nahm transform of triply periodic instantonshamanaka/20190118yoshino.pdfthe kobayashi-hitchin...

Nahm transform of triply periodic instantons

Masaki Yoshino

RIMSKyoto University

Graduate School of Science

2019/1/20

Masaki Yoshino (RIMS) Nahm transform of triply periodic instantons 2019/1/20 1 / 32

Contents

1 The Nahm transform

2 The Fourier-Mukai transform

3 Main Result (1): Asymptotic behavior of L2-finite instantons

4 Main Result (2): The construction of the Nahm transform

5 Main Result (3): Correspondence between weights


The definition of instanton

Definition(X, g) : an oriented Riemannian 4-fold.(V, h) : a Hermitian vector bundle on X .

A : a connection on (V, h).1 A tuple (V, h,A) is called an instanton.

def⇐⇒ The curvature F (A) satisfies the ASD equation F (A) = − ∗ F (A).2 An instanton (V, h,A) on X is called an L2-finite instanton.

def⇐⇒ |F (A)| ∈ L2(X, g).

RemarkIf we assume that (X, g) is a compact connected Kähler surface with the

Kähler form ω, then the ASD equation for (V, h,A) is equivalent to thefollowing equations:

∂A ∂A = 0F (A) ∧ ω = 0. (The Hermite-Einstein condition)


The Nahm transform

"Definition"Λ ⊂ R4 : a closed subgroup.Λ∗ := ξ ∈ Hom(R4,R) | ξ(Λ) ⊂ Z: the dual subgroup of Λ.

(It is believed that) There exists a transform between the following.Λ-invariant instantons on R4 (with some conditions)Λ∗-invariant instantons on Hom(R4,R) (with some conditions)

This transform is called the Nahm transform.

ExampleΛ = R4 , Λ∗ = 0 : the ADHM transform

Λ ≃ Λ∗ ≃ Z4 : the Fourier-Mukai transformΛ ≃ R , Λ∗ ≃ R3 : Nahm, Hitchin, Nakajima, etc


The Nahm transform of instantons on R× T 3

Today : Λ ≃ Z3 , Λ∗ ≃ R× Z3.

By taking the quotient spaces, we obtain R4/Λ = R× T 3 andHom(R4,R)/Λ∗ = T 3, where T 3 is a 3-dimensional torus and T 3 is the dualtorus.From

• L2-finite instantons on R× T 3,we will construct• Dirac-type singular monopoles on the dual torus T 3.

Previous studiesCharbonneau constructed this Nahm transform for the case that instantonsare of rank 2 and satisfy a certain genericity.


The Kobayashi-Hitchin correspondence

The Kobayashi-Hitchin Correspondence is the relation between (the modulispaces of) some algebraic geometric objects and (hyper-)Kähler geometricobjects.

Fact (Donaldson, Uhlenbeck-Yau)(X,ω) : a connected compact Kähler surface.Then, the following objects correspond each other unique up to gaugetransformations.

(irreducible) instantons on X

(poly-)stable holomorphic vector bundles of degree 0 on X

Considering similar relation with both L2-finite instantons on R× T 3 andDirac-type singular monopoles on T 3, we obtain

stable filtered bundle on P1 × T 2

stable singular mini-holomorphic bundle on T 3

respectively.


The algebraic Nahm transform

As a variant of the Fourier-Mukai transform, we will construct singularmini-holomorphic bundle on T 3 from stable filtered bundle on P1 × T 2. wecall this construction the algebraic Nahm transform.

Hence we obtain the following commutative diagram.

L2-finite instantonon R× T 3

Dirac-type singularmonopole on T 3

poly-stable filtered bundleon (P1 × T 2, 0,∞× T 2)

(poly-stable) singularmini-hol. bundle on T 3

Nahm //

K-H corres.

OO

K-H corres.(already exists.)

