Centro de Estudios CientíficosCentro de Estudios CientíficosCECS-Valdivia-CECS-Valdivia-ChileChile

Uses of TransgressionsUses of Transgressions

J. Zanelli, CECS - ValdiviaJ. Zanelli, CECS - Valdivia

Modern Trends in Field TheoryModern Trends in Field Theory

JoJoãão Pessoao Pessoa, Brasil September 2006September 2006

OutlineOutline

1.1. Gauge theories Gauge theories

2. 2. GravityGravity

3.3. Chern-Simons Chern-Simons AdS AdS gravigravityty

4.4. RegularizationRegularization

5.5. Transgressions Transgressions

6.6. InterpretationInterpretation

7.7. Other usesOther uses

8.8. Summary Summary

1.1. Gauge theories Gauge theories

• Maxwell, Yang-MillsMaxwell, Yang-Mills• Chern-SimonsChern-Simons• Diffeo-invariant systems (GR, strings,Diffeo-invariant systems (GR, strings, membranes,…)

Fiber bundle structureFiber bundle structure

ManifoldManifold

Fibers (group)Fibers (group)

a.a. Yang-Mills (including Maxwell) Yang-Mills (including Maxwell)

ActionAction

xdFFgggAI Dbαβ

aab

M

][

ManifoldManifold dim. dim. DD

MetricMetric Curvature 2-formCurvature 2-form

Curvature:Curvature: ],[ AAAAF

KKA GA Connection (Lie-algebra valued 1-form) Connection (Lie-algebra valued 1-form)

Thus, a YM type action requiresThus, a YM type action requires • A manifold A manifold MM• A metric A metric gg• A Lie algebra A Lie algebra GG • Killing form Killing form γγ

Lie alg. generatorLie alg. generator

Killing formKilling form

Invertible, non dynamical, preexistentInvertible, non dynamical, preexistent

xdAAAI λνμ

M

3

3

][ 3M

dAA

C-S theories are more economical: no invertible g, γ required

Invariant (Invariant (up to surface termsup to surface terms) under ) under

• Lorentz transformations,Lorentz transformations, AA '

• Gauge transformations, Gauge transformations, )(' xAA

• Diffeomorphisms,Diffeomorphisms,

Ax

xA

'

'

3D 3D ManifoldManifold Connection (not necessarily abelian)Connection (not necessarily abelian)

b.b. Chern-Simons Chern-Simons ((D=3D=3))

3

12131 ))][ ((

M

kk

kk cdcdI AAAAAA

wherewhere <<∙∙∙∙∙∙>> stands for some appropriately symmetrized trace.stands for some appropriately symmetrized trace.

NKLNLK ...... GGGSymmetricSymmetric invariant tensor:invariant tensor:

,1

0

12 )1()( k

tCS

k dtkL FAA

The The cc11, … , … cckk are dimensionless combinatorialare dimensionless combinatorial coefficients. coefficients.

The CS lagrangian readsThe CS lagrangian reads

22AAF ttdt where where

More dimensions, non abelian algebra

CS lagrangians are “potentials” for characteristic classes:CS lagrangians are “potentials” for characteristic classes:

FFFFdAAd )( 4D4D-Pontryagin density-Pontryagin density

• Gauge transformations Gauge transformations ggAggA d11'

• Diffeomorphisms (trivial)Diffeomorphisms (trivial)

AA

'

'x

x

Gauge invariance:Gauge invariance: 10 kF CSkdL 12 )( 12

CSkLd

)something(12 ddLCSk

=> => The action is invariant underThe action is invariant under

.112

kCSk FdLIn general,In general, (2k+2)D(2k+2)D-Topological density-Topological density

…… up to boundary termsup to boundary terms

No free parametersNo free parameters

2.2. Gravity Gravity

4

,)2(][ 4

M

xdRggIEinstein-Hilbert theory (Einstein-Hilbert theory (4D4D))

2l cosm. const.cosm. const. 221 )( gggR = = Ricci scalar,Ricci scalar,wherewhere

4

01 ],[M

dcbadcababcd eeeeeeReI

Equivalently,Equivalently,

wherewhere cbc

aabab dR == Curvature 2-formCurvature 2-form

dxee

dxaa

ba

ba

== Lorentz connectionLorentz connection

== VielbeinVielbein

ee11

TTxxee22

ee33

Isomorphism between the local orthonormal frame on Isomorphism between the local orthonormal frame on the tangent space and the coordinate basis:the tangent space and the coordinate basis:

