1. Confinement mechanism and topological defects Review+ papers by A.V.Kovalenko, MIP, S.V. Syritsyn, V.I. Zakharov hep-lat/0408014, hep-lat/0402017, hep-lat/0212003 Dubna, 2 February 2005

2. Confinment in Abelian theories Compact electrodynamics Compact electrodynamics Confinement is due to monopoles, which are condensed, and vacuum is a dual superconductor Confinement is due to monopoles, which are condensed, and vacuum is a dual superconductor Z(2) gauge theory Z(2) gauge theory Confinement is due to Z(2) vortices. Confinement is due to Z(2) vortices.

3. Monopole confinement Compact Electrodynamics Vortex Confinement Z(2) gauge theory

4. Confinement in compact electrodynamics Partition function:

5. Confinement in compact electrodynamics Various representations of partition function Dual superconductor Action

6. Vacuum of compact QED is dual to supercoductor Monopole-antimonopole in superconductor Charge-anticharge in cQED vacuum

7. MONOPOLE CONFINEMENT hep-lat/0302006, Y. Koma, M. Koma, E.M. Ilgenfritz, T. Suzuki, M.I.P. Dual Abelian Higgs Model Abelian Projection of SU(2) of SU(2) gluodynamics gluodynamics

8. Monopole is a topological defect EXAMPLE: 2D topological defect Topology: from real numbers we get integer, thus small, but finite variation of angles does not change topological number m !!!

9. Monopole is a topological defect For each cube we have integer valued current j

10. These monopoles are responsible for the formation of electric string

11. Confinement in Z(2) Gauge Theory 2D example of “vortices” (they are points connected by “Dirac line”) In 3D vortices are closed lines, Dirac strings are 2d surfaces spanned on vortices In 4D vortices are closed surfaces, Dirac strings are 3d volumes spanned on vortices 1

12. Confinement in 3D Z(2) Gauge Theory If p is the probability that the plaquette is pierced by vortex then the expectation value of the plaquette is: If vortices are uncorrelated (random) then expectation value of the Wilson loop is:

13. Linking number 3D 4D

14. Confinement in 4D Z(2) Gauge Theory 3D4D Vortex in 4D is a closed surface, 3d Dirac volume is enclosed by the vortex. If vortices are random then the string tension is the same as in 3D:

15. Confinement in 4D SU(2) Gauge Theory by Center Vortices 4D Each piercing of the surface spanned on the Wilson loop by center vortex give (-1) to W. Center projection: partial gauge fixing

16. Monopole confinement Compact Electrodynamics Vortex Confinement Z(2) gauge theory

17. In SU(2) gluodynamics we observe monopoles, center vortices and 3d Dirac volumes as physical objects Physical object 1. Carry action (monopoles and vortices) 2. Scales (3d volumes, monopoles and vortices)

18. The gauge fixing is a detector of gauge invariant objects Monopoles Center vortices 3d volumes (Dirac volume)

19. Length of IR monopole cluster scales, Length of IR monopole cluster scales,

20. P-VORTEX density, Area/(6*V 4 ), scales: P-VORTEX density, Area/(6*V 4 ), scales:

21. Minimal 3D Volumes bounded by P-vortices scale

22. P-VORTEX has UV divergent action density: (S-S vac )=Const./ a 2 In lattice units

23. Monopoles have fine tuned action density: Monopoles have fine tuned action density:

24. “Usual” explanation of confinement Monopoles Center vortices Monopoles Center vortices

25. Usually people are interesting in confinement mechanism Usually people are interesting in confinement mechanism (monopoles and vortices) Now inverse logic How to change gauge fields to get

26. Field strength and gauge fields which are responsible for the confinement are singular!

27. Monopoles belong to surfaces (center vortices). Monopoles belong to surfaces (center vortices). Surfaces are bounds of minimal 3d volumes in Z(2) Landau gauge 3D analogue: center vortex Minimal 3d volume corresponds to minimal surface spanned on center vortex monopole 3d volume

28. 1.Removing P-vortices ( after Z(2) gauge fixing) we get zero string tension and we get zero string tension and zero quark condensate zero quark condensate P. de Forcrand and M. D'Elia, Phys.Rev.Lett. 82 (1999) 4582 2.Removing P-vortices we also remove 3d volumes 3.Wilson loop intersects with 3d volumes, not with P-vortices

29. 3. Wilson loop intersects in points in 4D with 3d volumes, not with P-vortices 4. We get changing strong, fields in points, the average distance between these points is.

30. 5. Note that we changed very small fraction of links to get, since we use Z(2) Landau gauge. De Forcrand and d’Elia had to change 50% of links 6. All information about confinement, quark condensate and any Wilson loop is encoded in 3d branes Holography

31. Low dimensional structures in lattice gluodynamics 1. I. Horvath et al. hep-lat/0410046, hep-lat/0308029, Phys.Rev.D68:114505,2003 I. Horvath I. Horvath 2. MILC collaboration hep-lat/0410024

32. http:\www.lattice.itep.ru\seminars\ e-mail: polykarp@itep.ru e-mail: polykarp@itep.rupolykarp@itep.ru ИТЭФ, Б. Черемушкинская 25, Москва 117259