Centro de Estudios Científicos CECS-Valdivia-Chile

Uses of Transgressions J. Zanelli, CECS - Valdivia Modern Trends in Field Theory João Pessoa João Pessoa, Brasil September 2006

Outline 1. Gauge theories 2. Gravity 3. Chern-Simons AdS gravity 4. Regularization 5. Transgressions 6. Interpretation 7. Other uses 8. Summary

1. Gauge theories Maxwell, Yang-Mills Maxwell, Yang-Mills Chern-Simons Chern-Simons Diffeo-invariant systems (GR, strings, Diffeo-invariant systems (GR, strings, membranes,…) Fiber bundle structure Manifold Fibers (group)

a. Yang-Mills (including Maxwell) Action Action Manifold Manifold dim. D dim. D Metric Metric Curvature 2-form Curvature: Connection (Lie-algebra valued 1-form) Thus, a YM type action requires A manifold M A manifold M A metric g A metric g A Lie algebra G A Lie algebra G Killing form γ Killing form γ Lie alg. generator Killing form Invertible, non dynamical, preexistent

C-S theories are more economical: no invertible g, γ required Invariant (up to surface terms) under Lorentz transformations, Lorentz transformations, Gauge transformations, Gauge transformations, Diffeomorphisms, Diffeomorphisms, 3D Manifold 3D Manifold Connection (not necessarily abelian) b. Chern-Simons (D=3)

where stands for some appropriately symmetrized trace. Symmetricinvariant tensor: Symmetric invariant tensor: The c 1, … c k are dimensionless combinatorialcoefficients. The c 1, … c k are dimensionless combinatorial coefficients. The CS lagrangian reads where More dimensions, non abelian algebra

CS lagrangians are “potentials” for characteristic classes: 4D-Pontryagin density Gauge transformations Gauge transformations Diffeomorphisms (trivial) Diffeomorphisms (trivial) Gauge invariance: => The action is invariant under In general, (2k+2)D-Topological density … up to boundary terms No free parameters

2. Gravity Einstein-Hilbert theory (4D) cosm. const. cosm. const. = Ricci scalar, where Equivalently, where = Curvature 2-form = Lorentz connection = Vielbein

e1e1e1e1 TxTxTxTx e2e2e2e2 e3e3e3e3 Isomorphism between the local orthonormal frame on the tangent space and the coordinate basis: Vielbein Vielbein Vielbein This induces a metric structure on spacetime.The metric is not a fundamental field, but a composite: This induces a metric structure on spacetime. The metric is not a fundamental field, but a composite: where = Lorentz metric in the tangent space of the manifold.

Gauge symmetry: This is an indication that gravitation is gauge theory, invariant under local Lorentz transformations at each point in spacetime Spin connection The notion of parallelism, necessary for differentiation, is encoded in the connection  a b  (x). Equivalence Principle

Gravity for D≥4 arbitrary These theories are invariant under - Local Lorentz transformations, S=(D-1,1) - Gen. coord. Transf. [Trivial, as for any well posed theory] D. Lovelock, 1970: Unfortunately, this theory has [(D-1)/2] arbitrary coefficients  p - Dimensionful - Give rise to [(D-1)/2] different, arbitrary cosmological constants p - Not protected by symmetry (they get renormalized).

For D=2n+1there is a special choice of which makes all the cosmological constants p equal. Then, For D=2n+1 there is a special choice of which makes all the cosmological constants p equal. Then, Special choice: The theory becomes invariant under the AdS group, Symmetry enhancement All the dimensions can be absorbed in a rescaling of the vielbein, the theory becomes scale invariant, and the  p are rational numbers. Conformal invariance This is a Chern-Simons theory of gravity for the AdS group

For the special choice, the vielbein and the spin connection can be combined as parts of a single connection 1-form, AdS algebra where Lorentz generator generators of AdS boosts generators of AdS boosts 3. Chern-Simons AdS gravity The action describes a gauge theory for the AdS group. It is a Chern-Simons theory, and can be written as where.

AdS radius AdS radius In terms of geometric fields, the CS lagrangian for the AdS group in D=2n+1 dimensions reads C-S – AdS gravities No dimensionful couplings No dimensionful couplings No arbitrary (renormalizable) parameters No arbitrary (renormalizable) parameters Admit SUSY extensions for (spin ≤ 2) Admit SUSY extensions for (spin ≤ 2) Asymptotically AdS Asymptotically AdS Possess black holes Possess black holes Admit, and Admit, and Useful to do physics: black holes, thermodynamics, cosmology, etc.

