The Blowup Formula for Higher Rank Donaldson
Invariants
by
Lucas Howard Culler
Submitted to the Department of Mathematics
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2014
NAASSAr'HUSETTS INSiiE
STECHNOLOGY
@Lucas Culler 2014. All rights reserved.
JUN 1 201
LBRARIES
Signature redacted
Author.......
Department of Mathematics
May 1, 2014
<Signature redacted
Certified by.
Tomasz Mrowka
Singer Professor of Mathematics
Thesis Supervisor
Signature redacted
Accepted by...
.......................
Alexei Borodin
Chairman, Department Committee on Graduate Theses
2
The Blowup Formula for Higher Rank Donaldson Invariants
by
Lucas Howard Culler
Submitted to the Department of Mathematics
on May 1, 2014, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Abstract
In this thesis, I study the relationship between the higher rank Donaldson invariants
of a smooth 4-manifold X and the invariants of its blowup X#CP2 . This relationship
can be expressed in terms of a formal power series in several variables, called the blowup
function. I compute the restriction of the blowup function to one of its variables, by solving a
special system of ordinary differential equations. I also compute the SU(3) blowup function
completely, and show that it is a theta function on a family of genus 2 hyperelliptic Jacobians.
Finally, I give a formal argument to explain the appearance of Abelian varieties and theta
functions in four dimensional topological field theories.
Thesis Supervisor: Tomasz Mrowka
Title: Singer Professor of Mathematics
The author was partially supported by NSF grant DMS-0943787.
3
4
Acknowledgments
First and foremost I would like to thank my advisor Tom Mrowka, for introducing me
to gauge theory, providing me with an excellent thesis problem, and patiently listening to
me as I blundered my way through it. I would also like to thank the other two members
of my thesis committee, Peter Kronheimer and Paul Seidel, for their useful input and for
organizing stimulating courses and seminars.
I have had an uncountable number of interesting and stimulating mathematical conversations with my peers during the last few years. My discussions on gauge theory and Floer
homology with Josh Batson, Aliakbar Daemi, and Jon Bloom were particularly useful and
relevant to my thesis, as were dicussions with Sam Raskin and Matt Wolfe on algebraic
geometry. Many other friends have influenced my view of mathematics over the years, especially Raju Krishnamoorthy, Philip Engel, Ailsa Keating, Tiankai Liu, Aaron Silberstein,
Dustin Clausen, and Peter Smillie. I should also point out that this thesis would have been
impossible without Dan Gardner, who initiated me in the mysteries of theta functions as an
undergraduate.
I owe special gratitude to Glenn Stevens and the PROMYS program, which introduced
me to mathematics as a high school student. This wonderful program taught me that there
was truth in the world, and that I could find it myself if I only knew how to look.
Finally, I would like to thank my parents, who have always supported me in every way,
and without whom I would be nothing at all.
5
6
Contents
9
1
Introduction
2
13
Higher Rank Invariants
. 13
2.1 ASD Connections ................................
13
2.1.1 The Configuration Space .........................
15
.
.
.
.
.
.
.
.
.
.
.
2.1.2 The Moduli Space . . . . . . . . . . . . . . . . .
Index Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Reducibles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
19
.
.
.
.
.
20
22
22
23
25
Cylindrical Ends and Gluing . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Cylindrical Ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
29
2.3.2
The Equivariant Universal Bundle . . . . . . . . . . . . . . . . . . . .
31
2.3.3
2.3.4
Index Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Gluing Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
34
Higher Rank Blowup Formulas
3.1 Solutions Near a Negative Sphere . . . . . . . . . . . . . . . . . . . . . . . .
37
37
Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
3.1.2 Completely Reducible Connections . . . . . . . . . . . . . . . . . . .
The Blowup Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
44
3.2.1
3.2.2
Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A Simple Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
46
3.2.3
3.2.4
Partial Blowup Functions . . . . . . . . . . . . . . . . . . . . . . . .
Full Blowup Functions . . . . . . . . . . . . . . . . . . . . . . . . . .
49
51
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
2.1.3
2.1.4
2.2
2.3
3
2.1.5 Perturbations . . .
2.1.6 Orientations . . . .
Donaldson Invariants . . .
2.2.1 Cohomology Classes
2.2.2 Invariants . . . . .
3.1.1
3.2
3.3
4
The Case N=3
. . . .
. . . .
. . . .
. . .
. . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
3.3.1
Theta Functions on a Genus 2 Jacobian
. . . . . . . . . . . . . . . .
54
3.3.2
The Blowup Function . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
59
The Formal Picture
4.1
Axiom s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
4.2
The Formal Blowup Formula . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
7
4.3
The Blowup Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
4.4
4.5
4.6
Geometry of the Blowup Algebra . . . . . . . . . . . . . . . . . . . . . . . .
Analytic Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Periods and Theta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
67
70
8
Chapter 1
Introduction
Donaldson's invariants of smooth 4-manifolds were originally defined using moduli spaces
of anti-self dual connections on principal bundles with structure group SU(2) or SO(3) =
PU(2). These invariants and their corresponding generating functions have many interesting
structural features which have been described in a mathematically rigorous way. For instance,
a paper of Fintushel and Stern [7] shows that the invariants of a 4-manifold X and its blowup
X = X#Cp2 are related in a predictable way, and that the relationship can be expressed in
terms of a theta function on the elliptic curve
3 +a2+X
y2 ==x+ax2+x
y2
over the ring Q[a]. More generally, the structure theorem of Kronheimer-Mrowka expresses
the Donaldson invariants of a 4-manifold of simple type in terms of exponential functions
(theta functions on a degenerate genus 1 curve), and there is a conjectural structure theorem
for 4-manifolds not of simple type, which expresses the invariants of such a manifold in terms
of Jacobi elliptic functions.
It is natural to ask if Donaldson invariants can be defined for higher rank bundles, and if so,
whether or not these structural features have some natural generalization. Indeed, physicists
have achieved results in this direction (see for example [3] and [12]), which indicate that the
blow-up formula and structure theorem for SU(N) invariants can be expressed in terms of
function theory on the hyperelliptic curve
EN: y=
(N
+ aN-1X + aN)2
+ a2N-2 +
(.)
over the ring
A = H*(BPU(N);
Q)
=
Q[a2 ,
. . , aN].
In particular it is shown in [3] that the blow-up function is a theta function on the Jacobian of this curve, generalizing the result of Fintushel and Stern. Unfortunately, their
derivation relies on the use of quantum field theory techniques, which do not appear to be
mathematically rigorous.
One route to overcoming these difficulties, pursued by Nakajima and Yoshioka in [14],
is through Nekrasov's partition function. Using the localization formula in equivariant cohomology, they rigorously define certain integrals which are analogous to the Donaldson
9
invariants of the plane blown up at a single point. They compute these integrals using techniques of algebraic geometry, and thereby arrive at a blow-up formula which agrees with the
one from the physics literature.
It is not clear, however, what either of the methods above has to do with moduli spaces
of instantons on general 4-manifolds. In this paper we take a more direct route. In
[8],
Peter Kronheimer has given a rigorous definition of PU(N) invariants for general N. Using
his definition we directly attempt to compute the blowup function, using the method of
Fintushel and Stern. For small values of N we compute the blowup function recursively, and
for N = 3 we explicitly identify it with a theta function on the Jacobian of the curve E3For general N we can partially compute the blowup function. More precisely, we restrict the
blowup function to one of its variables and characterize the resulting single variable function
by a system of ordinary differential equations. After a change of variables, this system of
equations reduces to the Toda chain, an integrable system well-known to physicists, whose
relation with the blowup formula was predicted in [12].
Theta functions also arise in Seiberg-Witten theory, where the analogous blowup formula
[9] is encoded by an infinite Laurent series over the formal power series ring Z[[U]]:
O(T) =
(-1)"Un T
nEZ
This formal sum can be viewed as a theta function on the Tate curve, a special family of
elliptic curves defined over a formal disk. Going further in this direction, a paper of Lekili
and Perutz [10] uses mirror symmetry to relate Heegaard Floer theory to the derived category
of the Tate curve. Their paper can be viewed either as an explanation or generalization of
the blowup formula, depending on one's point of view.
One might hope for a conceptual explanation of the appearance of theta functions in
topological field theory. We have taken some partial steps in this direction. Given a Floer
homology theory satisfying certain formal properties, we construct a graded ring A, which
in the case of Seiberg-Witten theory is the homogeneous coordinate ring of the Tate curve.
From A we build a projective variety A, and explain how to interpret the blowup function
as a section of a line bundle on A. In favorable circumstances, we can use the geometry of
A to give a formal proof that the blowup function is necessarily a theta function.
This thesis is divided into three chapters. The first chapter deals with general background
on Donaldson invariants, and the definition of higher rank invariants due to Kronheimer. It
mostly follows standard references on Donaldson invariants, specifically [4], [5], and [13], as
well as Kronheimer's paper [8] on higher rank invariants. The only original material in this
chapter is the definition of extra cohomology classes not considered by Kronheimer, which
were implicit in the physics literature on the blowup formula, but had not been defined
explicitly in the mathematical literature.
The second chapter contains a proof of the partial blowup formula for SU(N) invariants
and the complete blowup formula for SU(3) invariants. It begins with a general discussion
of instantons on a tubular neighborhood of an embedded sphere of negative self-intersection.
To verify transversality of the higher rank instanton moduli spaces on these manifolds we
produce special metrics with anti-self-dual Weyl curvature and positive scalar curvature. We
give a detailed analysis of reducible connections in these moduli spaces, and use some special
10
cases to derive certain relations that imply the existence of a blowup formula. In the case
N = 3 we use classical results from the function theory of genus two Jacobians to verify that
the blowup function is a theta function on the curve 1.1.
Finally, in the third chapter we axiomatically define a notion of formal Floer theory,
which assigns vector spaces to 3-manifolds and maps to suitably decorated cobordisms.
Such a theory has a tautologically defined blowup function, which spans the first graded
piece of the ring referred to above. Given some nondegeneracy conditions on the theory, we
show that the blowup function is entire and has a natural set of quasiperiods, such that the
automorphy factor corresponding to any nontrivial quasiperiod is the exponential of a linear
function. If the quasiperiods span a full lattice, we conclude that the formal blowup function
is a theta function on a complex torus.
11
12
Chapter 2
Higher Rank Invariants
2.1
ASD Connections
In this section we set up the basic analytic framework needed to study the moduli space
of anti-self-dual connections on a 4-manifold.
2.1.1
The Configuration Space
Let X be a compact, connected, Riemannian 4-manifold, let E be a rank N vector bundle
on X, and let P = P(E) be the associated principal PU(N) bundle. Fix a smooth connection
AO on P and an integer 1 > 2.
Definition 1. We define the configuration space of connections on P to be
A(P) = {Ao+ a I a E L2(X, A' 0 adP)}
Since the difference of any two smooth connections is a smooth section of A1 (X) 9 adP,
the definition does not depend on the choice of A 0 . To explain the condition 1 > 2, recall
that in dimension 4 there is a Sobolev embedding L+ -+ C0, for 0 < e < 1. Thus our
definition gaurantees that every element of A(P) is continuous.
Definition 2. We define the determinant one gauge group of P to be
2 1 (X, G(P))}
9(P) = { E L+
where G(P) = P x PU(N) SU(N) is the bundle of groups associated to the conjugation action
of PU(N) on SU(N).
To explain the use of the index 1 + 1 rather than 1, consider the explicit formula for the
action of 9(P) on A(P),
y - (Ao + a) = Ao + yd
0
~1 + ya--i.
(2.1)
From this formula it is clear that elements of 9(P) must have at least one more derivative
than do elements of A(P), in order for the action to extend over both completions.
13
Definition 3. We define the quotient configuration space to be B(P) = A(P)/g(P), with
the natural quotient topology.
We would like to say that B(P) is a smooth Banach manifold. However, this is not the
case, due to the existence of connections with nontrivial stabilizers.
Proposition 4. For any connection A E A(P), its stabilizer IPA is a finite dimensional
Lie group. In fact, evaluation at any point x E X defines an isomorphism from FA to the
centralizer of the holonomy group
HA = Holx(A) C G(P)x
consisting of holonomies around all piecewise smooth curves based at x.
Proof. Suppose that A is stabilized by 7 E g(P). Then we have:
A =7
- A = A +7dAY
(2.2)
Hence '-ydA-y = 0, so y is parallel when considered as a section of G(P). Since Z is
discrete, this implies that -y is a parallel section of G(P). Conversely, any parallel section of
G(P) defines a gauge transformation that stabilizes A.
Now, for connected X a parallel section -y is determined by its value at any point x E X,
so PA injects into G(P)x = PU(N). Moreover, a given -yo E G(P)x will extend to a parallel
section if and only if it returns to itself under parallel transport around any closed loop based
at x, which is equivalent to saying that it centralizes HA.
Definition 5. We say that a connection A E A(P) is reducible if its stabilizer rA is strictly
larger than Z. Otherwise we say that A is irreducible. We write A*(P) for the set of
irreducible connections, and we write B*(P) for the quotient A*(P)/9(P).
To determine the local structure of B(P) we need to construct equivariant tubular neighborhoods for every gauge orbit in A(P). The key input for this construction is the following
proposition:
Proposition 6. Let A E A(P), let dA : L1+1(X, ad(P)) -+ LI(X, A' 0 ad(P)) be the associated covariant derivative, and let d* be its formal adjoint. Then there is a direct sum
decomposition into L 2 -orthogonal subspaces:
L/(X, A' 0 ad(P)) = ker d* E im dA
Proof. By definition im dA and ker d* are L 2-orthogonal. To show that they span, we must
solve the elliptic equation
d* dA = d* a
for any given a E L,(X, A' 0 ad(P)). To produce a solution, consider the inner product
QA(,
V) = (dA4, dAV).
14
on L,(X, ad(P)). Modulo the finite dimensional kernel of dA the corresponding quadratic
form is nondegenerate and equivalent to the Li norm. Hence we can find E Li(X, ad(P))
such that
(a, dAp) = QA(, 4) = (dAd, dAO)
for all cG Li. In other words,
as desired.
that E L
is a weak solution to (2.1.1). By elliptic regularity we know
We therefore define
TA,E
=
{ A+a d*a=O,aL <E} .
The stabilizer FA acts on TA,,, so we can form a g(P)-equivariant vector bundle over the
orbit !(P) - A = g(P)/FA:
NA,r
=
9(P) XFA
TAE
NA,, provides a local model for A(P) in a neighborhood of 9(P) - A, and allows us to
determine the local structure of B(P).
Proposition 7. Let A be an arbitrary connection in A(P). Then for some e > 0 there is a
neighborhood of [A] in B(P) homeomorphic to TA,,/J'A.
Proof. There is an evident map a : NA,, -+ A(P), whose differential is invertible along the
zero section by virtue of Proposition 6. In fact, a is a diffeomorphism onto its image for E
sufficiently small, and its image is an open neighborhood of the orbit of A. The quotient
0
NA,,/9(P) is homeomorphic to TA,,/JFA, which proves the claim.
Corollary 8. B*(P) is a smooth Banach manifold.
Proof. If PA = Z then it acts trivially on TA,,, hence Corollary 7 produces topological
charts on B*(P). The fact that these charts overlap smoothly follows from the fact that
E
a: NA,E -+ A(P) is smooth for any A and E.
Note that most of the above discussion goes unchanged when X is a manifold with boundary. However, we must modify our construction of slices by adding suitable boundary conditions. A natural choice is:
TA = {A + aldga = 0, *alax = 0}.
To check that this is a slice we must solve
d* (a - dA() =0
subject to the boundary condition *dA Iox = 0. We can do this by simply reiterating
the argument of Proposition 6, and being careful about boundary conditions whenever we
integrate by parts.
2.1.2
The Moduli Space
Let X be a smooth oriented 4-manifold with a Riemannian metric. The hodge star * acts
on the bundle A2(X), and satsifies *2 = 1. Therefore we can write A 2 (X) = A+(X)E A-(X)
15
where * acts on the bundles A±(X) by ±1. In particular, any 2-form W on X can be expressed
uniquely as w+ + w-, where w± is a section of A'(X).
Definition 9. We say that w is anti-self-dual if *w = -w, or equivalently, if w+ = 0.
Now let P -+ X be a principal G-bundle, and let A be a connection on P. The curvature
of A is a section of A 2 (X) 0 adP, hence it can be written uniquely as
FA = FZ + Fwhere F+is a section of A'(X) 0 adP.
Definition 10. We say that A is anti-self-dual if *FA = -FA,
the anti-self-duality (ASD) equations:
or equivalently if it satisfies
F+= 0
Note that the ASD equations are invariant under gauge transformations, so each solution
gives rise to an infinite dimensional space of gauge equivalent solutions. In particular, antiself-duality can be viewed as a property of the gauge equivalence class [A] E 3(P).
Definition 11. We define the moduli space M(P) to be the set of all equivalence classes of
ASD connections [A] E B(P).
To find the dimension of M (P)we need to compute the linearization of the ASD equations.
Let A be an ASD connection, and let A + a be a nearby connection. Then we have
F
+A=FZ +dAa+(a A a)+=0,
where dia is the self-dual part of dA. Dropping quadratic terms and setting FZ = 0, we see
that the linearization is
d+a = 0.
