type inequality

This section is entirely devoted to the proof of estimate (5.1.2) of Theorem 5.1.3. Recall
the tubular neighborhood N and the chart (3.1.1). N is a Riemannian manifold with

89

two equivalent metrics: g X and g Z × g N . These induce two different densities |dvol|
and dzdy which, under the exponential map, are related by |dvol| = kdzdy for some
map k : N → R+ . Henceforth we will be using the density dzdy and the constants
of equivalence will be suppressed in the calculations. Note that when w =

√

sy then

n−m

dzdw = s 2 dzdy.
For the proof of estimate (5.1.2) Theorem 5.1.3 we need the following lemma:
Lemma 5.2.1. If estimate (5.1.2) is true for η ∈ Vs⊥ supported in N = N (2 ), a tubular
neighborhood of Z , then it is true for every η ∈ Vs⊥ .
Proof. Let ρ4 : X → [0, 1] a bump function supported in B(Z , 2 ) with ρ4 ≡ 1 in B(Z , ).
Write η = ρ4 η + (1 − ρ4 )η = η1 + η2 with supp η1 ⊂ B(Z , 2 ) and supp η2 ⊂ X\B(Z , ).
Then

Ds η 2 = Ds η1 2 + Ds η2 2 + 2 Ds η1 , Ds η2 .

(5.2.1)

Since ρ4 · ρ = ρ we have η1 ∈ Vs⊥ and by assumption there exist C0 = C0 ( ) > 0 and
s0 = s0 ( ) > 0 so that
Ds η1 2 2 ≥ C0 η1 2 2
L
L
for every s > s0 . Also since η2 is supported away of ZA , by Proposition 1.0.2

Ds η2 2 ≥ s2 Aη2 2 − s| η2 , BA η2 | ≥ (s2 k 2 2 − sM ) η2 2 .

90

To estimate the cross terms we calculate

Ds η1 = ρ4 Ds η + (dρ4 )· η,

Ds η2 = (1 − ρ4 )Ds η − (dρ4 )· η

and hence

Ds η1 , Ds η2

2

=
X

ρ4 (1 − ρ4 ) Ds η dvg +
2

−
X

(dρ4 )· η dvg ≥ −

X

(1 − 2ρ4 ) (dρ4 )· η, Ds η dvg

1
3
2
Ds η 2 −
(dρ4 )· η dvg
2
2 X

for every s, > 0. We used that |ab| ≤ 21 (a2 + b2 ) and that (1 − 2ρ4 )2 ≤ 1. But (dρ4 )· η
is supported in X B(Z, ) hence by Proposition 1.0.2 applied again

2

X

(dρ4 )· η dvg ≤ C

C
1
2
|η dvg ≤ 2 2 2
Ds η 22 ≤ Ds η 22
3
s k
− sM
B(Z , )c

for s large enough. Hence
Ds η1 , Ds η2 ≥ − Ds η 22 .
Substituting to (5.2.1) and absorbing the first term in the left hand side there is an
s1 = s1 ( ) with

3 Ds η 2 ≥

Ds η1 2 + (s2 k 2 2 − sM ) η2 2

≥ C0 ( η1 2 2 + η2 2 2 )
L
L
≥ C0 η 2

91

✷

for every s ≥ s1 .

Since L2 -norms are additive on sections with disjoint supports it is clear that we can
work with η ∈ Vs⊥ so that supp η ⊂ N for some individual tubular neighborhood of some
individual singular component Z = Z.
Proof of estimate (5.1.2) in Theorem 5.1.3. This is a Poincar´e type inequality and we
prove it by contradiction. Suppose there exist s sequence {sj } → ∞ of positive numbers
sj

with no accumulation point and a sequence {ηj } ⊂ L1,2 (N , E|N ) so that ηj ∈ VZ has
ηj 2 = 1 and Dsj ηj 22 → 0 as sj → ∞.
Recall the tubular neighborhood N = B(Z, 2 ) of Z, the chart (NU , {zi , yα }) in
(3.1.1). In this chart we have the frames {∂i |U = ei } and {∂α |U = eα }, the parallel
ˆ ⊕K
ˆ ⊥ )|Z
transport map τ , and the decompositions E|Z = (K ⊕ K ⊥ )|Z and F |Z = (K
from Chapter 2. Introduce
m−n

τ ξj (z, w) = sj 4 ηj

for T > 0 in w =

√

√ws

j

and ξjT (z, w) = ξj (z, w)γ(z, w) with γ(z, w) = ρ Tw

T + ξ T where
sj y coordinates. We have the decomposition ξjT = ξj1
j2

T : B(Z, T ) → π ∗ (K| ) and ξ T : B(Z, T ) → π ∗ (K ⊥ | )
ξj1
Z
Z
j2

both supported on B(Z, T ).

