end-periodic dirac operators and seiberg-witten theory...

Introduction Seiberg-Witten theory Periodic manifolds and Dirac operators Seiberg-Witten invariant of X Periodic index theory

End-periodic Dirac operators andSeiberg-Witten theory∗

Tomasz Mrowka1 Daniel Ruberman2 Nikolai Saveliev3

1Department of MathematicsMassachusetts Institute of Technology

2Department of MathematicsBrandeis University

3Department of MathematicsUniversity of Miami

Duke University, December 2009

∗http://arxiv.org/pdf/0905.4319


The simplest smooth 4-manifolds

Simply connected: S4, CP2, S2 × S2.

Non-simply connected: S1 × S3.

Will concentrate on invariants of manifolds X with the homologyof S1 × S3. Classical Z2-valued invariant ρ(X ) arising fromRohlin’s signature theorem.

Choose oriented M3 ⊂ X generating H3(X ).

Choose spin 4-manifold W with ∂W = M

ρ(X ) = ρ(M) = 18σ(W )


Long-term goal: find Z-valued lift of ρ(X ).

Applications to classification of manifolds.

Applications to homology cobordism and triangulation ofhigh-dimensional manifolds.

Approach: calculate ρ(X ) analytically via gauge theory:

Yang-Mills and Seiberg-Witten theory.


Seiberg-Witten equations

Seiberg-Witten theory assigns to a 4-manifold Y and Spinc

structure s, a number SW (Y , s), by counting irreduciblesolutions (up to gauge equivalence) to the Seiberg-Wittenequations.

Variables: Spinc connection A, spinor ψ ∈ C∞(S+), andr ∈ R

+

D+A (g)ψ = 0

∫

Y|ψ|2 = 1

F+A + r2q(ψ) = µ

where g is a metric on Y , and µ ∈ Ω2+(Y ; iR).

A solution is irreducible if r 6= 0.



Equations depend on metric on Y and 2-form µ.Generic perturbation µ makes moduli space smooth,oriented 0-manifold.

Version of equations with r yield ‘blown-up’ moduli space ofKronheimer-Mrowka.

Count irreducible (r 6= 0) solutions to µ-perturbedSeiberg-Witten equations.

Independent from g and µ if b+2 Y > 1.

Specialize to case of X with homology of S1 × S3, and writeµ = d+β. The algebraic count of irreducible solutions isdenoted SW(X ,g, β).



Key problem: Dependence of SW(X ,g, β) on g and β.

Consider SW(X ,gt , βt ) for 1-parameter family (gt , βt).

Since b+2 (X ) = 0, may have solutions (At , rt , ψt) with

rt → 0 as t → t0, so count can change.

Want some other metric-dependent term with similar jump.

For X = S1 × M3, done by Chen (1997) and Lim (2000).

Counter-term from η-invariants of Dirac operator andsignature operator on M3.


Periodic Dirac operators

Proposed counter-term in non-product case:

Index of end-periodic Dirac operator.

Setup: Closed spin manifold X with a map f : X → S1,surjective on π1. This gives

Connected Z-cover X → X , and lift t : X → R of f .

Dirac operator D+ : C∞(S+) → C∞(S−).

For any regular value θ ∈ S1 for f , a submanifoldf−1θ = M ⊂ X .

Question: When is D+ a Fredholm operator?



To make sense of this, need to complete C∞0 (S±) in some

norm. Pick δ ∈ R, and define

L2δ(S

±) = s |

∫

Xetδ|s|2 <∞

as well as Sobolev spaces L2k , δ(S

±).Finiteness of dimensions of the kernel/cokernel of

D+ : L2k , δ(S

±) → L2k−1, δ(S

±)

summarized as: D+ is Fredholm on L2δ .

The most useful case for us is δ = 0.



Taubes’ idea: Fourier-Laplace transform

s ⇒ sµ = e µt(x)∞∑

n=−∞

e µns(x + n) for µ ∈ C

yields family of problems on compact X . For each z ∈ C, havethe twisted Dirac operator D+

z : C∞(S+) → C∞(S−):

D+z s = D+s − log(z) dt · s.

Theorem 1 (Taubes, 1987)

Fix δ ∈ R. Suppose that ker D+z = 0 for all z ∈ C

∗ with|z| = e

δ2 . Then D+ is Fredholm on L2

δ .



Theorem 2 (R-Saveliev, 2006)

For a generic metric on X, the operator D+ is Fredholm on L2.

Suffices to find one metric with Dz invertible ∀z ∈ S1.We apply technique of Ammann-Dahl-Humbert (2006).

Invertibility of Dz , ∀z ∈ S1, can be pushed across acobordism.


End-periodic manifolds

End-periodic manifolds are periodic in finitely manydirections, each modeled on a Z covering X → X . Let M ⊂ Xbe non-separating; it lifts to a compact submanifold M0 ⊂ X .

X

X

M0

X0

M



Let X0 be everything to the right of M0, and choose a compactoriented spin manifold W with (oriented) boundary −M. Fromthese pieces, form the end-periodic manifold with end modeledon X :

Z = W ∪M0X0

M0

X0

W



Excision principle: Everything we said about Dirac operatorson X holds for Dirac operators on Z .

For metric g on X , extending to metric on Z , get Diracoperator D+(Z ,g) and twisted version D+

β (Z ,g) forβ ∈ Ω1(X ; iR).

Fredholm on L2 for generic (g, β).

ind(D+β (Z ,g)) depends on choice of W in simple way.