Alg. Nahm//


The Fourier-Mukai Transform

DefinitionT 4 := C2/Λ : 4-dimensional torus.T 4 ≃ Pic0(T 4) : the dual torus of T 4.

1 The Poincaré bundle L on T 4 × T 4 is characterized by the following.∀ξ ∈ T 4, L|T4×ξ ≃ L−ξ := (C, h, d− 2π

√−1⟨ξ, dx⟩).

∀z ∈ T 4, L|z×T4 ≃ Lx := (C, h, d+ 2π√−1⟨x, dξ⟩).

2 The Fourier-Mukai transform • : Db(Coh(T 4)) → Db(Coh(T 4)) isdefined by V · := Rp2∗(p

∗1V

· ⊗ L), where pi is the projection fromT 4 × T 4 to the i-th component.

3 The inverse transform • : Db(Coh(T 4)) → Db(Coh(T 4)) is defined byW · := Rp1∗(p

∗2W

· ⊗ L−1).

Theorem (Mukai)We have isomorphisms ˆ• ≃ IdDb(Coh(T 4))[−2] and ˇ• ≃ IdDb(Coh(T 4))[−2].


The FM transform of irreducible instantons

(V, h,A) : an irreducible instanton on T 4 of rank r > 1.

Proposition

Hi(V ) = 0 for i = 0, 2.H1(V ) is a locally free sheaf of rank

∫T 4 ch2(V ) = (8π2)−1||F (A)||2L2 .

Under the identifications S+ ≃ Ω0,0T 4 ⊕ Ω0,2

T 4 and S− ≃ Ω0,1T 4 , the Dirac operator

/∂±A : Γ(T 4, V ⊗ S±) → Γ(T 4, V ⊗ S∓) can be written as /∂

±A =

√2(∂A + ∂

∗A).

TheoremFor ξ ∈ T 4, let (V, h,Aξ) denote the twisted instanton (V, h,A)⊗ L−ξ. Thenwe have

H1(V ) ≃⨿ξ∈T 4

Ker(/∂−A) ⊂ L2(T 4, V ⊗ S−).

Moreover, H1(V ) become an irreducible instanton by the induced metric.


Model solutions of the Nahm equations

DefinitionThe Nahm equation is the following u(r)-valued ordinary differentialeuqation:

∂Ai

∂t+ [α,Ai] = −sgn(ijk)[Aj , Ak] ((ijk) is any permuation of (123)).

This equation is the 3-dimensional reduction of the ASD equation.

For commuting matrices Γ = (Γi)i=1,2,3 ⊂ u(r) and N = (Ni)i=1,2,3 ⊂ C(Γ)satisfying the relations Ni = [Nj , Nk] for any even permutation (ijk) of (123),the tuple of α = 0 and Ai = Γi +Ni/t forms a solution of the Nahm equation.

DefinitionA pair of tuples (Γ = (Γi), N = (Ni)) is called a model solution of the Nahmequation.


Asymptotic behavior of L2-finite instantons

Theorem (Main Result (1) )(V, h,A) : an L2-finite instanton on (0,∞)× T 3 of rank r.∃R ≫ 0, ∃ δ > 0, ∃σ : (V, h)|(R,∞)×T 3 ≃ (Cr, h) : a trivialization, ∃ (Γ, N) : amodel solution of the Nahm equation s.t . we have the following for theconnection form αdt+

∑i Aidx

i of A with respect to σ.1 σ is a temporal gauge of A, i.e. α = 0.2 The following estimate holds at t → ∞ for 1 ≤ i, j ≤ 3 :

|Ai − (Γi +Ni/t)| = O(t−(1+δ)).

| [Ai,Γj ] | = O(− exp(−δt)).

CorollaryFor any L2-finite instanton on R× T 3, there exist model solutions (Γ±, N±)which approximate (V, h,A) at t → ±∞.


Sketch of proof of main result (1) I

Theorem (Morgan-Mrowka-Ruberman)(V, h,A) : an L2-finite instanton on (0,∞)× T 3.

∃R ≫ 0, ∃ δ > 0, ∃σ : (V, h)|(R,∞)×T 3 ≃ (Cr, h): a trivialization, ∃ (α, Ai): asolution of the Nahm equation on (R,∞) s.t . for the connection formαdt+

∑i Aidx

i of A with respect to σ, we have

|α− α|+3∑

i=1

|Ai − Ai| = O(exp(−δt)).

Briefly speaking, Any L2-finite instanton on (0,∞)× T 3 can beapproximated by a solution of the Nahm equation.


Sketch of proof of main result (1) II

Theorem (Biquard)(α(t), Ai(t)) : a solution of the Nahm equation on (0,∞).

∃ (Γ, N) : a model solution of the Nahm equation, ∃ g : (0,∞) → U(r) : agauge transformation s.t . we have the following.

g is a temporal gauge, i.e. g−1(∂t + α)g = 0.∃ δ > 0 s.t . for 1 ≤ i, j ≤ 3 we have

|g−1Aig − (Γi +Ni/t)| = O(t−(1+δ)).

| [g−1Aig,Γj ] | = O(− exp(−δt)).

This Biquard’s result can be summarized as "any solution of the Nahmequation on (0,∞) can be approximated by a model solution".


Singularity set

(V, h,A) : an L2-finite instanton on R× T 3.(Γ±, N±) : the model solutions which approximate (V, h,A).

Definition

Let ˜Spec(Γ±) ⊂ R3 denote the simultaneous eigenvalues of ((2π√−1)−1Γ±,i).

We define Spec(Γ±) := q( ˜Spec(Γ±)), where q is the quotient mapR3 → T 3.We define the singularity set Sing(V, h,A) ⊂ T 3 to beSing(V, h,A) := Spec(Γ+) ∪ Spec(Γ−).

RemarkTake ξ ∈ T 3. For the twisted instanton (V, h,Aξ) = (V, h,A)⊗ L−ξ,we have Sing(V, h,Aξ) = Sing(V, h,A)− ξ.


Propaties of Dirac operators and harmonic spinors

(V, h,A) : an L2-finite instanton on R× T 3.From Main result (1), we obtain the following corollaries.

Corollary

∀ ξ ∈ T 3 \ Sing(V, h,A), the Dirac operator

/∂−Aξ

: L2(R× T 3, V × S−) → L2(R× T 3, V × S+)

is a surjective closed Fredholm operator of index (8π2)−1||F (A)||2L2 .

Corollary

∃C > 0, ∀ ξ ∈ T 3 \ Sing(V, h,A), ∀ f ∈ Ker(/∂−Aξ

)s.t .

|f | decays exponentially at t → ±∞.We have ||tf ||L2 ≤ C · dist(ξ,Sing(V, h,A))−1||f ||L2 .


The definition of monopole

Definition(Y, g) : an oriented Riemannian 3-fold.

(V, h,A) : a Hermitian vector bundle with a connection on Y .Φ : a skew-Hermitian endomorphism of (V, h).

A tuple (V, h,A,Φ) is a monopole.def⇐⇒ It satitfies the Bogomolny equation F (A) = ∗∇A(Φ).

RemarkLet π : R× Y → Y be the projection.

A tuple (V, h,A,Φ) is called a monopole on Y⇐⇒ (π∗V, π∗h, π∗A+ π∗Φdt) is an instanton on R× Y .


Dirac type singularities of monopoles

Definition(Y, g) : an oriented Riemannian 3-fold.

p ∈ Y .(V, h,A,Φ) : a monopole of rank r / Y \ p.

The point p is called a Dirac-type singularity with weight k = (ki) ∈ Zr

if the following hold:

1 ∃B ⊂ Y : a nbd. of p , ∃V |B\p =

r⊕i=1

Li :a unitary decomposition into

line bundles Li s.t .∫∂B

c1(Li) = ki.2 In above decomposition, we have the following estimates: Φ =

√−1

2R

r∑i=1

ki · IdLi +O(1)

∇A(RΦ) = O(1).