92

Now in w-coordinates

T
/ s ξjT )(z, y) = (eα
(D
· ∂yα + sj yα ∇α A)ξj (z, y)
j

=
=

√

T
sj (eα
· ∂wα + wα ∇α A)ξj (z, w)

√

/ jT )(z, w).
sj (Dξ

Hence in w-coordinates Corollary 3.2.4 shows that

1
2 V
H
T
Z T
T
T ) + τ (√s D
sj A0 (ξj2
j / + D )ξj = Dsj (ξj ) + √ O |w| ∂ + |w|∂ + |w| ξj (5.2.2)
sj
for every |w| < T . The L2 norm of the left hand side is

T ) + (√s D
Z T 2
|sj A0 (ξj2
j / + D )ξj | dwdz =

+

|(sj A0 +

√

T |2 dwdz
/ + DZ )ξj2
sj D

√ /
T |2 dwdz.
|( sj D
+ DZ )ξj1

(5.2.3)

But by the concentration condition (1.0.1) for A0 there is a C1 so that

√ /
√ /
T |2 ≤ s C |ξ T |2 .
|( sj D
+ DZ )∗ A0 + A∗0 ( sj D
+ DZ )ξj2
j 1 j2

T |2 ≥ C |ξ T |2 . Hence there exist C > 0 so that
Also there exist C2 > 0 so that |A0 ξj2
2 j2
3

for all large sj
√ /
Z ))ξ T |2 dwdz ≥ s2 C
|(sj A0 + ( sj D+D
j2
j 3

93

T |2 dwdx +
|ξj2

√ /
Z )ξ T |2 dwdz.
|( sj D+D
j2

Substituting back to (5.2.3)

2
T ) + (√s D
Z T 2
|sj A0 (ξj2
j / + D )ξj | dwdz ≥ sj C3

+

T |2 dwdz
|ξj2

(5.2.4)

√ /
|( sj D
+ DZ )ξjT |2 dwdz.

/ + DZ the L2 norm of the error term of (5.2.2) is bounded by
By ellipticity of D

O |w|2 ∂ V + |w|∂ H + |w| ξjT 22 ≤ CT

/ + DZ )ξjT 22 + ξjT 22 .
(D

Therefore, taking L2 norms of (5.2.2), substituting (5.2.4), using Proposition 4.2.1 and
absorbing terms in the left hand side, one obtains

s2j C3

B(Z,T )

T |2 dwdz + (s − CT ) Dξ
/ jT 22 + (1 −
|ξj2
j
sj
2

≤

T2 X

CT
sj

) DZ ξjT 22

(5.2.5)

|dρ· ξj |2 dzdw + Dsj ηj 22 .

T
/ jT 2 → 0, and
By assumption, the right hand side is bounded in j, hence ξj2
2 → 0, Dξ

the sequence

/ + DZ )ξjT 22
(D

(5.2.6)

/ + DZ , the sequence
is uniformly bounded in j. By elliptic regularity for the operator D
{ξjT } ⊂ L1,2 is bounded. By Rellich Theorem there is a subsequence, denoted again as
{ξjT }, that converges to ξ T : B(Z, T ) → π ∗ (K|Z ),where ξ T is a compactly supported
section with ξ T 2 ≤ 5 ξi 2 = 5 . By the weak compactness of the unit ball in L1,2 ,
94

we can also assume that ξiT → ξ T weakly in L1,2 . In this context, (5.2.5) shows that for
every smooth section ψ

DZ ξ T , ψ dwdz =
B(Z,T )

ˆ Z ψ dwdz = lim
ξT , D
i

B(Z,T )

= lim
i

≤

B(Z,T )

B(Z,T )

