Marathe K. Topics in Physical Mathematics

Подождите немного. Документ загружается.

298 9 4-Manifold Invarian t s

This is the ﬁrst step in the deﬁnition of q

. The second step is to pair the

cohomology class on the right hand side of (9.16) with a suitable homology

cycle. We now describe this homology cycle. Observe ﬁrst that the formal

(or virtual) dimension, of the moduli space of anti-instantons of instanton

number k is given by

dim M

(M,SU(2)) = 8k − 3(1 + b

). (9.17)

We note that the instanton equations depend on the choice of a metric g

on M through the Hodge ∗-operator. It follows that the instanton moduli

space also depends on this choice. We indicate this dependence of M

the metric g by writing M

((M,g),SU(2)) or simply M

(g) for the moduli

space. If b

=2a +1,a>0, then for a generic metric g the moduli space

(g) has even dimension 2d,whered =4k − 3(1 + a). This requirement is

incorporated in the following deﬁnition.

Deﬁnition 9.1 A C-manifold is a pair (M, β) where

1. M is a smooth, compact, 1-connected, oriented 4-manifold with b

(M)=

2a +1,a>0;

2. β is an orientation of a maximal positive subspace H

⊂ H

(M,R) for

the intersection form of M.

In the following theorem we collect some properties of the moduli spaces

over a C-manifold that play a fundamental role in the deﬁnition of Donaldson

polynomials.

Theorem 9.15 Let M be a C-manifold. Let RM(M) denote the set of

Riemannian metrics on M . Then there exists a second category subset

GM(M) ⊂RM(M) (called the set of generic metrics) such that

1. For g ∈GM(M) and k>0 the moduli space M

(g) is a manifold with

boundary. The part M

∗

(g) corresponding to irreducible instantons is a

smooth submanifold of the space B

∗

of all irreducible gauge potentials, cut

out transversely by the instanton equations and hence is of dimension equal

to the virtual dimension 8k − 6(1 + a).

2. For g

∈GM(M ) there exists a smooth path of generic metrics g

,t∈

[0, 1] joining g

and g

such that the parametrized moduli space N

deﬁned by

N = {(ω,t) ∈B

∗

× [0, 1] | ω ∈M

∗

)}

is a smooth manifold with boundary, the boundary consisting of the disjoint

union of M

∗

) and M

∗

). The manifold N has dimension 8k −6a −5

and is a cobordism between M

∗

) and M

∗

3. The orientation β induces an orientation on all the moduli spaces M

(g)

so that changing β to −β reverses the orientation of all the moduli spaces.

Given an orientation on all the moduli spaces M

(g),theparametrized

moduli space N can be given an orientation so that it provides an oriented

cobordism between M

∗

) and M

∗

9.4 Donaldson Polynomials 299

In the rest of this section we take M to be a C-manifold. There are a

number of technical points to be addressed before we give the deﬁnition

of Donaldson polynomials. Brieﬂy, we know that M

(g)isamanifoldof

dimension 2d. If we can associate a 2d-dimensional homology class h

to this

manifold then we can evaluate the 2d-form deﬁned in (9.16)ontheclassh

deﬁne Donaldson polynomials. Theorem 9.15 can then be used to show that

the deﬁnition is independent of the metric and the orientation of M

(g). In

view of this we drop the reference to the metric and orientation of the moduli

spaces in what follows. For a compact manifold, the fundamental homology

class is well-deﬁned. However, the moduli spaces M

are, in general, not

compact. They do have a natural compactiﬁcation

deﬁned as follows.

