Marathe K. Topics in Physical Mathematics

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308 9 4-Manifold Invarian t s

the induced connection on L. An important property of the Dirac operator

that is used in the study of the SW equations is given by the following

theorem.

Theorem 9.23 (Lichnerowicz–Weitzenb¨ock formula) Let ∇

∗

be the formal

adjoint of the spin connection (covariant derivative operator) ∇.LetR be the

scalar curvature of the manifold (M,g). Then the Dirac operator satisﬁes the

following Lichnerowicz–Weitzenb¨ock formula:

(ψ)=



∇

∗

∇+

(R − iF

)



ψ, ∀ψ ∈ Γ (S

In the case we are considering, the manifold M admits Spin

structures

and they are classiﬁed by H

(M,Z). Given α ∈ H

(M,Z) there is a principal

U(1)-bundle P

with connection A and curvature F

whose ﬁrst Chern class

equals α. The associated complex line bundle L is the determinant bundle

of the positive and negative spinors. Given a positive spinor ψ, the Seiberg–

Witten or SW equations are a system of diﬀerential equations for the pair

(A, ψ)givenby

Dψ =0,F

σ(ψ), (9.31)

where D is the Dirac operator of the spin connection, F

is the self-dual part

of the curvature F

,andσ(ψ) is a pure imaginary valued self-dual 2-form

with local coordinate expression

σ(ψ)=e

ψ, ψe

∧ e

The SW equations are the absolute minima of the variational problem for

the SW functional given by

(A, ψ)=





||Dψ||

+ ||F

−

σ(ψ)||



. (9.32)

Using the Lichnerowicz–Weitzenb¨ock formula the SW functional can be writ-

ten as

(A, ψ)=





||∇ψ||

+ ||F

||ψ||



. (9.33)

This second form of the SW functional is useful in the study of the topology

of the moduli space of the solutions of the SW equations. The group of gauge

transformations given by G = F(M,U(1)) acts on the pair (A, ψ) as follows:

(A, ψ)=(A −f

−1

df , f ψ ), ∀f ∈G. (9.34)

This action induces an action on the solutions (A, ψ) of the SW equations.

The moduli space M

of the solutions of the SW equations is deﬁned by

:= {(A, ψ) | (A, ψ) ∈ C}/G,

9.5 Seiberg–Witten Theory 309

where C is the space of solutions of the SW equation. The moduli space

depends on the choice of α ∈ H

(M,Z) and we indicate this by writing

(α) for the moduli space. To simplify our considerations we now take

(M) > 1. Then for a generic metric on M at a regular point (i.e., where

ψ = 0) the dimension of the tangent space to M

(α) can be computed

by applying index theory to the linearization of the SW equations. This

dimension d

is given by

(L) − (2χ +3σ)).

It can be shown that the space M

(α) (or the space obtained by perturba-

tion) when non-empty is a smooth compact orientable manifold. We choose

an orientation and deﬁne the SW-invariants as follows.

Deﬁnition 9.3 (SW Invariants) Suppose that α ∈ H

(M,Z) be such that

=0. Then the space M

(α) consists of a ﬁnite number p

, 1 ≤ i ≤ k,

of signed points. We deﬁne the SW-invariant ι

(α) of α by

(α):=



i=1

s(p

) , (9.35)

where s(p

)=±1 is the sign of p

Theorem 9.24 The SW invariants ι

(α) are independent of the metric

and hence are diﬀerential topological invariants of M.

It is possible to deﬁne SW-invariants when d

> 0 by a construction similar

to that used in Donaldson theory. However, their signiﬁcance is not clear yet.

In most of the examples the SW moduli space turns out to be 0-dimensional.

We now restrict the class of manifolds to those of SW simple type deﬁned

below.

Deﬁnition 9.4 Aclassα is called an SW basic class if the invariant

(α) =0. It can be shown that there are only ﬁnitely many basic classes.

We say that M is of SW si m ple type if d

=0for every basic class α.

Simple Type Conjecture: Let M be a closed simply connected oriented

Riemannian 4-manifold with b

> 1. Then the simple type conjecture says

that every such manifold is of KM and SW simple type. It is known that

the conjecture holds if M is also a symplectic manifold. There are no known

examples of manifolds that are not simple type. If M has non-negative scalar

curvature then all the SW invariants are zero. The following theorem is similar

to the one on invariants of connected sums in Donaldson theory.