ˆ Z ψ dwdz
ξiT , D

DZ ξiT , ψ dwdz ≤ lim DZ ξiT 2 ψ 2
i

C
ψ 2.
T

Therefore ξ T satisfies

C
DZ ξ T 2 ≤
,
T

/ T =0
Dξ

and

ξT 2 ≤ 5

for every T > 0. Now notice that when T > T then ξ T agrees with ξ T in B(Z, T2 ). Hence
there is a well-defined section ξ : N → π ∗ (K|Z ) with ξ = ξ T on every neighborhood
B(Z, T2 ) ⊂ N . Furthermore, by Lemma 5.2.2 below, there is an estimate

/ + DZ )ξ T 2 + ξ T 2 ) ≤ C
ξ T 1,2 ≤ C( (D

C
+5
T

(5.2.7)

where the constant C is independent of T . Letting T → ∞ we see that ξ is in fact an
L1,2 -section satisfying

/ = 0 and DZ ξ = 0.
Dξ

(5.2.8)

Claim : ξ ≡ 0
s

By assumption ηi ⊥ VZ i . Using Definition 5.1.2 this condition translates in w =
95

√
si y

coordinates as
n−m

1

1

2
2
ηi ⊥ ρ · si 4 e− 2 si µ|y| τ ψ ⇐⇒ ξi ⊥ ρsi · e− 2 µ|w| ψ

for every section ψ : Z → K with DZ ψ = 0.
The sequence ξiT is supported in B(Z, T ) and since ξi 2 = 1
1

B(Z,T )

2

ρsi · e− 2 µ|w| ξiT , ψ dwdz =

B(Z,T )

=
B(Z,T )

1

2

1

2

ρsi · e− 2 µ|w| ξiT − ξi , ψ dwdz
ρsi · e− 2 µ|w| (γ − 1)ξi , ψ dwdz
1

≤

B(Z,T /2)c

2

e− 2 µ|w| |ψ|2 dzdw

for every i, T . Hence passing to the L2 limit as i → ∞

1

1

2

e− 2 µ|w| ξ T , ψ dwdz ≤
B(Z,T )

B(Z,T /2)c
1

2

e− 2 µ|w| |ψ|2 dzdw.

2

Letting T → ∞ we see that ξ is L2 orthogonal to e− 2 µ|w| ψ. But ξ is in L1,2 and satisfies
(5.2.8) therefore, by (4.2.1) ξ ≡ 0. This proves the claim.
It now follows that for every T > 0, as i → ∞

lim

T
i→∞ B(Z, √
2 si )

|ηi |2 dydz = lim

i→∞ B(Z,T /2)

|ξi |2 dwdz =

|ξ|2 dwdz = 0.
B(Z,T /2)

Finally, we obtain a contradiction from the concentration estimate. By the nondegeneracy
assumption (1.0.9) the bundle map A satisfies A∗ A|K = |y|2 M + O(|y|3 ). Hence by the

96

proof of Theorem 1.0.4 we estimate

Dsi ηi 22 ≥ s2i

T )c
B(Z, 2√
si

|A(ηi )|2 dydz − C1 si

T )c
B(Z, 2√
si

|ηi |2 dydz

C
≥ si ( 0 T 2 − C 1 )
|ηi |2 dydz
T
c
4
√
B(Z,
)
2 s
C
|ηi |2 dydz .
= si ( 0 T 2 − C 1 ) 1 −
T
4
√
B(Z,
)
2 si

But then for a fixed T > 0 large enough, as i → ∞ we have lim inf i Dsi ηi 2 = ∞
contrary to our original assumption that Dsi ηi 2 → 0.

✷

The following Lemma was used above to obtain estimate (5.2.7).
Lemma 5.2.2. For any compactly supported section ξ : B(Z, T ) → π ∗ (K|Z ), there is an
elliptic estimate
/ + DZ )ξ 2 + ξ 2 )
ξ 1,2 ≤ C( (D
for some constant C > 0 independent of T .
Proof. Recall the calculation (4.1.12)

/ ∗ Dξ
/ =−
D

∂α2 ξ −
α

2 M 2ξ
wα
α

Mα ξ +
α

97

α

where Mα are self-adjoint. Taking L2 inner products with ξ, we obtain

∂α ξ 22 −

/ 22 =
Dξ

α

α

∂α ξ 22 −

≥

(5.2.9)

