Up and Down
the Bow Construction

Sergey Cherkis(University of Arizona)

Photo by Bernhard Witz

RIMS, Kyoto
Feb. 20, 2023

2

These results are obtained in collaboration with Mark Stern and Andres Larrain-Hubach:• Instantons on multi-Taub-NUT spaces I: Asymptotic form and index theorem,J.Diff.Geom. 119 (2021) 1.

• Instantons on multi-Taub-NUT Spaces II: Bow Construction, J.Diff.Geom. 2023.
• Instantons on multi-Taub-NUT Spaces III: Down Transform, Completeness, and Isometry,
to appear in 2023

This work derives from:• M. F. Atiyah, N. J. Hitchin, V. G. Drinfeld and Y. I. Manin, “Construction of Instantons,”
Phys. Lett. A 65, 185 (1978)• W. Nahm, ”The Construction of All Self-dual Multimonopoles by the ADHM Method,”in Proceedings of Monopoles in Quantum Field Theory meeting Trieste, 1981.

• E. Corrigan and P. Goddard, “Construction of Instanton and Monopole Solutions and Reciprocity,”
Annals Phys. 154, 253 (1984).

• P.B. Kronheimer and H. Nakajima, “Yang-Mills instantons on ALE gravitational instantons,”
Math. Ann., 288(2):263– 307, 1990.

• Hiraku Nakajima, “Monopoles and Nahm’s equations,” in Einstein metrics and Yang-Mills connections(Sanda, 1990), volume 145 of Lecture Notes in Pure and Appl. Math. Dekker, New York, 1993.

3Intro:“Bows form only the rst step in generalizing quivers.”• Instanton on ALE<—> Quiver• Instanton on ALF<—> Bow• Instanton on ALG<—> Sling• Instanton on ALH<—> WallComment:

Kronheimer - Nakajima ’90

4 real dim. hyperkähler space with L2 Riemann curvature = Tesseron
All tesserons come in ALE, ALF, ALG, ALG*, ALH, and ALH* types.
Ak-1 ALF = TNk

ALF spaces:(dτ + πk * ω)2g = V | dt | +Vk1
V=ℓ+, dω = *3 dV∑ 2 | t − νσ |σ=1TNνk ← S 12πk ↓ℝ3 = Im ℍ

Yang-Mills Instanton:(D0=AH and D1=Double cover(AH)

Hermitian bundle with connection (ℰ → TNνk, A) satisfying:

1. *FA=-FA
2. FA is L2fi

Dk ALF are asymptotic to A2k-5 / ℤ2

⇔Im DA†DA = 0

Dirac operator DA

Bows
A bow representation ℜ is
1. collection Λ of λ-points on the bow,
2. hermitian line bundles E on (Bow −Λ)(matching at λ points),
3. space Wλ for each constant rank λ point.

A bow is a collection of
1.oriented intervals and
2. arrows connecting their ends
(respecting the orientation).

Bow representation data Dat(ℜ) consists of
2. Bifundamental data:
1. Nahm data:
For each arrowconnection ∇ d on E and

Bp : Ep− →dsthree endomorphisms
T1, T2, T3 of E.

⊗ Ep+2-dim rep. of quaternions.

3. Fundamental data:
for each const. rankλ-point:

Qλ : Wλ →⊗ Eλ:= I1 ⊗ T1 + I2 ⊗ T2 + I3 ⊗ T3

Let

Bow solution:

Dat(ℜ) is hyperkähler and the gauge group of ℜ acts triholomorphically.
Its bow moment map equations (at level ν) are𝒮ds𝒮i ∇ d T1 = [T2, T3],𝕋𝕋𝕋𝕋𝒢

4(p−) + Im iBp Bp† = νp,(λ+) − (λ−) = Im iQλQλ†5Bow Moduli SpaceHKQ level set:

μ −1(ν) := {t − Im bb † = ν} ↶Hyperkähler
af ne spaceDat( ) ∋ ( ∇ ∂s∂ , t, b)

A (small) bow
representation(Herm. line bundle)e ↶c.f. all ALE tesseronsas quiver moduli spaces:
Kronheimer ‘89

Moduli space ℳof the bow rep.

TNνk = μ −1(ν)/• Moduli space of a bow rep. is hyperkähler:ℳ = Dat ( )///∈Bowpoint ss

Performing HKQ in stages, presents ℳ
as a nite HKQ.