4. Regularization The mass and other Noether charges for all gravitation theories in asymptotically AdS spaces are generically divergent. The trouble is that the metric and Killing fields do not fall off sufficiently fast at infinity ( in standard coordinates): The standard approach to cope with this problem is to subtract the value of the charge of some background: Thiscan be practical, but: This can be practical, but: It is arbitrary, requires prior knowledge of the answer It is arbitrary, requires prior knowledge of the answer Is there an action principle that yields this? Is there an action principle that yields this? Is it gauge invariant? Is it gauge invariant?

The importance of gauge invariance A finite but not gauge invariant charge is meaningless. It may have a finite value in one gauge and it might diverge in another. Suppose Q(Â) is a conserved charge that takes a finite value for a particular value of the connection  at the boundary. Under a gauge transformation, Even if the function   is well behaved at the boundary, the integral might diverge is the boundary  happens to be noncompact. This is exactly the case faced by CS gravity in asymptotically AdS spaces. This problem is similar to that of a gauge theory in a finite box where the fields at the boundary spoil gauge invariance. The solution is to put sources at the boundary (currents) to restore gauge invariance.

In hep-th/ , a general solution is proposed: It is gauge invariant, It is gauge invariant, It comes from an action principle, and It comes from an action principle, and It yields finite values for the action, the Noether charges and It yields finite values for the action, the Noether charges and for the thermodynamic functions for black holes in AdS. for the thermodynamic functions for black holes in AdS. Second fundamental form The reference field is defined only at the boundary. The reference field is defined only at the boundary. This is a transgression. This is a transgression. What is that?

5. Transgressions Its exterior derivative is the difference of two characteristic classes, This is gauge invariant (and not just quasi-invariant), provided both A and A change in the same way under a gauge transformation at the boundary, where B is given by It is a simple exercise to show that the regularized action is a transgression form where and.

6. Interpretation This is puzzling: two connections, that only interact at the boundary, and one of the Lagrangians has the wrong sign (phantom field). What happens if one uses the transgression instead of the CS form as a Lagrangian? At least gauge invariance is ensured, but… Consider the following action

Σ : ∂M M A, A Two connections living together on the same manifold, but only seeing each other at the boundary…

MA A M Σ : ∂M= ∂M Perhaps not one manifold, but two… If one now reverses the orientation of the second manifold…

M A A M Σ: ∂M = -∂M There is no paradox if the two connections live on two distinct but cobordant manifolds (properly oriented). We live in a subsystem! Voila!

M3M3M3M3 M2M2M2M2 M1M1M1M1 Σ 12 : ∂M 1 = -∂M 2 Σ 23 : ∂M 2 = -∂M 3 Σ 12 : ∂M 1 = -∂M 2 ∏ 132 ∏ 123 Other possibilities…

Spacetime polymers... M1M1M1M1 M3M3M3M3 M4M4M4M4 M2M2M2M2 Σ 13 Σ 41 Σ 12 Σ 34 Σ 23 Σ 31 ∏ 123 ∏ 132 ∏ 134 ∏ 143

7. Other uses? which can be recognized as the two-dimensional gauged WZW system, where M 2 =  M 3, and  =g -1 dg. The transgression forms are not exceptional gauge invariants related to exotic systems. Consider the case in which A is related to A by a gauge transformation, A=g -1 Ag + g -1 dg. For example, for D=3, This idea can be extended to higher dimensions, T(A,A g ) …..

8. Conclusions  Chern-Simons forms for the AdS group [SO(D-1,2)] define gravitation theories in all odd dimensions. - No background geometry assumed - No free parameters, only dimensionless rational coefficients - Metric and affine structures on equal footing  Generically, the mass and other Noether charges in asymptotically AdS spacetimes are divergent and not gauge invariant.  A gauge-invariant regularization requires the addition of surface terms that turn the Lagrangian into a transgression form.

 The fact that transgressions have two connections that interact only at the boundary can be reinterpreted to mean that the spacetime manifold is a subsystem of a larger structure.  There seems to be no limitation to the kind of topologies that can be produced in this fashion, so long as some matching conditions be produced in this fashion, so long as some matching conditions are satisfied. are satisfied.  Holography?  If the two connections are related by a gauge transformation g, the transgression is the gauged WZW action for g and the transgression is the gauged WZW action for g and A.  Quantum mechanics?...  Gribov problem?...

 P. Mora, R.Olea, R. Troncoso and J. Zanelli, JHEP. 06, 036 (2004). hep-th/  P. Mora, R.Olea, R. Troncoso and J. Zanelli, JHEP. 02, 067 (2006). hep-th/  Andrés Anabalón, Steve Willison and J.Zanelli, (in preparation) REFERENCES