Theorem 9.25 (Connected sum theorem) Let M

, i =1, 2, be compact ori-

ented 4-manifolds with b

) ≥ 1;thenSW(M

,s)=0,wheres is an

arbitrary Spin

structure on (M

310 9 4-Manifold Invarian t s

A symplectic manifold M has a canonical Spin

structure for which the

SW invariant is non-zero. It follows from the above theorem that such M

cannot be a connected sum of two manifolds each with b

≥ 1ifb

(M) ≥ 1.

In his theory of pseudo-holomorphic curves in symplectic manifolds, Gromov

deﬁned a set of invariants, which were given a physical interpretation in terms

of sigma models by Witten. They are called the Gromov–Witten or GW-

invariants. The following theorem due to Taubes gives a surprising relation

between the SW- and the GW-invariants.

Theorem 9.26 (Taubes) Let M be a compact symplectic manifold with

> 1.OrientM and its positive determinant line bundle by using the sym-

plectic structure. Deﬁne Seiberg–Witten invariants SW and Gromov–Witten

invariants Gr as maps from H

(M; Z) to Λ

∗

(M; Z).ThenSW = Gr.

For a complete treatment of this theorem and relevant deﬁnitions we refer

the reader to Taubes’ book [362].

One of the earliest applications of SW-invariants was the proof by Kron-

heimer and Mr´owka of the classical Thom conjecture about embedded sur-

faces in CP

. This was beyond reach of the regular or singular gauge theory

methods. There is a version of SW equations for 3-manifolds and the corre-

sponding Seiberg–Witten Floer homology (see, for example, Marcolli [277]).

For other applications of the SW-invariants see [249,293, 329,367,379].

9.6 Relation between SW and Donaldson Invariants

As we saw earlier, Donaldson used the moduli spaces of instantons to deﬁne

a new set of invariants of M , which can be regarded as polynomials on the

second homology H

(M). In [239] Kronheimer and Mr´owka obtained a struc-

ture theorem for the Donaldson invariants in terms of their basic classes and

introduced a technical property called “KM-simple type” for a closed simply

connected 4-manifold M . Then, in 1994, the Seiberg–Witten (SW) equations

were obtained as a byproduct of the study of super Yang–Mills equations. As

we discussed earlier these equations are deﬁned using a U(1) monopole gauge

theory, and the Dirac operator obtained by coupling to a Spin

structure.

The moduli space of solutions of SW equations is used to deﬁne the SW-

invariants. The structure of this SW moduli space is much simpler than the

instanton moduli spaces used to deﬁne Donaldson’s polynomial invariants.

This led to a number of new results that met with insurmountable diﬃculties

in the Donaldson theory (see, for example, [240,368]). We also discussed the

simple type condition and basic classes in SW theory. Witten used the idea

of taking ultraviolet and infrared limits of N = 2 supersymmetric quantum

Yang–Mills theory and metric independence of correlation functions to relate

Donaldson and SW-invariants. The precise form of Witten’s conjecture can

be expressed as follows.

9.6 Relation betw een SW and Donaldson In variants 311

Witten’s conjecture for relation between Donaldson and SW-

invariants: A closed, simply connected 4-manifold M has KM-simple type

if and only if it has SW-simple type. If M has simple type and if D(α)

(resp., SW(α)) denotes the generating function series for the Donaldson

(resp., Seiberg–Witten) invariants with α ∈ H

(M; R), then we have

D(α)=2

(α)/2

SW(α), ∀α ∈ H

(M; R). (9.36)

In the above formula ι

is the intersection form of M and c is a constant

given by

c =2+

7χ(M)+11σ(M)

. (9.37)

A mathematical approach to a proof of Witten’s conjecture was proposed

by Pidstrigatch and Tyurin (see dg-ga/9507004). In a series of papers, Feehan

and Leness (see [122,123] and references therein) used similar ideas employing

an SO(3) monopole gauge theory that generalizes both the instanton and

U(1) monopole theories to prove the Witten conjecture for a large class of 4-

manifolds. This result is an important ingredient in the proof of the property

P conjecture that we next discuss. The problem of relating Feehan and Leness’

proof to Witten’s TQFT argument remains open.