VielbeinVielbein

aaa edxxedz )(

VielbeinVielbein

, baab eeg

This induces a metric structure on spacetime.This induces a metric structure on spacetime. The metric is not a The metric is not a fundamental field, but a composite:fundamental field, but a composite:

where = Lorentz metric in the tangent space of the manifold. where = Lorentz metric in the tangent space of the manifold. ab

Gauge Gauge symmetry:symmetry:

bc

bd

dc

ca

ba

ba

bb

aaa

dxxx

xexxexe

)()()(')(

)()()(')( 1

)1,3()( SOxba

This is an indication that gravitation This is an indication that gravitation is gauge theory, invariant under is gauge theory, invariant under local Lorentz transformations at each point in spacetimelocal Lorentz transformations at each point in spacetime

Spin connectionSpin connection

The notion of parallelism, necessary for differentiation, is encodedThe notion of parallelism, necessary for differentiation, is encodedin the connection in the connection aa

bb(x)(x)..

Equivalence PrincipleEquivalence Principle

Gravity for Gravity for D≥4D≥4

arbitraryarbitrary

These theories are invariant underThese theories are invariant under- Local Lorentz transformations, Local Lorentz transformations, S=(D-1,1)S=(D-1,1)- Gen. coord. Transf. Gen. coord. Transf. [Trivial, as for any well posed theory][Trivial, as for any well posed theory]

D

pp

D

M

Dpaaaaaaa

D

pp eeRRgI ,][ 12

...

]2/[

0

21221

21

D. Lovelock, 1970:D. Lovelock, 1970:

Unfortunately, this theory has Unfortunately, this theory has [(D-1)/2][(D-1)/2] arbitrary coefficients arbitrary coefficients p p

- DimensionfulDimensionful- Give rise to Give rise to [(D-1)/2][(D-1)/2] different, arbitrary cosmological constants different, arbitrary cosmological constants pp - Not protected by symmetry (they get renormalized).Not protected by symmetry (they get renormalized).

For For D=2n+1D=2n+1 there is a there is a special choicespecial choice of which makes all the of which makes all the cosmological constantscosmological constants pp equal. Then,equal. Then,

p

Special choice:Special choice:

The theory becomes invariant under the AdS group,The theory becomes invariant under the AdS group,

)2,1( DSO)1,1( DSO

Symmetry enhancementSymmetry enhancement

All the dimensions can be absorbed in a rescaling of the vielbein, All the dimensions can be absorbed in a rescaling of the vielbein, the theory becomes scale invariant, and the the theory becomes scale invariant, and the pp are rational numbers. are rational numbers.

Conformal invarianceConformal invariance

This is a Chern-Simons theory of gravity for the AdS group

For the special choice, the vielbein and the spin For the special choice, the vielbein and the spin connection can be connection can be combinedcombined as parts of a single as parts of a single connection 1-form,connection 1-form,

, 2

11ab

aba

ael JJA

AdSAdS algebra algebrawhere Lorentz where Lorentz generator generator

generators of generators of AdS boostsAdS boosts

a

ab

J

J

3.3. Chern-Simons AdS gravity Chern-Simons AdS gravity

nAdSCS dteL n tFA

1

0

)1(),(

The action describes a gauge theory for the AdS The action describes a gauge theory for the AdS group. It is a Chern-Simons theory, and can be group. It is a Chern-Simons theory, and can be written aswritten as

tttt AAAF dwhere .where .

DDD aaaaaaa 11321

#JJJ

][),( 12)12(

313

1231

nn

nnAdSCS eeReReL n

lnll

l AdS radiusAdS radius

In terms of geometric fields, the CS lagrangian for the AdS groupIn terms of geometric fields, the CS lagrangian for the AdS groupin in D=2n+1 D=2n+1 dimensions reads dimensions reads

C-S – AdS gravitiesC-S – AdS gravities

• No dimensionful couplingsNo dimensionful couplings• No arbitrary (renormalizable) parametersNo arbitrary (renormalizable) parameters• Admit SUSY extensions for (Admit SUSY extensions for (spin spin ≤ 2≤ 2))• Asymptotically Asymptotically AdSAdS• Possess black holesPossess black holes• Admit , and Admit , and )( 0 l0

0

Useful to do physics: black holes, thermodynamics, cosmology, etc.Useful to do physics: black holes, thermodynamics, cosmology, etc.

