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2015/11/14 Noncommutative Instantons 1. Introduction and Reciprocity Non Commutative (NC) spaces are defined by noncommutativity of spatial coordinates: Masashi Hamanaka [ , ] x x i : NC

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SLIDE 1

2015/11/14 1

Noncommutative Instantons and Reciprocity

・We give a proof of one‐to‐one correspondence between

moduli space of instantons and moduli space of ADHM data in noncommutative spaces.

・ MH&Toshio Nakatsu(Setsunan), NC Instantons and Reciprocity,

to appear, [cf. arXiv:1311.5227] ・MH&TN, work in progress.

Masashi Hamanaka (Nagoya U, Japan)

YI TP Workshop QFT2015 November 10th 2015

  • 1. Introduction
  • Non‐Commutative (NC) spaces are defined

by noncommutativity 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)

 i p q  ] , [

 ~

  • Com. space

NC Space



: NC parameter (real const.)

[Nekrasov-Schwarz]   

 i x x  ] , [

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)

. 23 14 , 42 13 34 12

, F F F F F F      

        

A A A A A A F       :

4 , 3 , 2 , 1 ,   

:

A Gauge field (N×N anti-Hermitian) Field strength

.)) . ( , (

2 1 2 2 1 1

rep cpx F F F

z z z z z z

   

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

  • NC ASDYM eq. (real rep.)

     

     

12 03 31 02 23 01

, , F F F F F F

) : (

        

A A A A A A F        

              

2 2 1 1

      O O

(Spell:All products are Moyal products.)

) ( ) ( ) ( 2 ) ( ) ( ) ( 2 exp ) ( : ) ( ) (

2

  

     

O x g x f i x g x f x g i x f x g x f                  

Under the spell, we can calculate : Note: Coordinates and functions themselves are c-number-valued usual ones

        

   i i x x i x x x x       

) 2 ( 2 : ] , [

G=U(N) NC ASDYM in operator formalism

  • Take coordinates as operators (in 2dim):

field (infinite matrix): integration

  • NC ASDYM eq. (real rep.)

12 03 31 02 23 01

ˆ ˆ , ˆ ˆ , ˆ ˆ F F F F F F      

              

2 2 1 1

      O O

1 ] ˆ , ˆ [ 2 ] ˆ , ˆ [ ] ˆ , ˆ [           

a a z z i y x

rescale complex

 

ann op. cre op. acting on Fock space:

n m F z z F

n m mn

,

) ˆ , ˆ ( ˆ ,... 2 , 1 ,    n n C H

Occupation number basis

1

H

2

H

2 1 2 1 , , , , , ,

, , ˆ

2 1 2 1 2 1 2 1

n n m m F F

n n m m n n m m

) ˆ , ˆ ( ˆ 2 z z F Tr

H



  • 2. Atiyah‐Drinfeld‐Hitchin‐Manin Construction

based on duality for the instanton moduli space

2 1 2 2 1 1

  

z z z z z z

F F F

k N N k k k

J I B

  

: , : , :

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

N N A  :

] , [ ] , [ ] , [

2 1 2 2 1 1

     

   

IJ B B J J I I B B B B

k × k Matrix eqs. N × N PDE 2

Gauge trf.: ) (

1 1

N U g g g g A g A   

     

Gauge trf.: g J J I g I k U g g B g B ~ , ~ ) ( ~ , ~ ~

1 2 , 1 1 2 , 1

  

 

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SLIDE 2

2015/11/14 2

Fourier‐Mukai‐Nahm transformation Beautiful duality between instanton moduli on 4‐tori and instanton moduli on the dual tori

2 1 2 2 1 1

  

z z z z z z

F F F

  • 4dim. ASD Yang-Mills eq.
  • n the dual torus
  • 4dim. ASDYang-Mills eq.
  • n a 4-torus

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

1:1

N N A  :

k × k PDE N × N PDE 2 2

~ ~ ~

2 1 2 2 1 1

  

     

F F F k k A  : ~

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 transformation Beautiful duality between instanton moduli on 4‐tori and instanton moduli on the dual tori

2 1 2 2 1 1

  

z z z z z z

F F F

  • 4dim. ASD Yang-Mills eq.
  • n the dual torus
  • 4dim. ASDYang-Mills eq.
  • n a 4-torus

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

1:1

  

V V x A   , ) (

k × k PDE N × N PDE 2 2

~ ~ ~

2 1 2 2 1 1

  

