ICMP 09, Prague, August 3, 2009
Noncommutativity of Rn
NC parameter , Comm. Lim. , Moyal Product.
NC parameter , ℏ→0 comm. lim.
Moyal product
1
NC Instanton
Curvature 2-form
NC
NC Instanton
Instanton Eq.
Eq.
Nekrasov Schwarz discovered the ADHM method.
Many studies are done but we did not know if
there exist an Instanton smoothly deformed from a
commutative one. Let ’s look for it!!
2
ℏ-expansion
formal expansion
l-th orderNC
l-th orderInstanton
InstantonEq.
Eq.
where
Given fun.
We solve
recursively
3
Elliptic Diff. Eq.
gauge condition
Main Eq.
where
4
Solution & Asymp. Behavior
Using this fact, we can prove
There exists the formal solution that is
smoothly NC deformation of Instanton.
5
Instanton # indep. of ℏ
Theorem In R4 ,
Instanton #
after NC
deformation
Instanton #
before 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+.
:
n-th order
Hi(n) is a given fun.
The homogeneous part has k zero modes:
7
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
9
Instanton ⇒ ADHM
Completeness
Def.
relation
of ADHM data
10
Using
the Completeness relation and the Definition
11
2nd and 4th terms vanish at Ry→∞
The 5th term vanishes in
The
Asymptotic
3rd
behavior
term becomes
NC ADHM Eq.
12
Completeness and
Uniqueness
Completeness
Uniqueness
Instanton → ADHM → Instanton
ADHM → Instanton → ADHM
One to One correspondence between the ADHM data and
Instantons up to
zero mode is shown.
13
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
Conclusions
15
References
Yoshiaki 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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