witten's laplacian and the morse...

Witten’sLaplacian and

the MorseInequalities

GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation

Weak MorseInequalities

Strong andPolynomialMorseInequalities

References

Witten’s Laplacian and the Morse Inequalities

Gianmarco Molino

University of Connecticut

December 1, 2017



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References

1 Background

2 Morse Inequalities

3 Witten’s Idea

4 Local Approximation

5 Weak Morse Inequalities

6 Strong and Polynomial Morse Inequalities



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References

Morse Theory

Morse Theory is the study of critical points of a smoothfunction f : M → R.

A smooth manifold M is a topological manifold withcompatible smooth atlas (in the following all manifolds areassumed to be n-dimensional, smooth, oriented, closed,and without boundary.)

A critical point q ∈ M of a smooth function f : M → R isa zero of the differential df .

The Hessian Hf (q) of f at a critical point q ∈ M is thematrix of second derivatives. (Independent of coordinatesystem at critical points.)



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References

Morse Functions

A smooth function f : M → R is called Morse if its criticalpoints are isolated and nondegenerate (that is, the Hessianof f is nonsingular.)

Remark: Nondegenerate critical points are necessarilyisolated.

The Morse index mq of a critical point q is the dimensionof the negative eigenspace of Hf (q).

The i-th Morse number Mi is the number of critical pointswith Morse index i .

Remark: The Morse numbers are invariant underdiffeomorphism.



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References

Betti Numbers

Associated to every smooth manifold is the sequence ofBetti numbers βi , 0 ≤ i ≤ n defined as

βi = dimH idR(M) = dim

α ∈ Ωi : dα = 0dβ : β ∈ Ωi−1

where Ωi is the space of differential i-forms. This sequence is atopological invariant, and notably

χ(M) =n∑

i=0

(−1)iβi



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References

Morse Inequalities

The Weak Morse Inequalities are a classical result, proved usinggeometric techniques by Milnor [1].

Theorem (Weak Morse Inequalities)

Let f : M → R be Morse. Then for any 0 ≤ i ≤ n

βi ≤ Mi

and moreover

χ(M) =n∑

i=0

(−1)iMi



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References

Morse Inequalities

Theorem (Polynomial Morse Inequalities)

Let f : M → R be Morse. Then for any t ∈ R there exists asequence of nonnegative integers Qi such that

Mt − Pt :=n∑

i=0

Mi ti −

n∑i=0

βi ti = (1 + t)

n−1∑i=0

Qi ti

Theorem (Strong Morse Inequalities)

Let f : M → R be Morse. Then for any 0 ≤ k ≤ n

k∑i=0

(−1)i+kβi ≤k∑

i=0

(−1)i+kMi



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References

Edward Witten

In 1982, Edward Witten published a proof [2] of the MorseInequalities, essentially using the idea of the flow generated bya Morse functions, with an intuition deriving from QuantumMechanics. He was awarded a Fields Medal in 1990, partiallyfor this work.



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References

Supersymmetry

A Hilbert space H is called supersymmetric if there exists adecomposition H = H+ ⊕H− and maps

Q1 : H+ → H−

Q2 : H− → H+

H, (−1)F : H → H

that obey certain symmetry rules.



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References

Supersymmetry

Notice that the space of differential forms is supersymmetric,splitting into even and odd forms, with

Ω∗ =

n/2⊕i=0

Ω2p ⊕n/2−1⊕i=0

Ω2p+1

Q1 = d + δ

Q2 = i(d − δ)

H = ∆ = dδ + δd



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References

Witten Deformation

Witten generalizes this idea, conjugating d with the flow etf

for t ≥ 0, f Morse.

dt = e−tf detf

δt = etf δe−tf

∆t = dtδt + δtdt

It is easily verifiable that Ω∗ is still a supersymmetric spaceusing these deformed operators.



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References

Hodge Theory

To understand the motivation for the Witten Laplacian, weneed to look to Hodge Theory.

Theorem (Hodge Theorem)

For 0 ≤ i ≤ n, the maps

hi : ker ∆i → H idR(M)

ω 7→ [ω]

are isomorphisms.