Main Result (2) Construction of monopoles I

(V, h,A) : an L2-finite instanton on R× T 3.

ConstructionLet V denote the product Hermitian vector bundle(L2(R× T 3, V ⊗ S−), || · ||L2

)on T 3 \ Sing(V, h,A).

We take a Hermitian subbundle (V , h) to be

(Vξ, hξ) := Ker(/∂−Aξ

) ∩ L2(R× T 3, V ⊗ S−)

for any ξ ∈ T 3 \ Sing(V, h,A).

Using the orthogonal projection P : V → V and the trivial connection d

on V , we take a connection A to be

A := Pd.


Main Result (2) Construction of monopoles II

ConstructionWe take a skew-Hermitian endomorphism Φ : V → V to be

Φ(f) := P (2π√−1tf)

for f ∈ Vξ = Ker(/∂−Aξ

).

Theorem (Main Result (2) )

The tuple (V , h, A,Φ) is well-defined and a monopole onT 3 \ Sing(V, h,A).

Each point of Sing(V, h,A) is a Dirac-type singularity of (V , h, A, Φ).


Sketch of proof of main result (2)

Well-definedness of (V , h, A) follows from Fredholmness of the Diracoperator /∂

−Aξ

for ξ ∈ T 3 \ Sing(V, h,A).

One of Φ follows from exponentially decay of harmonic spinors.the last assertion is an easy consequence of the following theorem.

Theorem (Mochizuki-Y.)U ⊂ R3 : a nbd. of 0 ∈ R3.

(V, h,A,Φ) : a monopole on U \ 0.

Then the following are equivalent :The point 0 is a Dirac-type singularity of (V, h,A,Φ).The estimate |Φ(x)| = O(|x|−1) is satisfied.


An assumption of the torus T 3

For simplicity, we assume the following condition.

AssumptionThe 3-dimensional torus T 3 is isometric to the product of a circle S1 = R× Zand a 2-dimensional torus T 2.

Under this assumption, we take holomorphic coordinates

τ = t+√−1x1 ∈ R× S1

w = x2 +√−1x3 ∈ T 2

on R× T 3 = (R× S1)× T 2 ≃ C∗ × T 2.

RemarkFor general T 3, we fix a 2-dimensional subtorus T 2 ⊂ T 3 and regard R× T 3

as a principal T 2-bundle on R× S1 ≃ C∗. Then, T 3 is regarded as a principalS1-bundle on T 2.


Prolongation of instantons I

∆ ⊂ C : the unit disk, and put ∆∗ := ∆ \ 0.X : a complex manifold.

i : ∆∗ ×X → ∆×X : the inclusion map.

Definition(V, ∂V , h) : a holomorphic Hermitian vector bundle on ∆∗ ×X .

For a ∈ R, we take an OX×∆-submodule PaV of i∗V as follows:

f ∈ Γ(U, i∗V ) belongs to Γ(U,PaV )⇐⇒ ∀ p ∈ U ∩ (X × 0), we have |f |h = O(|z|−a−ε) (∀ε > 0) on a nbd. of p.

The family P∗V := PaV a∈R is called the prolongation of V .

For an L2-finite instanton (V, h,A) on R× T 3 ≃ C∗ × T 2, we take theprolongation P∗∗V = PabV a,b∈R on P1 × T 2. Here the first and secondindices represent the prolongation over 0 × T 2 and ∞ × T 2 resprectively.