Let s

(M)denotetheith symmetric power of M. Deﬁne the topology T

:= M



i=1



k−i

× s

(M)



(9.18)

by the following notion of convergence. We say that a sequence [ω

]inM

converges to a point ([ω], (x

,...,x

)) ∈M

k−i

× s

(M)if

lim

n→∞

[ω

]=[ω]onM \{x

,...,x

}

and the corresponding sequence of Yang–Mills integrands

converges to |F

+8π



j=1

By extending the analytical results of Uhlenbeck on the compactiﬁcation of

Yang–Mills moduli spaces, Donaldson has proved the following theorem:

Theorem 9.16 Let M be a C-manifold. Then the closure

of M

in the

topological space (M

∪

i=1

k−i

×s

(M), T

) isacompactstratiﬁedspace

with principal stratum M

. Moreover, if

4k>3(b

(M)+1) (9.19)

then the secondary strata

∩ (M

k−i

× s

(M)),i>1,

have codimension at least 2 in

. In particular, this implies that the com-

pactiﬁed moduli space

carries the fundamental homology class.

The compactiﬁcation

is called the Donaldson compactiﬁcation

of M

.Thevaluesofk satisfying the inequality (9.19) are said to be in

the stable range. The fundamental class of the space M

deﬁnes a 2d-

dimensional homology class [M

] in the homology H

∗

). The Donaldson

polynomial q

300 9 4-Manifold Invarian t s

: H

(M) ×···×H

(M)



 

d times

→ Q

is then deﬁned by

,...,a

)=(μ

,...,a

))[M

]. (9.20)

An important alternative deﬁnition of the map μ can be given by con-

sidering a representative of the homology class a ∈ H

(M) by an embedded

surface S as follows. The embedding map ι : S → M deﬁnes the map

r : A

(M) →A(S)

obtained by restricting an irreducible connection over M to a connection

over S. Coupling the Dirac operator D on S, to the restriction of gauge ﬁelds

on M to S gives a family of coupled Dirac operators. This family deﬁnes a

determinant line bundle L over A(S). We denote by L

the pull-back r

∗

L of

this bundle to A

(M). We then deﬁne

μ(a):=c

−1

) ∈ H

(M)).

One can show that this deﬁnition of μ leads to the same cohomology class as

the previous deﬁnition.

In [237] the deﬁnition of Donaldson polynomial invariants has been ex-

tended beyond the stable range as well as to manifolds M with b

(M)=1

(see also [113]).

In [404] it is shown that the Donaldson polynomial invariants of a 4-

manifold M appear as expectation values of certain observables in a topolog-

ical QFT. As we have discussed in Chapter 7, this is a physical interpretation

of Donaldson’s polynomial invariants. We note that this TQFT formulation

or its later variants have not led to any new insight into the structure of these

invariants. As we have indicated above, Donaldson polynomials are indepen-

dentofthemetriconM and depend only on its diﬀerential structure. Thus

we have the following theorem.

Theorem 9.17 Let M be a smooth, oriented, compact, simply connected

4-manifold with b

(M)=2a +1,a≥ 1.Foreachk in the stable range

(4k>3(b

(M)+1)), the polynomial q

deﬁned by equation (9.20)isadiﬀer-

ential topological invariant of M, which is natural with respect to orientation-

preserving diﬀeomorphisms.

When M is a connected sum of manifolds M

and M

, one can construct

instantons on M by gluing instantons on M

and M

. By carefully analyzing

the corresponding moduli spaces Donaldson proved the following important

vanishing theorem.

Theorem 9.18 Let M be a smooth, oriented, compact, simply connected 4-

manifold with b

(M)=2a +1,a≥ 1.IfM can be written as a smooth,

9.4 Donaldson Polynomials 301

oriented connected sum M = M

and b

) > 0,fori =1, 2, then the

polynomial invariant q

is identically zero for all k.

The case when the base manifold carries a compatible complex structure is

usually referred as the integrable case. In particular, when M is a compact

K¨ahler manifold with K¨ahler form ω, there exists a one-to-one correspondence

between the following two spaces of equaivalence classes:

1. space of irreducible SU(2) anti-instantons,

2. space of holomorphic SL(2, C) bundles that satisfy a certain stability con-

dition with respect to the polarization induced by the Ka

hler form.

In the integrable case one has the following positivity property of the in-

variants, which has strong inﬂuence on the diﬀerential topology of complex

surfaces even in cases where the invariants cannot be explicitly calculated.

Theorem 9.19 Let M be a compact K¨ahler manifold with K¨ahler form ω.

Let Σ ∈ H

(M; Z) be the Poincar´edualoftheK¨ahler class [ω] ∈ H

(M; Z).

Then q

(Σ,...,Σ) > 0 for all suﬃciently large k.

The integrable case has been studied extensively (see, for example,

[108, 113], [141]and[150]). For algebro-geometric analogues of Donaldson

polynomials see J. Morgan and K. O’Grady [291]. For applications to com-

plex manifolds and algebraic varieties see R. Friedman and J. Morgan [142]

and [290].

9.4.1 Structure of Polynomial Invariants

Donaldson’s polynomial invariants are deﬁned in terms of the 2-dimensional

cohomology classes in the image of the μ map. The moduli space O

also

carries a distinguished 4-dimensional cohomology class ν deﬁned by the image

of the generator 1 ∈ H

(M) under the map

(μ

(P)

)

: H

(M) → H

). (9.21)

We note that the cohomology class ν is essentially the ﬁrst Pontryagin class

of the principal SO(3)-bundle

over O

generated by the base point ﬁ-

bration. Using the class ν we can deﬁne a larger class q

(i,j)

: H

(M) ×···×H

(M)



 

i times

→ Q

of invariants of M by

(i,j)

,...,a

)=(μ

,...,a

) ∧ν

)[M

], (9.22)

302 9 4-Manifold Invarian t s

where i, j are non-negative integers such that i +2j = d(k). We say that M

is of KM simple type, or just simple type, if

(i,j)

=4q

(i,j+2)

(9.23)

for all i, j for which the invariants are deﬁned. It is known that condi-

tion (9.23) holds for several classes of 4-manifolds including simply connected

elliptic surfaces, complete intersections, and some branched covers. It is not

known whether the condition holds for all simply connected 4-manifolds.

In the rest of this section we restrict ourselves to manifolds of simple type.

Then the evaluation of the invariants q

(i,j)

is reduced to two cases. Case one

has j = 0 and corresponds to the original polynomial invariants. In case two

j = 1 and in this case, following [238, 239] we deﬁne new invariants q

d(k)−2

(d(k)−2,1)

(9.24)

For non-negative n we deﬁne q

by equation (9.20)ifn = d(k), by equa-

tion (9.24)ifn = d(k) − 2, and set q

=0forallothern. Then we deﬁne a

function q by a series involving q

q : H

(M; Z) → R by q(a):=

∞



n=0

(a)

. (9.25)

The following structure theorem for the invariant q was proved by Kronheimer

and Mr´owka [239].

Theorem 9.20 Let M be a simply connected 4-manifold of simple type. Then

there exist ﬁnitely many cohomology classes K

,...,K

∈ H

(M; Z) and

non-zero rational numbers r

,...,r

such that

q = e

(ι

)/2



i=1

, (9.26)

where the equality is to be understood as equality of analytic functions on

(M; R). The classes K

, 1 ≤ i ≤ p, are called the KM basic classes

and each K

is an integral lift of the second Stiefel–Whitney class w

(M) ∈

(M; Z).

The proof of the above theorem depends in an essential way on studying

the moduli spaces of singular gauge ﬁelds on M with codimension two singular

surface Σ. One starts by specifying the following data:

1. aprincipalSU(2) bundle over M \ Σ with instanton number k;

2. a complex line bundle L over Σ with monopole number l = −c

(L)[Σ];

3. a non-trivial holonomy α around Σ such that α ∈ (0, 1/2).

Fix a background connection A

and deﬁne the conﬁguration space A

9.4 Donaldson Polynomials 303

= {A

+ a |∃p>2 such that ∇

a ∈ L

Then deﬁne the moduli space:

k,l

= {A ∈A

|∗F

= −F

}/G,

where G is the group of gauge transformations. We call M

k,l

a moduli space

of connections on M with ﬁxed singularity along Σ. In general, M

k,l

is a

manifold with singularities, and

dim M

k,l

=8k − 3(b

(M) − b

(M)+1)+4l − (2g − 2),

where g is the genus of Σ. One can deﬁne a family of invariants of Don-

aldson type using these moduli spaces. These invariants interpolate between

the usual Donaldson invariants to provide the recurrence relations and also

furnish information on their structure. In particular, we have the following

theorem:

Theorem 9.21 Let M be a simply connected 4-manifold of simple type and

let K

,...,K

∈ H

(M; Z) be a complete set of KM basic classes given by

Theorem 9.20.LetΣ be any smoothly embedded, connected, essential surface

in M of genus g with normal bundle of non-negative degree. Then g satisﬁes

the inequality

(Σ)+max

K

,Σ≤2g − 2. (9.27)

This result is closely related to the classical Thom conjecture about embedded

surfaces in CP

. While the singular gauge theory methods do not apply to

, they do lead to interesting new results. As an example of such a result

we state the following theorem due to Kronheimer and Mr´owka.

Theorem 9.22 Let M be a manifold with b

≥ 3 and non-zero Donaldson

invariants. Let Σ ⊂ M be an essential surface with g>0.Then

2g − 2 ≥ ι

(Σ).

The condition that M be of simple type is very mysterious. It is not known

whether every simply connected 4-manifold is of simple type. Simple type

and basic classes also arise in the Seiberg–Witten theory, as we will see later

in this chapter.

9.4.2 Relative Invariants and Gluing

Let Y be an oriented compact manifold for which the Fukaya–Floer homol-

ogy FF

∗

(Y ) is deﬁned. Let M be an oriented, simply connected compact

manifold with boundary ∂M = Y . Then one can deﬁne a relative version

of Donaldson’s polynomial invariants of M with values in the Fukaya–Floer

304 9 4-Manifold Invarian t s

homology FF

∗

(Y ). This deﬁnition can be used to obtain gluing formulas

for Donaldson polynomials for 4-manifolds, which can be written as general-

ized connected sums split along a common boundary Y .If

Y is Y with the

opposite orientation, then there is a dual pairing between the graded homol-

ogy groups FF

∗

(Y )andFF

∗

(

Y ). Let N be an oriented simply connected

closed 4-manifold, which can be written as the generalized connected sum

N = N

.Ifq

(resp., q

) is a relative polynomial invariant with values

in FF

∗

(Y )(resp.,FF

∗

(

Y )) then q

can be glued to obtain a polynomial

invariant q of N by the “formula”

q = q

.

The deﬁnition of relative invariants and detailed structure of the gluing for-

mula in this generality is not yet known. Some speciﬁc examples have been

considered in [110, 369].

We give a deﬁnition of relative invariants in the special case when Y is

an integral homology sphere and the representation space R

∗

(Y )isregular.

Let M be an oriented simply connected 4-manifold with boundary Y .Let

P be a trivial SU(2) bundle over M and ﬁx a trivialization of P to write

P = M × SU(2) and let θ

be the corresponding trivial connection. Let θ

denote the induced trivial connection on P

= Y × SU(2). Let α ∈R

∗

(Y )

and let M(M; α) denote the moduli space of self-dual connections on P which

equal α on the boundary Y of M. The index of the corresponding instanton

deformation complex gives the formal (or virtual) dimension of the moduli

space M(M; α). Let n be the dimension of a non-empty component M

(α)

of M

(M; α) for a suitable generic metric g on M.Then

n ≡−3(1 + b

(M)) − sf(α, θ)(mod8). (9.28)

If n =2m we can deﬁne the relative polynomial invariants q

: H

(M,Y ; Z) ×···H

(M,Y ; Z)



 

m times

→R

∗

(Y )

,...,a

)=(μ

,...,a

)[M

(α)])α. (9.29)

Let n

∈ Z

be the congruence class of the spectral ﬂow sf(α, θ). Then one

can show that the right hand side of equation (9.29) depends only on the

Floer homology class of α ∈ HF

. Thus the relative polynomial invariants

of M take values in the Floer homology of the boundary ∂M = Y .

9.5 Seiberg–Witten Theory 305

9.5 Seiberg–Witten Theory

I ﬁrst learned about the Seiberg–Witten theory from Witten’s lecture at

the 1994 International Congress on Mathematical Physics (ICMP 1994) in

Paris. His earlier formulation of the Jones polynomial using quantum ﬁeld

theory had given a new and geometrical way of looking at the Jones poly-

nomial and it led to the WRT invariants of 3-manifolds. It was thus natu-

ral to consider a similar interpretation of the Donaldson polynomials. Wit-

ten’s results were given a more mathematical reformulation in [21]. However,

these results provided no new insight or method of computation beyond the

well known-methods used in physics and mathematics. Another idea Witten

used successfully was a one-parameter family of supersymmetric Hamiltoni-

ans (H

,t≥ 0) to relate Morse theory and de Rham cohomology. Large

values of t lead to Morse theory while small t give the de Rham cohomol-

ogy. In physics these limiting theories are considered dual theories and are

referred to as the infrared limit and ultraviolate limit, respectively. ,

Witten used the ultraviolet limit of N = 2 supersymmetric Yang–Mills the-

ory to write the Donaldson invariants as QFT correlation functions. As these

invariants do not depend on the choice of a generic metric, they could also

be calculated in the infrared limit. The infrared behavior of the N =2su-

persymmetric Yang–Mills theory was determined by Seiberg and Witten in

1994. The equations of the theory dual to the SU(2) gauge theory are the

Seiberg–Witten or the monopole equations. They involve U(1) gauge ﬁelds

coupled to monopoles. Thus the new theory should be expected to give in-

formation on the Donaldson invariants of 4-manifolds. Witten told me that

his paper [407] giving further details should appear soon.

After returning to America, Witten gave a lecture at MIT discussing his

new work. He remarked that the monopole invariants (now known as the

Seiberg–Witten invariants or the SW-invariants) could be used to com-

pute the Donaldson invariants and vice versa. Gauge theory underlying

the SW-invariants has gauge group U (1), the Abelian group of Maxwell’s

electromagnetic ﬁeld theory. The moduli space of the solutions of the SW-

equations has a much simpler structure than the instanton moduli spaces

used in the Donaldson theory. A great deal of hard analysis is required to

deal with the problem of compactiﬁcation of moduli spaces and the existence

of reducible connections to extract topological information about the base

manifold in Donaldson’s theory. It was (and still is) hard to believe that one

can bypass the hard analysis if one uses the SW-moduli space and that doing

so gives a much simpler approach to the known results. It also leads to proofs

of several results that have seemed intractable via the Donaldson theory.

Taubes described what happened after the lecture: Several people working in geometric

topology gathered at Bott’s house. Most of us were thinking of obtaining a coun terexample

to Witten’s assertion of equivalence of the monopole and instanton invariants. No such

example was found by the time we broke up late that night. It was agreed that anyone

who ﬁnds a counterexample would communicate it to the others. (Personal communication)

306 9 4-Manifold Invarian t s

Witten’s assertion of equivalence of the monopole and instanton invari-

ants seems incredible. However, as of this writing no counterexample has

been found to Witten’s assertion that monopole and instanton invariants are

equivalent. In fact, the equivalence of the monopole and instanton invariants

has been established for a large class of 4-manifolds. SW-equations arose

as a byproduct of the solution of N = 2 supersymmetric Yang–Mills equa-

tions in theoretical physics. It was the physical concept of S-duality that led

Witten to his formula relating the SW and Donaldson invariants. Thus, SW

theory provides the most exciting recent example of physical mathematics.

SW theory is an active area of current research with vast literature including

monographs and texts. The book [277] by Marcolli gives a very nice introduc-

tion to various aspects of SW theory. A comprehensive introduction can be

found in Nicolaescu [299]. The relation between the Seiberg–Witten theory

and the Gromov–Witten theory is discussed in detail by Taubes in [362].

Spin structures and Dirac operators on spinor bundles are now reviewed.

These are used to deﬁne the SW-equations, SW-moduli space, and SW-

invariants. We then indicate some applications of the SW-invariants and state

their relation to Donaldson’s polynomial invariants.

