1. Duality and Confinement in the Dual Ginzburg-Landau Superconductor Physics Beyond Standard Model – 1st Meeting Rio-Saclay 2006 Leonardo de Sousa Grigorio - Advisor: Clovis Wotzasek Universidade Federal do Rio de Janeiro

2. This seminar is organized as follows: Duality in usual Maxwell Eletrodynamics; The Ginzburg-Landau model of a Dual Superconductor; Confinement between static electric charges; Julia-Toulouse Mechanism; Duality between GLDS(Higgs) and JT.

3. Let us review the symmetry between charges and electromagnetic fields. Let the Maxwell’s equations: Dirac introduced magnetic charges exploring the symmetry of Maxwell’s equations. After that The symmetry presented by these equations is

4. In a covariant form these equations read where And, as usual

5. However if we want to describe the fields in terms of potencials we get a problem. So, if we didn’t have magnetic monopoles which is the Bianchi identity. This one can be solved In order to introduce monopoles we have to violate Bianchi identity by rewrinting the field strenght as where the last term is a source, defined by

6. Let the current created by one electric charge where is the world line of the particle. While associated withthere is a world sheet. Applying a divergence and settingto infinity

7. The Dirac string defines a region in space where the gauge potencial becomes ill defined. Summarizing the duality described above could be seen at the level of Maxwell’s equations. How do we describe duality in a most fundamental way?

8. And the other Maxwell equation comes from the definition of So, what is the dual of that Lagrangean? The answer is Minimizing the action we obtain the equation of motion,

9. We may get a picture of the couplings So, lowering the order by a Legendre transformation This one can be obtained as follows:

10. This is the dual of the original one. * By inserting this into the Lagrangean we obtain Eliminating the vector field we get

11. GLDS Let the dual Abelian Higgs model where we have a covariant derivative coupling minimally the vector and matter field and coupling non-minimally the vector field to electric charges.

12. Let. We may write the Lagrangean in the following manner and with an adequate choice of the gauge All work as if the vector potential absorbed one of the degrees of freedom of the complex scalar field and became massive. Let us freeze the remaining degree of freedom and define

13. After solving forwe obtain Going back to the Lagrangean we find The confinement properties are present in this form.

14. Confinement between static electric charges Substituting it in the Lagrangean this becomes It can be shown that the previous Lagrangean provides confinement between opposite charges.

15. In order to find the energy we look at the Hamiltonean We perform a Fourier transformation and arrive at

16. After performing this integral the energy reads where a cutoff was introduced. It’s physical meaning is related to a length scale: the size of the vortex core.

17. Julia-Toulouse Mechanism Let us start with this situation The corresponding Lagrangean is Field that describes the condensate.

18. As it turns a field, it must arise three modifications: -A kinetic term for the condensate; -The vector field is absorbed by the condensate; -An interaction that couples the new field to the charges. Let us work theese ideas through the following symmetry, that is already present. A kinetic term wich respects this symmetry is

19. We can choosesuch thatdesapears, or in other words, is absorbed by the condensate. In order to preserve the symmetry we should have eats and gets massive

20. So Redefining It yelds If we solve, not surprisingly

21. Duality between GLDS(Higgs) and JT We start with the GLDS Lagrangean By the same methods above we have Eliminatingand reescaling, we obtain