9.6.1 Property P Conjecture

In early 1960s Bing tried to ﬁnd a counterexample to the Poincar´e conjec-

ture by constructing 3-manifolds by surgery on knots. Bing and Martin later

formalized this search by deﬁning property P of a knot as follows:

Deﬁnition 9.5 A non-trivial knot K has property P if every 3-manifold

Y obtained by a non-trivial Dehn surgery on K has a non-trivial fundamental

group.

Using the above deﬁnition we can state the property P conjecture as

follows: Every non-trivial knot K has property P. In particular, Y is not a

homotopy 3-sphere.

Thus, to verify the property P conjecture, it is enough to show that π

(Y )

is non-trivial if the knot K is non-trivial. It can be shown by topological

arguments that π

(Y ) is non-trivial if it is obtained by Dehn surgery with

coeﬃcient other than ±1. The proof of the remaining cases was recently

given by Kronheimer and Mr´owka [241], thereby proving the property P

conjecture. In fact, they showed that in these cases π

(Y ) admits a non-trivial

homomorphism to the group SO(3). The proof uses several results from gauge

theory, symplectic and contact geometry, and the proof of Witten’s conjecture

relating the Seiberg–Witten and Donaldson invariants for a special class of

manifolds. The proof is quite complicated as it requires that methods and

312 9 4-Manifold Invarian t s

results from many diﬀerent areas be ﬁt together in a precise way. According

to Gromov

a simpliﬁed proof of the theorem should be possible. We note

that it is known that surgery on a non-trivial knot can never produce the

manifold S

. It follows that the Poincar´e conjecture implies the property P

conjecture. Thanks to Perelman’s proof of the Poincar´e conjecture, we now

have a diﬀerent proof of the property P conjecture closely related to ideas

from gravitational physics.

Gromov made this remark after the seminar by Mr´owka at Columbia University explain-

ing this work.

Chapter 10

3-Manifold Invariants

10.1 Introduction

In Chapter 9 we discussed the geometry and topology of moduli spaces of

gauge ﬁelds on a manifold. In recent years these moduli spaces have been

extensively studied for manifolds of dimensions 2, 3, and 4 (collectively re-

ferred to as low-dimensional manifolds). This study was initiated for the

2-dimensional case in [17]. Even in this classical case, the gauge theory per-

spective provided fresh insights as well as new results and links with physical

theories. We make only a passing reference to this case in the context of

Chern–Simons theory. In this chapter, we mainly study various instanton

invariants of 3-manifolds. The material of this chapter is based in part on

[263]. The basic ideas come from Witten’s work on supersymmetric Morse

theory. We discuss this work in Section 10.2. In Section 10.3 we consider gauge

ﬁelds on a 3-dimensional manifold. The ﬁeld equations are obtained from the

Chern–Simons action functional and correspond to ﬂat connections. Casson

invariant is discussed in Section 10.4. In Section 10.5 we discuss the Z

-graded

instanton homology theory due to Floer and its relation to the Casson in-

variant. Floer’s theory was extended to arbitrary closed oriented 3-manifolds

by Fukaya. When the ﬁrst homology of such a manifold is torsion-free, but

not necessarily zero, Fukaya also deﬁnes a class of invariants indexed by the

integer s,0≤ s<3, where s is the rank of the ﬁrst integral homology group

of the manifold. These invariants include, in particular, the Floer homology

groups in the case s = 0. The construction of these invariants is closely re-

lated to that of Donaldson polynomials of 4-manifolds, which we considered in

Chapter 9. As with the deﬁnition of Donaldson polynomials a careful analysis

of the singular locus (the set of reducible connections) is required in deﬁning

the Fukaya invariants. Section 10.6 is a brief introduction to an extension of

Floer homology to a Z-graded homology theory, due to Fintushel and Stern,

for homology 3-spheres. Floer also deﬁned a homology theory for symplec-

tic manifolds using Lagrangian submanifolds and used it in his proof of the

K. Marathe, Topics in Physical Mathematics, DOI 10.1007/978-1-84882-939-8 10, 313

 Springer-Verlag London Limited 2010

314 10 3-Manifold Invariants

Arnold conjecture. We do not discuss this theory. For general information on

various Morse homologies, see, for example, [29]. The WRT invariants, which

arise as a byproduct of Witten’s TQFT interpretation of the Jones polyno-

mial are discussed in Section 10.7. Section 10.8 is devoted to a special case of

the question of relating gauge theory and string theory where exact results

are available. Geometric transition that is used to interpolate between these

theories is also considered here. Some of the material of this chapter is taken

from [263].