9.5.1 Spin Structures and Dirac Operators

Dirac’s 1928 discovery of his relativistic equation for the electron is consid-

ered one of the great achievements in theoretical physics in the ﬁrst half of the

twentieth century. This equation introduces a ﬁrst order diﬀerential operator

acting on spinors deﬁned over the Minkowski space. This is the original Dirac

operator, which is a square root of the D’Alembertian (or the Minkowski

space Laplacian). The equation also had a solution corresponding to a pos-

itively charged electron. No such particle was observed at that time. The

Dirac gamma matrices in the deﬁnition of the Dirac operator have a natural

interpretation in the setting of the Cliﬀord algebra of the Minkowski space.

In fact, given a ﬁnite-dimensional real vector space V with a pseudo-inner

product g, we can construct a unique (up to isomorphism) Cliﬀord algebra

associated to the pair (V,g). The deﬁnitions of the gamma matrices, spinors

and the Dirac operator all have natural extensions to this general setting.

For mathematical applications we want to consider Riemannian manifolds

which admit spinor bundles. The Dirac operator is then deﬁned on sec-

tions of these bundles. As we discussed in Chapter 2 and Appendix D, there

are topological obstructions to a manifold admitting a spin structure. How-

ever, a compact oriented Riemannian manifold of even dimension admits a

Spin

-structure which can be used to construct spinor bundles on it. We now

discuss this structure on a ﬁxed compact oriented Riemannian manifold M

of dimension m. The group Spin

(m) is deﬁned by

9.5 Seiberg–Witten Theory 307

Spin

(m):=(Spin(m) × U(1))/Z

. (9.30)

Deﬁnition 9.2 A Spin

-structure on a compact oriented Riemannian

manifold M is a lift of the principal SO(m)-bundle of oriented orthonormal

frames of M toaprincipalSpin

(m)-bundle over M .

As with the existence of Spin-structure, that of the Spin

-structure has topo-

logical obstruction that can be expressed in terms of the second Stiefel–

Whitney class w

(M). Using standard homology theory argument, it can be

shown that a manifold M admits a Spin

-structure if w

(M) is the reduction

of an integral homology class (mod 2). Moreover, a compact oriented even

dimensional Riemannian manifold admits a Spin

-structure. Recall that the

Cliﬀord algebra of the Euclidean space R

has a unique irreducible com-

plex representation on a Hermitian vector space S of rank 2

. It is called

the (complex) spinor representation . The elements of S are called (com-

plex) spinors .ThespaceS has a natural Z

-graded or superspace structure

under which it can be written as a vector space direct sum S = S

⊕ S

−

The elements of S

(resp. S

−

) are called positive spinors (resp. negative

spinors) . Cliﬀord multiplication by v ∈ R

acts as a skew-Hermitian op-

erator on the vector space S interchanging the positive and negative spinors.

9.5.2 The Seiberg–Witten (SW) Invariants

In what follows we restrict ourselves to a compact oriented Riemannian 4-

manifold M. The manifold M admits a Spin

-structure. We choose a ﬁxed

Spin

-structure and the corresponding S as discussed above. A spin con-

nection ∇ on S is a connection compatible with the Levi-Civita connection

∇

on the orthonormal frame bundle of M ; i.e.,

∇

(Y · ψ)=∇

(Y )+∇

(ψ),

where X, Y are vector ﬁelds on M and ψ is a section of the spinor bundle S.

The corresponding Dirac operator D acting on a spinor ψ is the ﬁrst order

operator whose local expression is given by

D(ψ)=



k=1

·∇

(ψ),

where e

, 1 ≤ k ≤ 4 is a local orthonormal basis for the tangent bundle TM.

The Dirac operator maps positive spinors to negative spinors and vice versa.

The determinant line bundles of S

and S

−

are canonically isomorphic. We

denote this complex line bundle by L. The spin connection ∇ on S leaves

the bundles S

and S

−

invariant and hence induces a connection on L.We

denote by A the gauge potential and by F

the corresponding gauge ﬁeld of