     

F F F k k A  : ) ( ~ 

map F (Dirac eq.) On a 4-torus On the dual 4-torus

x :

 :

) ~ (          V ix A e V

   

N k V  2 :

Family index thm. N×N ) 1 , ( :

2 a

i e 

 

Fourier‐Mukai‐Nahm transformation Beautiful duality between instanton moduli on 4‐tori and instanton moduli on the dual tori

2 1 2 2 1 1

  

z z z z z z

F F F

  • 4dim. ASD Yang-Mills eq.
  • n the dual torus
  • 4dim. ASDYang-Mills eq.
  • n a 4-torus

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

1:1

N N x A  : ) (

k × k PDE N × N PDE 2 2

~ ~ ~

2 1 2 2 1 1

  

     

F F F

  

     ~ , ) ( ~ A map G (Dirac eq.) On a 4-torus On the dual 4-torus

x :

 :

) (          

     

i A x e D e

k N  2 : 

Family index thm. k×k

Fourier‐Mukai‐Nahm trf. (radii of the torus∞) Duality between instanton moduli on R and instanton moduli on ``1pt.’’ [cf. van Baal, hep‐th/9512223]

2 1 2 2 1 1

  

z z z z z z

F F F

  • 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

V V A

 

 

k × k PDE N × N PDE 2 2

~ ~ ~

2 1 2 2 1 1

  

     

F F F k k A  : ~

map F (0dim Dirac eq.) On a 4-torusR On the dual 4-torus1 pt.

) ~ (          V ix A e V

   

N k V  2 :

Linear alg. 4

] ~ , ~ [ ~ ~ : ~

      

A A A A F      Matrix eq.！ Matrix eq. !

Atiyah‐Drinfeld‐Hitchin‐Manin (ADHM) Construction based on the following duality

2 1 2 2 1 1

  

z z z z z z

F F F k N N k k k

J I B

  

: , : , :

2 , 1

ADHM eq.. (≒0dim. ASDYM)

  • 4dim. ASDYang-Mills eq.

Sol.=ADHM data (G=`U(k)’) Sol.=instantons (G=U(N), C =k) N N A  :

] , [ ] , [ ] , [

2 1 2 2 1 1

     

   

IJ B B J J I I B B B B

k × k matrix eq. N × N PDE 2

Proved in the same way as the Nahm trf. ] , [ ] , [

2 2 1 1

z z z z 

RHS is in fact

1:1

G(4dim D.eq.) F(0dim D.eq.)

D‐brane interpretation of ADHM Construction

2 1 2 2 1 1

  

z z z z z z

F F F

ADHM eq. (≒0dim. ASDYM) (G=`U(k)’)

  • 4dim. ASDYang-Mills eq.

(G=U(N), C =k)

) , (

       

     F F diag F

ASD SD

   ] , [ ] , [ ] , [

2 1 2 2 1 1

     

   

IJ B B J J I I B B B B

k × k matrix eq. N × N PDE 2

G(4dim D.eq.) F(0dim D.eq.)

  • N D4 branes

k D0 branes D-term conditions SUSY trf. of gaugino

0-0 strings  k×k: B 0-4 strings  k×N：I,J [Witten, Douglas]

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SLIDE 3

2015/11/14 3

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

2 1 2 2 1 1

  

z z z z z z

F F F

ADHM eq. (≒0dim. ASDYM)

  • 4dim. ASDYang-Mills eq.

Sol.=ADHM data (G=`U(1)’) BPST instanton (G=U(2), C =1)

] , [ ] , [ ] , [

2 1 2 2 1 1

     

   

IJ B B J J I I B B B B

k × k matrix eq. N × N PDE 2

ASD ASD

z i F z b x i A

    

      

2 2 2 2 2 2

) ) (( 2 2 2 , ) ( ) (        

      

      (

), , 1 1 ,

2 , 1 2 , 1

J I B 

i

position size

 

: singular

M

 

singularity

 

C R

 

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

2 1 2 2 1 1

  

z z z z z z

F F F

NC ADHM eq. NC ASDYang-Mills eq. Sol.：ADHM data (G=`U(1)’) NC BPST instanton (G=U(2), C =1)

] , [ ] , [ ] , [

2 1 2 2 1 1

     

   

IJ B B J J I I B B B B 

k × k marix eq. N × N PDE 2

i

position size

 

: regular!      