Solutions of this equation are tangent vectors to the space of all ASD connections, which
sits inside in the configuration space A(P). By definition, the moduli space M(P) is the
image of this infinite dimensional space in B(P). To compute the tangent space to M(P)
we restrict our attention to the slice
daa = 0,
so as to select a single ASD connection from each gauge equivalence class. Combining the
two conditions, we can identify the tangent space of M(P) with the kernel of the Dirac
operator,
DA = d* + d+ : A1(X, ad(P)) - A0 (X, ad(P)) D A2 (X, ad(P)).
We will show in the next section that DA is an elliptic operator, and therefore has a finite
index. Modifying A amounts to a compact perturbation, and does not change the index.
Definition 12. We define the virtual dimension of M(P) to be the index of the Dirac
operator DA, for any A E M(P).
16
If DA is surjective, then the moduli space will be cut out transversely, and will therefore
be a smooth manifold of dimension equal to the index of DA, in which case the notions of
dimension and virtual dimension coincide. On the other hand, if DA has nontrivial cokernel,
then M (P) may fail to be a smooth manifold near [A]. The cokernel can be understood
precisely in terms of the "deformation complex",
0
> AO(X, ad(P))
^> A'(X, ad(P))
^ > A+(X, ad(P))
> 0,
(2.3)
which is genuinely a complex if A is anti-self-dual, due to the identity
d+ o dA = F+ = 0.
General theory of elliptic complexes shows that the cokernel of DA is the direct sum of the
two even dimensional cohomology groups Ho and HA. Thus there are two separate ways
that M(P) can fail to be regular, and we will eventually need to explain how to avoid or
deal with both kinds of failure.
2.1.3
Index Theory
We now compute the virtual dimension of ASD moduli spaces on compact manifolds,
using the Atiyah-Singer index theorem for Dirac operators.
Proposition 13. Let X be a compact Riemannian 4-manifold and let V -+ X be a unitary
complex vector bundle of rank r. If A is any unitary connection on V, then
DA = d* +d+ : Q(XV) -+C (XV) D Q+(XV)
is an elliptic operator, whose index (as a complex linear operator) is given by
ind(DA) = cI(V) 2 - 2c 2 (V) - r (X + 0-).
2
Here d* denotes the adjoint of the operator dA, and x, o are the Euler characteristicand
signature of X.
Proof. Ellipticity is a local condition, and only depends on the highest order part of the
operator, hence it suffices to show that the operator D = d* + d+ is elliptic, for V a trivial
®'
D
bundle with the trivial connection. To see this consider the operator D* = d E 2d* :
1
Q+(X) -+ Q (X). Composing the two operators,
D*D
= dd*+2d*
(
2
) d
= d*d*+*d*(l+*)d
= d*d*+*d*d
we get the Laplace operator, which has an invertible symbol. We conclude that D has an
invertible symbol as well, hence it is elliptic.
17
To calculate the index, we first treat the case where V is the trivial complex line bundle.
Modifying A does not change the index, so in this case ker D can be identified with the
harmonic 1-forms on X, and cokerD can be identified with the sum of the harmonic 0 forms
and the self-dual harmonic 2-forms. Hence
ind(D) = -1+
bi - b+-
X
(2.4)
2
In general, the Atiyah-Singer index theorem for Dirac operators allows us to express the
index of DA as
ind(DA) =
(2 + a)Ch(V)
-
(2.5)
where a is a degree 4 cohomology class that is independent of V, and
Ch(V) = r + c1(V) +
1
1c,(V) 2
2
-
c2 (V) + -
is the Chern character of V. Some explanation of the factor 2 + a is required. The reason
for the 2 is that D is a Dirac operator on a rank 2 Clifford module. The absence of a degree
2 cohomology class is due to the fact that this Clifford module is real. The exact nature of
a is not important to us, because we only need to compute its integral over X, which can
be done by comparing equations (2.4) and (2.5). Hence
ind(DA)
=
=-
(2 + a)(r + cc(V) +
-
fjc(V)2 -2c
(2c 2 (V)
-
2 (V)
c1(V) 2 ) [X]
1(V)
2
+ c2 (V))
+ ra
-
r(X + -)
as desired.
Corollary 14. Let E be a U(N) bundle on a compact 4-manifold X and let P be the
associated PU(N) bundle. Then the moduli space M (P) has virtual dimension
2
+
dim M(P) = 4Nc 2 (E) + (2 - 2N)c1(E) + (1 - N2)X
a
2
Proof. We apply Proposition 13 to the complexified adjoint bundle
zf(E) = C 0 ad(P).
Note that zl(E) G C = E 0 E*. Therefore,
2c 2 (s[(E)) - ci(s[(E))2
=
-2Ch 2 (E 0 E*)
-2NCh 2(E) - 2Chi(E)Chi(E*) - 2NCh 2 (E*)
=
4Nc2 (E)+ (2 - 2N)c1(E) 2 ,
=
which completes the proof.
El
18
2.1.4
Reducibles
Let E be a rank N vector bundle on a connected, oriented 4-manifold X. Let P = P(E)
be the corresponding PU(N) bundle. Observe that if A is any connection on E, we can
uniquely write it as
-1
A=A+ N
where A E A(P) is a PU(N) connection and 6 = tr(A) is the connection induced by A on
det(E). Keeping this in mind, we can fix a connection 6, and view elements of A(P) as
connections on E rather than on P.
The moduli space M(P) can fail to be smooth at a connection A if the stabilizer FA is
nontrivial. In particular, HO = ker dA is the Lie algebra of the stabilizer FA, so DA fails to
be surjective if FA has positive dimension. It is therefore worth understanding the stabilizers
in more detail.
Proposition 15. Let H C U(N) be any subgroup and let F C U(N) be its centralizer. Then
_j71 U(ni), and it acts on CN through a decomposition
F is isomorphic to a group of the form
n10
CN
i=1
Proof. By basic representation theory, the action of H on CN decomposes as a sum of
irreducible representations
r
r.
ni(g V
n =
CN
i=1
i=1
By Schur's lemma any y E End(CN) that commutes with H lies in the subspace
g
End(C"n) & jwi C End(CN).
Intersecting this with U(N) gives the desired decomposition F = j
1 U(ni).
E
Corollary 16. If A E A(P) is reducible, then FA has positive dimension.
Proof. Let H be the holonomy group of A and let F be its centralizer. Then FA = FnSU(N).
If there are two nontrivial factors U(ni) x U(n 2 ) c F, then F has dimension at least two. Since
SU(N) has codimension one, the intersection F n SU(N) has positive dimension. The only
other possibility is that F = U(N), but then FA = SU(N) still has positive dimension. E
In many cases it is not possible to rule out reducible connections. In particular, if E is
the trivial bundle then it always admits the product connection. However, there is a simple
condition we can impose on E to avoid reducibles.
Definition 17. We say that a class c E H 2 (X; Z) is coprime to N if there is a homology
class a E H 2 (X; Z) such that c(-) = 1 mod N.
19
Proposition 18. If E is a rank N bundle admitting a reducible ASD connection, then there
is an integral class c such that Nc - nci(E) is represented by an anti-self-dual 2-form.
Proof. Suppose A is a reducible connection on P(E) and let A be a lift to E with trace
6. Then there is a splitting E = El D E 2 compatible with A. Write A = A 1 E A 2 and let
trAj = 6J. Then we have
Since the curvature of A - -
1-j
= -51
N
ni
1
1
=J-2.
n2
is anti-self-dual, we have
NN1
= -F+
F+Al
ni J, = -Fj+
N
In particular the class Nc1(E 1 ) - nici(E) is represented by an anti-self-dual 2-form.
L
Corollary 19. If c1 (E) is coprime to N then the set of metrics for which there exists a
reducible ASD connection on E is a countable union of submanifolds of codimension bf (X).
Proof. If c1 (E) is coprime to N then evaluating on a shows that Nc - nc1 (E) is nonzero
for any integral class c. The corollary then follows from the fact that the space of metrics
for which a given nonzero cohomology class is anti-self-dual is a submanifold of codimension
L
b(X) in the space of metrics.
In particular, if b(X) > 1 then we do not find reducible connections in generic 1parameter families of metrics.
2.1.5
Perturbations
The second way that M (P) can fail to be smooth is if HA = cokerd is nonzero. To
rule out this possibility, we need to artificially perturb the ASD equations. Acceptable
perturbations take the form
FZ =V(A)
where V(A) is a section of the bundle A+ 0 ad(P), depending on the connection A. To be
gauge invariant, the perturbated equations must transform like the curvature under gauge
transformations:
V(-y - A) = -yV(A)-1.
Note that if q : S' -+ X is a loop based at a point xO, then the holonomy Holq(A) transforms
in this way:
Holq(y - A) = -yHol(A),y 1 .
However, Holq(A) lies in G(P)., rather than ad(P)x.. To remedy this, we choose a Gequivariant map q : G -+ g with the property that # inverts the exponential map in a
neighborhood of the identity. Then O(Holq(A)) is an element of ad(P)x, that transforms
appropriately under gauge transformations.
To produce a section V(A), we choose a map q : S' x B 4 -+ X, with the property that
each map q(t, -) is a diffeomorphism onto its image, and a section w of A+ whose support
20
is contained in the open set U = q(O, B 4 ). We then define
V,w(A) = w 0 #(Holq(A))
where Holq(A) is regarded as a section of ad(P) on U. Given a countable collection 7r
{(qi, wi)}ic of loops and anti-self-dual 2-forms, we can define a perturbation
=
V(A) = E Vq,,w,(A).
iEI
Definition 20. Let X be a compact 4-manifold, let E be a rank N bundle, and let V, be a
perturbation of the ASD equations on P = P(E). We define the perturbed moduli space to
be
Mr(P)={ [A]
B(P) IF++Vr(A) = 0}
Because we have chosen V,(A) to be gauge invariant, its derivative DV, vanishes on the
tangent space to any gauge orbit. Hence we obtain a deformation complex
0
-
Ll+ 1 (X, ad(P))
^> L (X A' 0 ad(P))
^'i> L j 1 (X, A+ 0 ad(P))
0-
-
(2.6)
whose cohomology groups we denote by H. . The operator d,
is defined to be
d7 = d+ + DVr(A),
the linearization of the perturbed ASD equations.
Definition 21. We say that A is a regular solution of the perturbed equations if H2,
=
0
Proposition 22. Suppose that A is regular and irreducible. Then in a neighborhood of A,
the moduli space M(P) is a smooth manifold whose tangent space is isomorphic to HA,.
Proof. In a neighborhood of A, every connection in the moduli space is equivalent to one in
the slice
TA,f = A + kerd*,
which is a chart for B(P). Thus it suffices to show that the set of ASD connections in this
slice is a smooth manifold. If A is regular then the map
d' : ker d* --+ L1+1(X, A+ 0 ad(P))
is surjective, hence by the implicit function theorem M(P) is a smooth manifold near A and
its tangent space is ker d,, n ker d* = HA,7.
The use of holonomy perturbations is justified by the fact that generic perturbations yield
regular moduli spaces.
Proposition 23. Let E be a rank N bundle over a compact Riemannian 4-manjifold X.
Then there exists a residual set of small perturbations ,r, such that every A e M,(P) is
regular.
21
The ASD moduli spaces M (P) are not compact in general, due to bubbling. However,
we do have the following compactness theorem for the perturbed equations
Proposition 24. Let P = P(E) and let [A,] be a sequence in M(P(E)). Then there are
points X1,...,Xk
X, a bundle F with c2 (F) = c 2 (E) - k, a point [A,] E M(P(F)), a
subsequence of the A.,
and gauge transformationsyn : P(E) -+ P(F)such that yn- A, -+ A,
, X,} in the LP topology.
on every compact K C X \ {x 1 ,...
For the proofs of Propositions 23 and 24, we refer to Kronheimer's paper [8].
2.1.6
Orientations
It is important for the definition of Donaldson invariants that we give the moduli space
of ASD connections an orientation.
Proposition 25. Let E be a rank N bundle and let P = P(E). Then the moduli space
M(P) is orientable.
Proof. We refer to Donaldson [6], but we outline his proof. Orientations are induced by
sections of the determinant line bundle of the family of Dirac operators DA over M(P).
This is the restriction of a universal determinant line bundle A over B(P). Therefore it
suffices to show that A is trivial, or equivalently that wi(A) pairs trivially with any loop in
B(E). Given a class [-y] E H 1 (B(E)), Donaldson writes down an explicit loop of connections
representing [7] and checks that its determinant line is trivial.
E
To define invariants in Z rather than Z/{t1}, one must single out a preferred trivialization
of A. To do this it suffices to trivialize the determinant line of any particular connection
A E A(E), and we briefly describe how to do this.
For a bundle with c2 (E) = 0 one has E = L e CN-1 for a unique complex line bundle L.
Choosing a connection A on L and taking A = A D 1 ... E 1, one sees that the adjoint bundle
ad(P) splits as a sum of N 2 -N complex line bundles and N - 1 trivial real line bundles.
Using the complex orientation of the complex line bundles, ordering the real line bundles,
and trivializing the determinant line of the operator d* +d+, one obtains a trivialization of the
determinant of DA. The result is clearly independent of the choice of A, hence only depends
on a homology orientation of X (which is equivalent to a trivialization of the determinant of
d* + d+). Reversing the homology orientation of X changes the orientation when N is even
but preserves it when N is odd.
For a bundle with c2 (E) = k, one chooses k points x 1, ... , Xk E X. Starting with a
connection of the form A D CN one can glue in a pointlike instanton at each xi, obtaining
a connection A. Trivializing the determinant of DA is equivalent (by excision, see [6]) to
orienting the moduli space of ASD SU(2) connections on S4 with c2 (E) = 1. This reduces
us to an arbitrary choice, which we make according to the conventions in [8].
2.2
Donaldson Invariants
In this section we define various cohomology classes on the quotient configuration space
B(X), and study when these classes can be localized on compact subsets of the moduli
22
space of ASD connections. In this section X will always be a compact, connected, oriented
4-manifold.
2.2.1
Cohomology Classes
We now begin to study the topology of the quotient configuration space B(P). Because
the concepts in this section are fairly general, we write G for SU(N) and Gad for PU(N).
We begin by constructing a principal G' bundle over X x B(P).
Definition 26. We define the universal bundle to be
P = P xg(P) A*(P)
where 9(P) acts from the left on A(P) by pullback and from the right on P by automorphisms.
Note that there is a natural quotient map P -+ X x B(P), whose fibers admit an action
of Gad.
bundle over B*(P) x X.
Proposition 27. P = P xg(p) A*(P) is a prinicipalG'
Proof. We only need to produce local trivializations of P. If A is a connection with slice
and q$: U x G' -+ PI is a local trivialization on X, then the map
TA,,
# x id: (U x Gad) x TA,, -* P
0
defines a trivialization near ([A], x).
The universal bundle allows us to construct an abundance of rational cohomology classes
on B*(P). In particular, its classifying map 3*(P) x X -+ BG induces a map on rational
cohomology
p* : H*(BG; Q)
> H*(B*(P); Q) 0 H*(X; Q)
and hence by duality a map on rational (co)homology
p : H*(BGd; Q) 0 H, (X; Q)
> H*(13(P);
)
In place of p(a 0 o-) we will usually write pa(o-). For the special case where o- is a point we
will write pa(pt) = a. This notation is unambiguous when X is connected.
Note that the classes pa(-) necessarily satisfy some relations, due to the fact that /,* is an
algebra homomorphism. In general, given a graded-commutative algebra A and a topological
space X there is an algebra A(X; A) that is universal among all pairs (A, P), where A is a
graded-commutative algebra and
p : A -+ H*(X; Q) ®A
is a grading-preserving homomorphism. We can construct A(X; A) explicitly, given generators and relations for the algebras A and H*(X; Q).
23
Proposition 28. If A is a polynomial algebra on generators a1 , ... ,an of even degree, and
H,(X) has basis -1,... ,-n,
then A(X; A) is a free graded-commutative algebra on the ele-
ments Mai(c-j)Proof. Let m* : H,(X) -+ H,,(X) 0 H.(X) be the comultiplication induced by the cup
product on H*(X). Then the relations that must be satisfied by the classes pa(o-) take the
form
p'aUb(U) = Pa 0 Ib(mn(c)) = (
mijPa(oi),b(cj)
iji
(2.7)
for constants mij. These relations merely say that any Pa(-)can be expressed as a polynomial
in the Pa(-).
Definition 29. We write A(X; G) = A(X; H*(BGI;Q)). When context is clear we omit
G.
In general A(X; G) can be described quite concretely, due to the following description of
H*(BGad;Q).
Proposition 30. Let G be a semisimple compact Lie group. Then H*(BG; Q) is a polynomial algebra on finitely many even generators, and the map BG -+ BGad induces an
isomorphism on rational cohomology.
Proof. It is well known that the map BG -+ BT induces an isomorphism H*(BG; Q) +
H*(BT; Q)W, for any compact Lie group G, and the latter is a polynomial algebra on finitely
many generators. To prove the second part, note that the map BG -+ BG' has homotopy
fiber BZ, where Z is the center of G. Since G is semisimple, Z is finite, so BZ has the
rational cohomology of a point. Hence the Serre spectral sequence
H*(BGa; H*(BZ; Q)) ==> H*(BG; Q)
collapses on the E 2 page, which means that H*(BGad; Q) -+ H*(BG; Q) is an isomorphism.
For example, if G = U(N) we might take as generators the Chern classes c 2 , ... , cN- On
the other hand, Newton's theorem on symmetric polynomials implies that we could equally
well use the first N components of the Chern character, Chi,... , ChN- In the context of
higher rank invariants we need a set of generators for the group PU(N).