Corollary

For 0 ≤ i ≤ n,βi = dim ker ∆i



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References

Hodge Theory

The corollary is the starting point for analytic approaches tothe Betti numbers. We will prove the Hodge Theorem using aheat flow argument, the following lemma will be necessary.

Lemma

For all smooth differential forms ω,

∆e−t∆ω = e−t∆∆ω

andde−t∆ω = e−t∆dω

The first claim follows from self-adjointness of ∆, while thesecond can be proved using the uniqueness of solutions to theheat equation in L2.



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References

Hodge Theory

Proof (Hodge Theorem).

Let ωii∈N be an orthonormal basis for Ωp with ∆ωi = λiωi .This can be done since M is compact, and follows from theSpectral Theorem for compact, self-adjoint operators applied tothe heat operator e−t∆. Then

limt→∞

e−t∆ω = limt→∞

∑i

aie−tλiωi =

N∑i=0

aiωi

where ωi , . . . , ωN is an orthonormal basis for ker ∆p. Thusas t →∞, ω flows to its harmonic component.



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References

Hodge Theory


Then for any closed differential form ω,

e−t∆ω − ω =

∫ t

0∂t(e

−t∆ω)dt

= d

(−∫ t

0e−t∆δωdt

)so

e−t∆ω = ω + d

(−∫ t

0e−t∆δωdt

)∈ [ω]

which implies that heat flow preserves the cohomology class ofa form.



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References

Hodge Theory


We see that each cohomology class contains a harmonic form,

limt→∞

e−t∆ω = ω − limt→∞

d

∫ t

0e−t∆δω dt = ω − d∆−1δω

(which is well-defined, after showing δω is independent ofker ∆) , now finally we show that the form is unique. Assumethere exist harmonic forms η1 6= η2 with [η1] = [η2] Then

η1 = η2 + dθ

δη1 = δη2 + δdθ

0 = δdθ

so0 = 〈θ, δdθ〉 = 〈dθ, dθ〉 = ‖dθ‖2



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References

Witten Laplacian

Lemma

For any t ≥ 0,βi = dim ker ∆i

t

Proof.

Observe that dte−tf = (e−tf detf )e−tf = e−tf d , which implies

that e−tf : Ωi → Ωi is an isomorphism making the followingdiagram commute,

· · · Ωi Ωi+1 · · ·

· · · Ωi Ωi+1 · · ·

d d

e−tf

d

e−tf

dt dt dt

so βi = dim ker ∆i = dim ker ∆it .



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References

Witten Laplacian

Now we will see Witten’s key insight: that the kernel of ∆it is

much simpler to understand as t →∞. It is necessary toexpand the Witten Laplacian directly.



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References

Witten Laplacian

For ω1, ω2 ∈ Ωi

dtω1 = e−tf detf ω1

= e−tf (etf dω1 + tetf df ∧ ω1)

= (d + tdf ∧)ω1

and

〈δtω1, ω2〉 = 〈ω1, dtω2〉= 〈ω1, (d + tdf ∧)ω2〉= 〈ω1, dω2〉+ 〈ω1, tdf ∧ ω2〉= 〈δω1, ω2〉+ 〈tι∇f ω1, ω2〉= 〈(δ + tι∇f )ω1, ω2〉



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References

Witten Laplacian

Expanding ∆t ,

∆t = dtδt + δtdt

= (d + tdf ∧)(δ + tι∇f ) + (δ + tι∇f )(d + tdf ∧)

= dδ + tdf ∧ δ + tdι∇f + t2df ∧ ι∇f+ δd + tι∇f d + tδdf ∧+t2ι∇f df ∧

= ∆ + t2‖df ‖2 + th

whereh = L∇f + L∗∇f



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References

Witten Laplacian

Then as t becomes large, ω ∈ ker ∆t implies that ω can benonzero only on small neighborhoods of the critical points of f .We will now consider a neighborhood Uq of a critical point q off , and compute ∆t in a local coordinate system on Uq.



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References

Morse Lemma

We will use the Morse Lemma to provide a coordinate system.

Theorem (Morse Lemma)

Let q be an isolated, nondegenerate critical point forf ∈ C∞(M,R). Then there exists a coordinate systemx1, . . . , xn on a neighborhood Uq of q such that forx = (x1, . . . , xn) ∈ Uq,

f (x) = f (q)−mq∑i=1

x2i +

n∑i=mq+1

x2i

where mq is the Morse index of q.



