NONCOMMUTATIVE DEFORMATION OF INSTANTONS,

INSTANTON NUMBERS AND ADHM CONSTRUCTION

　

ICMP 09, Prague, August 3, 2009

Akifumi SakoKushiro National College of Technology

Noncommutativity of Rn

NC parameter , Comm. Lim. , Moyal Product.

NC parameter , ℏ→0 comm. lim.

Moyal product

1

NC Instanton Curvature 2-form

NC Instanton Eq.

Nekrasov Schwarz discovered the ADHM method.Many studies are done but we did not know if there exist an Instanton smoothly deformed from acommutative one. Let ’s look for it!!

NC Instanton Eq.

2

ℏ-expansion formal expansion

l-th order Instanton Eq.

whereGiven fun.We solve

recursively

l-th orderNC Instanton Eq.

3

Elliptic Diff. Eq. gauge condition

where

Main Eq.

4

Using this fact, we can prove

Solution & Asymp. Behavior

There exists the formal solution that is smoothly NC deformation of Instanton.

5

Instanton # indep. of ℏTheorem In R4 ,

Instanton # before NC

deformationInstanton # after NC deformation

We can prove this theorem by using the asymptotic behavior of A(l) .

6

Index of the Dirac Operator

There is no Zero mode in S+.

Hi(n) is a given fun.

The homogeneous part has k zero modes:

:

:

n-th order

7

Solution

where an is arbitrary coefficient. Determined uniquely up to zero mode

Theorem when we fix the ambiguity an

8

Green's Function

n-th order ℏ-expansion

9

Instanton ⇒ ADHM Completeness relation

Def. of ADHM data

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11

Using the Completeness relation and the Definition

The 2nd and 4th terms vanish at Ry→∞The 5th term vanishes in Asymptotic behavior

3rd term becomes

NC ADHM Eq.

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Completeness andUniqueness

Completeness Instanton 　→　 ADHM 　→　 Instanton

Uniqueness ADHM 　→　 Instanton 　→　 ADHM

One to One correspondence between the ADHM data and Instantons up to zero mode is shown.

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Vortex Case

The k-th order Eq. reduces to Schrödinger Eq.

and the solution is uniquely determined .The Vortex number is not deformed as well

as the instanton number in R2. 14

ConclusionsThe Smooth NC Deformation of Instanton

exists.The Instanton # is not deformed in R4.The Index theorem is not deformed up to zero modes.The Green's function exists.The ADHN construction exists. 1 to 1 between ADHM ⇔Instanton 　 exists up to zero modes.The Smooth NC Deformation of Vortex exists

and it’s uniquely Determined .The Vortex # is not deformed in R2.

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ReferencesYoshiaki Maeda, Akifumi Sako

" Are Vortex Numbers Preserved? " J.Geom.Phys. 58 (2008) 967-978 e-Print Archive: math-ph/0612041

Yoshiaki Maeda, Akifumi Sako " Noncommutative Deformation of Instantons " J.Geom.Phys. 58 (2008) 1784-1791 e-Print Archive: arXiv:0805.3373

Akifumi Sako " Noncommutative Deformation of Instantons and Vortexes " JGSP 14 (2009) 85-96

Yoshiaki Maeda, Akifumi Sako "Noncommutative Deformation of ADHM Constructions" e-Print Archive: arXiv:0908.XXXX coming soon

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