Noncommutative (=NC) Instantons and Reciprocity

・MH&T.Nakatsu, ADHM Construction and Group Actions for NC Instantons, to appear (Moyal‐product formalism)

・MH&T.Nakatsu, Gauge instantons in NC space, work in progress, … (operator formalism)

cf.MH&Nakatsu, ADHM const. of NC Instantons, arXiv:1311.5227,…

Masashi Hamanaka (Nagoya U, Japan)

Based on

Group30, July17th 2014 in Gent@0.05μSv/h

1. Introduction• Non‐Commutative (NC) spaces are defined bynoncommutativity of spatial coordinates:

(cf. CCR in QM :                      )( ``space‐space uncertainty relation’’)

Resolution of singularity( new physical objects)

Ex) Resolution of small instanton singularity( U(1) instantons)  

ipq ],[

~

Com. space NC Space

: NC parameter (real const.)

[Nekrasov-Schwarz]

ixx ],[

Anti‐Self‐Dual Yang‐Mills (ASDYM) eqs. play  important roles in elementary particle theory, geometry and integrable systems.• Finite‐action solutions (instantons) reveal non‐perturbative effects in QFT ADHM is essential.

• Noncommutative instantons are used to obtain the Nekrasov partition fcn.

• We want to understand the ADHM fully.• (NC extension  background flux)

1. Introduction

ASDYM eq. in 4‐dim. with G=U(N)• ASDYM eq. (real rep.)

• There are two descripitions of NC extension:‐Moyal‐product formalism (deformation quantization)

‐ Operator formalism (Connes’ theory)

.2314

,4213

3412 ,

FFFFFF

AAAAAAF :

4,3,2,1,

:A Gauge field(N×N anti-Hermitian)

Field strength

NC ASDYM eq. with G=U(N) in Moyal• NC ASDYM eq. (real rep.)

1203

3102

2301

,

,

FF

FF

FF ):( AAAAAAF

00

00

2

2

1

1

O

O

(Spell:All products are Moyal products.)

)()()(2

)()(

)(2

exp)(:)()(

2

Oxgxfixgxf

xgixfxgxf

Under the spell, we geta theory on NC spaces:

i

xxxxxx

:],[

,)()()(),( )2(2)1()0( xAxAxAxA Under the spell, the solution is deformed:

Note: Coordinates and functions themselvesare c-number-valued usual ones

2. Atiyah‐Drinfeld‐Hitchin‐Manin Constructionbased on reciprocity for the instanton moduli space

0

0

21

2211

zz

zzzz

F

FF

kNNkkk JIB :,:,:2,1

ADHM eq. (≒0dim. ASDYM)(Easy)

4dim. ASDYang-Mills eq.(Difficult)

Sol.=ADHM data(G=`U(k)’)

Sol.= instantons(G=U(N), C =k)

1:1

NNA :

0],[0],[],[

21

2211

IJBBJJIIBBBB

k × k Matrix eqs.N × N PDE

2

Gauge trf.:

)(

11

NUggggAgA

Gauge trf.:

gJJIgI

kUggBgB~,~

)(~,~~

1

2,11

2,1

Fourier‐Mukai‐Nahm transformationBeautiful reciprocity between instanton moduli on 4‐

tori and instanton moduli on the dual tori

0

0

21

2211

zz

zzzz

F

FF

4dim. ASD Yang-Mills eq.on the dual torus

4dim. ASDYang-Mills eq.on a 4-torus

Sol.=the dual instantons(G=U(k), C =N)

Sol.=instantons(G=U(N), C =k)

1:1

NNA :

k × k PDEN × N PDE

2 2

0~0~~

21

2211

F

FF

kkA :~

On a 4-torus On the dual 4-torus

Define the maps F & G,& G◦F=id. & F◦G=id.