• Focussing on one complex structure:

𝒢𝒢𝒢𝔰

𝒢𝒢

fifi𝔰𝔰𝔰𝔰𝔰𝔰𝔰

𝒫𝔰𝔰𝔰𝔰𝔰ℳ = μℂ−1(νℂ)/ℂ

One can cut the bow into pieces to present ℳℜ
as a nite HK quotient.

This can be used to study its asymptotic,Ch ‘10or realize it as a quiver variety in any given complex structure,
Nakajima & Takayama ‘16
or adjust the level to obtain a non-commutative deformation of ℳℜ
R. Bielawski, Y. Borchard and S. A. Ch ‘22

fiin which case the Bow construction delivers Instantons on non-commutative ℳℜ.

7

TNk as a Bow Rep. Moduli Space
Taub-NUT (A0 ALF)

multi-Taub-NUT (Ak ALF)

Dk ALF

Note: there are other bows with representations with the same moduli space.
E.g. Cheshire bow rep:

Ch-Blair ‘10

and Ray bow rep.

Ch-Kapustin ‘98
all have TNk as their moduli space.

Level set as a Poincaré bundle• For any point on the bow s ∈ Bow, one has exact sequence 1 →whereand the partial quotient μ −1(ν)/

s := {g ∈s→→ U(es) → 1| g(s) = 1}

s forms a U(es) principal bundle over the moduli space:μ −1(ν)/s ← U(es)↓TNνkThis is the Tautological principal bundle at s.
Its associated bundle is Ls → TNνk, andit comes equipped with a tautological instanton connection as!

νTogether,
thefamilyoftautologicalbundles
formabundleoverTN•
k × Bow:

e := π*e2↷

μ −1(ν) × BowLTNνk × Bow
Section of L =π1π2e
Bow-equivariant section of e.

𝒢𝒢𝒢𝒢𝒢𝒢

Note: all of the above applies to ANY bow rep. (except, its moduli space might not be TN).
𝒢𝒢

8

9Down Transform“A space knows its bow;
an instanton knows its bow representation.”
CLHS’21
Analytic results:
1. Any instanton (with asymp. holonomy) on ALF space has asymptotic formmλ ω−2||A = ⊕ π*
η−i(λ+)+O(t),λk2|t| V )λ∈Λ (λwhere exp(2πi ) are the asymptotic values of the holonomy eigenvalues andℓmλ are the rst Chern numbers of the corresponding eigen-linebundles.
2. Index of the associated Dirac operator isfi

1k1
2
indL2 DA =({λ/l} − )(mλ − k⌊λ/l⌋) − {λ/l} + 2 tr F ∧ F.
∑() 8π ∫22λ∈Λ
3. Harmonic spinors decay exponentially, if no λ = 0.

Given an instanton (ℰ, A), its Dirac operator is DA : Γ(S ⊗ ℰ) → Γ(S ⊗ ℰ)

The spin bundle S splits into chiral parts S = S + ⊕ S −.

Hyperkählerity => S + is trivial. Moreover, the Clifford action of the three Kähler forms Ii := Cl(ωi)
is covariantly constant and I1, I2, I3 form quaternionic units!

Twisting by tautological bundles, we have a family of Dirac operators parameterized by the bow:
Ds := DA⊗1L +1ℰ⊗as : Γ(S ⊗ ℰ ⊗ Ls) → Γ(S ⊗ ℰ ⊗ Ls)s

Importantly: anti-self-duality of both A and as is equivalent toDs2 |S+ = ∇*s ∇s ⊗ 1S+−If (ℰ, A) has no trivial factors, then ∇*∇>0,thuskerDisentirelyinS.
2ssLs• Index bundle:

Bounded harmonic sections:ℰs := kerL2 Ds

Wλ := kerL∞ ∇*∇λ λ• Induced Bow Solution: T, B, Q
1) Wλ comes with a map

Q : Wλ ⊗ S + → ℰλf ↦ Dλ fequivalently, Q ∈ ℰλ ⊗ (S +)* ⊗ W*
λSince Ds Ds |S + = ∇*s ∇s ⊗ 1S +, such Dλ f is indeed in ker Ds.
2) Using the orthogonal projection Πs : L 2(S − ⊗ ℰ ⊗ Ls) → kerL 2 Ds
Multiplication by t and b induces

Tj := ΠstjΠsandBα = ΠsbαΠsProp: The resulting (T,B,Q) solves the bow moment map equations at level ν (determined by TNνk) !