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References

Morse Lemma

To prove the Morse Lemma, we follow the following outline:

1 Prove Hadamard’s Lemma (1st order Taylor expansion).

2 Show that if f (x) = f (q)−∑λ

i=1 x2i +

∑ni=λ+1 x

2i , then

λ = mq.

3 Choose a coordinate system x1(q) = · · · = xn(q) = 0, andapply Hadamard’s Lemma twice (using the fact thatdf (q) = 0.)

4 Argue inductively on the coordinates xi .



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References

Local Coordinate Expansion

In the coordinate system provided by the Morse Lemma,

‖df ‖2 = 4n∑

i=1

x2i ,

∇f = −mq∑i=1

2xi∂

∂xi+

n∑i=mq+1

2xi∂

∂xi,

and by choosing the metric in local coordinates to beg =

∑ni=1(dx i )2 at q,

∆ = −n∑

i=1

∂2

∂x2i

.



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References


A (long) computation for h = L∇f + L∗∇f gives

h = 2n∑

i=1

ηi [dxi , ι ∂

∂xi

]

where

ηi =

−1 i ≤ mq,

1, i > mq.



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References


Thus on Uq we can approximate ∆t by,

Hq,t =n∑

i=1

− ∂2

∂x2i

+ 4t2x2i + 2ηi t[dx i∧, ι ∂

∂xi

]

=n∑

i=1

Ji + 2tKi

Where Ji := − ∂2

∂x2i

+ 4t2x2i and Ki := ηi [dx

i , ι ∂∂xi

].



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References


Notice, for a differential form ω = fIdxI (with I a multiindex,)

Kiω =

−ω (i ≤ mq and i ∈ I ) or (i > mq and i /∈ I )

ω otherwise

so Ki = ±1 =⇒ [Ji ,Ki ] = 0 which implies that they can besimultaneously diagonalized.



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References

Local Coordinate Approximation

Proposition (Kernel of Hq,t)

For any q ∈ Cr(f ), the map Hq,t : Uq → R defined in the localcoordinates xi given by the Morse lemma on theneighborhood Uq of q by

Hq,t =n∑

i=1

− ∂2

∂x2i

+ 4t2x2i + 2ηi t[dx i∧, ι ∂

∂xi

]

has kernel of dimension one, and is generated by the eigenform

e−t|x |2dx1 ∧ · · · ∧ dxmq .

and moreover all of the nonzero eigenvalues of Hq,t are greaterthan Ct for some fixed C > 0.



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References

Eigenvalues of the Quantum Harmonic Oscillator

The operator Ji is a scaling of the simple quantum harmonicoscillator from physics. It’s spectrum is well-known.

Proposition

The eigenvalues of Ji are precisely 2t(1 + 2j) for non-negativeintegers j . Moreover, the 2t-eigenfunction of Ji is

e−tx2i



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References

Eigenvalues of the Quantum Harmonic Oscillator I

To determine the eigenvalues of the quantum harmonicoscillator, argue using the ‘Dirac Ladder Operator’ method:

1 Define p = −i ∂∂xi and a =√t(xi + i

2t p)

so that

Ji = 4t2x2i + p2

2 Then

2t(1 + 2a†a) = Ji

so an eigenvalue of N = a†a is an eigenvalue of Ji .

3 Show that

Nafλ = (λ− 1)afλ

Na†fλ = (λ+ 1)a†fλ.



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References

Eigenvalues of the Quantum Harmonic Oscillator II

4 Argue that eigenvalues must be nonnegative since

λ〈fλ, fλ〉 = 〈fλ, a†afλ〉= ‖afλ‖2 ≥ 0

5 Argue that if λ is not a nonnegative integer, applying asufficiently many times to fλ would result in a functionwith negative eigenvalue, a contradiction.

6 Conclude the first claim using 2t(1 + 2N) = Ji .

7 Finally, show that e−tx2i is the 2t-eigenfunction of Ji

directly, solving Naf0 = 0.