4.4. Regularization Regularization

The mass and other Noether charges for all gravitation theories in The mass and other Noether charges for all gravitation theories in asymptotically AdS spaces are generically divergent. asymptotically AdS spaces are generically divergent.

The trouble is that the metric and Killing fields do not fall off The trouble is that the metric and Killing fields do not fall off sufficiently fast at infinity ( in standard coordinates):sufficiently fast at infinity ( in standard coordinates):r

)1(11 21

132

2

2

00 OrrqMg nD

l

r

The standard approach to cope with this problem is to subtract the The standard approach to cope with this problem is to subtract the value of the charge of some background: value of the charge of some background:

BckgndQQQ 0ˆ

ThisThis can be practical, butcan be practical, but::

• It is arbitrary, requires prior knowledge of the answerIt 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 A finite but notnot gauge invariant charge is meaningless. gauge invariant charge is meaningless. It may have a It may have a finite value in one gauge and it might diverge in another. Suppose finite value in one gauge and it might diverge in another. Suppose QQ((ÂÂ)) is a conserved charge that takes a finite value for a particular value of is a conserved charge that takes a finite value for a particular value of the connection the connection ÂÂ at the boundary. Under a gauge transformation, at the boundary. Under a gauge transformation,

.),ˆ()ˆ()'ˆ(')ˆ(

gAAQAQAQ

Even if the function Even if the function is well behaved at the boundary, the integral is well behaved at the boundary, the integral might diverge is the boundary might diverge is the boundary happens to be noncompact. This is happens to be noncompact. This is exactly the case faced by CS gravity in asymptotically AdS spaces.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 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 the fields at the boundary spoil gauge invariance. The solution is to put sources at the boundary (currents) to restore gauge invariance.sources at the boundary (currents) to restore gauge invariance.

MM

CSn eBeLeI ),(),(],[12

In In hep-th/0405267hep-th/0405267, a general solution is proposed:, a general solution is proposed:

• It is gauge invariant,It is gauge invariant,

• It comes from an action principle, andIt 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.

ababab Second fundamental Second fundamental formform

ab• 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?

5.5. Transgressions Transgressions

)()( 1112 AFAFd nnnT

Its exterior derivative is the difference of two characteristic classes,Its exterior derivative is the difference of two characteristic classes,

),,()()(),( 2121212 AAdBALALAA n

nCS

nCS

n T

.)( )1( 11

0

1

0

2 nsttn FAAAsdtdsnnB

This is gauge invariant (and not just quasi-invariant), provided This is gauge invariant (and not just quasi-invariant), provided both both AA and and AA change in the same way under a gauge transformation change in the same way under a gauge transformation at the boundary, where at the boundary, where BB is given by is given by

It is a simple exercise to show that the regularized action is a It is a simple exercise to show that the regularized action is a

transgression form where transgression form where and . and .ababA J

2

1 0ae

6.6. Interpretation Interpretation

This is puzzling: This is puzzling: two connectionstwo connections, that only , that only interact at the boundaryinteract at the boundary, , and one of the Lagrangians has the wrong sign (and one of the Lagrangians has the wrong sign (phantom fieldphantom field).).

What happens if one uses the transgression instead of the CS form What happens if one uses the transgression instead of the CS form as a Lagrangian? as a Lagrangian?

At least gauge invariance is ensured, but…At least gauge invariance is ensured, but…

),()]()([],[ 21212 AABALALAAI

M

nn

CS

M

nCS

T

Consider the following actionConsider the following action

ΣΣ : : ∂∂MM

MM

A, AA, A

AA,BALALIMM

CS

M

CS

Two connections living together on the same manifold, but Two connections living together on the same manifold, but only seeing each other at the boundary…only seeing each other at the boundary…

MM

A A AA

MM

ΣΣ : : ∂∂M= ∂MM= ∂M

AA,BALALIMM

CS

M

CS

Perhaps not one manifold, but two…Perhaps not one manifold, but two…

If one now reverses the orientation of the second manifold…If one now reverses the orientation of the second manifold…

MM

A A

AAMM

ΣΣ: : ∂∂MM= -∂M= -∂M

AA,BALALIM

CS

M

CS

There is no paradox if the two connections live on two distinct but There is no paradox if the two connections live on two distinct but

cobordant manifolds (properly oriented).cobordant manifolds (properly oriented). We live in a subsystem!We live in a subsystem!