Definition 31. We denote by a2 ,..., aN the rational cohomology classes corresponding to
c2 ,... , cN under the isomorphism H*(BPU(N), Q) -+ H*(BSU(N); Q). Given a homology
class o E H,(X) we write pi(o-) for the element Pai(-) E A(X; PU(N)). We also make the
abuse of notation pj(pt) = aj.
Note that classes ai are not the Chern classes of any bundle over BPU(N), and in fact
they are not integral classes in general. However, we can construct them as rational linear
combinations of Chern classes, using the following general principle.
24
Proposition 32. Let G be a compact Lie group, and let P -+ BG be the universal Gbundle. Then H*(BG; Q) is generated by classes of the form ci(V(P)), where V is a finite
dimensional representation of G.
Proof. It is useful to think in terms of the Chern character Ch(V(P)). This gives us a ring
homomorphism
ChG : R(G) -+ flHn(BG; Q),
n
where R(G) is the virtual representation ring with rational coefficients, and fJn Hn(BG; Q)
is the completion of the usual cohomology ring, with the product topology. When G = T
is a torus, the map ChG is injective and has dense image (in the product of the discrete
topologies). In general, restricting to a maximal torus T C G gives us a commutative
diagram:
R(G; Q)
HH"(BG;Q)
ChG
resit
I
Hn H(BT; Q)W
1ilT
R(T; Q)w
Both the left and right arrows are isomorphisms, and since ChT is W-equivariant, the bottom
arrow is an injection with dense image. Hence so is the top arrow. Since the Chern character
can be explicitly written as a formal power series in the Chern classes, this shows that Chern
l
classes of associated bundles generate H*(BG; Q).
In the case G = SU(N) we can be even more explicit. Let V = CN be the defining
representation of SU(N), and let W = VON. The center of SU(N) acts trivially on W, hence
it descends to a representation of PU(N). Viewing V and W as bundles over BSU(N),
we have Ch(W) = Ch(V)N, hence taking an appropriate branch of the N-th root power
Since the Chern classes ck(V) can be expressed
series we can write Ch(V) = N/Ch(W).
as rational polynomials in the components of Ch(V) and vice versa, we see that the classes
ak E H*(BPU(N); Q) can be expressed as rational polynomials in the Chern classes ck(W),
with 2 < k < N. Likewise, the classes pi(o-) can be represented as rational polynomials in
the classes ,ck(W)(rj), where T1 , ... , r, form a basis for H*(X).
2.2.2
Invariants
In the previous section, we defined a map p : A(X; G) -+ H*(B(X); Q). To produce
this map, however, we needed to use the existence of a classifying map for the universal
bundle. Strictly speaking, we only know that such a map exists because B(P) is a separable
Banach manifold and therefore has the homotopy type of a CW complex. If we only wish
to evaluate cohomology classes unambiguously on compact submanifolds M c B(P), such a
construction is valid but quite inexplicit.
On the other hand, the ASD moduli space M(P) is noncompact in general. It does have
a fundamental class in compactly supported cohomology, but in order to make use of this
we must have some notion of support for classes in A(X; G). Therefore, we now give a more
explicit recipe for computing these classes. Our starting point is an explicit description of
the Chern classes of a complex vector bundle.
25
Proposition 33. Let X be a finite dimensional compact manifold, let E -+ X be a vector
bundle of rank N, and let s X -+ Hom(Cn-k+l, E) be a section that is transverse to the
stratification of Hom(Cn~k+1, E) by rank. Define
V(s){x E X1rk s(x) < N - k}
Then V is a stratified subspace of X, with strata of even codimension. It therefore has a
fundamental class [V], which is Poincaridual to ck(E).
Proof. The rank stratification of Hom(CN-k+, CN) is a stratification by complex algebraic
varieties. In particular, each stratum has even codimension. Since V is (locally) the pullback
of the rank stratification under a transverse map, it also has strata of even codimension.
To show that [V] represents ck(E) it suffices to check the case where E = L 1 E ... G L®
is a sum of line bundles. In this case let o- be transverse sections of Li, whose zero loci
intersect transversely. Now define n - k + 1 sections by
si= Zcio-j,
where C = (cij) is a constant matrix of size (N - k +1) x n, whose (N - k +1) x (N - k +1)
minors all have full rank. The locus of points x E X where the vectors si(x) become linear
dependent is then the same as the locus where k of the o- (x) vanish simultaneously. In other
words, the class represented by V(si, ... , sN-k+1) is
PD[V]
=
3
c1 (Li1 ) U
...
U cl(Lik) = ck(E)
21.1
Now consider the universal bundle P -+ B*(P) x X. It is a principal PU(N) bundle,
hence we can take the faithful representation W = (CN)®N and form its associated bundle
W -+ B*(P) x X.
Proposition 34. Let M be a finite dimensional manifold and let f : M -+ B*(P) x X be
a smooth map. Then for each integer k and every even integer p > 2 there exists a section
s : B*(P) x X -+ Hom(C,-k+l, W), continuous in the L, topology on B*(P), such that the
pullback s o f is smooth and transverse to the stratification of Hom(Cn-k+1, f*W) by rank.
Proof. We refer to Kronheimer [8].
l
Let o E H 2 (X) be an integral homology class of dimension at most 2. It is always
possible to represent o by an immersed submanifold E C X. Let v(E) C X be a connected 4dimensional submanifold with boundary whose interior contains E. Let Pr be the restriction
of P to v(E). Then there is a partially defined restriction map r : B*(P) -+ B*(P,) whose
domain of definition consists of those connections which remain irreducible when restricted
to v(E). Denote this domain of definition by B*(P, E).
Now let M C B*(P,E) be a finite dimensional submanifold. Let 1 , ... , a-,
a be homology
classes on X and let ai = ck(W) E H*(BPU(N)). Suppose that there is a well-defined
26
restriction map
r = jri:
M
-*
llB*(PE),
1
i
i
or in other words that every connection in M remains irreducible when restricted to each
v(E). Consider the associated bundles Wr, over B*(P,) x v(Ei). Each of these pulls back
to a bundle Wi on the product
B= fJB*(P)
x Ej.
i
As above we can choose sections si : B -+ Hom(Cnki+l, W,) such that the restriction
s=
f
si o r : M x El x - -.x E,
-
r*Hom(Cn-k±,+W)
is transverse to the product of the rank stratifications of Hom(Cn-ki+l, Wi). If M is compact
and oriented then the pairing
-- Pa,(o-r), M)
(P-1 (o-1) ..
is equal to the number of points m E M, counted with sign, such that all components of
s(m) have strictly less than full rank.
Proposition 35. Let P = P(E), and let aj, o- as above. Then there are:
such that the restrictionmaps ri : M,(P(F)) -4 B*(v(Ei), P(F))
(1) Neighborhoodsv(i)
F with c1 (F)= c1 (E) and c 2 (E) < c 2 (F).
bundles
U(N)
all
for
are defined
(2) Sections si such that 1 si o r is transverse to M,(P(F)) x E1 x -
x E,.
l
Proof. We again refer to Kronheimer [8].
Proposition 36. Let aj, a-, EY as above, and let Vk C B(EA) x Ei be the the representatives
) as in Proposition 35. Let M (P) have dimension d, and suppose that the
of zi = pa,t,
sum of the degrees of the classes zk is less than or equal to d. Then the intersection
M(P) x Ei x ...x E, n Vi n -..nVr
(2.8)
is 0-dimensional and compact. The signed count of points in this intersection is independent
of the choice of metric, perturbations,and sections si. It only depends on the homology classes
o-1 represented by E1, and this dependence is multilinear (over Z). It is graded-symmetric in
its arguments in the sense that swapping two classes pa, (o-) and Pa, (oj), of degrees di and
dj respectively, changes the sign of the count by a factor of (-l)didj.
Proof. This is a standard dimension counting argument. Choose the neighborhoods v(E.)
such that all triple intersections
v(Eri) n V(Ei2) nv(-i3)
27
are empty, and every double intersection involving a point class is empty as well. Any point
x E X is contained in at most two neighborhoods v(E), and at most one neighborhood such
that E is a point. Thus for any finite set of points {X 1, ... , Xk} C X, the intersection of all
Vi such that {y} n v(Ei) = 0 has codimension strictly larger than d - 4Nk. By Proposition
35, such an intersection is disjoint from M(P(F)), where F is the bundle with c1(F) = ci(E)
and c 2 (F) = c 2 (E) - k.
Now let (An, yi,..., y,) be a sequence of points in the intersection (2.8). The connections
An must limit, after passing to a subsequence and bubbling at a finite set of points x,, ... Xk,
to a connection Ao on P(F). But the moduli space M(P(F)) is transverse to the intersection
of all Vi such that {x} n v(Ei) = 0. In particular the set of possible limiting connections A,
is empty, unless k = 0, in which case the limit lies in (2.8). This shows that the intersection
is compact.
To show graded symmetry, just observe that the only effect of swapping two factors is
to change the orientation on El x ... x Er, and the effect of this change is to modify the
orientation by a factor of
(-1)didi.
To show independence of metric and perturbation, sections, and representatives of the
Ei, we apply the same argument to a 1-parameter family of moduli spaces obtained by
varying the metric and perturbation, restricting the family to neighborhoods of various
cobordisms Ei ==> E', and taking transverse sections on the configuration spaces of these
neighborhoods. We thereby obtain a compact, oriented cobordism between two sets of signed
points, which verifies that the signed counts are equal.
The same argument also shows that the invariant is linear, because whenever we can write
- = 0-'+ a" there is a homology (more precisely, a stratified cobordism) from E to the union
of E' and E".
Definition 37. Given ac
Donaldson invariant
and c-i as above, and a class c E H 2 (X), we define the SU(N)
Dx,c(pi0 1 (- 1 )
...
(-,))
to be the signed count of points defined in Proposition 36, viewed as a linear map
Dx,c : A(X; SU(N)) -+ Q
defined on the image of p: H*(PU(N)) 0 H<2 (X) -+ A(X; SU(N)).
There is something a bit arbitrary about our definition of Dx,c, in that we chose a particular associated bundle W in the construction. We could have made the same definition
as above, replacing p, (o-x) with a combination of classes 1p1 33 (-rj), where the 6j are Chern
classes of a different associated bundle and the Tj are different homology classes on X (see
equation 2.7). We would like to know that such a modification results in the same map
Dx,c : A(X; SU(N)) -+ Q. We can use the following refinement of our dimension counting
argument to handle some cases:
Proposition 38. Let ak and U-k as above. Suppose further that all geometric representatives
E,... , E,_1 are mutually disjoint (but not necessarily disjoint from E,). Then the signed
count of points in the intersection (2.8) depends only on pa,(Er) as an abstract cohomology
class on B(P).
28
Proof. For a fixed metric and perturbation, it suffices to show that the intersection
X
K = M(P) x Ei X ..
ErnVi
n ---n v_
is compact, because then the signed count of points in Vr n K is just given by the cohomological pairing (Par-(ar), [K]), which does not depend on the choice of geometric representative
for pa,(Ur).
As in Proposition 36, consider a finite set of points X1 , ... , Xk E X. Because E1,... Er1
are mutually disjoint, each point xj can be contained in at most one neighborhood v(E).
Hence the intersection of all V such that {x} n v (Ei) = 0 has codimension at least d - 2Nk.
In particular, it is disjoint from the moduli space M(P(F)) with c 2 (F)= c 2 (E) - k. To see
that K is compact, we now proceed exactly as in Proposition 36.
Clearly we can do better in cases where there are not "too many" intersections between
the Ei. However, in the simplest cases it is impossible to do better using the methods above.
For example, we might try to evaluate classes pa, (O-), [a, (0-2 ), and Pa, (0 3 ) on a 12-dimension
moduli space of SU(3) connections. If the representatives Ei each have nontrivial pairwise
intersections, then we cannot show that V 2 n V 3 is compact, because there might be bubbling
on E2fn 3.
One way to resolve the issue is to use the blowup formula. In the next chapter we will prove
a vanishing result (Proposition 65) which implies that if E, is a sphere of self-intersection
-1, then the signed count of Proposition 38 automatically vanishes. As a consequence, we
can blow up X at all points where the Ei intersect, then replace each E with its proper
transform Ei = Ei + Ej +eij, without changing the signed count of points. We can then
replace each pa,(di) with a combination of pg (ii) in the blowup. Applying Proposition 65
again, we can eliminate all ri which intersect the exceptional curves. Blowing back down,
we find that the corresponding replacement is valid on X as well.
2.3
2.3.1
Cylindrical Ends and Gluing
Cylindrical Ends
Much of our discussion of ASD moduli spaces on compact manifolds carries over to more
general noncompact manifolds.
Definition 39. We say that an oriented Riemannian 4-manifold X has asymptotically cylindrical ends if there is a compact subset K c X whose complement X \ K is diffeomorphic
to a product (0, oo) x Y for some 3-manifold Y, such that the metric on X \ K is given by
gx = dt 2 + gy + e-Ath(t),
where gy is a fixed metric on Y, A is a positive constant, and h(t) is a smooth bounded,
t-dependent section of Sym 2 (T*Y). If h(t) is identically zero we say that W has cylindrical
ends.
To set up configuration spaces on manifolds with asymptotically cylindrical ends we need
to choose appropriate boundary conditions at infinity.
29
Definition 40. An adapted bundle (P,77) on a 4-manifold X with asymptotically cylindrical
ends is a principal bundle P -+ X, together with a choice of a flat connection 77 on Ply.
Given an adapted bundle, we choose connection Ao that restricts to 21 on the cylinder
Y x [0, oo). We would like to define the configuration space to be the space of all connections
on X that approach q asymptotically at infinity. If q is irreducible we can imitate the compact
case exactly and define
A(P; r) = {Ao + ala E L2(X, A' 0 adP)},
However, when 77 is reducible this definition is inadequate. The issue is that the operators
d*dA are not Fredholm, hence our construction of slices is invalid.
To remedy the difficulty, we need to introduce weighted norms on the space of sections of
A' 0 adP. Fix a small positive real number a, and a smooth nonvanishing function W on
X such that
W(y, t) = e"t
on the cylindrical end Y x (0, oo). Define a weighted norm
a2,-
|WaIL
=
and define L ',(X, A' 0 ad(P)) to be the completion of the space of smooth sections with
respect to this norm.
Definition 41. Let (P,2) be an adapted bundle on a 4-manifold X with cylindrical ends.
We define
Aa(P, 7 ) = {Ao + a a E L2',(X, Al 0 adP)}.
ga(P,27 ) = {'y: X -+ G(P)17-'dAO7 E L2'a(X, A' 0 ad(P))}
B'(P,)
=
(Pq1G(M
When the context is clear, we will drop a and 7 from the notation.
There is a homomorphism evc : G'(P) -+ r. given by "evaluation at infinity".
natural to consider the kernel of this homomorphism.
It is
Definition 42. We define the framed gauge group to be
GO"(P) =
{y E Ga(P)|7. = 1}
and we define the framed configuration space to be
B"(P) = A"(P)lGo(P)
The framed gauge group is in some ways a more natural group to consider. For example, its
tangent space is L2,(X,
ad(P)), which fits naturally into the framework of weighted spaces.
It also acts freely on Ac(P), because a nontrivial stabilizer would evaluate nontrivially at
infinity. Thus B(P, 2) has no quotient singularities, unlike the configuration space on a
compact manifold.
30
Proposition 43.
3(P,77) is a Banach manifold with a smooth action of
]FJ7.
Proof. Let d*" denote the formal adjoint of dA with respect to the weighted norm.
construct a slice for the action of g(P), we consider its kernel:
TA,E = {A
+ ald*aa = 0,|a2,.
To check that this is a slice we can imitate the compact
is now a Fredholm operator, provided that a is nonzero
the operator d,*d,7 . Since 90(P) acts freely we see that
smooth action of ga(P)/g (P).
It remains to be shown that the homomorphism ev..
<
To
}.
case. The key point is that d*'adA
and smaller than any eigenvalue of
B(P) is a Banach manifold with a
-+ F,, is surjective, so that
all of F, acts on g(P). The proof of Proposition 15 shows that F, is connected mod Z.
Since elements of Z extend over X, it suffices to show any y in the identity component of
F extends as well. But such an extension can be given explicitly, by homotoping Y to the
E
identity along the neck, then extending by the identity over the rest of X.
: !;(P)
Our discussion of moduli spaces and perturbations in the compact case carries over to
manifolds with asymptotically cylindrical ends.
Definition 44. Let 7r be a perturbation of the ASD equations on an adapted bundle (P,n).
We define the framed moduli space of solutions to the perturbed equations to be
M7(P) =
[A] E f(P) IFA=V(A)
Proposition 45. Let P be an adapted bundle over a Riemannian 4-manifold X with asymptotically cylindrical ends. Then there exists a residual set of small perturbationsir, supported
in a compact subset of X, such that every A G M,(P) is regular.
Proof. The connection A is regular if and only if the cokernel of di is trivial. The proof of
Proposition 23 shows that we can choose compactly supported perturbations such that any
element of the cokernel vanishes outside of the cylindrical end. But elements of the cokernel
satisfy a unique continuation property on the cylindrical end (because they are solutions of
0
an ODE), and hence must vanish there as well.
2.3.2
The Equivariant Universal Bundle
Because connections in B(P, q) all have trivial stabilizers, there is a universal PU(N)
bundle P defined over all of ff(P, 7) x X, not just over the space of irreducible connections.