F

G

Fourier‐Mukai‐Nahm transformationBeautiful reciprocity between instanton moduli on 4‐

tori and instanton moduli on the dual tori

0

0

21

2211

zz

zzzz

F

FF

4dim. ASD Yang-Mills eq.on the dual torus

4dim. ASDYang-Mills eq.on a 4-torus

Sol.=the dual instantons(G=U(k), C =N)

Sol.=instantons(G=U(N), C =k)

1:1

VVxA ,)(

k × k PDEN × N PDE

2 2

0~0~~

21

2211

F

FF

kkA :)(~

map F (Dirac eq.)

On a 4-torus On the dual 4-torusx: :

0)~(

VixAeV

NkV 2:Family index thm.

Fourier‐Mukai‐Nahm trf. (radii of the torus∞)Reciprocity between instanton moduli on R

and instanton moduli on ``1pt.’’  [cf. van Baal, hep‐th/9512223]

0

0

21

2211

zz

zzzz

F

FF

0dim. ASD Yang-Mills eq.4dim. ASDYang-Mills eq.

Sol.=``dual instantons’’(G=U(k), ``C =N’’)

Sol.=instantons(G=U(N), C =k)

1:1

VVA

k × k PDEN × N PDE

2 2

0~0~~

21

2211

F

FF

kkA :~

map F (0dim Dirac eq.)

On a 4-torusR On the dual 4-torus1 pt.

0)~(

VixAeV

NkV 2:Linear alg.

4

]~,~[~~:~ AAAAF

Matrix eq.！

Matrix eq. !

Atiyah‐Drinfeld‐Hitchin‐Manin (ADHM) Constructionbased on the following reciprocity

0

0

21

2211

zz

zzzz

F

FF

kNNk

kk

JIB

:,:,:2,1

ADHM dim. (≒0dim. ASDYM)4dim. ASDYang-Mills eq.

Sol.=ADHM data(G=`U(k)’)

Sol.=instantons(G=U(N), C =k)

NNA :

0],[0],[],[

21

2211

IJBBJJIIBBBB

k × k matrix eq.N × N PDE

2Proved in the same way as the Nahm trf.

],[],[ 2211 zzzz RHS is in fact

1:1

F(0dim D.eq.)

G(4dim D.eq.)

ADHM(Atiyah‐Drinfeld‐Hitchin‐Manin) constructionEx.) Commutative BPST instanton（N=2, k=1）

0

0

21

2211

zz

zzzz

F

FF

ADHM eq.(≒0dim. ASDYM)4dim. ASDYang-Mills eq.

Sol.=ADHM data(G=`U(1)’)

BPST instanton(G=U(2), C =1)

0],[0],[],[

21

2211

IJBBJJIIBBBB

k × k matrix eq.N × N PDE

2

)(222

2

22

)(

))((2

22,)()(

ziF

zbxi

A

0( ),0,

11,2,12,1

JI

B

i

position

size0 : singular

M0

singularity

C

R

ADHM(Atiyah‐Drinfeld‐Hitchin‐Manin) constructionEx.) NC BPST instanton（N=2, k=1）

0

0

21

2211

zz

zzzz

F

FF

NC ADHM eq.NC ASDYang-Mills eq.

Sol.：ADHM data(G=`U(1)’)

NC BPST instanton(G=U(2), C =1)

0],[],[],[

21

2211

IJBBJJIIBBBB

k × k marix eq.N × N PDE

2

i

position

size0 : regular!

02( ),0,

,2,12,1

JI

B

Fat by ζ !

FA , ：exact sol.

M0

Resolution of the singurarity!

C

R

ADHM(Atiyah‐Drinfeld‐Hitchin‐Manin) constructionEx.) NC BPST instanton（N=2, k=1）

0

0

21

2211

zz

zzzz

F

FF

NC ADHM eq.NC ASDYang-Mills eq.

Sol.=ADHM data(G=`U(1)’)

NC BPST instanton(G=U(2), C =1)

0],[],[],[

21

2211

IJBBJJIIBBBB

k × k matrix eq.N × N PDE

2

i

position

size0 : regular!

02( ),0,

,2,12,1

JI

B

Fat by ζ！

FA , ：exact sol.By calculation of TrFΛF

Do k×k ADHM data giveInstanton number k in general ? (We prove this.)