Prolongation of instantons II

The prolongation P∗∗V = PabV a,b∈R has the following propaties.Each PabV is a locally free sheaf on P1 × T 2.If a′ < a and b′ < b, then we have Pa′b′V ⊂ PabV .∀ a, b ∈ R, ∃ ε > 0 s.t . we have P(a+ε)bV ≃ Pa(b+ε)V ≃ PabV .We set P<a,bV :=

∪a′<a Pa′bV and 0Gr(P∗∗V ) := PabV/P<a,bV . Then

0Gra(P∗∗V ) := PabV/P<a,bV is a locally free sheaf on 0 × T 2. Asimilar condition holds for ∞ × T 2 ⊂ P1 × T 2.We have P(a+1)bV ≃ PabV (0 × T 2) and Pa(b+1)V ≃ PabV (∞ × T 2).In particular, we have P(a−1)(b+1)V ≃ PabV .

DefinitionA family of holomorphic bundles satisfying the above conditions is called afiltered bundle on (P1 × T 2, 0,∞× T 2).


Propaties of the prolongation P∗∗V

If (V, h,A) is irreducible and of rank r > 1, then the prolongation P∗∗Vsatisfies a kind of stability. From this, we obtain the following propaties.

Propositionp : P1 × T 2 → T 2 : the projection.

For any a ∈ R and any F ∈ Pic0(T 2), we have

H0(P1 × T 2, P−aaV ⊗ F ) = H2(P1 × T 2, P−a<a<V ⊗ F ) = 0.

PropositionFor any a ∈ R, the gradations 0Gra(P∗∗V ) and ∞Gra(P∗∗V ) are semi-stablebundles on T 2 of degree 0.


Gradations of the prolongation P∗∗V

Moreover, for the prolongation P∗∗V , the gradation can be explicitlyrepresented by the model solutions which approximate (V, h,A).

Proposition(Γ±, N±) : the model solutions that approximate (V, h,A) to t → ±∞.Xa

± ⊂ Cr : the eigenspace of Γ±,1 of eigenvalue 2π√−1a ∈

√−1R.

B± := (∑3

i=2(Γ±,i +N±,i)dxi)(0,1) ∈ Ω0,1

T 2 .

Then, we have isomorphisms

0Gra(P∗∗V ) ≃ (Xa−, ∂T 2 +B−|Xa

−),

∞Gra(P∗∗V ) ≃ (Xa+, ∂T 2 +B+|Xa

+).


Mini-holomorphic structure of monopoles

(Σ, gΣ): a compact Riemann surface. I ⊂ Rs: an open interval.(V, h,A,Φ): a monopole on I × Σ.

DefinitionWe take a decomposition ∇A = ∇A,s ds+ ∂A + ∂A.

For s ∈ I , we define a holomorphic bundle V s as V s := (V, ∂A)|s×Σ.

Let Ψs,s′ : Vs → V s′ denote the parallel transport by a differential

operator dA,s := ∇A,s −√−1Φ. We call Ψs,s′ the scattering map.

Proposition

We have the commutativity condition [dA,s, ∂A] = 0. In particular,Ψs,s′ : V

s → V s′ is a holomorphic isomorphism.

DefinitionA tuple (dA,s, ∂A) that satisfies the commutativity condition [dA,s, ∂A] = 0 iscalled a mini-holomorphic structure of V .


Mini-hol. structures on Dirac-type singularity I

(V, h,A,Φ) : a monopole on I × Σ \ (0, p).(s0, p) : a Dirac-type singularity of (V, h,A,Φ) of weight k = (ki) ∈ Zr.

PropositionAssume that [−ε, ε] ⊂ I .∃v± = (v±,i): a local frame of V ±ε on a nbd. of p s.t .

Ψ−ε,ε(v−) = v+ · diag(zki),

where z is a local chart with z(p) = 0, and diag(ci) means the diagonalmatrix whose (i, i)-th entry is ci.

By this proposition, we obtain a mini-holomorphic bundle (V, dA,s, ∂A) onS1 × Σ with meromorphic singularities from a Dirac-type singular monopole(V, h,A,Φ) on S1 × Σ.


