The relationship between a topological Yang-Mills field and a magnetic monopole

1. The relationship between a topological Yang-Mills fieldand a magnetic monopole Nobuyuki Fukui (Chiba University, Japan) Kei-Ichi Kondo (Chiba University, Japan) Akihiro Shibata (Computing Research Center, KEK, Japan) Toru Shinohara (Chiba University, Japan) • Contents • Introduction • Reformulation of SU(2) Yang-Mills theory • Numerical calculation • Results • Summary Based on N. Fukui, K.-I. Kondo, A. Shibata, T. Shinohara, Phys. Rev. D82 045015 (2010) RCNP, Osaka, 7 Dec. 2010.

2. 1. Introduction Magnetic monopoles are indispensable to quark confinement from a viewpoint of the dual superconductor picture. We proposed a reformulation of Yang-Mills theory to extract the magnetic monopole from the theory with keeping gauge symmetry. K.-I. Kondo., T. Murakami and T. Shinohara, Prog. Theor. Phys. 115 201-216 (2006) K.-I. Kondo., T. Murakami and T. Shinohara, Prog. Theor. Phys. 120 1-50 (2008) Our purpose is to show that the magnetic monopole exists in Yang-Mills theory. In this talk, I show that the magnetic monopole comes out of instanton solutions based on the reformulation numerically.

3. 2. We impose the reduction condition to reduce the enlarged gauge symmetry to , 2. Reformulation of Yang-Mills theory 1. By introducing a color vector field with a unit length, We constructed “enlarged Yang-Mills” with the enlarged gauge symmetry . enlarged Yang-Mills Reduction condition reformulated Yang-Mills original Yang-Mills equipollent

4. The reduction condition is given by minimizing the reduction functional under the enlarged gauge transformation: The reduction condition and a definition of gauge-invariant magnetic monopole Reduction functional The local minima are given by the differential equation which we call the reduction differential equation (RDE): In the reformulated Yang-Mills, a composite field is very important. The field strength of is parallel to : So we can define the gauge-invariant field strength and the gauge-invariant monopole current as

5. We use a lattice regularization for numerical calculations. A link variable is computed by 3. Numerical calculation The reduction functional on a lattice is given by where is a unit color field on each site, boundary We introduce the Lagrange multiplier. Then, the stationary condition for the reduction functional is given by After a little calculation, we obtain a lattice version of the RDE:

6. The monopole current on a lattice is constructed as The V-part on a lattice is given by S. Ito, S. Kato, K.-I. Kondo, T. Murakami, A. Shibata and T. Shinohara, Phys. Lett. B645, 67-74 (2007). A. Shibata, K.-I. Kondo and T. Shinohara, Phys. Lett. B691, 91 (2010).

7. A boundary condition We recall that the instanton configuration approaches a pure gauge at infinity: Then, we assume that behaves asymptotically So, we adopt a boundary condition as

8. the procedure of a numerical calculation 1. We calculate for 2. We solve under the boundary condition. 3.

9. 4. Results • We calculate the magnetic monopole for • Regular one-instanton • Jackiw-Nohl-Rebbi (JNR) type two-instanton • Here, I give a detailed account of the result of JNR type two-instanton.

10. Jackiw-Nohl-Rebbi (JNR) type two-instanton size pole In this case, is Consequently, boundary condition is Hopf map In the calculation, we equate three size parameters and put three pole positions on plane, so that the three poles are located at the vertices of an equilateral triangle:

11. JNR two-instanton and the associated magnetic-monopole current for various choice of . The grid shows an instanton charge density on plane. These figure show that monopole currents form a circular loop. The circular loops are located on the plane specified by three poles of the JNR two-instanton.

12. The monopole current has a non-zero value on a small number of links. This table indicates that the size of the circular loop increases proportionally as r increases.

13. The relationship between and the magnetic-monopole loop The configuration of the color field and a circular loop of the magnetic monopole current obtained from the JNR two-instanton solution , viewed in (a) the plane which is off three poles, and (b) the plane which goes through a pole . Here the SU(2) color field is identified with a unit vector in the three-dimensional space. These figure show the color vector field is winding around the loop, and it’s direction is indeterminate on the loop.

14. 5. Summary conclusion ・We show that the magnetic monopole comes out of an instanton. The magnetic monopole for the JNR two-instantonshapes a circular loop. ・ We found the relationship between the magneticmonopole and the singular point of the color field . is winding around the loop. future problem • solving the RDE analytically. • computing contribution of the magnetic monopole configuration to physical quantities (ex. Wilson loop). • extending to finite-temperature field theory

15. Thank you for your attention!

16. regular one-instanton (BPST type) center size In this case, is and the boundary condition is Hopf map In the calculation, we fix the center on the origin and change the value of size .

17. One instanton in the regular gauge and the associated magnetic-monopole current for various choice of size parameter . The grid shows an instanton charge density on plane. These figures show that non-zero monopole currents form a small loop.

18. The monopole current has a non-zero value on a small number of links. The size of the loop hardly changes while the size parameter increases.

19. A numerical technique RDE on a lattice ・・・・・　(A) 1. We give a initial configuration for n. We recursively apply (A) to on each site and update it until converges. At this time, We fix on a boundary of a finite lattice. 2.

20. The monopole loop in a lattice simulation A. Shibata, K.-I. Kondo, S. Kato, S. Ito, T. Shinohara, and N. Fukui, Proceedings of the 27th International Symposium on Lattice Field Theory (Lattice 2009), Beijing, China, 2009(arXiv:0911.4533).