C

R

3．Origin of instanton number in ADHM construction

We prove the formula:

Then:

kTrd

ffTrxdFFTrxdC

kk

kN

116

816

116

1

2

1242

**422

[Corrigan-Goddard-Osborn-Templeton]

comes from the size of the ADHM data!

)(12 rrf kkk

1)(: f

22

11

11

22 )(BzBz

Bz

BzJI ：determined by the

ADHM data only：f always exists.

[Nakajima, Maeda-Sako]

)detlog( 220

124**4

fFFTr

ffTrxdFFTrxd

N

kN

)1),2((''1`` 22

kUr

f

Ex)

4. Proof of the reciprocity: (inst)↔(ADHM)

NC instanton NC ADHM

NNA :

kNNk

kk

JIB

:,:,:2,1

(i) ASD (ASDYM)(ii) C_2=k(iii) D^2 has inverse

(i) ASD (ADHM eq.)(ii) matrix size = k, N(iii) ∇^2 has inverse

F

G

Proof of the one-to-one ⇔ Define the maps F & G,& G◦F=id. & F◦G=id.

(iii) is automatically satisfied in the noncommutative situation [Maeda-Sako] [Nakajima]

F：(ADHM)→(inst):ADHM constructionNC instanton NC ADHM

NNVVA :

kNNk

kk

JIB

:,:,:2,1

NVVV 1,0

22

11

11

22 )(BzBz

Bz

BzJI

kkNopDirac 2)2.(dim0

0dim.Dirac eq.

(i) ASD (ASDYM)(ii) C_2=k

(i) ASD (ADHM eq.)(ii) matrix size= k, N←[MH Nakatsu]

[Nekasov-Schwarz]

G：(inst)→(ADHM): inverse constructionNC instanton NC ADHM

NNA :

kN

kk

rJI

zxdB

:,

,:

3

2,14

2,1

kxdDe 1,0 4

.dim4: opDiracDe

(i) ASD (ASDYM)(ii) C_2=k

(i) ASD (ADHM eq.)(ii) matrix size= k, N

4dim.Dirac eq.

[Maeda-Sako2009] proves the existenseof the Dirac zero-mode be a formalpower expansion of θ recursively. ,),(

,),()2(2)1()0(

)2(2)1()0(

AAAxAx

G◦F=id：(ADHM)→(inst)→(ADHM)NC instanton NC ADHM

??,?2,12,1

JJIIBB

4dim.Dirac eq.

The answerCfVVJIB

),,,(

0dim.Dirac eq.NC ADHM

VJIB ,,2,1 VVA

kxdDe 1,0 4

33

2,12,14

2,1

,,r

JIr

JI

BzxdB

[Maeda-Sako] are shown

find in terms of

the original B, I, J, Vto give the new data

F◦G=id：(inst)→(ADHM)→(inst)NC instanton

0dim.Dirac eq.

),( AVV

4dim.Dirac eq. NC ADHM

VJIB ,,2,1A

[Maeda-Sako] assume the existence of V.[MH-Nakatsu] prove it

NC instanton

? AA

CVD 42 :the answer

AVVA

NVVV 1,0

are shown (existence proof is also made by us)

find in terms of

the original instanton Aμto give the new instanton

Main result：We prove the ADHM one‐to‐one correspondence in the formal power series ofθ-expansion.

5. Conclusion and Discussion• We prove the one‐to‐one correspondence between NC instantons and NC ADHM data in the Moyal‐product formalism. [to appear soon…]

• This is valid only in the region that the θ‐expansions converge.

• We proceed to reveal the reciprocity in operator formalism. (mostly completed [work in progress] )

• We are also interested in non‐finite action solutions (soliton‐like) of NC ASDYM (twistor theory and Ward’s conejecture) in relation to lower‐dimensional integrable systems such as KdV. [See e.g. MH, arXiv:1101.0005]           (and perhaps the AGT corresp.)