Effective Action on Manifolds with Branes and Boundaries

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Effective Action on Manifolds with Branes and
Boundaries
Effective action in DGP brane models
Andrei Barvinsky Dmitry Nesterov

• Lebedev Physics Institute, Moscow
• Quarks-2008

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References

• A.O.Barvinsky and D.V.Nesterovin 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
• arXivhep-th/0611326
• A.O.BarvinskyQuantum effective action in
  spacetimes with branes and boundaries
  diffeomorphism invariancePhys.Rev. D74 (2006)
  084033arXivhep-th/0608004
• A.O.Barvinsky and D.V.NesterovQuantum effective
  action in spacetimes with branes and
  boundariesPhys.Rev. D73 (2006)
  066012arXivhep-th/0512291
• A.O.Barvinsky and D.V.NesterovDuality of
  boundary value problems and braneworld action in
  curved brane modelsNucl.Phys. B654 (2003)
  225-247arXivhep-th/0210005

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Map

Braneworld Field Theory
Braneworld effective action

Gauge aspects
weak field expansion
tree level
one-loop
higher loops


Neumann-to-Dirichlet reduction
Induced (brane) contribution
F.-P. ghostcontributions,
Dirichlet (Bulk) contribution


type of boundary (junction) conditions
DGP sector
Standard fields
Heat Kernelon manifolds with boundaries/singular
ities
Oblique fields
Heat Kernel Theory
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Braneworld Setup(field theory framework)
Topological content
Field content
Next slide effective action approach
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Effective action approach

• starting point of braneworld field theory
• local functional of brane and Bulk fields

• functional of brane (induced) fields
• essencially nonlocal functional
• encorporates Bulk dynamics

• functional of brane mean fields
• essencially nonlocal functional
• encorporates brane quantum effects

Next slide weak field (loop) expansion
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Perturbative loop expansion
Tree
1-loop
Next slide Neumann-to-Dirichlet reduction
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NeumanntoDirichlet Reduction
Systematic expression of Neumann quantities in
terms of Dirichlet quantities (based on
dualities between complementary boundary value
problems)

• brane-to-brane inverse propagatortree-level
  duality relation(Dirichlet-to-Neumann map)

• functional determinants1-loop duality
  relation(A.Barvinsky, D.N. 2005)

Next slide 1-loop effective action
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1loop effective action
Next slide Brane contribution. DGP fields
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DGP sector
1-loop brane contribution
-- purely brane functional --
essentially nonlocal
Key idea

• well-known object
• UV behavior Schwinger-DeWitt technique
• IR behavior A.Barvinsky, D.N. (2003)
• Nonperturbative curvature expansion
  A.Barvinsky, G.Vilkovisky (1987)
• incorporates internal geometry of brane
• incorporates the bundle geometry of induced
  brane field

• canonical theory
• pure Neumann junction (RS, )
• Robin junction (generalized RS,)

• DGP

Next slide generic braneworlds
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General algorithm. Leading order.
Example UV expansion braneworld
generalization of Schwinger-DeWitt expansion.
Next slide subleading orders
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General algorithm. Subleading orders.
Next slide Heat Kernel applications or
Conclusions
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Heat Kernel Theory Applications
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

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.)

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Applying Neumann-to-Dirichlet Reduction
Generalized Neumann boundary conditions
Neumann boundary operator
large mass expansionof effective action
duality relation


reference coefficients (known)
1/M expansion (expansion method forintegrals
with weak peculiarity)
- are obtained!
for arbitrary generalized Neumann cases
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Conclusion

• Effective action approach is still effective for
  braneworlds!

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Multibulk scenario
Bulks
braneworld configuration
three Bulks glued along single brane
determinant duality
one-loop brane effective action
brane
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Duality Relations
tree-level
1-loop
where (case of codimension 1 brane in
(d1)-dimensional Bulk )
-- induced brane operator
-- brane-to-brane propagator
Path integral motivation
integration splitting
(generalized) Neumann propagator (Green function)
Dirichlet propagator (Green function)