α

Mα ξ, ξ dwdz
α

α

≥

2 |M ξ|2 dwdz
wα
α

Mα ξ, ξ dwdz +

∂α ξ

2
2

− C1 ξ 22

α

with C1 independent of T . Also, there is a Weitzenbock identity for DZ : Γ(Z, K|Z ) →
ˆ Z ) of the form
Γ(Z, K|
ˆ Z DZ ξ = ∇
¯ ∗ ∇ξ
¯ + Rξ
D
for some curvature term R. Again taking L2 inner products with ξ, there is a constant
C2 > 0 with
(|DZ ξ|2 + |ξ|2 )dz.

¯ 2 dz ≤ C
|∇ξ|
Z

Z

When ξ is instead, a section supported in B(Z, T ) we integrate this inequality with respect
to the w-variable to get

∂ H ξ 2 ≤ C( DZ ξ 2 + ξ 2 )

Combining (5.2.9) and (5.2.10) we get the result.

98

✷

(5.2.10)

Chapter 6
Nonlinear concentration

6.1

The nonlinear model

There is a simple way to “nonlinearize” concentrating Dirac operators. Suppose

Ds = D + sAψ : Γ(W ) → Γ(W )

is a concentrating Dirac operator determined by ψ ∈ Γ(L). Suppose that W decomposes
as a sum ⊕j Wj of bundles, one of which, say W1 = L. Then we can convert the linear
PDE
Dξ + sAψ ξ = ξ0

ξ = (ξ1 , . . . , ξk )

to a non-linear equation simply by taking ψ = ξ1 . The resulting equation

Dξ + sAξ ξ = ξ0

is quadratic in ξ. It is interesting to study the concentrating properties of the solutions
to this problem.
For example, let (X, g, W + ⊕W − , c) a 2n-dimensional Spinc Riemannian manifold with
99

determinant bundle L = detC (W + ). A fixed Hermitian connection A0 on L determines
a Dirac operator D0 acting on spinors. Fixing additionally ψ ∈ Γ(W + ), special instance
of the Clliford pairs example is

Dξ +

1
A ξ = ξ0
2 ψ

(6.1.1)

 
ϕ
where ξ =   ∈ Γ(W + ⊕ Λodd X) and ξ0 ∈ Γ(W − ⊕ Λev X). Since ψ, ϕ ∈ Γ(W + ) we
α
can take ψ = ϕ to get a nonlinear example.

6.2

Examples

In examples 8-10 below, we take W1 to be a spin bundle and W2 to be a bundle whose
sections are connections, as in the linear examples 4 and 5 of Chapter 2. However, we
ev/odd

restrict the operators Pψ

to the real subbundle Λev/odd X ⊗ iR of Λ∗C X and we treat

the term α of ξ as a connection 1-form. Introduce a new connection A of L satisfying

A − A0 = α,

FA − FA0 = dα

1
DA − DA0 = α·
2

and

for α ∈ Ω1 (X, iR).
ev/odd

Finally it worth to be noted than when we restrict Pψ


 0
W + → End(E + ⊕ E − ) : ψ → 
Aψ
100

to Λev/odd X the map



A∗ψ 
0



defines a real Clifford representation of W + to E.

Example 8: Let (Σ, ω, J) be a closed Riemman surface with canonical bundle KΣ and a
−1 ⊗L
holomorphic line bundle L giving a Spinc structure on Σ. Then W + = L, W − = KΣ


0


and DA0 = ∂¯A0 . For ξ0 = 
 equation (6.1.1) is rewritten as
ir ω − F
A0
2

1
∂¯A0 ψ + α· ψ = 0
2
1
ir
(d + d∗ )α − Pψev∗ ψ − ω + FA0 = 0.
2
2

But for every ρ ∈ R
Pψev∗ ψ, (ρ, ω) = ρ|ψ|2 − i|ψ|2
since ω· ψ = iψ. So Pψev∗ ψ = |ψ|2 −i|ψ|2 ω and the projection to imaginary part is −i|ψ|2 ω.
Therefore the above equations are rewritten as

∂¯A ψ = 0
i
FA = − (|ψ|2 − r)ω
2
d∗ α = 0.