Mini-hol. structures on Dirac-type singularity II

A stability condition of singular mini-holomorphic bundles is defined byCharbonneau and Hurtubise, and they proved the following theorem.

Theorem (Charbonneau-Hurtubise)∀ (V, dA,s, ∂A): stable mini-hol. bundle with meromorphic singularities,∃ (V, h,A,Φ): a Dirac-type singular monopoles.t . the underlying mini-hol. bundle of (V, h,A,Φ) is isomorphic with(V, dA,s, ∂A), and the monopole (V, h,A,Φ) is unique up to suitable gaugetransformations.


The algebraic Nahm transform I

As a variant of the Fourier-Mukai transform, we introduce the algebraicNahm transform, which construct a mini-holomorphic bundle on S1 × T 2

from a stable filtered bundle on (P1 × T 2, 0,∞× T 2).

PropositionP∗∗V : a stable filtered bundle on (P1 × T 2, 0,∞× T 2).

L : the Poincaré bundle on T 2 × T 2.We take coherent sheaves ANs(P∗∗V )s∈R on T 2 to be

ANs(P∗∗V ) := R1p3(p∗12(P−ssV × p∗23L)),

where pij is the projection from P1 × T 2 × T 2 to the product of the i-th andj-th components.Then, ANs(P∗∗V ) is a locally-free sheaf for any s ∈ R.

We have an isomorphism ANs(P∗∗V ) ≃ ANs+1(P∗∗V ).Hence we regard the parameter s as a element of S1 = R/Z.


The algebraic Nahm transform II

By the cohomology vanishing of P∗∗V and semi-stability of the gradationmodule of P∗∗V , ANs(P∗∗V )s∈S1 forms a mini-holomorphic bundle(AN(P∗∗V ), dAN,s, ∂AN) on T 3 with meromorphic singularities.

DefinitionWe call (AN(P∗∗V ), dAN,s, ∂AN) the algebraic Nahm transform of P∗∗V .

Then we have the following theorem.

Theorem(V, h,A) : an irreducible L2-finite instanton on R× T 3.

P∗∗V : the prolongation of (V, h,A).(V , h, A, Φ) : the Nahm transform of (V, h,A).

(AN(P∗∗V ), dAN,s, ∂AN) : the algebraic Nahm transform of P∗∗V .

Then the algebraic Nahm transform (AN(P∗∗V ), dAN,s, ∂AN) is isomorphicwith the underlying mini-holomorphic bundle of (V , h, A, Φ).


Main Result (3) Correspondence between weights I

By the algebraic Nahm transform, we can calculate weight of Dirac-typesingularities of the transformed monopole (V , h, A, Φ) from the gradations0Gr∗(P∗∗V ) and ∞Gr∗(P∗∗V ).Moreover, we showed that the gradations can be explicitly represented by

the model solutions (Γ±, N±).Hence we can combine weight of singularities of (V , h, A, Φ) with (Γ±, N±).

We make a notational preparation.

NotationWe take a representation ρ± : su(2) → u(r) to be ρ±(ei) = N±,i, where ei isa basis of su(2) satisfying the relation [ei, ej ] = sgn(ijk)ek. SinceN±,i ∈ C(Γ±), we obtain the following decomposition:

ρ± =⊕

ξ∈Sing(V,h,A)

ρ±,ξ.

Let w±,ξ denote dimensions of irreducible representations contained by ρ±,ξ.


Main Result (3) Correspondence between weights II

Theorem (Main result (3) )(V, h,A) : an irreducible L2-finite instanton of rank r on R× T 3.(Γ±, N±) : the model solutions which approximate (V, h,A).

k±,ξ : the positive (negative) part of weight of(V , h, A, Φ) at ξ ∈ Sing(V, h,A).

For any ξ ∈ Sing(V, h,A),k±,ξ agrees with ±w±,ξ under a suitable permutation.


nahm transform of triply periodic instantonshamanaka/20190118yoshino.pdfthe kobayashi-hitchin...

Documents