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Effective action in DGP brane models
Effective Action on Manifolds with Branes and Boundaries
Andrei Barvinsky
Dmitry Nesterov
Lebedev Physics Institute, Moscow
Quarks-2008
References
•
A.O.Barvinsky and D.V.Nesterov
in preparation
•
A.O.Barvinsky, A.Yu.Kamenshchik, C.Kiefer and D.V.Nesterov
Effective action and heat kernel in a toy model of brane-induced gravity
Phys.Rev. D75 (2007) 044010
arXiv:hep-th/0611326
•
A.O.Barvinsky
Quantum effective action in spacetimes with branes and boundaries: diffeomorphism invariance
Phys.Rev. D74 (2006) 084033
arXiv:hep-th/0608004
•
A.O.Barvinsky and D.V.Nesterov
Quantum effective action in spacetimes with branes and boundaries
Phys.Rev. D73 (2006) 066012
arXiv:hep-th/0512291
•
A.O.Barvinsky and D.V.Nesterov
Duality of boundary value problems and braneworld action in curved brane models
Nucl.Phys. B654 (2003) 225-247
arXiv:hep-th/0210005
…
Braneworld Field Theory
Map
Braneworld effective
action
…
tree level
+
one-loop
Dirichlet (Bulk)
contribution
+
Gauge aspects
+
higher loops…
Induced (brane)
contribution
Standard fields
Heat Kernel
on manifolds with
boundaries/singularities
Heat Kernel Theory
+
F.-P. ghost
contributions, …
DGP sector
“Oblique” fields
Braneworld Setup
(field theory framework)
Starting point:
classical “fundamental” action (low energy limit of string theory)
e.g.:
Topological content:
“glued” space
set of manifolds of
different dimensions
Field content:
-- Bulk fields
-- induced (brane) fields
what we are mostly interested in!
Next slide:
effective action approach
Effective action approach
 multidimensional action
(“fundamental” action in braneworld field theory)
• starting point of braneworld field theory
• local functional of brane and Bulk fields
 brane effective action
(“classical” action for induced fields
as seen by brane measurements of induced fields)
• functional of brane (induced) fields
• essencially nonlocal functional
• encorporates Bulk dynamics
 quantum brane effective action
(full quantum action for induced fields)
• functional of brane mean fields
• essencially nonlocal functional
• encorporates brane quantum effects
Next slide:
weak field (loop) expansion
Perturbative loop expansion
Tree
where
which satisfies
1-loop
-?
( Determinant of some operator? )
dependence on background
local second-order symmetric differential Bulk operator
local self-adjoint differential brane operator
(generalized) Neumann Bulk propagator
Next slide:
Neumann-to-Dirichlet reduction
-- Bulk “background”,
NeumanntoDirichlet Reduction
Systematic expression of Neumann quantities in terms of Dirichlet quantities
(based on dualities between complementary boundary value problems)
 brane-to-brane inverse propagator
tree-level duality relation
(Dirichlet-to-Neumann map)
 functional determinants
1-loop duality relation
(A.Barvinsky, D.N. 2005)
Neumann Bulk propagator:
-- nonlocal brane operator
-- disentangle contributions from different Bulks
Dirichlet Bulk propagator:
-- local brane operator
-- parameterizes Neumann-type boundary conditions
Next slide:
1-loop effective action
1loop effective action
Bulk contribution
Bulk contribution:
sum of logarithms of determinants
of Dirichlet operators on each Bulk
Brane contribution
Brane contribution:
trace of logarithm of brane-to-brane propagator
(matrix structure if multibrane scenario)
-- nonlocal brane
“induced” operator
-- local brane operator encoding (generalized)
Neumann boundary / junction conditions
• unique objects (does not depend
on brane physics – only on Bulk
geometry)
• existence of elaborated
algorithmized calculational
techniques
• incorporates the brane content of initial theory
• brane operator (we assume branes – boundaryless manifolds)
• essentially nonlocal
• encodes near-brane Bulk geometry and brane embedding
Next slide:
Brane contribution. DGP fields:
DGP sector
1-loop brane contribution:
“+” -- purely brane functional
“–” -- essentially nonlocal
Key idea:
echo of braneworld
standard heat kernel trace !
• well-known object
• UV behavior: Schwinger-DeWitt technique
• canonical theory
• IR behavior: A.Barvinsky, D.N. (2003)
• pure Neumann junction
• Nonperturbative curvature expansion:
A.Barvinsky, G.Vilkovisky (1987)
(RS, …)
• Robin junction
(generalized RS,…)
• incorporates internal geometry of brane
• DGP :
where
Next slide:
generic braneworlds
,
-- error function
• incorporates the bundle geometry of
induced brane field
General algorithm. Leading order.
This DGP-type structure describes most general leading contribution when:
 Bulk curvatures – small
 extinsic curvatures – small
 interbrane distances – large
Example:
where
,
-- error function
UV expansion
braneworld generalization of Schwinger-DeWitt expansion.
, where
Substituting
and performing integration one obtains answer in the following form:
Next slide:
subleading orders
-- standard DeWitt coefficients,
General algorithm. Subleading orders.
 A straightforward algorithmized (but a bit involved) perturbation procedure can be performed:
where
-- leading DGP-type structure,
-- perturbation of brane operator, and
-- some highly involved but perturbatively calculable
brane operator with coefficients depending on powers of curvatures and their derivatives.
 After commuting (in the last term) dependence on brane D’Alembertian to the right one faces the
following structures:
 Again, exploiting trick with Laplace transform in subleading orders one comes to a bit more
general structure compared to the leading order:
where formfactors now are differential operators of finite order with coefficients dependent on geometric
invariants.
Next slide:
Heat Kernel applications or Conclusions
Heat Kernel Theory Applications
Heat Kernel for operator
:
Schwinger-DeWitt expansion:
Heat Kernel – suitable tool for
 one-loop divergences,
 counterterms,
 quantum anomalies,
 Casimir effect,
 UV asymptotics of 1-loop effective action and propagator
-- second-order differential operator
-- proper time
-- square of geodesic distance between two points
-- Schwinger-DeWitt coefficients
(at coincidence limit -- local geometric and gauge invariants)
Heat Kernel on manifolds with boundaries:
New ingredients – boundary S-DW coefficients:
 Essentially depend on boundary conditions!
 Can be regularly found for: only Dirichlet and homogeneous Neumann cases
using: method of images, conformal properties (Branson, Gilkey, Dowker, Kirsten, Vassilevich, etc.)
Generalized Neumann boundary conditions
Neumann boundary operator:
-?
Boundary S-DW coefficients:
 Essentially depend on boundary conditions!
 Can be regularly found for: only Dirichlet and homogeneous Neumann
 No regular techniques for
containing tangential derivatives!
Applying Neumann-to-Dirichlet Reduction:
large mass expansion
of effective action :
duality relation
=
=
reference coefficients (known)
1/M expansion
(expansion method for
integrals with weak peculiarity)
- are obtained!
for arbitrary generalized
Neumann cases:
Conclusion
Effective action approach is still effective for
braneworlds!