The first two equations are the r-vortex equations and the third one is a Coulomb condition
defining a slice of the U (1)- gauge action on Γ(u(L)). Vortex equations have been studied
thoroughly over C by C. Taubes (see [JT]) and over closed Riemann Surfaces and more
general line bundles over closed K¨ahler manifolds by O. Garcia-Prada (see [O]) and S.

101

Bradlow (see [B]).

Example 9: On a symplectic 4-manifold (X, ω, J), adopting the definitions of Example


0


4 of the linear case for D and Aψ , using as ξ0 = 
 for r ∈ R, equation
√
+
ir
√
− ω − 2FA
2

0

(6.1.1) is rewritten as

DA0 ψ +

1
α· ψ = 0
2

√
√
1
ir
( 2d+ + d∗ )α − Pψev∗ ψ + √ ω + 2FA+ = 0.
0
2
2
However a simple calculation shows that Pψev∗ ψ =

√

¯ therefore
2τ (ψ ⊗ ψ)

DA ψ = 0
FA+ =

(6.2.1)

1
¯ − irω)
(τ (ψ ⊗ ψ)
2

d∗ α = 0.

The first two are the perturbed symplectic Seiberg-Witten equations and the last one is
defining a slice of the U (1)- gauge action on Γ(u(L)) as in the previous example. If (X, g)


0


we get the
is a 4-manifold with out a symplectic structure then for ξ0 = 
√ +
− 2FA
0
unperturbed Seiberg-Witten equations.

Example 10: The following example is the next interesting perturbation on a non
K¨ahler complex surface with positive geometric genus. This case has been studied by
2,0

O. Biquard (see [OL]). If (X, J, g) is a complex surface with dim HC (X) > 0 then
102

−1 of X and take
we can choose a holomorphic section b of the anticanonical bundle KX


0


ξ0 = 
. Equation (6.1.1) will then correspond to the equations
√
+
r
¯
√ (b + b) − 2F
A
2

0

DA ψ = 0
FA+ =

1
¯ + r (b + ¯b)
τ (ψ ⊗ ψ)
2
2

d∗ α = 0.

Remark:(L2 -concentration for the symplectic SW equations.) The perturbation irω by
the symplectic form was introduced by Witten for K¨ahler manifolds, then studied in detail
by Taubes, who showed solutions localize as r → ∞. We include this remark with the
L2 -nonlinear concentration (see also [D]). For the subtle pointwise estimates see [T2] and
for the subtle behavior of the concentrating set see [T1] and [T2] This perturbation is
different from the perturbation sAψ in the concentration Theorem 1.0.4.
In the following calculation we basically use a parameter δ to interpolate between the
Weitzenbock formulas of nonlinear Examples 7 and 8. This enables one to analyze the
nonlinear concentration for the pair of equations (6.2.1).
z1 ∧ d¯
z2 }, we denote by (FA )ω the part
With respect to an orthonormal basis {1, 12 d¯
0,2

of FA parallel to ω, FA = (FA )0,2 d¯
z1 ∧ d¯
z2 , β = 21 (β)d¯
z1 ∧ d¯
z2 and one can write
√
¯ Then
ψ = r(a, (β)) on W + = L ⊕ LK.

ψ ⊗ ψ¯ −

1
2
 2 (|a|

1 2
|ψ| Id = 
2

− |β|2 )

a
¯(β)

103

¯
a(β)





1 (|β|2 − |a|2 )
2

and


)2,0



−2i(FA )ω −4(FA

c(FA+ ) = 
.
0,2
4(FA )
2i(FA )ω
Equations (6.2.1) are then rewritten as

i(FA )ω

=

0,2

=

FA

r
(2 − |a|2 + |β|2 )
4
r
a
¯β.
2

If ∇A and ∂¯A are the connection with it’s (0,1) - part on L → X, then DA =

(6.2.2)
(6.2.3)
√

∗)
2(∂¯A , ∂¯A

is the Dirac operator on W + and we have the identities:

2a =
∂¯A
∗ ∂¯ a =
∂¯A
A
∗β
∂¯A ∂¯A

=

c1 (L)[ω]

=

1 0,2
1
FA a − ∂A a ◦ NJ
2
4
1 ∗
(∇ ∇ a − i(FA )ω a)
2 A A
1 ∗
1
∇A ∇A β + i(R + FA )ω β
2
2
i
i
1
(FA )ω ω ∧ ω =
(F )ω dvg
2π X 2
2π X A