This bundle is equivariant with respect to the action of F,, so it has equivariant characterTo compute these characteristic classes we will need a
istic classes ak(P) E Hr (B(P,i)).
description of the restriction of P to the orbit of a reducible connection A E 1(P,
M
).
Proposition 46. Suppose that A admits a reduction of structure to a subgroup H C PU(N),
so that P = Q XH PU(N) for some H-bundle Q and A is induced by a connection on Q.
31
Then the restriction of P to the orbit 9 A
X
(X
as a r 7 -equivariant bundle over
9
= r./FA
Q)
in
(P) x X is isomorphic to
XrAxHPU(N)
A.
Proof. By definition, the universal bundle is
P=
g 0 (P)\B(P)
x
P=
x
g(p) (A(P) X P)
Restricting to the orbit of A, we have an isomorphism
, x rA P -+ P
given by sending (z, p) to (z, A, p) for any z E r,, and any p E P. Applying P = Q x H PU(N)
gives
Ji, XrA P = (F77 X Q) XrAXGPU(N)
as desired.
Corollary 47. If o- C X is a cycle of positive dimension then for any a E H*(BPU(N))
the equivariant classes fA&(o-) are trivial in a neighborhood of the trivial connection E.
Proof. There is a neighborhood U of the orbit Oe that retracts equivariantly onto Oe, so
it suffices to show that A,,(o-) is trivial on Oe. Because re = PU(N) we have H = {id},
hence Proposition 46 shows that P is pulled back from an equivariant bundle on a point.
Therefore,
f.(u-) = a(P)/o-= 0
as desired.
2.3.3
Index Theory
We can compute the virtual dimensions of cylindrical end moduli spaces using the excision
principle for indices.
Proposition 48. Let X 12 = X 1 UX 2 and X 3 4 = X 3 UX 4 be 4-manifolds such that X 1 n X 2 =
X 3 n X 4 = Y x (0,1) for some 3-manifold Y. Write X 14 = X 1 U X 4 and X 3 2 = X 3 U X 2 .
If Di are elliptic operators on Xi that agree on Y x (0, 1), and Dij are the elliptic operators
obtained by identifying Di and Dj over Y x (0,1), then we have
ind(D 1 2 ) + ind(D3 4 ) = ind(D 14 ) + ind(D3 2 )
Proposition 49. Let V± be adapted U(N) bundles over 4-manifolds X±, with limiting flat
connections 77. Suppose that X+ has a cylindrical end Y x (0, oo), that X- has a cylindrical
end Y x (-oo, 0) and that there is an isomorphism y : q+ -+ i.
Let X be the closed 4manifolds obtained by gluing together X+ and X- and let V be the bundle obtained by gluing
V+ and V. Let A± be connections on X±, and let DA± be the corresponded Dirac operator,
dA: L 'a(V 0 A1 ) -+ L '"(V e V o A2 ).
32
viewed as an operator on weighted spaces. Then for all sufficiently small a > 0 and any
connection A on V we have
ind(DA,) + ind(DA ) = ind(DA) + ho(r) - hl(?)
where h'(71) are the ranks of the cohomology groups of Y with coefficients twisted by r.
Proof. By the excision principle, we can reduce to computing the index of the operator
(9
Da"= a+
D77~
W'(t)
H77 +
Wt
W(t)
Hat
on the cylinder Y x R, where H. is the self-adjoint operator
=(0
-dn
d* \
*dq
'
E is the operator which acts as +1 on A' and as -1 on A', and the weight function W is
equal to e-at for negative t and eat for positive t. But the index of D' is the spectral flow
of H,7 + 4dlW e, which is equal to ho(rq) - h'(7).
dt
Proposition 50. Let X be a 4-manifold with cylindrical ends, and let V be an adapted U(N)
bundle over X, with limiting flat connection i. Then there is a constant p(rq), depending only
on q, such that
ind(DA) = c(V) 2 - 2c 2 (V) + N
2
2
+ h2()
-
h(i 7 ) + p(q)
2
2
where A is any connection that restricts to r on the ends of X and c2 (V) and c1(V) are
defined by Chern-Weil integrals of the curvature of A.
Proof. Given an adapted bundle V define a quantity p(V) by
p(V) = 2indDAo + cI(V) 2 - 2c 2 (V) + N(x + a) + ho(rq) - hl(,q)
Note that this formula is additive under gluing adapted bundles, by virtue of Proposition
??. It is zero for a bundle over a closed 4-manifold, by Proposition ??. Thus if W is an
adapted bundle that can be glued to V, we have
p(V) + p(W) = 0
But then if V' is another adapted bundle with limiting connection r, we have
p(V) = -p(W) = p(V'),
so p(V) only really depends on the limiting connection 77.
Note that p(7) changes sign if we reverse the orientation on the underlying 3-manifold Y.
In particular, p(rq) vanishes if r is invariant under an orientation reversing diffeomorphism.
33
Corollary 51. Let X be a 4-manifold with cylindrical ends, let P = P(V) be an adapted
PU(N) bundle over X, with limiting flat connection q. Then the formal dimension of the
ASD moduli space on P is
dim M(P) = 4Nc2 (V)+ (2 - 2N)c (V) 2 + (1 - N 2 ) x+
2
+ h'(7) + h() + p(ad(T)) (2.9)
2
and the formal dimension of the framed moduli space is
2
dim -(P) = 4Nc2 (V) + (2 - 2N)c1(V) + (1 - N2) X + a + h()
-
2
h'(71) + p(ad(77))
2
Proof. Simply apply Proposition 2.9 to the complexification of ad(P).
Finally, we observe that the results above remain valid for manifolds with asymptotically
cylindrical ends, because the operators in that context are homotopic to the ones used above.
2.3.4
The Gluing Theorem
To prove the blowup formula we need to have a general understanding of ASD moduli
spaces on a 4-manifold that is decomposed along separating 3-manifold Y. In general suppose
that we can write X = X+ Uy X-, where X+ and X- are joined along their oriented
boundaries OX* = tY.
Let X be the 4-manifolds obtained by attaching infinite cylinders
(0, oo) x Y and (-oo, 0) x Y to X+ and X-, and let XT be obtained by similarly attaching
cylinders of length 2T. Given asymptotically cylindrical metrics on X
(with the same
limiting metric gy) and a partition of unity /1 + #2 = 1 on the cylinder ZT = [-T, T] x Y
we can form a Riemannian 4-manifold XT = XT+ UzT XI, by joining X along ZT and
superimposing their metrics using #. The rough idea of gluing is that as T tends to infinity,
most ASD connections on XT can be described as a pair of ASD connections on the manifolds
X , plus an isomorphism of their limiting flat connections. The connections which cannot
be described in this way admit an analogous decomposition into a series of ASD connections
on infinite cylinders Zo (with the product metric dt 2 + gy), plus a series of identifications
of their limiting flat connections.
Proposition 52. Let XT = X+ Uy ZT Uy X- as above, with b+(XT) > 2, and let P = P(E)
be a principalbundle over XT, such that M (XT; P) has dimension less than 4N (so that it is
compact). Suppose that the 3-manifold Y carries only finitely many flat PU(N) connections
up to gauge equivalence, each of which is regular (but possibly reducible), and that the (possibly
perturbed) cylindrical end moduli spaces M (X+; a) are regular. Then for T sufficiently large,
the moduli space M (XT; P) can be covered by finitely many open sets, each of which is
diffeomorphic to an equivariantproduct of the form
M(XI, a,) xr, M(Zo, ai,a 2 ) Xr 2 -
Xrk_1 M(Zo, ak_1, ak) X r, M(Xe, ak),
where the ac are flat connections on Y, and have stabilizers J'. Moreover, if 0- is a subspace
of X±, then the restriction of the universal bundle over M(XT) x a to such an open set is
obtained by pulling back the equivariant universal bundle over M(Xc,, ak).
34
We will not try to give a full exposition of this theorem here. References include [16], [9],
[13].
35
36
Chapter 3
Higher Rank Blowup Formulas
3.1
Solutions Near a Negative Sphere
To understand the blowup formula it is useful to have a more general understanding of
how the SU(N) invariants of a 4-manifold X behave in the presence of an embedded sphere
o-, C X of negative self intersection -p. Following general principals of gluing, we consider
an open tubular neighborhood Np of such a sphere. This noncompact manifold has a single
2
3
end diffeomorphic to LP x (0, oo), where Lp = L(p, 1) is the quotient of S c C by the group
of p-th roots of unity. In this section, we will study moduli spaces of ASD connections on
the manifolds Np.
3.1.1
Regularity
In general one needs perturbations to achieve regularity of the ASD equations. However,
in the case of the manifold Np, for sufficiently small p we can find a metric for which
transversality is gauranteed automatically.
Proposition 53. Let p be a sufficiently small integer (for our purposes, p K; 2). Then the
manifold Np admits a metric gp with the following properties:
(1) gp has positive scalar curvature and anti-self-dual Weyl curvature.
(2) gp decays exponentially to a product metric on (0, oo) x Lp.
Proof. Let O-1 ,o-2 , and O-3 be an orthonormal coframe field on S3, invariant under the left
action of SU(2), such that -3 is invariant under U(2). Define a metric on (0, 00) x S3 by
g = dr 2 +
l
+U
2
3
Because both o-3 and of + o2 are invariant under U(2), and L, is the quotient of S by a
cyclic subgroup of U(2), g descends to a metric on (0, oo) x Lp. If we furthermore demand
that A extend to a smooth odd function on all of R, satisfying A(0) = 0 and A'(0) = p, then
g will extend to a smooth metric gp on Np.
37
We now compute the Weyl curvature of the metric g, using the method of Cartan. We
choose as an orthonormal coframe
6 = (60, 61, 62,
6 3 )T
= (dr, Oc-,
-2 , A- 3 )T,
and compute its differential using the identities
da,
=
20-2 A a 3
dO-2
=
20- A -1
dO-3 =
2o A c- 2.
We find the matrix w of the Levi-Civita connection from dO by imposing the condition
d6 + w A 0 = 0. The result is:
0
0
0
_0
-Ac-
3
(2 - A ).3
-AO-
2
0
Ac0
0
2
0
(A2 - 2)o-3
A'9-3
Ac- 2
-Ao-
The curvature matrix Q = dw + w A w is given by
0
A/023
)
A1031
(003 - 2A'612
- A2 031
A'901 + A262 3
--
0
-2A6 03 + (4 - 3A 2 )0 12
2
2A'6 03 + (3A - 4)012
-A/031
0
2
A'6 02 + A
-A'og - A2623
0603 - 2A'012
A-23
-A/002
0
)
where to save space we have used the shorthand 6ij = 9, A O. We then compute the self-dual
part of the 2-forms Q01 + Q23 , Q02 + Q31, and Q03 + Q12. The result can be written as a
square matrix,
W++
=
12
A'+ }A2
0
0
A'+ 'A2
0
0
0
0
-A-
2A'+2-!A 2 )
/
whose traceless part is the self-dual Weyl curvature W+. Demanding that it vanish leads us
to the ordinary differential equation
A" + 6AA' + 4A3 - 4A = 0,
(3.1)
and the positive scalar curvature condition leads us to the inequality
A2
2
A'+ - > 0.
(3.2)
The equation (3.1) has two special solutions. Namely, if A satisfies one of the ordinary
38
differential equations
A' =
A'
=
1-A 2
2 - 2A 2,
(3.3)
(3.4)
then it automatically satisfies (3.1). We can use these special solutions to find A(r) in the
cases p = 1 and p
=
2, respectively. When p = 1 integrating (3.3) gives
A(r) = tanh(r)
and in the case p = 2, integrating (3.4) gives
A(r) = tanh(2r).
Both of these solutions are increasing, so the corresponding metrics have positive scalar
curvature by (3.2). They rise exponentially to the limiting value A = 1, so the corresponding
metrics are asymptotically cylindrical and limit on the round metric on Lp.
I have not tried to solve the equation (3.1) explicitly for larger values of p, but numerical
solutions indicate that A is increasing if and only if p K 2, and satisfies the positive scalar
0
curvature inequality (3.2) if and only if p 13.
In fact, the computation above can be modified to show that there is a unique U(2)invariant metric on N, with vanishing anti-self-dual Weyl curvature, up to conformal rescaling and reparameterization of the radial coordinate. In particular, the metrics gp are conformally equivalent to those described by LeBrun in [11]. It is conceivable that g, is conformally
equivalent to an asymptotically cylindrical metric with positive scalar curvature, but I have
not investigated this possibility.
Proposition 54. Every finite energy ASD connection on (Np, gp) is regular.
Proof. Consider the Dirac operator
DA = d' + d* : Q1(X, ad(P)) -> f 0 (X, ad(P)) D Q+(X, ad(P)).
Given any w E Q2 (X, ad(P)), the Weitzenbock formula for DA gives us:
dA(d+)*w - VAV* W + (S, W) + (W-, w) + (FA, w)
where VA is the connection on A+(X)®ad(P) obtained by combining the Levi-Civita connection on A+ with the connection A on ad(P). Since FjA = W- = 0 and S > 0, we conclude
that any w in the cokernel of d+ must vanish identically. Thus for any such connection
HA = 0, as desired.
In the future we will implicitly use the metric gp when referring to ASD moduli spaces on
Np. Therefore, any time we use regularity of the moduli space we will be assuming implicitly
that p is small enough so that gp has positive scalar curvature, or that there is a conformally
equivalent metric with positive scalar curvature.
39
3.1.2
Completely Reducible Connections
On a negative definite manifold like Np, one cannot avoid the presence of reducible connections, even when the bundles in question are coprime to N. In this section we'll describe
some reducible connections on Np and the moduli spaces they inhabit.
First we need to briefly discuss the topology of Np. Since it deformation retracts onto o-,
we have H 2 (Np) = Z. Since Lp is the quotient of S 3 by a fixed point free action of Z/pZ,
we have ir1(L,) = H1 (Lp) = Z/pZ. Geometrically, let Dp be a fiber of N - o-p, oriented so
that Dp and u, intersect positively, and let -, be the oriented boundary of Dp. Then -Y is a
generator of H 1 (Lp) and Dp is Poincare dual to a generator of H 2 (Np).
Now let L be the complex line bundle whose Chern class ci(L) is Poincar6 dual to -Dp.
Since Np is negative definite, the class c 1 (L) can be represented by a square integrable
anti-self-dual 2-form w, and there is a connection A on L with curvature -27riw, which is
unique up to gauge equivalence. At infinity, the connection A decays exponentially to a flat
connection X on LP, which we can also think of as a character X : 7ri(Lp) -+ U(1). The fact
that c1 (L) is Poincar6 dual to -Dp means that
X(7y)
= Hol(y(-)
= exp
= exp
27ri D
hence X generates the character group of
e
,
7ri(Lp).
Recall that when talking about ASD connections on a bundle P(E), we fixed a connection
6 on the determinant det E. For the manifold Np it is useful to take 6 to be the unique (up
to gauge equivalence) finite energy ASD connection Ad, where d is the degree of det E. Then
we can view any ASD connection on P(E) as an ASD connection on E and vice versa.
Definition 55. We say that a connection A on a U(N) bundle E is completely reducible if
it is gauge equivalent to a connection of the form A,
... @ A,,, where each Ai is a connection
on a line bundle Li.
There is a general principle that minimal energy moduli spaces on negative definite manifolds contain completely reducible connections. The remainder of this section is dedicated
to making this principle precise for the manifolds Np.
Proposition 56. Let k1,.... k,, be integers such that Iki - k1 5 p for all pairs i, j. Then
the ASD moduli space containing the completely reducible connection
Ak
... e AkN
= Aki
over Np has formal dimension
dimM(Ak) = 1 - N2 +
40
Iki - kj|
Proof. By the index formula (2.9), we have
dim(AkdimM(Ag)
=
1:(ki ____
kj )2
+1
+
- N2 + ho(l) + p(l) + E, -ho0(xki-kj)
2N +91
2..h~ki
Iki
It therefore suffices to show that for any integer k with
-
p(xki-kj
(
p we have
2k 2
ho (xk) + pxk) = 1 - 21k
p
This fact was proved by Atiyah, Patodi, and Singer in [1].
Then the moduli space
Corollary 57. Let k 1 ,. . ., k,, as above, and suppose that 2p < N +1.
M(Ak) has negative virtual dimension, and therefore contains no irreducible connections.
In particular,every irreducible ASD connection on Ek - Lk, E ... E LkN has energy strictly
then we can replace "strictlygreater than" with
greater than that of Ak. If 2p < N +1 + 4
"greaterthan or equal to".
Proof. Without loss of generality, we have k 1
Z |kj -
ki|
=
kn. Then for any fixed i, we have
--
k2
.
|kn - kil
p
j>i
Therefore,
dim M(Ak)
=
1 -N
2
2|k - kil
+Z
i
2
j>i
+ 2p(N - 1)
<
1 -N
=
(2p - N - 1)(N - 1)
The right hand side of this equation is negative if 2p < N + 1, hence M (Ak) has negative
virtual dimension. If 2p < N + 1 + N, then the right hand side is less than 4N, so any
moduli space of strictly smaller energy has negative virtual dimension, hence contains no
irreducible connections.
2
As an application, we can compute all the minimal energy moduli spaces on CP .