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References

Local Coordinate Approximation

Now we can prove the proposition. For an eigenform ω of Hq,t ,

Hq,tω = t

(2

n∑i=1

(1 + 2j + Ki )

)ω

so if ω = g(x)dx I ∈ kerHq,t is nontrivial, then it must be that

j = 0 which forces g(x) = e−t|x |2

and also i ∈ I if and only ifi ≤ mq. In conclusion,

ω = e−t|x |2dx1 ∧ · · · ∧ dxmq

generates the kernel of Hq,t .



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References

Physical Intuition

There is now a good intuition as to why the Weak MorseInequalities should hold.

For each q ∈ Cr(f ) with mq = p, the kernel of Hq,t isgenerated by a single p-form locally.

There will be precisely Mp critical points of f with p-formsgenerating the kernel of Hq,t |pΩ locally.

Globally it seems reasonable that for ω ∈ ker ∆pt it must

be that ω ∈ Hq,t so Mp ≥ dim ker ∆pt = βp.

Witten argues along these lines. Here is presented ajustification by global analysis of the low-lying eigenvaluesof ∆p

t , adapted from [3].



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References

Proof of the Weak Morse Inequalities

Denote by Ept (c) the eigenspace of ∆p

t with eigenvalues in theinterval [0, c]. The following key theorem will give the WeakMorse Inequalities as a corollary.

Theorem (Key Theorem)

For any c > 0, there exists a t0 > 0 such that for any t > t0,

dimEpt (c) = Mp

where Mp is the p-th Morse number, 0 ≤ p ≤ n.

Corollary (Weak Morse Inequalities)

For 0 ≤ p ≤ n,βp ≤ Mp



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References


To prove the key theorem, we will need estimates on Sobolevnorms. Without loss of generality assume Uq is an open ballcentered at critical point q with radius 4a, and chooseγ ∈ C∞(R, [0, 1]) to be such that

γ(x) =

1 |x | < a

0 |x | > 2a



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References


Define

αq,t =∥∥∥γ(|x |)e−t|x |2

∥∥∥2

0=

∫Uq

γ(|x |)2e−2t|x |2 dx1 ∧ · · · ∧ dxn

ρq,t =γ(|x |)√αq,t

e−t|x |2dx1 ∧ · · · ∧ dxmq

The ρq,t will have unit length, and are motivated by the localgenerators for the kernel of Hq,t . Denote by Et the vectorspace genersated by the ρq,t for q ∈ Cr(f ).



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References

Projection Lemma

The following lemma estimating the orthogonal projectionsPt(c) : H0(M)→ Et(c) will allow us to prove the key theorem.

Lemma

There exist constants C , t3 > 0 such that for any t ≥ t3 andany σ ∈ Et ,

‖Pt(c)σ − σ‖0 ≤C

t‖σ‖0



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References

Deformed Witten Operator

Denote by Dt the ‘deformed Witten operator.’

Dt = dt + δt

and observe that

∆t = dtδt + δtdt = (dt + δt)(dt + δt) = D2t

since d2t = δ2

t = 0.



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References

Deformed Witten Operator

Since the ρq,t are compactly supported, Et is a finitedimensional subspace of H0(M), and so there is an orthogonaldecomposition

H0(M) = Et ⊕ E⊥t

with projections

pt : H0(M)→ Et

p⊥t : H0(M)→ E⊥t



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References

Proof of Key Theorem

Proof (Key Theorem).

By the lemma, when t is large enough the Pt(c)ρq,t will belinearly independent, so

dimEt(c) ≥ dimEt .

Assume for contradiction that dimEt(c) > dimEt . Then theremust be a nonzero s ∈ Et(c) orthogonal to Pt(c)Et . That is

〈s,Pt(c)ρq,t〉0 = 0

for all q ∈ Cr(f ).



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References



Then we can write the projection

pts =∑

q∈Cr(f )

〈s, ρq,t〉0ρq,t

=∑

q∈Cr(f )

〈s, ρq,t〉0(ρq,t − Pt(c)ρq,t)

+∑

q∈Cr(f )

〈s, ρq,t − Pt(c)ρq,t〉0Pt(c)ρq,t

so by the lemma

‖pts‖0 ≤C

t‖s‖0.



