Uses of Chern-Simons Actions

1. Uses of Chern-Simons Actions Ten Years of the AdS/CFT Conjecture Buenos Aires, December 2007 J. Zanelli CECS – Valdivia (Chile)

2. A typical Chern-Simons action is something like this: or this… A typical Yang-Mills action is something like this:

3. Chern-Simons lagrangians define gauge field theories in a class different from Yang-Mills: • They are explicit functions of the connection (A), • not local functions of the curvature (F) only. • Yet, they yield gauge-invariant field equations. • Related to homotopic/topological invariants on fiber • bundles: characteristic classes. • They require no metric; just a Lie algebra (not • necessarily semisimple); no adjustable parameters, • conformally invariant. More fundamental(?)

4. They are very sensitive to the dimension. • They naturally couple to branes. • The classical field equations are consistency • conditions for those couplings. • Their quantization corresponds to sum over • holonomies in an embedding space. • CS theories are not exotic but a rather common • occurrence in nature: Anomalies, quantum Hall effect, • 11D supergravity (CJS), superconductivity, 2+1 • gravity, 2n+1 gravity, all of classical mechanics,...

5. 1. CS action in 0+1 dimensions

6. Aμ(x) ∙ e z Γ E-Mcoupling Gauge invariance Aμ(x)→Aμ(x)+∂μ Ω(x) , is ensured by current conservation, ∂μjμ =0, provided Ω(z(+∞))=Ω(z(-∞)). Not quite gauge invariant, but quasi-invariant.

7. Gauge transformationsAA+dΩ(x) • Gen. coordinate transf. zμz’μ(z) This coupling is consistent with the minimal derivative substitution Good for quantization: • This expression is invariant under • Lorentz transformations, (Λμν)

8. Take for an abelian connection and set n=0: 0+1 CS theory ? ? ? not exactly… The simplest Chern-Simons action Is this a sensible action?

9. Varying the action, This only means δA=dΩwith Ω(-∞)=Ω(∞), or ∂Γ=0. The classical configurations are arbitrary U(1) connections with PBC or living in a periodic 1d spacetime Alternatively, I can also be viewed as an action for the embedding coordinates zμ,

10. ∙ ∙ Varying the action, The classical orbits are those with zero Lorentz force: the electric and magnetic forces cancel each other out. N.B.: In order to obtain the equation of motion, z must satisfy periodic boundary conditions.

11. A bit about the quantum theory Thus, the integral is dominated by those orbits for which the holonomies are quantized, Flux quantization Does this describe a physically sensible system? What are the degrees of freedom?

12. ∙ z0=t Let z0=t,zμ=(t, zi),i=1,2,∙∙∙,2s. ∙ e zμ zi ∙ This action describes a mechanical system where eA0(z)= -H and eAi(z)=pi Varying w.r.t. zi(t) yields Hamilton’sequations: where F=dA andEi= ∂iA(z).

13. Classical mechanics Chern-Simons Vanishing Lorentz force Hamilton’s equations Invariance under canonical transf. Gauge invariance Gen. coordinate transformations Invariance under time reparam. Bohr-Sommerfeld quantization Flux/holonomy quantization • Any mechanical system with s degrees of freedom can be described by a 0+1 C-S action in a (2s+1)-dimensional target space.

14. z(xi) More dimensions… A C-S action in 2+1 dimensions can be viewed as a coupling between a brane and an external gauge field • Invariant under: • Gen. coord. transf. on the worldvolumezμz’μ(z) • Lorentz transformations on the target space (Λμν) • Gauge transformationsAA+dΩ [quasi-invariant]

15. z(xi) For D=3, 5, 7,…New possibilities arise: • Nonabelian algebras • A(z) can be dynamical (propagating in the worldvolume) • Worldvolume dynamics (gravity) • Degeneracy (for D≥5) • Quantization? (open problem)

16. 2. CS action in 2n+1 dimensions

17. The coefficients c1, …cn are fixed rational numbers, where Invariant under gauge transformations(up to boundary terms) under Non abelian CS action in 2n+1 dimensions

18. C-S dynamics LδL=0 Comment 0+1 eA -- No dynamics for A 2+1 A=pure gauge nondegenerate 2n+1 Nontrivial, propagating, degenerate

19. The problem arises from the fact that for D=2n+1 with n≥2, the field equations are nonlinear in the curvature, where Gk are the generators of the Lie algebra. The linearized perturbations around a given classical configuration F0, obey Degeneracy of CS theories (D=2n+1≥5) The dynamics depends on the form ofF0 .

20. Consequences of the degeneracy • Unpredictability of evolution • Freezing out of degrees of freedom • Loss of information about the initial data • Irreversibility of evolution

21. D/sourceQuantizationComment Dirac charge quant. cond. point particle Flux/Holonomy (0-brane)Bohr-Sommerfeld quant. Finite, power- membrane counting renormalizable Anybody’s guess 2n-brane How?

22. For example, Pontryagin 4-form In general, • Homogeneous gauge-invariant polynomial • Characteristic class (Chern-Weil, Euler, Pontryagin, …) • Its integral defines a homotopic invariant of the manifold All the interesting features of CS forms originate from their relation with characteristic classes.

23. 3. CS Gravity actions in 2n+1 dimensions

24. 1. Equivalence Principle: • Spacetime is locally approximated by Minkowski • space and has the same (local) Lorentz symmetry. • GR is the oldest known nonabelian gauge theory for • the group SO(3,1). 2. Gravitation should be a theory whose output is the spacetime geometry. Therefore, it is best to start with a theory that makes no assumptions about the local geometry. 1&2  Chern-Simons theory is probably a better choice

25. Action (first order formalism ): Vielbein (metricity) Connection (parallelism) =Curvature 2-form =Torsion 2-form There are two characteristic classes associated to the rotation groups SO(s,t): the Euler and the Pontryagin classes. Associated with each of them there are the corresponding CS actions

26. Steps for constructing CS gravities: 1. Combine the vielbein and spin connection into a connection for the dS, AdS, or Poincaré group: 2. Select the bracket that corresponds to the invariant characteristic class to be used (Euler or Pontryagin), e.g., 3. Write down the lagrangian

27. No dimensionful constants (scale invariants) • No arbitrary adjustable/renormalizableconstants • Possess black hole solutions • Admit and limit • Admit SUSY extensions for and any odd D, • and yield field theories with spins ≤ 2 only • Give rise to acceptable D=4effective theories Some form of dimensional reduction is needed CS gravities (D=2n-1) (summary) However, they only exist in odd dimensions… A.Anabalón, S.Willison, J.Z: hep-th/0610136;hep-th/0702192

28. 4. Summary

29. CS actions have been used in physics much longer than • one usually thinks. • CS theories can be viewed as boundary theories coming • from topological field theories in even D=2n manifolds. • Theyhaveno free adjustable parameters and require no • metric structure. • Degeneracy for D≥5: limited predictability, irreversibleloss • of degrees of freedom, dynamical dimensional reduction. • There exist CS (super-) gravities with dimensionless • couplings, all fields have spins ≤ 2and the metric is not a • fundamental field but a condensate… • They are so exceptional, it’s at least worth studying them!

30. Thanks!