(6.2.4)
(6.2.5)
(6.2.6)
(6.2.7)

¯ . Observe that the difference in the i signs of formulas
where R denotes the curvature of K
¯ with eigenvalues
(6.2.5) and (6.2.6) comes from the fact that ω· has eigenspaces L and LK
−2i and 2i respectively. Finally we compute:





∗
∗
∗
∗
¯
¯
¯
¯
¯
∂ ∂ a + ∂A ∂A β 
1 ∗
∂A 
∗ β) =  A A
0 = DA
DA ψ =   (∂¯A a + ∂¯A

.
2
2
∗
¯
¯
¯
¯
∂A
∂A a + ∂A ∂A β

(6.2.8)

Taking L2 - product with a on the first row element of (6.2.8) and using (6.2.4) and
104

(6.2.5), for 0 < δ < 1:

∗ ∂¯ a, a + ∂¯∗ ∂¯∗ β, a = (1 − δ) ∂¯ a 2 + δ ∇ a 2
∂¯A
A
A
A A
2 A
iδ
2a
−
(F )ω |a|2 dvg + β, ∂¯A
2 X A
iδ
δ
(FA )ω dvg +
= (1 − δ) ∂¯A a 2 +
∇A a 2 − iδ
(FA )ω (2 − |a|2 )dvg
2
2 X
X
1
1
0,2
β, FA a − β, ∂A a ◦ NJ .
+
2
4

0

=

Using now equations (6.2.2), (6.2.3) and (6.2.7) and after a few calculations:

0

rδ
r
rδ
δ
(2 − |a|2 )2 dvg
∇A a 2 +
β 2 + (2 − δ) aβ 2 +
(1 − δ) ∂¯A a 2 +
2
4
8
8 X
1
− 2πδc1 (L)[ω] − β, ∂A a ◦ NJ .
(6.2.9)
4
=

Similarly taking L2 - product with β on the second row element of (6.2.8) and using
(6.2.4) and (6.2.6):

0

2 a, β + ∂¯ ∂¯∗ β, β = 1 F 0,2 a, β − 1 ∂ a ◦ N , β + 1 ∇ β 2
∂¯A
J
A A
2 A
4 A
2 A
i
(FA )ω |β|2 dvg .
+ i
Rω |β|2 dvg +
2
X
X

=

Using again equations (6.2.2) and (6.2.3) we conclude:

1
r
r
r
∇A β 2 + β 2 +
aβ 2 +
|β|4 dvg + i
Rω |β|2 dvg
2
4
8
8 X
X
1
−
∂ a ◦ NJ , β .
(6.2.10)
4 A

0 =

105

Finally, adding up (6.2.9) and (6.2.10), the Weitzenbock formula from (6.2.8) will read:

(1 − δ) ∂¯A a 2

δ
1
r
∇A a 2 + ∇A β 2 + (1 + δ) β 2
2
2
4
r
r
+ (3 − δ) aβ 2 +
(|β|4 + δ(2 − |a|2 )2 )dvg
8
8 X
1
Rω |β|2 dvg (6.2.11)
β, ∂A a ◦ NJ − i
= 2πδc1 (L)[ω] +
2
X
+

for every 0 < δ < 1. On the other hand, there exist C = C(X, ω, J) > 0 such that

1
2

β, ∂A a ◦ NJ

C
δ
β 2 +
∂A a 2 + C β 2
δ
2
X
C
δ
= (1 + δ) β 2 +
∂ a 2.
δ
2 A

− i

Rω |β|2 dvg ≤

Substituting to (6.2.11) and absorbing the terms to left hand side we get

(1 − δ) ∂¯A a 2

r C
r
1
∇A β 2 + (1 + δ)( − ) β 2 + (3 − δ) aβ 2
2
4
δ
8
r
≤
(|β|4 + δ(2 − |a|2 )2 )dvg + 2πδc1 (L)[ω].
8 X
+

For r large and δ = 5C
r , the terms in the left hand side are all positive and we get the
desired concentration when c1 (L)[ω] > 0 i.e. L → X is a nontrivial line bundle and the
zero set of it’s section a is nonempty. From this we see that as r → ∞

β L1,2 → 0

∂¯A α L2 → 0

106

2 − |α|2 L2 → 0.