Proposition 58. Let 0 < k < N and let Ek be the U(N) bundle over N 1 with (cl(Ek)
kc1 (L). Then the completely reducible connection
=
Ak = kA e (N - k)1
has energy strictly less than any other finite energy ASD connection on Ek. As a consequence, the framed moduli space M(Ak) is diffeomorphic to the GrassmannianGrk(CN).
Proof. Let B be an ASD connection on Ek, and suppose that B has minimal energy among
all ASD connections. Write B = Di Bi as a sum of irreducible ASD connections on bundles
41
FR. Write c1(F) = (ki+Ndc)c1(L) with 0 < ki < N. Then by Proposition 57, the completely
reducible connection
Ai = k Adi+1 e (N - ki)Adi
has strictly less energy than Bi. Substituting Ai for Bi, we obtain a completely reducible
connection A on Ek with strictly less energy than B. We conclude that if A has minimal
energy then it is completely reducible, so
A = Ak, e
Ak2 E ... E AkN
for some integers ki. If ki - kj > 1 for any i and j, then the connection A, E A has strictly
greater energy than Aki+1 E Aki-1. Since A has minimal energy, this implies that Iki - kyj
1
for all i and j, hence A = Ak.
The description of the framed moduli space then follows from the fact the stabilizer of
Ak is rk = S(U(k) x U(n - k)), hence its orbit in M(Ak) is Grk(CN) = SU(N)/S(U(k) x
U(N - k)).
l
Note that the argument above fails if p > 1, because the inequality 2p < N + 1 does not
hold for all N. However, the weaker inequality 2p < N + 1 + 4N is valid for all N as long
as p < 4. In this regime we can therefore hope to partially salvage Proposition 58.
Before proceeding, we need to establish some notation. Let 0 < r1 < ...
rN < p be
integers, and consider-the flat connection
7
r= X
e Xr 2
D
... DXrN
on LP. Let r = r1 + - + rN, so that c1(2jr) = rc1 (L). If E is an adapted bundle on Np with
limiting flat connection ?7r, then we necessarily have
(c,(E), o-,) =-r mod p.
If E satisfies this constraint, we say that it is admissible for 7r.
Proposition 59. Let E be a vector bundle over Np. Then E admits a completely reducible
ASD connection, satisfying the hypotheses of Proposition 57.
Proof. Define a connection
Ar's
=
Ara ®D
...
(1 ArN
eD A
l+p..
eD A
r.-i+p
and call the underlying bundle of this connection E,. Then c1(E,) = (r +ps)c1(L), so as s
runs from 1 to N we obtain all admissible values of ci(E) mod N. In particular, for some
s and some integer d we have E = Ld 0 E,, so E admits the completely reducible ASD
connection Ad 0 Ar,s.
El
Proposition 60. Suppose that p
4. Let E be a U(N) bundle on N, and let B be a
finite energy ASD connection on E, which limits on a flat connection Lyr. Then there is a
completely reducible ASD connection A on E, also with limiting value 7, whose energy is
less than or equal to that of B.
42
Proof. Write B = Ei Bi, where each Bi is an irreducible ASD connection on a bundle F of
dimension ni. Each F admits a completely reducible connection Ai = Ar. ,,,, which satisfies
the hypotheses of Proposition 57. Because p < 4, the inequality
2p < n + 1 +
4ii
4n
n -1
holds for all n > 2. Since Bi is irreducible, we conclude that it has energy greater than or
equal to that of Ai. Therefore the completely reducible connection A = Ei Ai has energy
less than or equal to that of B, as desired.
It is unknown to the author whether Proposition 60 holds for all p. As an example of the
difficulties involved, let E be the trivial SU(2) bundle over N 13 , and let A be the connection
A-3 @A 3 , which has limiting flat connection q = x 3 ( X"i. Then A has minimal energy among
all reducibles limiting on i, but the moduli space containing A has virtual dimension 9. It
is conceivable that the moduli space of virtual dimension 1 is always empty in this case, but
proving it would require more subtle methods and might depend on the choice of metric on
N 13 -
In the case p = 2 we can explicitly write down the completely reducible connections of
minimal energy.
Proposition 61. Let Ek be the U(N) bundle over N 2 with cl(Ek) = kc1(L). Then
k-
A= A ((N - r) E r-kA-1
2 E rA E (N -k
r)1
r >k
k> r
has minimal energy among all ASD connections on Ek with limiting flat connection r, =
rX E (N - r)1, and it is the unique completely reducible connection with this property.
Proof. By Proposition 60 we only need to show that this connection has less energy than
any other completely reducible connection on Ek limiting on 7,.. So, let
A = A kiD.A kn
be completely reducible. Write ri = 1 for i < r and ri = 0 for i > r, and without loss of
generality write ki = ri + 2di. With k = E ki fixed, the energy of A is proportional to
c2 (A) - 2c 1 (A) 2
(3.5)
= Eki.
In the case k > r, we rewrite (3.5) using the identity r2
=
ri:
c2 (A) - 2c(A) 2 = r + Zk2 + (ri - 1)2 - 1
(3.6)
Because k2 and ri are congruent mod 2, each term on the right hand side of (3.6) is positive,
hence
c2 (A) - 2c 1 (A) 2 > r.
43
Equality holds precisely when (ri, ki) = (0, 0), (1, 1),or (1, -1)
A = Ak,r.
for every i, in which case
In the case r > k, we can rewrite (3.5) in a different way:
c2 (A) - 2c 1 (A) 2 = r + E k2 + (ri - 1)2 - 1 > 2k - r
(3.7)
Again, equality holds precisely when (ri, ki) = (0, 0), (0, 2),or (1, 1) for every i, in which case
A
= Ak,r.
3.2
The Blowup Formula
In this section we prove the existence of a blowup formula for higher rank invariants.
3.2.1
Initial Conditions
Let Y be a 4-manifold, let X = Y#Cp 2 , and let e C X be the exceptional sphere.
Proposition 62. Let k and d be integers satisfying 0 ; k < N and d < k(N - k) + 2N.
There is a class 8 k,dE H*(BPU(N); Q) such that for any z E A(Y) and any c E H 2 (y; Z)
we have
Dx,c+ke(A 2 (e)dz) = Dyc(/k,dz).
Moreover, the class Ok,d is homogeneous and has degree 2d - 2k(N - k), so it vanishes if
d < k(N - k).
Proof. Suppose that z has degree 2r. As we stretch the neck between Y and CAP , transversality implies that every connection in Mc+ke(X) n V(z) must eventually lie in
X SU(N)
fM~2r+2d-2k(N-k)(y)
k2 -(N-k)
2
(Ni)
where Mke(Ni) = Grk(CN) is the minimal energy moduli space on the bundle Ek - Lkk
CN-k. In particular, G = Mc+ke(X) n V(z) is the bundle of Grassmannians associated to a
PU(N) bundle P = Mc(Y) n V(z) over a compact stratified space B = Mc(Y) n V(z). By
definition, the number Dx,c+ke(p 2 (e)dz) is obtained by evaluating P 2 (e)d on the fundamental
class of G. We can therefore calculate it in two stages, by first pushing forward to B, then
pushing forward to a point.
Observe that p 2 (e)d is the pullback of an equivariant class A 2 (e)d E H U(N)(Grk(CN).
Hence we can compute its pushforward to B by first applying the equivariant pushforward
7r* : H U(N)(Grk(CN))
PU(N)(pt) =
-
H*(BPU(N))
to the class ft(e)', then pulling back to B along the classifying map of P. Setting
7r*(A2(e)d), we conclude that Dx,c+ke(p 2 (e)dz) = Dyc(k,dz), as desired.
fk,d
=
In fact, we can be much more explicit about the classes 6k,d. On the Grassmannian
Grk(CN) we have a splitting of equivariant bundles CN = Fk E FN-k. By Proposition 46,
44
the universal bundle over Mc+ke(X) x e is the projectivization of the vector bundle
V = (L 0 Fk) ( FN-kWe can easily compute the Chern character of this bundle,
Ch(V) = exp(L)Ch(F) + Ch(F_-k)
and from this we can obtain the classes a 2 (P), from the formula
a2 (P)
=
Ch(V)] 2 -
-[exp (kL)
Slanting with e, we get
N -k
=2(e)
=
N Ncl(Fk)
-
k
c1 (Fnk) = c1(Fk).
N
We are therefore reduced to computing the equivariant pushforward
#6k,d =
c1(Fed
IGrk(CN)
which is straightforward for any particular values of k and d. For example, Kronheimer
proved that 3 1,N-1 = 1 and used this to define higher rank invariants in the case where c is
not coprime to N. In more generality, we have:
Proposition 63. The class /k,d
have OO = 81,N-1 = 1.
is a nonzero integer when d
k(N - k). In particularwe
=
Proof. The class cl(F) is represented by a section of an ample line bundle on the complex
projective variety Grk(CN), hence the integral of cl(Fk)k(N-k) is nonzero. In the case k = 0
we have #o,0 = +1 trivially. In the case k - 1 we are evaluating the top power of the
hyperplane class on CpN-1, hence we get 01,N-1 = ±1. For evaluation of the sign we refer
E
Kronheimer [8].
The vanishing criterion in Proposition 62 generalizes as follows, with the same proof:
Proposition 64. Let 0 < k < N and let I = (i 2 , - - -, iN) be a multi-index satisfying
0 < dim(I) = 2i 2 + 4i 3+ -.. + (2N - 2)iN < 2k(N - k) + 4N.
Then there is a class
H 2 (Y; Z) we have
fk,I
E H*(BPU(N); Q) such that for any z E A(Y), and any c E
Dx,c+ke(p(e)'z)
If dim(I) < 2k(N - k) then
= Dx,c+ke (/2(e)i2 ...
fA,I =
[N(e)iNz)
0.
There is another vanishing condition in the case k = 0.
45
=
Dyc(#k,Jz)
Proposition 65. Let I = (i 2 , - - -, iN) be a multi-index satisfying
0 < dim(I) = 2i 2 +4i
Then /k,I
=
+(2N - 2)iN< 4N.
3+
0.
Proof. The classes pi(e) vanish in a neighborhood of the trivial connection by Proposition
46 , hence do not intersect the moduli space Ii-N 2 (N 1 ). Thus for a sufficiently long neck
they do not intersect the moduli space Mdim(I) (X).
El
3.2.2
A Simple Relation
In this section, we consider a 4-manifold X that contains a sphere a of self-intersection
-2. Such a 4-manifold can be written as X = Y UL 2 N 2 for some Y with DY = L 2 . In this
section we will use our understanding of the minimal energy moduli spaces on N 2 derive a
simple relationship between the invariants Dx,,+, and Dx,c.
Proposition 66. Let MO,2 be the moduli space containing the connection AO, 2
(N - 2)1 of Proposition 61. Then every connection A E MO,2 takes the form
=
A e A~ 1
e
A = B E (N - 2)1
for some rank 2 connection B.
Proof. Write A = Ei miAi as a sum of irreducible connections of dimension ni, each having
multiplicity mi. Then the orbit of A in M 0 ,2 has dimension
d = 22 + (N - 2)2
-
m2
On the other hand, by Proposition 56, the moduli space M0O,2 has dimension 4. Therefore,
mi > (N - 2)2
If one of the Ai has dimension at least 3, or if two of the Ai have dimension at least 2, then
EZ mi
N - 2, hence
Zmi
(Zmi)
2
< (N-2)2
The first inequality is strict unless there is a single term in the sum, in which case the second
inequality is strict. We conclude that A = BD C, where C is completely reducible. We know
that there is a reducible Bo = Ar E A' with the same energy as B. By Proposition 61, the
connection A' D A' D C has to be AO, 2 , so either B 0 = A E A- or B 0 = A'' ( 1. In the latter
case, B cannot be irreducible, because it lives in a moduli space of negative dimension. We
conclude that either A = A 2 ,0 or C is trivial, which completes the proof.
l
Proposition 67. The SU(2) moduli space MO,2 consists of a half-infinite segment [0, oo),
plus a union of finitely many circles and infinite segments (-cc, oo). The corresponding
46
components of the framed moduli space MO,2 are equivariantly diffeomorphic to T*(S 2 ),
SO(3) x S', and SO(3) x R, respectively.
Proof. The space of irreducible connections M0,2 is a smooth 1-manifold with two kinds of
ends. First, there is the reducible connection A E A- 1 . Second, there are limits where energy
where E is the trivial connection
slides off the end of N 2 . The latter take the form F#6,
and F is one of finitely many connections F 1 , .. . , F,, on L 2 x R. Exactly one component of
MO,2 joins A E A- 1 to one of the Fi#O. The other components are either compact (circles)
or they join two of the Fi#E).
To show that the component of the framed moduli space lying over [0, oo) is diffeomorphic
1
to T*(S 2 ), it suffices to show that the normal bundle to the orbit of A e A has degree ±2.
But this is clear, because the link of the orbit is SO(3) = L 2 .
We now prove the simple relation promised above. In the case N = 2 this result is due
to Ruberman [15].
Proposition 68. For any c E H 2 (Y) and any z E A(Y) we have
Dx,c(p2 (a)2Z) = ENDx,c,+(2z),
where
EN
is a universal sign given by
f+1
N = 2
-
N odd
EN
1
mod 4
Proof. First we describe how to compute the left hand side of the equation. From Proposi2
2
tions 61 and 56, we see that the framed moduli space MO,2 ,. has dimension 4r . Since A 2 (U-)
has degree 4, the only moduli spaces we encounter as we stretch the neck are Mo,o and M 0 ,2 .
Since M 0 ,0 only contains the trivial connection, it does not intersect any V(pi(o)). Hence
4
all intersections take place on an open subset of M2d+ (X) diffeomorphic to
U
=
MC[r 2] Xr MO, 2 ,
where IF = S(U(2) x U(N - 2)) is the stabilizer of rqO, 2 We can compute the universal bundle over MO,2 x -, using Proposition 46. Note that the
stabilizer of any connection in MO,2 is a subgroup of S(U(1) x U(1) x U(N - 2)). Hence
there are line bundles P and Q, and a rank N - 2 vector bundle F over Mo,2 , such that the
universal bundle is the projectivization of
V = (L ®P) ( (L-1 0 Q) e F.
Away from AO, 2 the stabilizers reduce to S(U(1) 2 x U(N)), hence the line bundles P and
Q become isomorphic. Writing p = c1 (P) and q = c 1 (Q), we conclude that p - q is supported
in a neighborhood of the orbit of AO, 2 . On the orbit of AO, 2 , which is a 2-sphere, P is a degree
one line bundle and Q is its inverse. Therefore,
((p - q) 2,7M o, 2 ) = +2
47
with the sign depending on how we orient MO,2 .
Now, the Chern character of V is given by
Ch(V) = exp(l) exp(p) + exp(-l) exp(q) + Ch(F).
From this formula we can compute a 2 (P) and pi2 (0). Slanting both sides by a, we obtain
12(a)=p
- q
from which it follows that Dx,,(pi2 (a)2 z) = ±2n, where c is a universal sign and n is the
signed count of points in M2dc[q 2] n V(z).
The left hand side is easier to compute. Indeed, the moduli spaces M2,2r all have positive
dimension, except for M 2 ,2 , which has dimension 0 and consists of a single connection A 2 ,2 .
Stretching the neck, we conclude that the moduli space Mid(X) n V(z) consists of a single
connection for each point in M2dc[772 ] f V(z). Therefore,
2Dc+,(z) = t2n = EDX,c(p2(U)2Z)
for some choice of sign e.
To compute E, let Y be a 4-manifold with Dyc(z) = 1 for some c and z. Let X be the
manifold obtained by blowing up Y twice. Then we have
Dx,c+ke1+le2 (
#k
for some nonzero integers
Dc+kel+ke2 (/L2(el
-
and
2
(eAk(N-k)
#1. The
2
(e)l(N-lz) =
k1
result we have proved so far shows that
2
e2 ) 2 pA
2 (el + e 2 ) k(N-k)-2)
=
2
=
EDc+(k+)e+(k-1)e
2
2EDc+(k+)el+(k-1)e
(/.12(el
2
(/p2(el
+ e 2 )k(N-k)-2)
+ e 2 )(k+1)(N-k-1)+(k-1)(N-k+1))_
(3.8)
Assume that k is strictly between 0 and N, and expand out both sides. For dimension
reasons there is only one nonzero term on either side. Examining this term shows that
-sign('32)
Since
#%and
= E - sign( 3 k+1k-1)
31 are both positive we conclude that
sign(#k)
=
On the other hand, we also have
3k
( )k(N-k)Z)
=
Dc+ke(P
=
Dc-ke( 1 2 (e)k(N-k)Z)
=
-ENDc+(N-k)e(P2 ek(N-k)Z)
=
(-1)(k(N-k)+l)
2
48
ENN-k-
Combining these two pieces of information we see that
(-e)J+[Y&J
(_l)(k(N-k)+l)EN-
for all k, hence E = 1. If N is congruent to 2 mod
If N is odd then the right hand side is +1
4 then setting k = 1 gives e = EN = -1.
3.2.3
=
Partial Blowup Functions
Using the simple relation of Proposition 68, we can extend Proposition 62 to arbitrary
values of d.