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References



Then‖p⊥t s‖0 ≥ ‖s‖0 − ‖pts‖0 ≥ C ′‖s‖0

and then using the proposition,

CC ′√t‖s‖0 ≤ ‖Dtp

⊥t s‖0 ≤ ‖Dts‖0 + ‖Dtpts‖0

≤ ‖Dts‖0 +1

t‖s‖0.

Rearranging,

‖Dts‖0 ≥CC ′t3/2 − 1

t‖s‖0

which as t →∞ contradicts s ∈ Et(c).



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References



Now

dimEt(c) = dimEt =n∑

i=0

Mi

and Et(c) is generated by Pt(c)ρq,t . Let Qi denote theprojection H0(M)→ L2Ωi . We have that

∆tQi s = Qi∆ts = µ2Qi s

so that Qi s is a µ2-eigenform of ∆t . We wish to show that fort large enough, dimQiEt(c) = Mi .



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References



By the lemma, for t large enough

‖QmqPt(c)ρq,t − ρq,t‖0 ≤C

t

thus the forms QmqPt(c)ρq,t are linearly independent anddimQiEt(c) ≥ Mi . However,

n∑i=0

dimQiEt(c) ≤ dimEt(c) =n∑

i=0

Mi

forcing dimQiEt(c) = Mi , and completing the proof.



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References

Polynomial Morse Inequalities I

Witten proves the Polynomial Morse Inequalities. The StrongMorse Inequalities follow by equivalence. We outline the proofof the Polynomial Morse Inequalties:

1 Let Cp(f ) be the free abelian group generated by thecritical points q ∈ M with Morse index mq = p. Denoteby dp

t the map Cp(f )→ Cp+1(f ) determined by dt ,identifying critical points with the associated eigenforms of∆t . Consider the Morse-Smale-Witten chain complex

0→ C1(f )→ · · · → Cn(f )→ 0.



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References

Polynomial Morse Inequalities II

2 The sequence

0→ ker dpt → Cp(f )

dpt−→ im dp

t → 0

is exact, so

Mp = rankCp(f ) = rank ker dpt + rank im dp

t

3 The sequence

0→ im dp−1t → ker dp

t → Hk(C∗(f ), d∗t )→ 0

is also exact, so

βp = rankHk(C∗(f ), d∗t ) = rank ker dpt − rank im dp−1

t



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References

Polynomial Morse Inequalities III

4 Then letting Qp = Mp − rank ker dpt ≥ 0,

Mt − Pt =n∑

p=0

(rank ker dpt + rank im dp

t )tp

−n∑

p=0

(rank ker dpt − rank im dp−1

t )tp

= (1 + t)n−1∑p=0

(Mp − rank ker dpt )tp

= (1 + t)n−1∑p=0

Qptp



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References

Strong Morse Inequalities I

Theorem

The Strong and Polynomial Morse Inequalities are equivalent.

This can be proved as follows:

1 Assume the Strong Inequalities. Then

M−1 =n∑

i=0

(−1)iMi =n∑

i=0

(−1)iβi = P−1

implies that Mt − Pt is divisible by (1 + t).

2 Then for some Qi ∈ Z,

Mt − Pt = (1 + t)n−1∑i=0

Qi ti



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References

Strong Morse Inequalities II

3 Arguing by induction using the Strong Inequalities, we canshow that the Qi must be nonnegative, proving thePolynomial Inequalities.

4 Assume the Polynomial Inequalities. Then by induction fork ∈ 0, 1, . . . , n − 1,

k∑i=0

(−1)i+kMi =k∑

i=0

(−1)i+kβi + Qk

5 Letting t = −1,

n∑i=0

(−1)iMi =n∑

i=0

(−1)iβi

completing the equivalence.



GianmarcoMolino

Background

MorseInequalities

Witten’s Idea

LocalApproximation



References

John W. Milnor. Morse Theory. Princeton UniversityPress, 1973. isbn: 978-0691080086.

Edward Witten. “Supersymmetry and Morse Theory”. In:J. Differential Geom. 17.4 (1982), pp. 661–692. doi:10.4310/jdg/1214437492.

Weiping Zhang. Lectures on Chern-Weil Theory andWitten Deformations. Singapore: World ScientificPublishing, 2001. isbn: 981-02-4685-4.

http://dx.doi.org/10.4310/jdg/1214437492

witten's laplacian and the morse...

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