Renormalization and the Holographic Cotton Tensor Recent Advances in Topological Quantum Field Theory University of Lisbon September 14 2012 Sebastian de Haro ITFA and ID: 298973
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Slide1
Holographic Renormalization and the Holographic Cotton Tensor
Recent Advances in Topological Quantum Field TheoryUniversity of Lisbon, September 14, 2012Sebastian de Haro (ITFA and AUC, University of Amsterdam)Slide2
OutlineAdS/CFT: generalities and holographic
renormalizationBoundary graviton: duality and CFT coupled to gravityInstanton
solutions
Holographic Cotton Tensor. SdH, UvASlide3
AdS/CFTHolographic duality that relates:Gravity (string theory, M-theory) in
-dimensional AdS space to: A CFT on the (conformal) boundary of this space (-dimensional).The duality works for both Euclidean and Lorentzian signatures.
Euclidean case much better understood.
Interested in
instanton solutions.
Holographic Cotton Tensor. SdH, UvASlide4
Euclidean AdSQuadric in
:One can choose coordinates:
There is a boundary
at
.
We will use local coordinates (half space
):
Induces (
conformally
) flat metric on the boundary.
The CFT lives on this conformal boundary.
Holographic Cotton Tensor. SdH, UvA
Slide5
Bulk Hilbert SpaceSetting up QFT in AdS space, fields may contain a classical and a quantum part:
Both satisfy equation of motion in the bulk.
is part of Hilbert space
normalizable,
Holographically
:
vev
of operator
need not be: background
Holographically
: source
Holographic Cotton Tensor. SdH, UvA
Slide6
AdS/CFT
If
is independent of
,
it
is the
coupling
constant
for
that
operator.
Holographic Cotton Tensor. SdH, UvA
Slide7
AdS/CFTSemi-classical approximation in the bulk:Boundary theory is strongly coupled in that case. Typically: large
and large
(for
): strong ‘t
Hooft
coupling.
Expectation values of operators:
bulk solution for
(
normalizable
mode
)
Holographic Cotton Tensor. SdH, UvA
Slide8
Example: 2-point function
, conformally coupled scalar field in bulk:
is
boundary
condition at
.
Regularity
at
imposes
:
Holographic Cotton Tensor. SdH, UvA
Slide9
Holographic RenormalizationWhen computing correlation functions from bulk, we encounter divergences as
.Need formalism for generic boundary metric not just flat.Allows computation correlation functions of stress-energy tensor.Allows computation CFT in any background.Take into account back-reaction.Holographic renormalization systematic method to do this.
Holographic Cotton Tensor. SdH, UvASlide10
Holographic RenormalizationBulk metric
:
Solve
Einstein’s
equations
perturbatively
in
for given boundary values of metric:
Holographic Cotton Tensor. SdH, UvA
Slide11
Holographic RenormalizationSolve Einstein’s equations
perturbatively in :
undetermined (=
b.c.
)
Higher
’s:
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Slide12
Holographic RenormalizationHolographic recipe:
Regularize the bulk action at Add counterterms that do not modify
eom
Send
, obtain finite result.
Holographic Cotton Tensor. SdH, UvA
=
induced
metric
Slide13
Holographic RenormalizationTo compute
, need to solve eom all the way to interior.
Only need its variation
Compute
the
regularized
action
with
counterterm
subtraction
,
vary
and take limit.It is enough to know the divergent terms.
Holographic Cotton Tensor. SdH, UvASlide14
Holographic RenormalizationThe expectation value
:
Result:
Holographic Cotton Tensor. SdH, UvA
Slide15
The Boundary GravitonGoal:
understand holography of graviton and whether CFT can be coupled to
dynamical
gravity. What does this
give in bulk?Standard normalizability
analysis:
normalizable, depends on a choice of solution:
vev
of stress-energy.
is non-
normalizable
:
b.c., corresponds to source on boundary. Or is it?Ishibashi-Wald (2004): both modes are normalizable. Holographic Cotton Tensor. SdH, UvASlide16
The Boundary GravitonThis means that
either or
can be interpreted as boundary gravitons.
Correspond to different CFT’s.
Since both modes can be normalized in the bulk, they need not be a priori fixed.
We may set up dynamical equation that selects certain solutions.May couple CFT to gravity.
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InstantonsInvestigate the dynamics of boundary graviton
in simple case: self-dual Weyl.Boundary graviton does not solve full Einstein equations, but more restrictive one.Physically, self-dual solutions are ‘gravitational instantons’ that signal instability of bulk under deformation of b.c. Decay towards new vacuum.
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Self-Dual SolutionsIn spaces without
cosmological constant, natural condition is self-duality of Riemann.It automatically implies
.
If cosmological constant non-zero, need to choose a different condition. Self-duality of
Weyl
tensor is compatible with cosmological constant and Euclidean signature:
Holographic Cotton Tensor. SdH, UvA
Slide19
Vanishing Weyl TensorSimplest case: vanishing
Weyl tensor.Bulk metric is conformally flat.On-shell, the Weyl tensor reduces to:Together with
Einstein’s
equations:
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Slide20
Vanishing Weyl TensorEquation
implies:Series terminates at order :
is last non-
vanishing
coefficient
.
can
be
integrated
:Equation implies that the Cotton tensor of vanishes:The boundary metric is conformally flat.Bulk metric conformally flat iff boundary metric is conformally flat. Holographic Cotton Tensor. SdH, UvA
Slide21
Vanishing Weyl TensorExample: BTZ black hole on the
boundary.
The
geometry
extends
to
the bulk.
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Slide22
Self-Dual SolutionsNext case: non-zero, self-dual
Weyl tensor
Solve
coupled
equations
asymptotically
:
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Slide23
Self-Dual SolutionsResult:
Combine
with
holographic
interpretation
of
as 1-point function of stress-energy:
Integrate stress-tensor to obtain boundary generating functional:
Holographic Cotton Tensor. SdH, UvA
Bulk Einstein
Slide24
Self-Dual SolutionsThis can
be integrated to yield the boundary generating function:
We get:
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Slide25
Self-Dual SolutionsAt next level
, we get a compatibility condition for the curvature:
Also:
Non-linear gravity theory in 3d. Needs to be studied further.
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SummaryHolographic dictionary:
boundary gravitonStress-energy tensor:Both modes are normalizable (linearized fluctuations).
is related to boundary graviton of CFT2 via Cotton tensor.
Solutions with zero bulk
Weyl
tensor have zero boundary Cotton tensor.Bulk solution with BTZ black hole on boundary.
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Slide27
SummarySolutions with self-dual Weyl tensor:Generating functional can be integrated and gives the gravitational Chern
-Simons action.At next level, we get a non-linear consistency condition for the curvature.Holographic Cotton Tensor. SdH, UvA
Slide28
ReferencesMaldacena (1997)Witten (1998)Skenderis, Solodukhin (2000)
SdH, Skenderis, Solodukhin (2000)SdH, Petkou (2008, 2012)Holographic Cotton Tensor. SdH, UvASlide29
Thank you!Holographic Cotton Tensor. SdH, UvA