Proposition 69. Let Y be a 4-manifold with bj > 1 and let X be its blowup. Then for
every integer k there exists a formal power series Bk(t) = Ed bk,dL E H*(BPU(N); Q[[til)
such that for any c E H2(Y) and any z E A(Y) we have
Dx,c+ke(exp(P2(e)t)z) = Dx,c(Bk(t)z)
Proof. Let X be the manifold obtained by blowing up Y twice, and let el, e 2 be the exceptional spheres. Then the simple relation tells us
Dx,c+kei+ke2 (I2(ei--e 2 ) 2 exp(tP 2 (e 1 )+tA2 (e 2))) =
2
ENDx,c+(k+1)e+(k-1)e 2 (exp(ty 2 (ei)+tp2 (e 2 )))
(3.9)
Expanding in powers of t gives us an infinite series of relations between the invariants
Dx,c+ke(2(e)d). We want to see that these relations can be solved recursively, by induction
on d.
To see why a solution exists, it is convenient to think of the relations as a system of differential equations to be satisfied by the functions Bk(t). Solving these differential equations
order by order in t, given suitable initial conditions, is equivalent to using (3.9) to inductively
find acceptable values for bk,d. Explicitly, the differential equations we must solve read as
follows:
B'Bk - B B1
= ENBk+1Bkl.
When k is a multiple of N we have Bk(O) = +1, so examining the coefficient of td always
allows us to solve for bo,d and bN,d in terms of coefficients bk,e with e < d. It therefore remains
to solve for the rest of the bk,d in terms of bo,d, bN,d, and lower order terms.
Unfortunately, for 0 < k < N the equations are degenerate, because Bk(O) = 0. To
remove the degeneracy, we can write
Bk(t)
bktk(N-k)fk(t)
=
for some power series fk satisfying fk(0) = 1. By Proposition 63, we can take the coefficients
bk to be nonzero integers. Rewriting our differential equations in terms of the fk we see that
b2t 2 (fk'fk
-
fklfkl)
=
k(N
-
k)b2 f2
49
-
ENbk+lbk-lfk+lfk-l-
Setting t = 0 in this equation gives the identity
k(N
so we can eliminate b2 and
EN
k)b2
-
= ENbk+1bk-1
from the equations, obtaining
t2(fl'fk
fklfi)
-
k(N - k)(fk
=
-
fk+lfk-1).
This equation is still degenerate, but we can understand the degeneracy in a precise way.
Collecting terms of order td, we get a system of equations
fkd
= k(N - k) 2
- fk+,d -
fkd
(d -2)!
1k-1,d
d
+ lower order terms
which can be rewritten as follows:
2
fk,d - fk+1,d -
fk-1,d
2
d(d - 1)
2k(N - k) k'd
=k.
The left hand side can be viewed as a linear operator L : RN+1 _+ RN-1, and we must show
that this operator is surjective for fixed values of fo and fN. To do so it is sufficient to
restrict to fo = fN = 0, in which case L becomes a symmetric operator with corresponding
quadratic form
N-1
QM=
N-1
N(fk+
-
k=O
Q
It therefore suffices to show that
k=1
is negative definite for sufficiently large d. But
n-1
Q(f)
-
so
Q
kk
fk)2
N-1
dd-1
E 2(f+1
+ f )
(
I:
k=1
d(d - 1)
2
2k(N - k)J k
4
k(N - k)
k
will be negative definite provided
d(d -1)
2k(N-k)
> 4
for all k between 0 and N. This inequality is automatically satisfied if d > N. In the range
d < N, Proposition 62 already provides us with values for fk,d = bk,k(N-k)+d- Using these
values as initial conditions, we can solve for a blowup function Bk(t) to all orders in t.
0
Definition 70. Any function Bk(t) which correctly computes the invariants Dx,c(p (e)dz)
2
as in Proposition 69 is called a partial blowup function.
Note that we make no claims about the uniqueness of the functions Bk(t). However, the
difference of two partial blowup functions vanishes inside the Donaldson invariant of any 4manifold, so one could in principle prove uniqueness by finding 4-manifolds that distinguish
50
all classes in H*(BPU(N)), or by viewing the blowup function as taking values in a suitable
Floer homology theory.
Finally, it is worth remarking on the differential equations satisfied by the partial blowup
functions,
BI'Bk - BIBL = ENBk+1Bk.
If we make the substitution
Uk = log( fEN
Bk
then this system of equations becomes
d2
dt 2 - ek1k-ekk
which is known in mathematical physics as the Toda lattice. It describes a periodic chain
of particles, each of which interacts with its neighbors via an exponential potential function.
A relationship between the Toda lattice and the SU(N) blowup function was predicted by
Marifio and Moore in [12], so Proposition 69 can be seen as a confirmation of physicists'
expectations about the behavior of higher rank invariants.
3.2.4
Full Blowup Functions
We call the Bk(t) partial blowup functions because they only allow us to calculate invariants of the form Dx,c(p 2 (e)dz). The full SU(N) blowup function would be a power series in
N - 1 variables t 2 ,..., tN such that
Dx,c(exp(t2 P 2 (e) + --- + tnin(e)) = Dx,c(B(t 2 , ...
, tN))
In this section we will prove that a full blowup function exists for N < 4, and study the full
blowup function in detail when N = 3 and k = 0.
To extend the recursion of Proposition 68 to arbitrary classes pi(o), we again consider a
4-manifold X containing a sphere o- of self-intersection -2, and we write X as Y UL2 N 2.
Proposition 71. Let I = (i 2 , - - -, iN) be a multi-index with dim(I) = 2p < min(16, 4N).
Then there exist universal aij E H*(BPU(N); Q) such that for any c
z E A(Y) we have
Dx,c(p(o)'z) = [
E H 2 (Y, Y) and any
Dx,c(ai,jpi(o-)p (o-)z)).
i+jip
Proof. We proceed as in 68. In the moduli space M 2d+ 2 P(X) all intersections with V(z) take
place in
iU
= M2d+2p(y) [,
2
]
where IF = S(U(2) x U(N - 2)) is the stabilizer of
the projectivization of the vector bundle
Xr
MO, 2 ,
NO,2.
V = LP D L-1 Q D F,
51
As before, the universal bundle is
where P and Q are line bundles that are isomorphic away from AO, 2 .
The Chern character of V is given by
Ch(LP D L~ 1Q E F) = exp(l) exp(p) + exp(-l) exp(q) + Ch(F).
From this formula we can extract concrete expressions for ak(P) and pk(u), by writing the
classes ak(P) in terms of Ch(V). A nice way to organize the computation is to introduce
classes
(k - 1)!Ch (V)/-.
Vk = p(k-1)!Chk(-) =
Because the classes Chk generate HsU(N) we know that each vk is a linear combination of the
classes pi(o), with coefficients in HSU(N). Hence we can replace pi(a) by vi in the statement
of the theorem.
Explicitly, Vk is given by
(k
-
1)!Chk(V)/-
= Pk-1 -
qk-1 =
(p - q)sk-2,
where Sk-2 is the sum of all monomials of degree k - 2. In particular, each product vk v is
divisible by (p - q) 2 , hence is a multiple (over Hf) of the equivariant fundamental class of
T*(S 2 )_
We must therefore show that the classes sisj form a basis for H* over HsU(N). To prove
this, observe that BF fibers over BSU(N), with fiber Gr 2 (CN). It therefore suffices to show
that the sisj restrict to a basis for the cohomology of Gr 2 (CN). But it is well known that
the Schur polynomials
,
p -pq
form a basis, and these can be expressed in terms of the sisj by the identity
sk,1 = skS+1
-
Sk+1Sl
Because of the restriction on dim(I), Proposition 71 only produces 9 identities, which can
be derived from the following relations between the vk:
Degree 8
v2
=
4v 2v 4
Degree 10
-v7-
=
3v 2v5 - 2v 3 v4
Degree 12
vTv 4 =
22
vl
v2v3v4
6v2v6 - 15v 3 v 5
=
v7-
2v2v7
=
-
-
2v4
3v 3 v 6 + v4v5
v 4 v5
-3v2v 7 - 9vv
32v1=
v2v3
v42
2v 2 v 6 + v3 v 5 +2z2
=
v5
3v3
U2v6- 3v v5 +
v2=
Degree 14
-
6
+ 9v 4 v5
5v2v7-9v3v6+55v4v5
=
52
We can replace each vi above with a combination of pi(o-) using the following table:
k
(k - 1)!Chk
Vk = /L(k-1)!Chk (a)
2
-a
-L
3
a3
A3(0r)
4
5
a - a4
-a 2a 3 + a5
a2 P2 (0) -
6
-- !a3 + 1a2 + a 2a 4 - a6
(a4 - a2)[ 2 (0) + a 3P 3 (0-) + a 2
2
2 (0-)
4(-)
a 3P 2 (0-) - a 2P3 (0-) +
s(0-)
4
(U) - /1
'
In the case N = 3 there only 2 identities, which we write out for convenience.
Corollary 72. Let N = 3. Then for all c
Dx,c(a 2 (o-)4 z) =
Dx,c(P 2 (0) 3 i' 3 ( u-)z) =
E H2(Y) and all z E A(Y) we have
-4Dx,c(a 2 i 2 (0-) 2z) - 3Dx,c(P 3 (0-)2 z)
-3Dx,c(a112 (o) 2z) - Dx,c(a2A2(0-)P3(U-)z)
(3.10)
(3.11)
We can now prove the existence of a full blowup function when N < 4.
Proposition 73. Let N be 3 or 4. Then there is a formal power series B = B(t 2 , ...
2
H*(BPU(N);Q[[t 2 , . . . ,tN1]) such that for all c E H (Y) and z G A(Y) we have
Dx,c(exp(t 2 [t2 (e)
+ - -- + tNIpN(e))z) = Dx,c(B(t 2 ,.
. .
,tN)
E
, tN)z)
Proof. We proceed as in Proposition 69, by blowing up twice and using the identities produced in Proposition 71. The strategy of the induction is to eliminate all powers of p3(e) and
N 4 (e), thereby expressing the full blowup function in terms of the partial blowup functions.
For example, suppose N = 3. Then B(0) = 1, and B must satisfy two differential
equations:
3(BB 33 - B 3 B 3 ) = -B
B2223B - 2B
223
B 2 - 2B
222
22 22
B + 4B 222 B 2 - 3B 22 B 22 - 4a 2 (B 22 B - B 2 B 2 )
B 3 + 3B 2 2 B 2 3
=
-3a
3 (B 2 2B
- B 2B 2 ) - a 2 (B 23 B - B 2 B 3 )
Expanding these equations to all orders in t, it becomes clear that we can solve for all
partial derivatives of B in terms of B 223 ,B 2 3 ,B 3 , and the coefficients of the partial blowup
function that are already known. In fact, the vanishing theorem of Proposition 65 shows
that B 2 2 3 = B 2 3 = B 3 = 0, so knowledge of the partial blowup function is sufficient.
The case N = 4 is similar, but involves using 8 of the 9 equations from Section 3.2.4. E
We have not attempted to prove the existence of a full blowup function Bk(t2, ... , tN) here
because the vanishing of Bk(0) means the equations are degenerate, and a precise analysis
of the degeneracy is quite complicated.
53
3.3
3.3.1
The Case N=3
Theta Functions on a Genus 2 Jacobian
Definition 74. Let A c C9 be a full lattice. We say that an entire function 0 : C9 -+ C
is quasiperiodic with respect to A if for every A E A there exist constants aA E Hom(C9, C)
and bA E C such that
O(x + A) = eax+b\O(x)
for all x E C9.
For example, any Gaussian f(x) = exTAx+bTx+c is periodic.
function
6(z) = E e"inTn e27inTz
So is the Riemann theta
nEZ9
if the lattice takes the form A = Z9 D 7Z9 for Q a symmetric matrix with positive definite
imaginary part. If two quasiperiodic functions have the same automorphy factors aA,bA,
then any linear combination of them is quasiperiodic, with the same automorphy factors.
The product of any two quasiperiodic functions is quasiperiodic, with the product of the
two automorphy factors. A linear map followed by a quasiperiodic function is quasiperiodic,
with respect to a different lattice.
Note that we can regard any quasiperiodic function as a section of a certain line bundle
over C/A, whose transition maps can be derived from the automorphy factors. We will use
capital letters to denote the line bundle corresponding to a set of automorphy factors, so if
6 is quasiperiodic, we will say that it is a section of the line bundle E.
Definition 75. Let 9(x) be an analytic function on C9. We define the Hirota differentials
0[I](x) to be the unique functions such that
(x +t)(x -t)
t0[I](x)
where the sum ranges over all multi-indices I=
0[I](x) =
(ii, ... , ig). In other words,
0(x +t)(x -t)t=0
Proposition 76. The Hirota differentials of an arbitrary analytic (or Co)
the following properties:
function 6 satisfy
1. 0[I] = 0 if I has odd degree.
2. 0 [ij] =2ai log 0
3. 0[ijkl] = 2 aijkI log 0+ 4 (O4j log 0 - Oki logO +
0
Proof. Straightforward computation.
54
log - O10
log 0+
log 0
k
log 0)
Proposition 77. Let 9 be a quasiperiodicfunction with respect to a full lattice A C C9, so
that it defines a section of a line bundle E. Then for any fixed t E C9 the function
#t(x)= (x +t)(x -t)
is a section of 2E.
Proof. By definition, sections of
tities
E
correspond to quasiperiodic functions 0 satisfying iden-
4(x + A) = ea\x+b\b(x)
for every A E A. Thus sections of 2E correspond to those satisfying
O4x + A)
=
e2 a,\x
2
b(OW
It is then straightforward to check that this identity is satisfied by
#t.
Corollary 78. If 9 is a quasiperiodic function, then its Hirota differentials 0[I] are all
sections of 2E.
Proof. Since 9(x + t)9(x - t) is quasiperiodic in x, so are all its derivatives with respect to
0
t.
Proposition 79. Let E be a curve of genus 2, and let 9 be the Riemann theta function on its
0
Jacobian JE. Then the Hirota differentials 0[I] with deg I = 0,2 form a basis of H (JE, 2E).
Proof. Translating and multiplying by a Gaussian, we can assume 9 is an even function
whose Hessian vanishes at the origin. In this case it suffices to prove that the 3 components
of the logarithmic Hessian of 9 are linearly independent. If there were a linear relation, then
we would have oo9, log 9 = 0 for some vectors v and w. But this implies that an irreducible
component of the zero locus of 9 is invariant under a 1-parameter group of translations,
U
which is absurd since the zero locus is a genus 2 curve.
Corollary 80. The Riemann theta function on JE satisfies a system of differential equations
of the form:
9[ijkl] = 2Aijkli
2
Ci
+
[pq]
pq
or equivalently,
Pijk + 2 (PijPki + PikPjil + PiPik) = Aijkl +
ij Pq
pq
where P, = a1 log 9.
where # is a
Proof. By Proposition 76, we know that the left hand side is a ratio
section of 2E. By Proposition 79, # is a linear combination of Hirota difkerentials of order
< 2. Dividing by 92 and applying Proposition 76 again, the system of equations follows. U
55
Observe that if 6(x) satisfies a system of the above form, then so does
eXTAx+bx+c
(Mx + t)
for any matrices A, C, covector b and constants c, t. In other words, multiplying by a
Gaussian and precomposing with an affine transformation preserves the form of the equations. Ignoring this ambiguity, we can show that the Riemann theta function is characterized
by the above system of equations. More precisely:
Proposition 81. Suppose that 0 is a function whose logarithmic derivatives satisfy a system
of equations of the form:
Pill, + 6P
1
+ 6P1 1 P 12
+ 2P 1 1 P 22 + 4P 2 = A11 2 2 + G 4 P1n - 2G 3 P 12 + G 2P 22
P1222 + 6P 12 P 22 = A 1 22 2 + G5 P1 - 2G 4 P 12 + G3 P 22
P 2222 + 6P22 = A 2 22 2 + G 6 Pnl - 2G 5 P 12 + G 4 P 22
P 11 12
P1 122
= All,, + G2 P - 2G 1 P 12 + GoP 22
= A11 12 + G3 P1 - 2G 2 P 12 + G 1 P 22
that the coefficients AijkI are given by:
All,, = 1 (GoG4 - 4G1G + 3G 2
1
(GoG5 - 3G 1 G4 + 2G 2 G3 )
6
1
1 (GoG 6 - 9G 2 G4 + 8G 2
1
- (G1G6 - 3G 2G5 + 2G 3 G 4 )
A 1112
= -
A 11 2 2
=
A1 2 2 2
=
A2 2 2 2 =
6
12
3 (G2G6-
2GG5+ 3G4)
and that 0 is not itself a Gaussian. Suppose, moreover, that the homogeneous polynomial
G(x,y)=
2
G ij
i+j=6
has exactly 6 nondegenerate zeroes in Pl. Then, up to a multiplying by a Gaussian and
precomposing with an affine transformation,6 is the Riemann theta function on the Jacobian
of the hyperelliptic curve
y2 = g(x) = G(x, 1)
Proof. This is proved in Baker [2]. To give some sense of what is involved in the argument, we
will explain how to recover the hyperelliptic curve. The idea is as follows. First differentiate
each of the 5 equations above with respect to each of the two variables. This yields a set of
equations like:
P11112 + 12P1 1 P11 2 = G 2Pn2 - 2G 1 P122 + G0 P222
56
If we eliminate all 5th derivatives, we are left with system of 4 equations of the form:
MilP111 + Mi2P11 2 + Mi 3P122 + Mi4 P222 = 0
where the coefficients Mij lie in H 0 (JE, 26). Since 0 is not a Gaussian, at least one of
the Pik is nonzero. Hence the 4 sections of H 0 (JE, 26) satisfy a nondegenerate degree 4
equation,
det M = 0
This is the equation of the Kummer surface in P3 . When we intersect it with the plane at
infinity (i.e. the image of the theta divisor) we get a doubled conic. The Kummer surface
has six singular points along this conic, which an explicit computation shows are precisely
El
the roots of G(x, y).
