1. Instantons Toshikazu Negi on the Taub-NUT spaces (Ref. Edward Witten, JHEP 0906:067,2009.) KEK journal club on 10.26th 2009. 1Total 18 page

2. References Edward Witten, JHEP 0906:067,2009,(arXiv:0902.0948). Sergey Cherkis, “Instantons on the Taub-NUT Space. ”, (arXiv:0902.4724). Sergey Cherkis, Commun.Math.Phys.290:719-736,2009,(arXiv:0805.1245 ). 2

3. Introduction D-brane construction of the Taub-NUT space Gauge instantons on the Taub-NUT space Systematic approach are given for case. Concrete calculus for- instanton case.[S.Cherkis] [E.Witten(the first half)] (Hyper Kahler quotient construction) [E.Witten(the last half)] (physical interpretation of the system) Cf. For concrete description, 3

4. (Euclidean)Taub-NUT space Metric c.f. Multi-centered ( - centered) Taub-NUT space they has asymptotic For, structure. 4

5. How to construct the system Type IIB Type IIA T-duality ondirection.NS5-D5-D3 branes on D6-D2 branes on 5

6. Set up NS5-D5-D3 branes on NS5 D5 D3 01 Type IIB theory NS5 D5 23475689 D3Gauge instanton number on Taub-NUT space Gauge group on Taub-NUT space # of the centers of the Taub-NUT space We take T-dual on this direction. ○○○○○○ ○○○○○○ ○○○○. D6 D2 6

7. Set up II D6-D2 theory D6 D2 01 Type IIA theory 2 3475689 k centered (Euclidean) Taub-NUT space appears here. ○○○ ○○○ D2 branes seems as point-like object from Taub-NUT space. ○○○○ These branes generates effects of instantons in Taub-NUT space. 7

8. Hyper Kahler quotient construction Method of the construction for hyperKahler manifold from larger hyper-Kahler manifold Larger hyper Kahler manifold NS5 0123475689 ○○○○○○ D3○○○○ (p,q)5 –branes / D3 brane system [Hanany-Witten] NS5 / D3 8

9. Taub-NUT as hyper Kahler quotient Moment map: Residual U(1) symmetry We want Set Taub-NUT 9

10. ADHM data & ADHM construction Convenient on the quarternion basis. General construction for the instanton solution for the gauge-group of the A,B,C,D-type Dynkin diagrams on the 4-dim. flat Euclidean spacetime. ADHM construction We define For - instantons case gauge theory with complex-valued matrices,, where 4 – Pauli matrices [Dorrey-Hollowood-Khoze-Matis,Hamanaka] 10

11. ADHM construction Null space matrices for We denote gauge field by If the condition is satisfied, becomes instanton. ADHM constraints : Orthonormalized complex valued matrix Introduce as ∴.. self-dual 11

12. ADHM constraints(1) Null space condition & ADHM condition: This eq. has symmetry,,, Exploiting symmetry, where we can denote trivial ADHM data complex 12

13. ADHM constraints(2) Residual symmetry, Note that, really becomes the coordinates of moduli spaces of instantons. where Considering these symmetries, we finally get ADHM constraints. We can solve ADHM constraints when. 13

14. Nahm construction 1-dim reduction of the ADHM construction (3-dim. Yang-Mills + 1 adjoint matter) Construction for the 3-dim. BPS monopole solution Nahm equation 14

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16. Moment map Null space eq Instanton 16

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18. Conclusion and Prospects We can get Taub-NUT space from T-duality description. It is also interpreted as hyper-Kahler quotient construction. Systematic approach to construct gauge instantons on Taub-NUT space are given for case. ● ● Concrete description is made for - instanton case. General A_k, D_k type descriptions are desired. 18