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Topological Toys

Topological Toys. in Field Theory and Gravity. Gerard ’t Hooft, Utrecht University t.g.v . het afscheid van Sander Bais , Amsterdam, 16/06/2010. Solitons and Fermions in 1 D in 2 D in 3 D in 4 D. This is a membrane in the y - z direction.

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Topological Toys

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  1. Topological Toys in Field Theory and Gravity Gerard ’t Hooft, Utrecht University t.g.v. het afscheid van Sander Bais, Amsterdam, 16/06/2010

  2. Solitons and Fermions in 1 D in 2 D in 3 D in 4 D

  3. This is a membrane in the y - z direction

  4. Fermions: There will be a “bound state”:

  5. Dirac equation:

  6. At long distance scales: In the 2 dim. subspace we write Like the Pauli matrices, acts in a 2 dim. Spinor space, so, we have chiral fermions on the brane.

  7. + _

  8. The 2 dimensional soliton: Nielsen - Olesen The scalar field has 2 (real) components: it’s complex Needed: a local gauge field. Here: U (1)

  9. If: chiral fermions with different U(1) charges:

  10. Far from the origin: Depending on the value N of and there are N solutions of

  11. The 3 dimensional soliton: The Magnetic Monopole Here, the scalar field has 3 components Chiral fermions can be coupled in various ways, Isospin 0 and 1 : Isospin ½ :

  12. Isospin 0 and 1 : Isospin ½ : Isospin 0 and 1: Jackiw - Rebbi bound state, The monopole has “fermion number ½” The case isospin ½: Here, the fermion has electric charge ½ e , so that e·g= 2π → violation of spin-statistics addition rule : the fermion becomes a boson … There’s also the Rubakoveffect : due to the fermion’s magnetic moment it can approach the magnetic monopole very closely, and obey anomalous boundary conditions at the origin: violation of baryon number conservation !

  13. The 4 dimensional soliton: the instanton A scalar field with 4 components ? In 4 dimensions, the total classical action of the gauge theory sector is scale invariant. Therefore, a classical solution could exist without the need of a scalar field. A.Belavin, A.Polyakov, A.S.Schwartz, Y.Tyupkin:

  14. The Instanton Group of Gauge Transformations

  15. The case with massless fermions Instanton time LEFT Fermi level RIGHT

  16. The instanton breaks chiral symmetry explicitly

  17. And much more: Alice strings, Quantum groups, Etc.

  18. What about gravity ? 2-d fluxes, 3-d particles and 4-d instantons A compact boundary with antipodal point identification: C’ In 2-d and in 4-d: no solution in pure gravity (unstable against scaling); In 3-d: the Schwarzschild horizon ? B’ A’ A X X X X X X There may exist more solutions in gravity with matter ! B C

  19. ALE (asymptotically locally Euclidean) solutions: the Eguchi – Hanson Instanton Let If then This means that, in 4-d Euclidean space, the points and denote the same point in space-time. Can such an instanton be applied in physics?

  20. According to the index theorem, we can impose a boundary condition if we add an electromagnetic potential such that It so happens that ‰

  21. Best wishes Sander

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