12Up TransformBow Solution => Bow Dirac Operator => Index bundle over the level set => Instanton
Quotient bundle ℰ → TNνkwith induced connection A
is an Instanton

Given a bow solution(T, B, Q)of a bow rep. ℜℜ : Γ(⊗ E) → Γ( ⊗ E) ⊕ Ep+ ⊕ Ep− ⊕ Wλ( − ∇ d + i )ψAny (t, b) ∈ μ −1(ν) isa solution of the small bow rep.
and has associated Bow Dirac op.

dsBp†ψ(p−)ℜψ :=−Bpc†ψ(p+)−Qλ†ψ(λ)

Use its charge conjugate operator
cto form a family(t,b) :=ℜ ⊗ 1e* + 1E ⊗(− ∇ dsd + i − itt) ψ𝔰𝔇𝒮𝒢𝔇𝕋𝔇𝒮𝔰𝔰𝔇𝔇𝔇𝒢𝔇𝔰𝔰𝕋𝔇

𝔇(t,b)ψ :=Bp†ψ(p−) − bp†ψ(p+)−Bpc†ψ(p+) + bpc†ψ(p−)−Qλ†ψ(λ)

cIndex bundle over the level setisequivariant:.

ker

†
↶
(t,b)

↓μ −1(ν)

13Instanton Class in terms ofBow Rep. Ranks
1. rk ℰ = dim ker†
= − ind(t,b)(t,b) = | Λ | .

2. As | t | → ∞ solutions of the Bow Dirac eq. concentrate near the λ-points;
so the induces connection Â has the formm̂ λ∂−îA( ) = diagλ++ O(t −2),∂τV (2t )
λwhere the magnetic charges are m̂ λ = R(λ+) − R(λ−) + | {p | p < λ} | .
3. Chern numbers of the resulting connection areiλtrFA = R(p+) − R(p−) − | {λ | λ > p} | +,∑2π ∫Cpℓλ1i1λkλtrFÂ ∧ FÂ = −m̂ λ − R0 +m̂ λ −∑∑∑(ℓ)()∫2 2π2 λℓ
2λλ22𝔇𝔇ℳ

“Two Dirac equations know each other.”
Schematic Relation:

Instanton Dirac Operator Ds(parameterized by a s ∈ Bow)τ∇* ∇ χ ̂ = c ΨGiven an orthonormal frame of ker Ds
DsΨα = 0

Frame ofker(t,b)

on the Bow(parameterized by (t, b) ∈ μon TNdτ + ωdc = Cl() = [Ds, i ]Vdsτχ∇Aμ β =f† τ(t,b)Iμ c Ψκ C := χα(s) ⊗ Ψ*α (s) ∈ S + ⊗ ℰ ⊗ e* ⊗ E*sχ
Let κ c = βf⊗ Ψ†α, thenακ solves the Bow Dirac equation𝔰𝔇Analogous calculation using Up transform give schematic relation for χ ̂ ∈ kerΔBowΨ = χ ̂“Up harmony Eq”
𝔇𝔇(ν))

ker Ds“Down harmony Eq”

𝔇(t,b)−1frame ofConsider bounded solution χα of Poisson eqsbpτ∇*s ∇s χα = c Ψαc τ Ψαand ∇*s ∇s βα =2 | t − νp |d2Ds χα = i Ψα
dsd0Ds χα = Ψα + ΨβTβα
dsjDs Ij χα = − t jΨα + ΨβTβα

Bow Dirac Operator(t,b) :15

Completeness“The heaven and the Earth are in harmony.”Up ∘ Down = IdInst

Down ∘ Up = IdBow(ℰ, A) → (E; T, B, Q) → (ℰ,̂ A)̂ ≃gauge (ℰ, A)(E; T, B, Q) → (ℰ, A) → (E,̂ T,̂ B,̂ Q)̂ ≃gauge (E; T, B, Q)

χConsider κ c = β (t; s) ⊗ Ψ†α(t′; s) is valued in (S + ⊗ ℰ ⊗ L*s ) |t ⊗ E*s,fα

its charge conjugate κ is valued in S + ⊗ ℰ*t ⊗ Ls ⊗ Esand thus gives a map κ : ℰ |t → S + ⊗ E ⊗ e* with image in ℰ̂ := ker(t,b).

It maps a ber of ℰ at [(t,b)] to a ber of ℰ̂ at (t,b).
Bow index theorem => κ is bijective.

fifi𝔇

￼

Down harmony eq => Fiber isometry and covariantly constant.