Voila!Voila!

MM33 MM22

MM11 ΣΣ1212: : ∂∂MM11= = --∂∂MM22

ΣΣ2323: : ∂∂MM22= = --∂∂MM33

ΣΣ1212: ∂: ∂MM11= = --∂∂MM22

∏∏132132

∏∏123123

2313211332

21321

1321233123

12321

,A,AAB,A,AAB,AAB,AAB

,AABALALALIM

CS

M

CS

M

CS

Other possibilities…Other possibilities…

Spacetime Spacetime polymers...polymers...

341321

3221

41

143123

4112

41

,A,AAB,A,AAB

,AAB,AAB

ALALIM

CS

M

CS

MM11

MM33

MM44

MM22ΣΣ1313

ΣΣ4141

ΣΣ1212

ΣΣ3434

ΣΣ2323

ΣΣ3131∏123∏132

∏134

∏143

7.7. Other uses? Other uses?

which can be recognized as the two-dimensional which can be recognized as the two-dimensional gauged WZWgauged WZW system, where system, where MM2 2 == MM3 3 , and , and ==gg -1 -1dg.dg.

The transgression forms are not exceptional gauge invariants related The transgression forms are not exceptional gauge invariants related to exotic systems. Consider the case in which to exotic systems. Consider the case in which AA is related to is related to AA by a by a gauge transformation, gauge transformation, A=gA=g -1 -1Ag + g Ag + g -1-1dg. dg. For example, for For example, for D=3D=3,,

g

M MM

g AAAAgAI )(),(],[3 23

33

3

1 T

This idea can be extended to higher dimensions, This idea can be extended to higher dimensions, T(T(A,AA,Agg))……....

8.8. Conclusions Conclusions

Chern-Simons forms for the AdS group [Chern-Simons forms for the AdS group [SO(D-1,2)SO(D-1,2)] define ] define gravitation theories in all odd dimensions.gravitation theories in all odd dimensions.

- No background geometry assumed- No background geometry assumed

- No free parameters, only dimensionless rational coefficients- No free parameters, only dimensionless rational coefficients

- Metric and affine structures on equal footing- Metric and affine structures on equal footing

Generically, the mass and other Noether charges in asymptotically Generically, the mass and other Noether charges in asymptotically AdS spacetimes are divergent and not gauge invariant.AdS spacetimes are divergent and not gauge invariant.

A gauge-invariant regularization requires the addition of surface A gauge-invariant regularization requires the addition of surface terms that turn the Lagrangian into a terms that turn the Lagrangian into a transgression formtransgression form..

The fact that transgressions have two connections that interact The fact that transgressions have two connections that interact only at the boundary can be reinterpreted to mean that the only at the boundary can be reinterpreted to mean that the spacetime manifold is a subsystem of a larger structure.spacetime manifold is a subsystem of a larger structure.

There seems to be no limitation to the kind of topologies that canThere 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?Holography?

If the two connections are related by a gauge transformation If the two connections are related by a gauge transformation gg, , the transgression is the gauged WZW action for the transgression is the gauged WZW action for gg and and A.

Quantum mechanics?...Quantum mechanics?...

Gribov problem?...Gribov problem?...

P. Mora, R.Olea, R. Troncoso and J. Zanelli, P. Mora, R.Olea, R. Troncoso and J. Zanelli, JHEPJHEP.. 0606, 036 , 036 (2004). (2004). hep-th/0405267hep-th/0405267

P. Mora, R.Olea, R. Troncoso and J. Zanelli, P. Mora, R.Olea, R. Troncoso and J. Zanelli, JHEPJHEP.. 0202, 067 , 067 (2006). (2006). hep-th/0601081hep-th/0601081

Andrés AnabalAndrés Anabalóón, Steve Willison and J.Zanelli, (in preparation)n, Steve Willison and J.Zanelli, (in preparation)

REFERENCESREFERENCES