10.2 Witten Complex and Morse Theory

Classical Morse theory on a ﬁnite-dimensional, compact, diﬀerentiable man-

ifold M relates the behavior of critical points of a suitable function on M

with topological information about M . The relation is generally stated as an

equality of certain polynomials as follows. Recall ﬁrst that a smooth function

f : M → R is called a Morse function if its critical points are isolated and

non-degenerate. If x ∈ M is a critical point (i.e., df (x) = 0) then by Taylor

expansion of f around x we obtain the Hessian of f at x deﬁned by



∂

∂x

(x)



Then the non-degeneracy of the critical point x is equivalent to the non-

degeneracy of the quadratic form determined by the Hessian. The dimension

of the negative eigenspace of this form is called the Morse index,orsimply

index,off at x and is denoted by μ

(x), or simply μ(x)whenf is under-

stood. It can be veriﬁed that these deﬁnitions are independent of the choice

of the local coordinates. Let m

be the number of critical points with index

k. Then the Morse series of f is the formal power series



Recall that the Poincar´eseriesofM is given by



,whereb

≡ b

(M)

is the kth Betti number of M. The relation between the two series is given



+(1+t)



, (10.1)

where q

are non-negative integers. Comparing the coeﬃcients of the powers

of t in this relation leads to the well-known Morse inequalities



k=0

i−k

(−1)

≥



k=0

i−k

(−1)

, 0 ≤ i ≤ n − 1,

10.2 Witten Complex and Morse Theory 315

and to the equality



k=0

n−k

(−1)



k=0

n−k

(−1)

The Morse inequalities can also be obtained from the following observation:

Let C

∗

be the graded vector space over the set of critical points of f. Then the

Morse inequalities are equivalent to the existence of a coboundary operator

∂ : C

∗

→ C

∗

so that ∂

= 0 and the cohomology of the complex (C

∗

,∂)

coincides with the de Rham cohomology of M.

In his fundamental paper [403] Witten arrives at precisely such a complex

by considering a suitable supersymmetric quantum mechanical Hamiltonian.

Witten showed how the standard Morse theory (see Morse and Cairns [292]

and Milnor [284]) can be modiﬁed by consideration of the gradient ﬂow of the

Morse function f between pairs of critical points of f . One may think of this

as a sort of relative Morse theory. Witten was motivated by the phenomenon

of quantum mechanical tunneling. We now discuss this approach. From a

mathematical point of view, supersymmetry may be regarded as a theory

of operators on a Z

-graded Hilbert space. In recent years this theory has

attracted a great deal of interest even though as yet there is no physical

evidence for its existence. In our general formulation of a supersymmetric

theory we let E denote the Hilbert space of a supersymmetric theory, i.e.,

E = E

⊕ E

, where the even (resp., odd) space E

(resp., E

) is called

the space of bosonic states (resp. fermionic states). These spaces are

distinguished by the parity operator S : E → E deﬁned by

Su = u, ∀u ∈ E

Sv = −v, ∀v ∈ E

The operator S is interpreted as counting the number of fermions modulo

2. A supersymmetric theory begins with a collection {Q

| i =1,...,n} of

supercharge operators (or supersymmetry operators)onE that are

of odd degree, that is, they anti-commute with S,

+ Q

S =0, ∀i, (10.2)

and satisfy the following anti-commutation relations

+ Q

=0, ∀i, j with i = j. (10.3)

The mechanics is introduced by the Hamiltonian operator H,whichcom-

mutes with the supercharge operators and is usually required to satisfy ad-

ditional conditions. For example, in the simplest non-relativistic theory one

requires that

H = Q

, ∀i. (10.4)

316 10 3-Manifold Invariants

In fact, this simplest supersymmetric theory has surprising connections with

Morse theory, as was shown by Witten in his fundamental paper [403]. It gave

a new interpretation of this classical theory and paved the way for Floer’s

new homology theory. We consider Floer homology later in this chapter. We

now discuss Witten’s work.