        2 (

), , ,

2 , 1 2 , 1

J I B

Fat by ζ !

  F

A ,

：exact sol.

M

  Resolution of the singurarity!

 

C R

  Do k×k ADHM data give Instanton number k in general ? (We prove this.) By calculation of TrFΛF

  • 3. Proof of the duality: (inst)↔(ADHM)

NC instanton

NC ADHM N N A  :

k N N k k k

J I B

  

: , : , :

2 , 1

(i) ASD (ASDYM eq.) (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] (For any θ [MH, Nakatsu])

F：(ADHM)→(inst):ADHM construction

NC instanton

NC ADHM N N V V A    

:

 

k N N k k k

J I B

  

: , : , :

2 , 1

N

V V V 1 ,     

 

                

   2 2 1 1 1 1 2 2

) ( B z B z B z B z J I k k N

  • p

Dirac 2 ) 2 .( dim   0dim. Dirac eq.

(i) ASD(ASDYM eq) (ii) C_2=k (i) ASD (ADHM eq.) (ii) matrix size= k, N

←[MH Nakatsu],…

[Nekasov-Schwarz]

We prove the NC version of the formula: cf.[Atiyah, Hori]

2 / 1 * * 2 4 * * 4

) ( : ~ , ~ ~ : ] , [ :                        

  

 

           

     where Tr x d F F Tr x d

k N

2k×2k (N+2k)×2k

F：(ADHM)→(inst):ADHM construction

NC instanton

NC ADHM N N V V A    

:

 

k N N k k k

J I B

  

: , : , :

2 , 1

N

V V V 1 ,     

 

                

   2 2 1 1 1 1 2 2

) ( B z B z B z B z J I k k N

  • p

Dirac 2 ) 2 .( dim   0dim. Dirac eq.

Then:

r e x g k dg g Tr Tr Tr x d F F Tr x d C

k k S k N      

             

  

: , ) ( 1 24 1 16 1 16 1 :

3 1 2 2 * * 2 4 2 * * 4 2 2

3

comes from the size of the ADHM data!

Interpretation in operator formalism would be

  • insteresting. (The matrix g is a shift operator!)

G：(inst)→(ADHM): inverse construction

NC instanton

NC ADHM N N A  :

k N k k

r J I z x d B

 

 

     : , , :

3 2 , 1 4 2 , 1

  

k

x d D e 1 ,

4

   

  

 

. dim 4 :

  • p

Dirac D e

 

(i) ASD (ASDYM eq.) (ii) C_2=k (i) ASD (ADHM eq.) (ii) matrix size= k, N

4dim. Dirac eq.

[Maeda-Sako2009] proves existence

  • f the Dirac zero-mode by a formal

power expansion of θ, recursively.

, ) , ( , ) , (

) 2 ( 2 ) 1 ( ) ( ) 2 ( 2 ) 1 ( ) (

          A A A x A x          commutative input

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SLIDE 4

2015/11/14 4

G◦F=id：(ADHM)→(inst)→(ADHM)

NC instanton

NC ADHM

? ?, ?

2 , 1 2 , 1

J J I I B B      

 4dim. Dirac eq.

The answer

Cf V V J I B   

) , , , (  

0dim. Dirac eq.

NC ADHM

V

J I B , ,

2 , 1

V V A

 

  

k

x d D e 1 ,

4

   

  

 

3 3 2 , 1 2 , 1 4 2 , 1

, , r J I r J I B z x d B

  

         

    

[Maeda-Sako]

are shown

find in terms of the original B, I, J, V to give the new data

F◦G=id：(inst)→(ADHM)→(inst)

NC instanton

 0dim. Dirac eq.

) , ( 

A V V 

4dim. Dirac eq.

NC ADHM

V

J I B , ,

2 , 1 

A

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

NC instanton

?

 

A A  

C V D

    4

2

:the answer

  

A V V A      

N

V V V 1 ,     

 

are shown (some existence proofs is also made by us)

find in terms of the original instanton Aμ to give the new instanton

NC instanton

NC ADHM

(i) ASD (ASDYM eq.) (ii) C_2=k (i) ASD (ADHM eq.) (ii) matrix size = k, N

Main result：We prove the ADHM duality in the formal power series of θ-expansion for arbitrary noncommutativity (including ζ=0). 1:1

  • This is valid only in the region that the θ‐

expansions converge.

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