We also need to make a remark about initial conditions. Clearly the function P is determined by P1 (0) for Il < 3. But as a consequence of Proposition 81, these initial conditions
are subject to the constraint that the second derivatives must lie on the Kummer surface.
There is, of course, no constraint on the zeroth or first derivatives, since they can be modified by multiplying by an exponential, which does not change the equations. Finally, it is
proved in Baker that the third derivatives can be expressed in terms of the lower derivatives,
which gives some actual constraints on the set of valid initial conditions. In the case where
the initial conditions are finite and specify a singular point of the Kummer surface, these
constraints simply say that the function 9 is (up to third order) an even function times an
exponential.
3.3.2
The Blowup Function
Theorem 82. The SU(3) blowup function is a quasiperiodicfunction on the Jacobianof the
hyperelliptic curve
2
y2 = (X3 + a 2 x + a 3 ) - 4.
Proof. By Corollary 72
+ 3B22B22
- B2 22B 3 + 3B 22 B 2 3
B2222B - 4B 2 2 2 B 2
B2 223 B - 3B223B2
and setting
Q=
=
=
-4a 2 (B 22B - B 2 B 2 ) - 3(B 33 B - B3B3)
-3a 3 (B 22 - B 2B 2 ) - a2 (B 23 B - B2B3)
log B we get
+ 6Q2 2
= -
4a 2Q 2 2
+ 6Q22Q23
= -
3a 3Q 2 2
Q2222
Q2223
-
3Q33
a2Q23
which look similar to the first 2 of the 5 equations satisfied by the Riemann theta function.
In fact, if we suppose that Q satisfies such a system of 5 equations, there is a unique choice
of coefficients for the remaining three equations that is consistent with the initial conditions
derived in Section 4. In total the five equations read:
Q2
+ 6Q$2 =
-
57
4a 2 Q 2 2
-
3Q33
+ 6Q22Q23 =
Q2233 + 2Q22Q33 + 4Q23Q23 =
Q2333 + 6Q23Q33 =
Q2223
Q3333 + 6Q
+
a Q 23
-
2
a 2a 3Q 22
-
= - 2a
3
-
a2Q23
3a3Q 23 -
3a3 Q 22
-
(12 + 3a)Q
2-
+ 2a 2a 3Q 23
22
a 2 Q 33
- 3a3 Q 33
+ a2Q 33
These equations are not quite in the form required by Proposition 81. However, we can
bring them into that form if we multiply B(t 2 , t3 ) by a suitable Gaussian. Specifically, we
define
O(ti, t 2 ) = exp(
3
2
9
1- t
a 2t2 + -a 3t2 t 3 10
20
10 a2t )B(t 2 , t 3 ).
Writing P = log 0 and applying we get the system of equations
P2222
P 222 3
9 2-
+ 6P222 =
+ 6P 22 P 2 3
52 22
25a2 -
27
-a
50
=
P 22 33 + 2P 22 P33 + 4P22
3 =2 + -a
P=
P3333 + 6P33 =
8
a
- a 2a3 1 a_(
25
a
2
+
3
33
P2 2 +
4
4a 2 P 23
5
a2 P 22 +
2
3 a3 P 23 -
5
2 a 2 P 33
a2 a 3P22 +
2a2P23 -
0aP33
)P22 + 2a 2 a 3P23 -
5 2P33
-
25
P2333 + 6P 23P33 =
3
10
-a
2 a3 -
- -a3
50
3P 33
a225-
21a
3
5
The coefficients Gi now clearly take the form of Theorem 81, and a straightforward but
tedious computation shows that the coefficients Aijkl satisfy the necessary conditions as well.
It is also straightforward to check that the initial conditions for 9 correspond to a singular point of the Kummer surface, and since the first derivatives vanish the integrability
constraints simply say that the third derivatives vanish as well (which they do). Thus the
SU(3) blowup function can be equivalently defined as the unique solution of the complete
set of 5 equations above, with the initial conditions derived in Section 4.
From Proposition 81 we now conclude that the SU(3) blowup function is (the Taylor
expansion of) a quasiperiodic function on the Jacobian of a certain hyperelliptic curve E.
We can compute E explicitly using Theorem 81. It is given in hyperelliptic form by the
equation:
y
2
=
g(x)
=
-
6
- 6
4
X4
a 2 4 !2 !
x3
3
10O 3 3 !3 !
2
X2
10 a2 2 !4 !
-
2
x
a31!5!
-16
2 + 2a a X
= (x6+ 2a 2x 4 + 2a3 X3 + a2X
2 3 + a2+ 4)
= 240sult
p to
,
which is the desired result up to isomorphism.
58
3a2+12
6!
Chapter 4
The Formal Picture
4.1
Axioms
A formal Floer theory HF over the complex numbers consists of the following data:
D1 A graded-commutative, finitely generated, regular C-algebra A, supported in even
degrees.
D2 For each connected, oriented 3-manifold Y, a relatively Z-graded, absolutely Z/2Zgraded A-module HF(Y), on which A acts by even endomorphisms.
D3 For each connected, oriented, homology oriented 4-manifold W with oriented boundary
OW = Y - Y- - Y, and a graded, A-multilinear map
HFw :
0HF(Y
) 9 A(W; A)
HF(Y).
-
j=1
If n = 0 we interprete the empty tensor product as the 1-dimensional vector space k.
D4 For each connected, oriented, homology oriented 4-manifold X with oriented boundary
- Y, satisfying b'(X) ;> 2, a graded k-multilinear map
aX = -Y - Y2n
Dx:
0HF(Y)
9 A(X; A)
-
k
j=1
These data should satisfy the following axioms:
Al If # : HF(Y)
-+
HF(Y2) is a diffeomorphism, then the trivial cobordism from Y
to Y 2 induces an isomorphism HF4 : HF(Y) -+ HF(Y2 ). If # : W1 -+ W 2 is a
diffeomorphism of cobordisms, then HFw, and HFw
2 are intertwined by the action of
# on HF(Wi) and A(Wi, A).
A2 Reversing the homology orientation on W multiplies HFw by a factor of (-1)', where
E is the Z/2Z grading on Hom(O"_HF(Y), HF(Y)).
59
A3 Let OX = -Y-Y1-----Yn, OW =Y-Yni-- Ym, let[p E A(X;A), and
v E A(W; A). Let X be the 4-manifold obtained by gluing X and W along Y. Then
we have
HFy(pv) = HFx(p) o HFw(v)
and likewise
Dy(pv) = Dx(p) o HFw(v)
provided that X and W satisfy the conditions necessary to define both sides of these
equations.
A4 For every Y, HF(Y) is a finitely generated A-module.
A5 The map HF4 : A
-+
HF(S3 ) is an isomorphism of Z/2Z-graded A-modules.
It is worth clarifying some consequences of A5. There is an evident cobordism Z
3
_ S3 given by removing two 4-balls from B 4 . This cobordism induces an associative
multiplication HF(S3 ) 0 HF(S3) - HF(S3 ), giving HF(S3 ) the structure of an algebra
over the ring A. From A3 we conclude that the map HF4 is not just an isomorphism of
A-modules, it is also an algebra isomorphism. The fact that HFB4 respects the Z/2Z grading
shows that HF(S3 ) is supported in even degrees.
S3+S
4.2
The Formal Blowup Formula
The axioms of a formal Floer theory imply the existence of a blow-up formula for the
relative invariants Dx. To see how this comes about, we need to recall some general facts
from linear algebra.
Definition 83. Let V be a finite dimensional vector space over a field k. We define the ring
of regular functions on V to be the k-algebra
00
k[V] = Sym*(V*)
Sym"(V*)
=
n=O
and we define the ring of formal power series on V to be
00
k[[V]]
=
Symn(V*).
n=O
Equivalently, we could define k[[V]] to be the completion of k[V] at the ideal J generated
by V* = Syml(V*). Note that if t 1 , ... , t9 are a basis for V* then the evident map
k[t 1 ,... ,t] -+ k[V]
is an isomorphism of k-algebras, and likewise for k[[t 1,..., tg]]. This justifies viewing k[V]
as a ring of functions on V.
60
Proposition 84. If k has characteristic0, then the formal power series ring k[[V]] is naturally isomorphic to the dual of the vector space Sym*(V).
Proof. It suffices to give a duality between Symn(V*) and Symn(V)*. Define e : Symn(V*) -
Symn(V)* by
(e(fi
...
fn), v 1 - V2 -V)
fi(vo(i))
...
fn(Va(n))
To show that e is bijective, let vi,..., v be a basis for V and let v*, ... , v* be the dual basis.
Then for any pair of multi-indices I = (ii,..., in) and J= (ji, ... ,jn) we have
FI J
(6 (VAv*) = |I1
where IF, C S, is the stabilizer of I. Since E can be represented by a diagonal matrix with
0
nonzero entries, hence is invertible.
To understand the meaning of the map e, observe that
(f), v&
f(v) =
n!
0V
for any f E Sym"(V*) and any v E V, where the left hand side is defined by viewing
homogeneous polynomial on V. More generally, for f E k[[V]] we have
f(v) = (e(f), exp(v)) ,
f
as a
(4.1)
provided we interpret both sides as formally converging infinite sums.
We now generalize the above discussion slightly. Let A be a commutative Noetherian
k-algebra, and write B for the corresponding scheme Spec A. Let V be a finitely generated
projective module over A. Viewed as a sheaf of modules over B = Spec A, V is locally free,
hence elements of V can be viewed as global sections of a vector bundle V over B. Explicitly,
this vector bundle can be described as the spectrum of the k-algebra
A[VI
Sym'(HomA(V, A)),
=
n=o
and the ideal J generated by terms of positive degree cuts out the zero section B
Completing at J yields the formal power series ring
- V.
00
A[[V]] =
Symn (HomA(V, A)),
n=O
which we can view as the ring of functions defined on a formal neighborhood of the zero
section.
Proposition 85. If k has characteristiczero, then the map c : A[[V]] -+ HomA(Sym*(V), A)
is an isomorphism.
61
Proof. Locally on Spec A, V is free and the map e is an isomorphism by the proof given
above. Since e is a local isomorphism, it is an isomorphism of sheaves, so it induces an
isomorphism on global sections as well.
For example, let Qi(A) denote the module of differentials over A. By definition, QG(A) is
the target of a universal derivation
d: A -+Q(A).
If A is regular then Q (A) is a finitely generated projective module, and the spectrum of
the ring A[QV(A)] is the cotangent bundle T*B. The spectrum of the formal power series
ringA[[Q1 (A)]] is a formal neighborhood of the zero section in T*B, which we will denote
by T*B.
We will now show that any formal Floer theory admits a blowup function, which can be
naturally viewed as a function on T*B. Consider the 4-manifold B 4 obtained by blowing up
the origin, and let e denote a generator for H 2 (B 4 ). Because H 1 (B 4 ) = 0, we have a Liebniz
identity in A(B 4 ; A):
Pab(e) =
which induces a map Q1 (A)
-+
ia(e)b + apb(e),
A(B4; A). We can therefore interpret HFg as a map
HFg : Sym* (
1
(A)) -+ HF(S3 ).
By assumption D1, A is regular, hence Q1 (A) is a locally free A-module. By A5 the
natural map A -4 HF(S3 ) is an isomorphism, so we can view HFj as an element of
HomA(Sym* (Q1 (A), A). We can then use Proposition 85 to identify it with a formal power
series # E A[[Q'(A)]].
Definition 86. The blowup function of a formal floer theory HF is the formal power series
# obtained from HFi by the above identifications.
Proposition 87. Let X = X#p 2 be the 4-manifold obtained by blowing up a 4-manifold
X, let w E '7I(A), and let z E A(X; A). Then we have a formal blowup formula:
D2 (exp(we)z) = Dx(#(w)z)
Proof. View w as a section of the cotangent bundle T*B, where B = Spec A. Evaluating #
on w yields an element of A, or more precisely a formal sum with coefficients in A. Using
(4.1), we can characterize this formal sum by the property:
HFB4(83(w))
=
HFi (exp(w)).
Now write X = Xo o B 4 , where Xo is obtained by removing a 4-ball from X. Thus
Dg (exp(we)z)
= Dx0 (z) o HFij(exp(we)))
= Dx0 (z) o HFB4( (w))
=
Dx(0#(w)z)
62
4.3
The Blowup Algebra
Let Np be the disk bundle associated to a complex line bundle of degree -p over the
2-sphere S 2 . There is a unique orientation on Np that is compatible with the outward
orientation of S2 c R 3 and the orientation of the fibers induced by their complex structures.
Let Lp denote the inwardly oriented boundary of Np, so that Lp is diffeomorphic to the lens
space L(p, 1).
We can construct a cobordism Wp,q : Lp + Lq -+ Lp+q as follows. Let V denote the
3
3-manifold obtained by removing two small 3-balls from the unit 3-ball B . Orient the
boundary of V so as to make it a cobordism from S2 + S2 to S2. Then there is a unique
disk bundle Np,q over V that restricts to Np and Nq on the two incoming spheres. Note that
Np,q necessarily restricts to the disk bundle Np+q on the outgoing sphere. Let Wp,q denote
the inwardly oriented boundary of Np,q.
To be more precise in identifying Wp,q as a cobordism between Lp, Lq, and L,+q we
actually need to specify a diffeomorphism from each boundary component of Wp,q to the
corresponding lens space. However, the space of line bundle isomorphisms from Lk to itself
is just Maps(S 2 , Si), which is connected, so there is no ambiguity as long as we choose
diffeomorphisms which cover the identity map on S2 and act complex linearly on the fibers.
We also need a homology orientation on Wp,q. This is equivalent to picking an ordering
of the incoming 2-spheres in M, which we can do arbitrarily.
Proposition 88. The maps HFwpq : H F(Lp) 0 HF(Lq) -4 HF(L+q) induce the structure
of a nonunital graded-commutative A-algebra on
cc
HF(Lp).
A+ = (
p=i
Proof. By D3, HFwpq is A-multilinear, so we only need to check associativity and commutativity of multiplication. To verify associativity, observe that we can construct a cobordism
by
W,q,r : Lp + Lq + L, -+ Lp+q+r in perfect analogy with Wp,q. Associativity then follows
applying Al to the evident diffeomorphisms
Wp+q,r 0 Wp,q
> Wp,q,r
Wpq+r Wq,r.
To verify commutativity, observe that there is a diffeomorphism from V to itself which swaps
the two incoming spheres and restricts to an orientation preserving isometry on either one.
This diffeomorphism lifts to a diffeomorphism
> Wq,p.
Wp,q
Because it swaps the role of the incoming spheres, this diffeomorphism reverses the homology
orientation on Wp,q. By A2 this implies that the multiplication is graded commutative. LI
63
Note that we can turn the nonunital algebra A+ into an algebra by formally adjoining a
unit in degree 0. Let A denote the graded-commutative A-algebra obtained in this way.
Now let N* : Lp -+ S 3 be the cobordism obtained by removing a 4-ball from Np. The
direct sum of the maps HFN; induces a A-linear map i* : A+ -+ A.
Proposition 89. The map i* is an algebra homomorphism.
Proof. Consider the composition N*+q o Wp,q, which corresponds to first multiplying two
elements of A and then applying the map 0. We must show that this is diffeomorphic to
Z o (N* Li N*), where as above Z : S3 + S 3 -+ S3 is the cobordism obtained by removing
two 3-balls from B 4 . Equivalently, we must show that N+q 0 Wp,q is diffeomorphic to the
connect sum of Np and N.
To construct the diffeomorphism, let D 2 C V be a disk whose boundary is an equator of
the outgoing sphere, and which separates the two incoming spheres from each other. The
preimage of D 2 in the circle bundle W,q -+ V is the handlebody D 2 x S1. On the other hand,
the preimage of &D2 = S' in the disk bundle Np+q M S2 is the complementary handlebody
S' x D 2 . These two handlebodies glue together to form a 3-sphere, which separates NpqoWp,q
into two disjoint components. It is clear upon a bit of reflection that these components are
diffeomorphic to N* and N*. We conclude that Np+q 0 Wp,q is diffeomorphic to N#N,
as
desired.
Next we will explain a fundamental relationship between the algebra A and the formal
blowup formula. Before doing this, we need to make some remarks on the topology of the
cobordisms we have been using. Let o-, E H 2 (Np) be the homology class represented by the
zero section of the disk bundle Np -+ S 2 , and let 5, E H 2 (N, Np) be the relative homology
class represented by one of its fibers. By virtue of the decomposition Np#N = Np+q O Wp,q,
we can view Up+q as a homology class in H 2 (Np#N) = H 2 (Np) D H 2 (Nq). It is geometrically
evident that 3, - Op+q = q - U-, = 1, from which we conclude that U-+q = Up + Uq.
Imitating the construction of the blowup function, we can use op to map Q'(A) to
A(Np; A). Hence we can interpret HFN, as a map
HFN, : HF(Lp)
Let exp* : A+
-+
> Hom(Sym*(Ql(A)), A)= A[['(A)]]
A[[Q'(A)]] be the direct sum of these maps.
Proposition 90. exp* is an algebra homomorphism.
