Noncommutative Deformation of InstantonS , Instanton numbers and ADHM construction

1. Noncommutative Deformation of InstantonS, Instanton numbers and ADHM construction ICMP 09, Prague, August 3, 2009 Akifumi Sako Kushiro National College of Technology

2. NC parameter , Comm. Lim. , Moyal Product. NC parameter , ℏ→0 comm. lim. Moyal product Noncommutativity of Rn

3. Curvature 2-form NC Instanton Eq. 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 a commutative one. Let ’s look for it!! NC Instanton

4. formal expansion l-thorderNC Instanton Eq. l-th order Instanton Eq. where Given fun. We solve recursively ℏ-expansion

5. gauge condition Main Eq. where Elliptic Diff. Eq.

6. Using this fact, we can prove Solution & Asymp. Behavior There exists the formal solution that is smoothly NC deformation of Instanton.

7. Theorem In R4 , Instanton # after NC deformation Instanton # before NC deformation We can prove this theorem by using the asymptotic behavior of A(l) . Instanton # indep. of ℏ

8. : There is no Zero mode in S+. : n-th order Index of the Dirac Operator Hi(n)is a given fun. The homogeneous part has k zero modes:

9. Solution where an is arbitrary coefficient. Determined uniquely up to zero mode Theorem • when we fix the ambiguity an

10. n-th order ℏ-expansion Green's Function

11. Completeness relation • Def. of ADHM data Instanton ⇒ ADHM

12. Using the Completeness relation and the Definition

13. The 2nd and 4th terms vanish at Ry→∞ • The 5th term vanishes in • Asymptotic behavior • 3rd term becomes NC ADHM Eq.

14. Instanton　→　ADHM　→　Instanton ADHM　→　Instanton　→　ADHM • Completeness • Uniqueness • One to One correspondence between the ADHM data and Instantons up to zero mode is shown. Completeness andUniqueness

15. 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. Vortex Case

16. The 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 ⇔Instantonexists • up to zero modes. • The Smooth NC Deformation of Vortex exists and it’s uniquely Determined . • The Vortex # is not deformed in R2. Conclusions

17. Yoshiaki Maeda, AkifumiSako " Are Vortex Numbers Preserved? "J.Geom.Phys. 58 (2008) 967-978 e-Print Archive: math-ph/0612041 • Yoshiaki Maeda, AkifumiSako " Noncommutative Deformation of Instantons "J.Geom.Phys. 58 (2008) 1784-1791 e-Print Archive: arXiv:0805.3373 • AkifumiSako " Noncommutative Deformation of Instantons and Vortexes " JGSP 14 (2009) 85-96 • Yoshiaki Maeda, AkifumiSako • "Noncommutative Deformation of ADHM Constructions" e-Print Archive: arXiv:0908.XXXX coming soon References