APPENDIX

107

Manifold structures on the space of Fredholm operators
Let E, E be separable Hilbert spaces over the field K = R or C. Let also B := B(E, E )
be the set of bounded linear operators, F := F(E, E ) the space of Fredholm operators
and Fi := Fi (E, E ) the space of Fredholm operators of index i. Then Fi are the open
connected components of the set F, an open subset of the set B. Then Fi =

k
k≥0 Fi

where Fik are Fredholm operators with k - dimensional kernel and (k − i) - dimensional
cokernel (k ≥ i).
Proposition .0.1. Each Fik is a k(k − i) - codimensional submanifold of Fi and for
D ∈ Fik
TD Fik = {P ∈ B : P (ker D) ⊆ ImD}.
Proof. Let E = ker D ⊕ ImD∗ , E = ker D∗ ⊕ ImD where D∗ is the corresponding adjoint
of D. Then with respect to this decomposition D and an arbitrary D will be written as




0 0 
D=

0 d



α β 
and D = 

γ δ

with d ∈ F00 (ImD∗ , ImD). We would like to parametrize Fik near D. To do that suppose
there exist k1 : ker D → (ker D)⊥ = ImD∗ with adjoint k1∗ and k2 : coker D = ker D∗ →
ImD = (ker D∗ )⊥ with adjoint k2∗ . Then

{(x, k1 (x)) : x ∈ ker D} = ker D

{(y, k2 (y)) : y ∈ ker D∗ } = ker(D )∗

and
108

for some operator D ∈ Fik since given a pair of different finite dimensional subspaces
we can always construct operator having them as kernel and cokernel, a consequence of
Hahn -Banach Theorem. We also get the extra set theoretic relations

{(−k1∗ (x), x)) : x ∈ (ker D)⊥ } = (ker D )⊥

and {(−k2∗ (y), y)) : y ∈ ImD} = ImD .

Since D : (ker D )⊥ → ImD will be an isomorphism, for every y ∈ ImD there exist
unique x ∈ (ker D)⊥ with

 
 
∗
∗
α β  −k1 (x) −k2 (y)


=
·
y
γ δ
x


so we get

−α ◦ k1∗ (x) + β(x) = −k2∗ (y)

(.0.1)

−γ ◦ k1∗ (x) + δ(x) = y

(.0.2)

and furthermore −γ ◦ k1∗ + δ : (ker D)⊥ → ImD has to be isomorphism. Eliminating y
from (.0.1) and (.0.2) we get

−α ◦ k1∗ + β = k2∗ ◦ γ ◦ k1∗ − k2∗ ◦ δ

109

(.0.3)

on (ker D)⊥ . On the other hand, for every x ∈ ker D


 



α β   x 

·
=0
γ δ
k1 (x)
so

α + β ◦ k1 = 0

(.0.4)

γ + δ ◦ k1 = 0.

(.0.5)

Substituting γ from (.0.5) into (.0.2) we get

−γ ◦ k1∗ + δ = δ ◦ (k1 ◦ k1∗ + id)

(.0.6)

Furthermore substituting α and γ from (.0.4) and (.0.5) respectively to (.0.3) we get

β ◦ (k1 ◦ k1∗ + id) = −k2∗ ◦ δ ◦ (k1 ◦ k1∗ + id).

(.0.7)

However since E and E are separable and k1 ◦ k1∗ is selfadjoint, by spectral theory,
k1 ◦ k1∗ + id : (ker D)⊥ → (ker D)⊥ is an isomorphism. Therefore we get that
• using (.0.6) δ : (ker D)⊥ → ImD is an isomorphism
• using (.0.7) β = −k2∗ ◦ δ

110

Finally

D





∗
k2



◦ δ ◦ k1 −k2 ◦ δ 
α β 
= 
=

γ δ
−δ ◦ k1
δ

 
 

∗
∗
−k2 k2  δ 0 −k1 id
= 
·
·
.
id
0
0 0
0
0

If D is close to D in the operator topology then we can see that ker D and ImD
can be described uniquely as graphs via some k1 and k2 respectively giving the above
decomposition for D in terms of D. Conversely given
• k1 ∈ B(ker D, (ker D)⊥ ) ⊆ Gk (E)
• k2 ∈ B((ImD)⊥ , ImD) ⊆ Gk−i (E )
• δ ∈ F00 ((ker D)⊥ , ImD)
we can form the map Φ(k1 , k2 , δ) = D as described in the above decomposition with the
resulting D ∈ Fik . This will be the required chart of the space Fik (E, E ) around D.
Using this chart it is easy to verify the last description of the tangent space TD Fik .
Remark .0.2.