16

Isometry
Monopole case: Nakajima ‘93

Down harmony eq:∇Aμ χ ̂ =r
† τIμ(t,b) c ΨIts variation is···Aμ χ ̂ + ∇Aμ χ ̂ = Iμ†c τ Ψ +r
† τ·(t,b)Iμ c Ψ·

directly relates the tangent vector A to ℳInstanton·to the corresponding tangent vector of ℳBow
Using 1) the fact that Ψ is a frame of ker Ds,2) χ ĉ is a frame of ker (t,b), and3) some quaternionic identities
leads to the isometry relation

𝔇𝔇𝕋

𝔇𝕋·· · ·∥A∥2ALF = ∥(T, B, Q)∥2Bow

17c.f. Witten 0902.09481

Which Bow is Best?

(via brane considerations)• Distinct holonomy eigenvalues {e 2πiμ1, e 2πiμ2, …, e 2πiμn}

split the instanton bundle (on a complement of a compact set) ℰTNk∖B = W1 ⊕ W2 ⊕ … ⊕ Wn
Lemma: The eigenvalues have the formμj( t ⃗ ) =

λjl+

ϑj

2 | t ⃗|+ O( | t ⃗ |−2 )comparing to the asymptotic form of our connectionlϑj = mj +λj

mj dτ + ω mj
aj = (λj + )− ω.
2tVkklThis combination is Independent of any gauge choice!
• Consider the neighborhood of in nity:

TNk \ B contracts to the lens space S3/ k.

Pullback Bundle: Wj* ——————> Wj
is trivial
Covering space: S3 ——————> S3/ kfi𝕫𝕫𝕫fiThus, what distinguishes different line bundles Wj is k action on the ber.
There are only k types of line bundles Wj.
λj + lλjiτ̃/k3
Changing the trivialization of S acts by e :↦(mj) (mj − k)
• Moral: line bundles Wj are determined by k numbers {m̂ j | m̂ j = mj mod k,0 ≤ m̂ j < k}

Choice of Twisting Family
λj
1tr F =− ( σ + n σ ),∑∫2πi Cσljc1σ =• First Chern class: k real numbersσ ∈ℤσ ∈ {0,1,…, n − 1}

“Stiefel-Whitney classes” (obstructions of PSU(n) to SU(n) lifting)
Relabel the NUTs so that0≤ 1≤ 2≤… k<nThis labels the rows of a Young diagram that ts into an kxn rectangle
For example, for k=4, n=5 and ( 1, 2, 3, 4) = (1,1,3,4)

YBal=• View it as a closed path on an kxn torus.
Mark- horizontal segments λj and- vertical segments pσ
λ2p2𝔰

λ1
fi𝔠𝔠𝔠𝔠𝔠𝔠𝔠p1𝔠𝔠𝔰

18

λ5

λ4 p4
λ3 p3

This gives the balanced bow representation(the rank is continuous at each p-point):
p4

λ4p3

λ5
λ3
λ1p1p2λ2

19The set of magnetic charges also gives a Young diagram in nxk rectangle.

{m̂ 1, m̂ 2, …, m̂ n | m̂ j ∈ {1,2,…, k − 1}}

YCob=fiInterior:(Chern numbers)

YBal

In nity:(monopole charges)

YCob

20

Instanton Number
Since TNk is not compact, Chern character value ch2[E,A] does NOT have to be integer.
Need another de nition of the instanton number.
Let us focus on a single TN=TN1:
TN≃ℝ4 is contractible, so any bundle over it is trivial,and connection one-form A is globally de ned.
1. On a complement of a compact set, a gauge transformation g : TN∖B → U(n)transforms A to the diagonal form
λ1 + m1l
A g = − i diag

λ2 + m2l…

λn + mnldτ + ω
1+ O( 2 ) .
Vt

TN\B is contractible to S3, thus the homotopy class of g is in π3(U(n)) = ℤ .
Instanton Number is m0 := deg [g] ∈ π3(U(n)) .
2. Alternatively, holonomy splits the instanton bundle into orthogonal line bundles,
3
giving a map S∞→ U(n)/U(1)n = NnFlag spacenfifi

Instanton Number is an element of π3(U(n)/U(1) ) = ℤ .

21Summary• Quivers, Bows, Slings, and Walls give constructions of Instantons on
ALE, ALF, ALG, and ALH tesserons.

• Down transform: Instanton Dirac index bundle on the bow.

(The main problem is to identify ALL the necessary data to have a complete construction.)• Up transform: Bow Dirac index bundle.
• Completeness: Up∘Down = IdInst and Down ∘Up = IdBow• Isometry: ℳInst = ℳℜ.