Proof. Let f, = exp*(ap) and fq = exp*(aq) where ap E HF(Lp) and aq E HF(Lq). Then
for any w E Q1(A) we have
fp(w) = HFN (exp(wo-p) 0 a,)
By virtue of the identities N,q
0
Wp,q = Z o (N* u N*) and Jp+q = Up + Uq we have:
exp*(ap)exp*(aq) = fp(w)fq(W)
= HFz(fp(w) 0 fq(w))
= HFz (HFN; (exp(wop) 0 a,) 0 HFN; (exp(wrq) 0 aq))
64
=
HFzo(N;UN*)(exp(wop) exp(wOuq)) 0 ap 0 aq
=
HFN*q oWP,q(exp(wOap +
=
HFN-q (exp(wap+q) 0 apaq)
=
exp* (apaq)
Wcrq)
0 ap 0 aq)
We can now explain the connection to the blowup formula. Note that N 1 is diffeomorphic
3
to the blowup B 4 , and N* is a twice punctured CP 2 . Let E = HFB4(1) generate HF(S )
HF(L1 ). Then we have
exp* E(w) = HFNi*(exp(wal) 0 HFB4(1))
so the blowup function is the image of
4.4
E
=
HFN
1 (exp(wu1)) = /(w),
under exp*.
Geometry of the Blowup Algebra
In order to proceed any further we will need to make some sort of assumption about the
ring A.
Assumption 1. The map exp* : A -+ A[[Q1(A)]] is injective and has dense image.
Assumption 2. The ring A is finitely generated.
One consequence of the first assumption is that each HF(Lp) is supported in a single
mod 2 grading. Thus A is a commutative algebra (rather than just graded-commutative).
We can therefore give it a geometric interpretation, by viewing Ap = HF(Lp) as the space
of global sections of a line bundle LOP on a variety S. More precisely,
Definition 91. We define S = ProjA(A) to be the scheme obtained by applying the relative
Proj construction to the graded A-algebra A.
Note that injectivity of exp* implies that A is an integral domain, hence S is irreducible.
By definition F comes with a map 7r : E -+ B = Spec A. The assumption that A is finitely
generated tells us that each fiber Eb = 7r-'(B) is actually a projective variety. In fact a basis
of HF(Lp) defines a projective embedding, for sufficiently large p.
Proposition 92. The homomorphism exp* induces a map exp : T*B -+ 9 of schemes over
B, and the map i* induces a section i: B -+ S
Proof. Since exp* has dense image there is an element f E A+ such that exp* f is a unit.
Let Af be the localization of A at any such f. Then S has a distinguished open subset Uf
isomorphic to Spec A 1 . The fact that i*f is a unit implies that exp* f is a unit, hence we
obtain a map exp* : Af -+ A[[Q 1 A]]. This induces a map of schemes expf : T*B -+ Uf. If
i*g is also a unit, then expf and expg both factor through expfg, hence we get a well-defined
map exp : T*B -+ C, independent of the choice of f.
Because i* is just exp* followed by evaluation on the zero section, we see that it similarly
E
induces a map B -+ S, which is a section because i* is a map of A-algebras.
65
Proposition 93. E is regular on a Zariski open set containingi(B).
Proof. Let P denote the kernel of i*, let Ap be the localization at P, and let Ap be its
completion with respect to P. Because exp* has a dense image, it induces an isomorphism
Ap -+ A[[Q 1 A]]. In particular, Ap is regular, hence E is regular in a Zariski neighborhood of
V(P) = i(B) as desired.
0
We will now justify our notation by showing that exp is obtained by formally integrating
certain commuting vector fields on S.
Proposition 94. Let FC : Ap 0 Ap -+ A 2p be a family of A-multilinear maps satisfying
(1) (Antisymmetry) F
, 4) = -FC(4', #)
(2) (Liebniz) FC(# 1q 2 , 0 1 0 2 ) = FC(0 1,7
1 )0 2 0 2
+ 0 10 1F&(4
2 , 02)
Then there is a unique vertical vector field , defined on the smooth locus of S, such that
0( = 0- 2 Fe(#,9)
Proof. The fact that is a derivation follows from the Liebniz property of FC. However, we
must show that
is well-defined, or equivalently that ((g) = (()
for any homogeneous
s E A. But this follows from the identity
FC(sO, so) =
Fe(, 4)s2 + #OFC(s, s)
= Fg(#,4')s 2 .
Proposition 95. Let a E A and let Va be the vertical vector field on T*B whose value
at (b, w) is da(b). Then there is a globally defined vertical vector field a on C such that
exp*(V) = G .
Proof. To construct a we only need to produce F 0 : Ap0 Ap -+ A 2 p satisfying the conditions
of Proposition 94. We define it by the formula:
F&a(0,
4)
= H Fwp,p(A a(Op -
Tr)q00)
where o-, and
T, are a basis for the homology of N 2 p o W,, = Np#NP, so that
, - r, is an
integral generator of H 2 (W,,,).
Antisymmetry is clear from the definition, so we only need to prove the Liebniz property.
The proof is cumbersome to write down, but only involves applying the identity of cobordisms
Wp+q,p+q o (Wp,q U Wpq) = Wp,q,q = W2p,2q 0 (Wp,
and the identity of homology classes
Pa(cTp+q) = Pa(cTp) + Pa(Orq).
66
U W,q)
f
Lastly, we check that
= exp* #. Then
'Za
To begin, let 0 E HF(Lp) and let
is compatibile with Va.
Va(f)(W)
t=o
f(w + tda)
d
HFN (exp((w + tda)up)
dt
Writing
f
=
HFN; (da ueXp(wa,)b)
=
H FN- (P a(up) ex p(wao)
= exp* q and g = exp* 4, and applying the identity of cobordisms
N2p 0 WA, = Z o (N* L N*)
we therefore have
exp* F .(0, 0)
= Va(f)g -
from which we conclude that exp,(Va)
f Va(g) = FVa(f, 9)
= 'a.
Proposition 96. The vector fields 'a are mutually commuting.
Proof. Let a, b E A. In terms of F a and Fb we must show that
F
a(Fib(0,
4'),
4'2) = F6 (F&a(0,
4'), 42)
But the left hand side of this equation is equal to
HFw,,, (Aa ((
- U2) )Pb((
1
)-
(4) )OOV)'4
which is evidently symmetric in a and b.
4.5
Analytic Continuation
In this section we study the convergence properties of formal power series in the image
of exp*. More precisely, for a fixed point b E B and any f E HF(Lp) we consider the
function F = exp* f as formal power series on T*B, and show that it has an infinite radius
of convergence.
Since we will be working locally on B, we choose a set of functions a,, ... a,, E A such
that dai(b), ... , da,(b) form a basis for T*(B). We can then view each F as a formal power
series in n formal variables,
tI
+- -+
F(t da
1
where for each multi-index I
= (ii,,
tdan)=
ik) we
67
F1(O),
have written the I-th partial derivative of F
as
Ok F
&il
...
tik
Note that not all partial derivatives F, lie in the image of exp*, because
on E. However,
f is not
a function
Proposition 97. Let f E HF(Lp) such that F = exp* f is a unit. Then for any i and j,
the power series
G = FjF - FF
is equal to exp*g for some g - HF(L2p).
Proof. Simply take 2g = HFw,,,(1it(op - Tp)pj(op - Tp). Then
exp*(2g)
HFN
2, o HFw,,,(pi(up - Tp)Ij(0p - Tp) exp(- 2pw))
= 2HFN, (pi(up) j (o) exp (up))HFN, (exp(wTp))
=
-2HFN,(pi(up) exp(wo-p))HFN(tj (p) exp (wTp))
= 2FF- 2FFj = 2G
as desired.
Proposition 98. Let f E HF(Lp) and let F = exp* f. Then F has a positive radius of
convergence.
Proof. For fixed b E B the formal map exp has a positive radius of convergence, because it
is given by integrating some holomorphic vector fields. In particular, if g, h E Lq and exp* h
is a unit then the G/H = exp*(g/h) has a positive radius of convergence.
Assume F is a unit. In this case, for any i and j we have
j~j log(F) =
F-13F - 2 FFF
G
2
= exp*(g/f
2
),
for some g E HF(L2p). Therefore log(F) has a positive radius of convergence, hence so does
F.
Now drop the assumption that F is a unit. We know that for some n > 0 there is an
h E Lnp such that H = exp*(h) is a unit. But then F" - H = exp*(fn - h) has a positive
radius of convergence, and H has a positive radius of convergence, hence so does F", hence
so does F.
0
To show that each F has an infinite radius of convergence, we use a doubling formula,
which takes a form familiar from the theory of theta functions.
Proposition 99. Let fl, f2 E HF(Lp) and let F = exp* f2 . Let G 1 ,... , Gn be a basis for the
image of HF(L2p) in C[[T*(B)]]. Then there are constants ci3 such that for all t, s E T*B
we have
F1 (t + s)F 2 (t - s) = EcGi(t)Gj(s),
68
Proof. First let 91,... , gm be a generating set for HF(L2p) as a A-module. We can assume
without loss of generality that exp* gi = Gi for i < n, and exp* gi = 0 for i > n. Then
F(t + s)F(t - s) = HFN,#N,(exp(t(op+ irp)) exp(s(op - rp))fif 2 )
= HFN
2)
2,(exp(t(op + Tp)))HFwP,,(exp(s(-p - -r))fif
=
Hi(s)Gi(t),
where the formal power series H(s) are defined by the identity
HFw,,(exp(s op -
Tr)fif
2
)
=
H3 (s)gj.
(
Switching the role of s and t, we also see that
F 1 (t+ s)F 2 (t - s)
=
Gi(s)Ki(t)
for some formal power series K(t).
Now, because the functions Gi are linearly independent, we can find multi-indices Ij such
that the matrix
0
=
~i
is invertible. Therefore, the identity
0I'I
F(t + s)F(t - s)=
H (s)Mij
Gi(s) atj7
=
Ki
shows that each Hi(s) is a linear combination of the power series Gi(s). Therefore,
F(t + s)F(t - s) =
Gi(t)Hi(s) =
3 ciGit)Gj(s)
ij
i
as claimed.
Proposition 100. For any f E HF(Lp), and any b E B, the power series F = exp*(f) has
an infinite radius of convergence.
Proof. Let h, ... , hn be a generating set for the algebra A, and suppose that not all of the
power series H 1 , ... , Hn E C[[Tb*(B)]] have an infinite radius of convergence. Let r be the
minimal radius of convergence over all the Hi. Then every function in the image of exp* has
radius of convergence at least r. But for any F in the image of exp*, the doubling formula
shows
cGi (t)Gj (0),
F(2t)F(0) =
iji
from which we conclude that F has radius of convergence at least 2r, provided F(0) 5 0.
Using the trick from Proposition 98, we conclude that the same is true even if F(0) = 0. In
69
particular, each of the generators Hi has radius of convergence at least 2r. This contradicts
the definition of r, and shows that in fact each Hi has an infinite radius of convergence. El
Proposition 101. exp* induces an entire analytic map exp: TbB
-+ Sb
Proof. The fact that every function F = exp* f is entire shows that we have an entire
analytic map from TbB to the affine cone Spec A. The only issue is that there might be a
point to E TbB where all the functions F E A+ vanish simultaneously. But then the doubling
formula tells us
F(to + s)F(to - s) = 0
for all s, so one of F(to ± s) vanishes identically on T*B, so F vanishes identically on TbB.
But this contradicts the fact that some F is a unit in A[[Q 1 (A)]].
4.6
Periods and Theta Functions
We would like to say that every function in the image of exp* is a theta function. However,
this will not generally be the case because some fibers of 7r : S - B might be toric varieties,
or even compactifications of an affine space. However, when exp : T*B -+ S does have some
nontrivial periods, we can determine the automorphy factors of the functions exp* f with
respect to the periods.
Definition 102. For any b E B, we define Ab to be the set of all A E Tb*B such that
exp(A) = exp(0).
Proposition 103. Ab is a discrete subgroup of T*B. Moreover, the fiber exp- 1 (x) over any
x E Eb is a coset of Ab.
Proof. Suppose that exp(A) = exp(p) for some A, p E T*B. Then for all sufficiently large p
there is a nonzero constant mp such that G(A) = mpG(p) for any G = exp* g, g E HF(L2 p).
By the doubling formula we have
c 3G
Gi(t )Gj (A) = mpF 1 (t + p)F 2 (t -
F1 (t + A)F 2 (t - A) =
i~j
for any
fl, f2
E HF(Lp). Therefore,
F1 (t + A)
F1 (t + y)
_
F2 (t - p)
F2(t - A)
where up(t) is a meromorphic function that is independent of F. For any to there is some
choice of F such that F (to) 4 0, so u(t) is actually holomorphic, and there is also a
choice of F such that F(to + A) # 0, so up(t) is nonvanishing. We can therefore write
up(t) = exp(hp(t)) for some entire function hp(t).
Now drop the assumption that p is large. Then for any f E HF(Lp) we can still find an
integer N and an entire function h such that
F(t + A)N = exp(Nhp(t))F(t + p)N
70
and taking N-th roots of both sides we get
F(t + A) = exp(hp(t))F(t + p)
It is easy to see that the function hp(t) depends only on p and not on F, and that hp(t) = ph(t)
for some h(t).
To show that Ab is a subgroup, suppose that exp(A) = exp(pt) = exp(O). Then there is a
nonzero constants mA,p and m,,, such that for F = exp*(f), f E HF(Lp), we have
F(p + A) = mA,pF(p) = MApMppF(0),
hence exp(A + M) = exp(O).
To show that any fiber of exp is a coset, observe that
F(- p + A) = exp(hp(- p))F(0)
for any F, so A - p E AbThe fact that Ab is discrete is just a consequence of the fact that some F(t) is nonconstant.
In the course of proving the previous proposition we have also shown that for any A E Ab
there is an entire function hA (t) such that
F(t + A) = exp(hA(t))F(t)
for all F E exp(HF(Lp)).
Proposition 104. For every A E Ab the function hA(t) is linear.
Proof. For any F,G E exp(HF(Lp)), the ratio F(t)/G(t) is periodic, because the numerator
and denominator transform in the same way when we translate by a period A. In particular
the second logarithmic derivative of any F(t) is periodic.
Now, taking second logarithmic derivatives of both sides of the equation
F(t + A) = exp(phA(t))F(t)
we see that
F3 F - FF
F
a2h
2
&j
+ FjF - FF
+ti
F2
Because the second logarithmic derivative of F is periodic, we see that the Hessian of hA(t)
0
vanishes identically. We conclude that hA(t) is linear, as claimed.
Corollary 105. If Ab is a lattice of full rank, then the blowup function is a theta function
on the complex torus T*B/Ab.
Proof. By definition, a theta function is a quasiperiodic function whose automorphy factors
are exponentials of linear functions. Since the blowup function is in the image of exp*, its
L
automorphy factors are of the form exp(hA(t)) for some linear functions hA(t).
71
72
Bibliography
[1] M.F. Atiyah, V.K. Patodi, and I.M. Singer. Spectral Asymmetry and Riemannian Geometry, II. Mathematical Proceedings of the Cambridge Philosophical Society, 78(3):
405-432, 1975.
[2] H.F. Baker. Introduction to the Theory of Multiply Periodic Functions. Cambridge
University Press, Cambridge, 1907.
[3] J.D. Edelstein, M. Gomez-Reino and M. Marino. Blowup Formulae in Donaldson-Witten
theory and Integrable Hierarchies. Advances in Theoretical and Mathematical Physics,
4: 503-543, 2000.
[4] S.K. Donaldson and P.B. Kronheimer. The Geometry of 4-manifolds. Oxford University
Press, New York, 1990.
[5] Donaldson. Floer Homology Groups in Yang-Mills theory. Cambridge University Press,
Cambridge, 2002.
[6] Donaldson, The orientation of Yang-mills moduli spaces and 4-manifold topology. Jour-
nal of Differential Geometry, 26(3): 397-428, 1987.
[7] R. Fintushel and R. Stern. The Blowup Formula for Donaldson Invariants. Annals of
Mathematics, 143(3): 529-546, 1993.
[8] P.B. Kronheimer. Four Manifold Invariants from Higher Rank Bundles. Journal of Dif-
ferential Geometry, No. 70 (2005), 59-112.
[9] P.B. Kronheimer and T. Mrowka. Monopoles and Three-manifolds. Cambridge University Press, New York, 2007.
[10] Y. Lekili and T. Perutz. Arithmetic mirror symmetry for the 2-torus. Preprint:
arxiv.org/abs/1211.4632.
[11] C. LeBrun. Counter-Examples to the Generalized Positive Action Conjecture. Commu-
nications in Mathematical Physics 118(4): 591-596, 1988.
[12] M. Marinio and G. Moore. The Donaldson-Witten function for gauge groups of rank
larger than one. Communications in Mathematical Physics 199(1): 25-69, 1998.
[13] Morgan, Mrowka, and Ruberman. The L 2 Moduli Space and a Vanishing Theorem for
Donaldson Polynomial Invariants. International Press, Cambridge MA, 1994.
73
[14] H. Nakajima and K. Yoshioka. Instanton Counting on Blowup. I. 4-Dimensional Pure
Gauge Theory. Inventiones Mathematicae, 162(2):313-355, 2005.
[15] D. Ruberman. Relations among Donaldson invariants arising from negative 2-spheres
and tori. Duke Math Journal 83(3): 645-660, 1996.
[16] C. Taubes. L 2 Moduli Spaces on 4-Manifolds with Cylindrical Ends. International Press,
Somerville, 1993.
74