Unlike compact case, ind(D+β (Z ,g)) depends on (g, β).

Can jump in family gt if ker(D+z (X , g0)) 6= 0 for z ∈ S1.


Definition of λSW

Observation: ind(D+β (Z ,g)) jumps at the same place as

SW (X ,g, β). This suggests that we try to use one to balancethe other. Have to get rid of dependence of ind(D+

β (Z ,g)) oncompact manifold W .

Provisional definition: Consider the quantity

λSW (X ,g, β) = SW (X ,g, β) − ind(D+β (Z ,g)) −

18

sign(W )

Theorem 3 (Mrowka-R-Saveliev 2009)

λSW (X ,g, β) is independent of choice of W, metric, andperturbation, and gives a C∞ invariant of X .


Definition of λSW

Remark: Previous work (R-Saveliev 2004) defines λDon(X ) bycounting flat connections.

Conjecture 4

1 λSW (X ,g) = λDon(X ).2 λSW (−X ,g) = −λSW (X ,g)

Combining properties of λSW and λDon(X ), part (1) would solvean old problem in 4-manifold theory by showing that there is nohomotopy S1 × S3 with ρ 6= 0. Part (2) would contradict thetriangulation conjecture for high-dimensional manifolds!


Properties of λSW

1 Independence from various choicesChoice of slice M ⊂ X and lift M0 ⊂ X .Choice of W with ∂W = M, and extension of metric over W .

2 Reduction mod 2 of λSW is classical Rohlin invariant ρ(X ).3 Extension to some W with b−

2 6= 0.

Item 1: Excision principle.

Item 2: Two ingredients. Involution in Seiberg-Witten theorymakes SW (X ,g) even, and quaternionic nature of Diracoperator makes ind(D+(Z ,g)) even.

Item 3: Related to complex surfaces of type VII0.


Jumps in SW and ind(D)

Have seen that in a family gt , the invariants SW (X ,gt , βt) andind(D+

βt(Z ,gt)) jump at the same t . Change in SW (X ,g, β)

understood: wall-crossing phenomenon in gauge theory.

If X = S1 × M3, then change in index is ‘spectral flow’ of Diracoperators on M, studied by Atiyah-Patodi-Singer. Theorem 3proved in this situation independently by Chen and Lim.

General periodic case more subtle; there’s no operator on M orspectrum to flow.



Approach: Fix (g, β), and vary the exponential weight.Consider fixed operators D+ = D+

β (Z ,g) on L2δ , for δ ∈ [δ0, δ1].

When Fredholm, denote its index by indδ(D+).

Denote by S(δ0, δ1) the set of z ∈ C with ker(Dz) 6= 0 andeδ0/2 < |z| < eδ1/2. By Taubes’ theorem 1, this is a finite set. Toeach z ∈ S(δ0, δ1), we associate a ‘multiplicity’ d(z) in terms ofresolvent family (Dz)

−1.

Lemma 5

If dim ker(D+z ) = 1, then d(z) = 1.



Theorem 6

For generic metric g, the difference

indδ1(D+(Z ,g)) − indδ0(D

+(Z ,g)) =∑

z∈C(δ0,δ1)

d(z)

Geometric case–fix δ = 0, vary (g, β).

Translate back to fixed (g, β) and varying weights δ.


Proof of main theorem

Compare change in SW moduli space and change in index asg and β vary along a path.

Technical point: choose special path (gt , βt ).

Change in δ from local description of SW moduli space.

Genericity implies all d(z) = 1.

Relate d(z) to wall-crossing signs in SW theory.


Pictorial interpretation

Fix δ = 0, let (gt , βt) vary. Write C = S1 × [0,1]; this is wherechanges in SW and ind(D+

β (Z ,g)) occur.

Let S = (c, t) ∈ C × [0,1] | ker(D+βt

(X ,gt) − log(c) dt) 6= 0

C

t

S

Main Theorem: ∆SW = S · C.


The Atiyah-Patodi-Singer η-invariant

Work in progress: Analogue of Atiyah-Patodi-Singer theoremfor end-periodic DIrac operators.

APS theorem: D a Dirac operator on manifold W with productend M × R+. On end, D(W ) conjugate to

∂/∂t + D(M)

Then (on appropriate L2 spaces)

ind D(W ) =

∫

WA(W ) −

ηDirac(M) + dim ker(D(M))

2

The invariant ηDirac(M) is a spectral invariant; spectral flow isthe change in η-invariant.


End-periodic η-invariant

Want an APS theorem for end-periodic Dirac operators.

Following APS, study heat equation on W , with end modeledon X . Fourier-Laplace transform relates heat kernel on end toheat kernel on X . Proposed periodic-end η-invariant:

η(X ,g) = 1πi

∫ ∞0

∫|z|=1 Tr

(df · D+

z exp(−tD−z D+

z )) dz

z dt

Remarks:

η depends on f : X → S1.

D±z = D± − log z · f ∗dθ; family of operators on X .

Tr means integrate the pointwise trace over X .


End-periodic η-invariant

Proposed end-periodic index theorem.

Setup: W end-periodic spin manifold, end modeled on X .

Assume ind D(X ) = 0, so A(X ) = dα.

Choose f : X → S1, isomorphism on H1.

Choose regular value θ ∈ S1 and M = f−1θ.

Theorem in progress:

ind D+(W ,g) =

∫

WA (W ,g) +

∫

Mα−

∫

Xdf ∧ α −

12η(X ,g).

end-periodic dirac operators and seiberg-witten theory...

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