Let M be a compact diﬀerentiable manifold and deﬁne E by

E := Λ(M) ⊗ C.

The natural grading on Λ(M) induces a grading on E. We deﬁne



(M) ⊗ C (resp., E



2j+1

(M) ⊗ C),

the space of complex-valued even (resp., odd) forms on M.Theexterior

diﬀerential d and its formal adjoint δ have natural extension to odd operators

on E and thus satisfy (10.2). We deﬁne supercharge operators Q

,j=1, 2,

= d + δ, (10.5)

= i(d − δ). (10.6)

The Hamiltonian is taken to be the Hodge–de Rham operator extended to

E, i.e.,

H = dδ + δd. (10.7)

The relations d

= δ

= 0 imply the supersymmetry relations (10.3)

and (10.4). We note that in this case bosonic (resp., fermionic) states cor-

respond to even (resp., odd) forms. The relation to Morse theory arises in

the following way. If f is a Morse function on M , a one-parameter family of

operators is deﬁned:

= e

−ft

,δ

= e

δe

−ft

,t∈ R, (10.8)

and the corresponding supersymmetry operators

1,t

2,t

are deﬁned as in equations (10.5), (10.6), and (10.7). It is easy to verify

that d

= δ

= 0 and hence Q

1,t

2,t

satisfy the supersymmetry rela-

tions (10.3)and(10.4). The parameter t interpolates between the de Rham

cohomology and the Morse indices as t goes from 0 to +∞.Att =0the

number of linearly independent eigenvectors with zero eigenvalue is just the

kth Betti number b

when H

= H is restricted to act on k-forms. In fact,

these ground states of the Hamiltonian are just the harmonic forms. On the

other hand, for large t the spectrum of H

simpliﬁes greatly with the eigen-

functions concentrating near the critical points of the Morse function. It is

in this way that the Morse indices enter into this picture. We can write H

10.2 Witten Complex and Morse Theory 317

as a perturbation of H near the critical points. In fact, we have

= H + t



j,k

,jk

[α

k ]+t

df 

where α

= dx

acts by exterior multiplication, X

= ∂/∂x

,andi

k is the

usual action of inner multiplication by X

on forms and the norm df  is

the norm on Λ

(M) induced by the Riemannian metric on M. In a suitable

neighborhood of a ﬁxed critical point taken as origin, we can approximate

up to quadratic terms in x





−

∂

∂x

+ t

+ tλ

[α

j ]



where λ

are the eigenvalues of the Hessian of f. The ﬁrst two terms corre-

spond to the quantized Hamiltonian of a harmonic oscillator (see Appendix

B for a discussion of the classical and quantum harmonic oscillator) with

eigenvalues



| λ

| (1 + 2N

whereas the last term deﬁnes an operator with eigenvalues ±λ

.Itcommutes

with the ﬁrst and thus the spectrum of

is given by



[| λ

| (1 + 2N

)+λ

where N

are non-negative integers and n

= ±1. Restricting H to act on

k-forms we can ﬁnd the ground states by requiring all the N

to be 0 and

by choosing n

to be 1 whenever λ

is negative. Thus the ground states

(zero eigenvalues) of H correspond to the critical points of Morse index k.

All other eigenvalues are proportional to t with positive coeﬃcients. Starting

from this observation and using standard perturbation theory, one ﬁnds that

the number of k-form ground states equals the number of critical points of

Morse index k. Comparing this with the ground state for t = 0, we obtain the

weak Morse inequalities m

≥ b

. As we observed in the introduction, the

strong Morse inequalities are equivalent to the existence of a certain cochain

complex that has cohomology isomorphic to H

∗

(M), the cohomology of the

base manifold M. Witten deﬁnes C

,thesetofp-chains of this complex, to

be the free group generated by the critical points of Morse index p.Hethen

argues that the operator d

deﬁned in (10.8) deﬁnes in the limit as t →∞a

coboundary operator

∞

: C

→ C

p+1

and that the cohomology of this complex is isomorphic to the de Rham

cohomology of Y .