✷

1. For k = 1 a complement of the space TD F01 is the family of spaces

C = {P ∈ B(E, E ) : P (ker D) = C}
CD

for C any 1 - dimensional complement of ImD in E . If D ∈ F11 then TD F11 = B so
we have trivial complement in this case.
111

2. In a sufficiently small neighborhood U ⊂ F(E, E ) of D, using the decompositions
E = ker D ⊕ ImD∗ and E = ker D∗ ⊕ ImD, the map




A B 
−1
Fk : U → Hom(ker D, ker D∗ ) : D = 
→A−B◦∆ ◦Γ
Γ ∆
satisfies the relation Fk−1 ({0}) = U ∩ Fik giving a local model structure of a subvariety to Fik ⊂ F(E, E ). Actually the various {Fk : k ≥ 1} give to
structure of a stratified subvariety of B(E, E ).

112

k
k≥1 Fi

the

BIBLIOGRAPHY

113

BIBLIOGRAPHY
[B] S. Bradlow, Vortices in holomorphic line bundles over closed K¨ahler manifolds, Comm.
Math. Phys. 135 (1990), 1-17.
[D] D. Kotschick, The Seiberg-Witten invariants of symplectic four-manifolds [after C.H.
Taubes], Seminaire Bourbaki. 48eme annee no 812 (1995-96), 195-220.
[IP] E.N. Ionel and T.H. Parker, The Gromov invariants of Ruan-Tian and Taubes, Math.
Res. Lett. 4(1997), 521-532.
[JT] A. Jaffe and C. H. Taubes, Vortices and Monopoles: Structure of static gauge theories
(Progress in Physics - 2) Birkh¨auser, Boston, Mass (1980).
[K] U. Koschorke, Infinite dimensional K-theory and characteristic classes of Fredholm
bundle maps, in Proceedings of the Symposia in Pure Mathematics, Vol. XV, AMS,
Providence, R.I. 1970.
[LM] H. B. Lawson and M. L. Michelsohn, Spin Geometry, (1989) Princeton University
Press.
[LP1] J. Lee and T.H. Parker, A structure theorem for the Gromov-Witten invariants of
Kahler surfaces J. Differential Geom. 77 (2007), no. 3, 483 - 513
[LP2] J. Lee and T.H. Parker, An obstruction bundle relating Gromov-Witten invariants
of curves and K¨ahler surfaces, American J. of Math. 134 (2012), 453-506.
[OL] O. Biquard, Les ´equations de Seiberg-Witten sur une surface complexe non
K¨ahl´erienne Comm. Anal. and Geom. 6 (1998), 173-196.
[O] O. Garcia-Prada, A direct existence proof for the vortex equations over a compact
Riemann surface, Bull. London Math. Soc. 26 (1994) 88-96.
[Pal] R. Palais, Foundations of global nonlinear analysis, Benjamin, New York, 1968.

114

[PR] I. Prokhorenkov and K. Richardson, Perturbations of Dirac operators, J. Geom.
Phys. 57 (2006), no. 1, pp. 297-321.
[R] D. Rauch, Perturbations of the ∂¯ operator, Thesis, Harvard, January 2004.
[T1] C. H. Taubes, Counting pseudo-holomorphic curves in dimension four, J. Diff. Geom.
44(1996), 818-893.
[T2] C. H. Taubes, SW ⇒ Gr: From the Seiberg-Witten equations to pseudo-holomorphic
curves, J. Amer. Math. Soc. 9 (1996), no. 3, 845-918.
[W1] E. Witten, Supersymmetry and Morse theory, J. Diff. Geom. 17 (1982), 661-692.
[W2] E. Witten, Monopoles and 4- manifolds, Math. Res. Letters 1 (1994), 769-796.

115