Noncommutative Deformation of InstantonS , Instanton numbers and ADHM construction
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NONCOMMUTATIVE DEFORMATION OF INSTANTONS,
INSTANTON NUMBERS AND ADHM CONSTRUCTION
ICMP 09, Prague, August 3, 2009
Akifumi SakoKushiro National College of Technology
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Noncommutativity of Rn
NC parameter , Comm. Lim. , Moyal Product.
NC parameter , ℏ→0 comm. lim.
Moyal product
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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.
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ℏ-expansion formal expansion
l-th order Instanton Eq.
whereGiven fun.We solve
recursively
l-th orderNC Instanton Eq.
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Elliptic Diff. Eq. gauge condition
where
Main Eq.
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Using this fact, we can prove
Solution & Asymp. Behavior
There exists the formal solution that is smoothly NC deformation of Instanton.
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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) .
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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
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Solution
where an is arbitrary coefficient. Determined uniquely up to zero mode
Theorem when we fix the ambiguity an
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Green's Function
n-th order ℏ-expansion
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Instanton ⇒ ADHM Completeness relation
Def. of ADHM data
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Using the Completeness relation and the Definition
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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
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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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