1. Effective Action on Manifolds with Branes and Boundaries Lebedev Physics Institute, Moscow Quarks-2008 Andrei Barvinsky Dmitry Nesterov Effective action in DGP brane models

2. 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

3. Braneworld Field Theory Map DGP sector Braneworld effective action tree levelone-loophigher loops… Dirichlet (Bulk) contribution Induced (brane) contribution + + + Gauge aspects F.-P. ghost contributions, … + Standard fields “Oblique” fields Heat Kernel Theory … … Heat Kernel on manifolds with boundaries/singularities

4. Braneworld Setup (field theory framework) Next slide: effective action approach Field content: Topological content: Starting point: classical “fundamental” action (low energy limit of string theory) what we are mostly interested in! -- Bulk fields -- induced (brane) fields e.g.: set of manifolds of different dimensions “glued” space

5. Next slide: weak field (loop) expansion Effective action approach functional of brane (induced) fields essencially nonlocal functional encorporates Bulk dynamics brane effective action (“classical” action for induced fields as seen by brane measurements of induced fields) quantum brane effective action (full quantum action for induced fields) multidimensional action (“fundamental” action in braneworld field theory) functional of brane mean fields essencially nonlocal functional encorporates brane quantum effects starting point of braneworld field theory local functional of brane and Bulk fields

6. Perturbative loop expansion dependence on background Next slide: Neumann-to-Dirichlet reduction - ? ( Determinant of some operator? ) Tree 1-loop local second-order symmetric differential Bulk operator local self-adjoint differential brane operator (generalized) Neumann Bulk propagator which satisfies where -- Bulk “background”,

7. Neumann­to­Dirichlet Reduction Systematic expression of Neumann quantities in terms of Dirichlet quantities (based on dualities between complementary boundary value problems) Next slide: 1-loop effective action brane-to-brane inverse propagator tree-level duality relation (Dirichlet-to-Neumann map) functional determinants 1-loop duality relation (A.Barvinsky, D.N. 2005) -- nonlocal brane operator -- disentangle contributions from different Bulks -- local brane operator -- parameterizes Neumann-type boundary conditions Neumann Bulk propagator: Dirichlet Bulk propagator:

8. 1­loop effective action Next slide: Brane contribution. DGP fields: Brane contributionBulk contribution Bulk contribution: sum of logarithms of determinants of Dirichlet operators on each Bulk unique objects (does not depend on brane physics – only on Bulk geometry) existence of elaborated algorithmized calculational techniques -- nonlocal brane “induced” operator -- local brane operator encoding (generalized) Neumann boundary / junction conditions Brane contribution: trace of logarithm of brane-to-brane propagator (matrix structure if multibrane scenario) incorporates the brane content of initial theory brane operator (we assume branes – boundaryless manifolds) essentially nonlocal encodes near-brane Bulk geometry and brane embedding

9. DGP sector Next slide: generic braneworlds 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 standard heat kernel trace !echo of braneworld 1-loop brane contribution: “+” -- purely brane functional “–” -- essentially nonlocal DGP : canonical theory pure Neumann junction (RS, …) Robin junction (generalized RS,…) where -- error function,

10. General algorithm. Leading order. Next slide: subleading orders where -- error function,  Bulk curvatures – small  extinsic curvatures – small  interbrane distances – large This DGP-type structure describes most general leading contribution when: Example: UV expansion braneworld generalization of Schwinger-DeWitt expansion. -- standard DeWitt coefficients,, where Substituting and performing integration one obtains answer in the following form:

11. General algorithm. Subleading orders. Next slide: Heat Kernel applications or Conclusions  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.  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:

12. Heat Kernel – suitable tool for one-loop divergences, counterterms, quantum anomalies, Casimir effect, UV asymptotics of 1-loop effective action and propagator Heat Kernel Theory Applications Schwinger-DeWitt expansion: Heat Kernel on manifolds with boundaries: 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.) New ingredients – boundary S-DW coefficients: Heat Kernel for operator : -- second-order differential operator -- proper time -- square of geodesic distance between two points -- Schwinger-DeWitt coefficients (at coincidence limit -- local geometric and gauge invariants)

13. Applying Neumann-to-Dirichlet Reduction: duality relation = = Generalized Neumann boundary conditions Neumann boundary operator: 1/M expansion (expansion method for integrals with weak peculiarity) - are obtained! reference coefficients (known) large mass expansion of effective action : for arbitrary generalized Neumann cases: - ? Essentially depend on boundary conditions! Can be regularly found for: only Dirichlet and homogeneous Neumann No regular techniques for containing tangential derivatives! Boundary S-DW coefficients:

14. Conclusion Effective action approach is still effective for braneworlds!

15. Multibulk scenario one-loop brane effective action: Bulks brane determinant duality: three Bulks glued along single brane braneworld configuration:

16. -- induced brane operator Duality Relations Path integral motivation: where (case of codimension 1 brane in (d+1)-dimensional Bulk ): -- brane-to-brane propagator (generalized) Neumann propagator (Green function) Dirichlet propagator (Green function) integration splitting: “tree-level”: “1-loop”: