Download The boundary conditions

Boundary conditions for SU(2)
Yang-Mills on AdS4
Jae-Hyuk Oh
At ISM meeting in Puri, 2012.12.20.
Dileep P. Jatkar and Jae-Hyuk Oh,
Based on arXiv:1203.2106
Program
• 1. Alternative quantization for massive scalar
and U(1) fields.
• 2. Its extension to SU(2) Yang-Mills and
interesting boundary actions.
• 3. “Approximate” electric-magnetic duality.
• 4. Yang-Mills instanton in AdS and its
boundary condition.
Alternative quantization e.g. massive
scalar
In certain windows of the conformal dimensions of the boundary composite
operators corresponding their bulk excitations in AdSd+1 , there are two
possible quantization schemes for their boundary CFTs.
For massive scalar field theory on AdSd+1, in a certain window of its mass
square,
, which satisfies both unitarity bound
and BF bound , there are two possible quantization
schemes: so called Δ+-theory and Δ—-theory, where
the conformal dimension of the boundary operators in each theory.
Δ+-theory: near AdS boundary expansion of the scalar field is given by
where
is boundary value of the field ->an external source and
corresponds to a certain composite operator coupled to
in
dual CFT, where one imposes Dirichlet B.C. as
.
Δ—-theory: is obtained by imposing Neumann B.C. as
, where
the role of the external source and boundary composite operator is
switched. A(x μ) -> source and φ0(x μ) -> boundary operator.
Generalized boundary conditions
• In fact, these two boundary conditions are not the only possible boundary
conditions in this window. One can add a general form of the boundary
deformation term Sbdy to the r=0(AdS) boundary and obtain the
generalized boundary conditions as generalizations of Neumann boundary
condition.
• Boundary on-shell action at r=0(AdS) boundary, Ios will be defined as
where Sbulk is the boundary contributions from the bulk action.
• The boundary condition is obtained by saddle point approximation, δIos =0,
which corresponds to the classical vacuum of the boundary theory.
• For Dirichlet B.C, Sbdy =0, and For Neumann B.C.,
.
•
Δ+-theory and Δ—theory for U(1)
vector
field
in
AdS
Δ - and Δ -theories for bulk U(1) vector fields in AdS 4are well defined, in
+
—
4
which Δ+=2 and Δ-=1.
• The unitarity bound for the vector like observables in d-dimensional flat
space is given by
. Therefore Δ+=2 theory satisfies saturated
unitarity bound, but Δ-=1 theory looks like it does not.
• One interpretation for Δ—theory is that we interpret the corresponding
dual operator with conformal dimension “1” as U(1) vector gauge field in
the boundary theory. In fact, this operator has an ambiguity in its
expectation value from its gauge degrees of freedom. However, the field
strength made out of the gauge field has conformal dimension 2 and
which satisfies the unitarity bounds. The real observable -> field strength.
• The boundary conditions: Near boundary (r-> 0), Ai = A(0)i+E(0)i r… , Ai is
U(1) gauge fields in AdS4, A(0)iis boundary value of Ai and E(0)i boundary
value of the bulk electric field(“i” is boundary space time index, Ei=Fri).
A(0)i=0 is Dirichlet B.C. -> Standard quantization (Δ+- theory)
E(0)i=0 is Neumann B.C.-> Alternative quantization (Δ—-theory)
These two boundary conditions are related by electric magnetic duality.
SU(2) Yang-Mills in AdS4
• We want extend this discussion to SU(2) Yang-Mills in AdS4. Motivations
are three folds:
(1) U(1) theory is free theory. The most natural ways to introduce
interactions to boundary CFTs are considering SU(2) Yang-Mills in AdS.
(2) There is electric magnetic duality for U(1) case up to some
deformations to AdS boundary. What about such duality in SU(2) Yang-Mills?
(3) Investigating non-perturbative bulk solutions and their boundary
conditions are interesting. We want study Yang-Mills instanton solution in 4-d
Euclidean AdS space and its corresponding boundary conditions.
Yang-Mills action in AdS4 is given by
where GMN is Euclidean AdS4 metric, `a` is SU(2) index, and M,N={r,i}, where
i=1,2,3. Field strength is given by
Upto bulk equations of motion,
the bulk action is (in radial gauge as Aar=0)
• We define canonical momentum of Yang-Mills field Aai as Πai=∂rAai in radial
gauge .
• We construct the bulk partition function by deformations, Sbdy which is a
functional of boundary Yang-Mills field Aa(0)i and its canonical momentum
Πa(0)i . Now AdS/CFT correspondence becomes as follows:
=
𝐷 Aa(0)i, Π𝑎𝑖 exp −𝑆𝑏𝑑𝑦 Aa(0)i, Πa(0)i exp[-W(Aa(0)i )]
un-deformed CFT
generating functional
• This functional integration is evaluated at conformal boundary, r=ε -> 0,
and boundary condition independent since it contains a functional
integration measure with Aa(0)i and its canonical momentum Πa(0)i .
Appropriate boundary condition by such deformation is given by finding
its classical vacuum, for which we apply saddle point approximation,
δIos=0
• Now the source J is a generic function of Aa(0)i and Πa(0)i .(Here, we
indentify the boundary on-shell action with dual CFT generating functional
as
)
Bulk solution
• We evaluate the perturbative solutions order by order in small amplitudes
in Yang-Mills fields. Then, Yang-Mills fields are expanded as
• Where ε is a bookkeeping parameter for the expansion, which is
dimensionless small number. The solution of Yang-Mills equations of
motion up to O(ε2) is given by
where
Superscript, T denotes transverse parts of the solution and qi and pi are
momenta along boundary directions.
and
are boundary
momenta dependent functions and
are gauge degrees of freedom. We
will choose radial gauge by setting
.
has the same
form of solution
(bar -> tilde). This is homogenous part of the bulk YangMills equations of motion.(
will provide source terms to the second order
equations of motion in O(ε2) and one can sort O(ε2) solutions into
homogenous and inhomogeneous parts with respect to such source terms.)
•
and
are transverse and longitudianl parts of inhomogeneous bulk
equations of motion, which are given by
where
and
is an
integration const.
To construct boundary on-shell action, we need to know the boundary expansion of
the bulk solution(
):
,
where
Canonical
momenta
and
where
=
,
•
Boundary actions (Dirichlet)
With such a solution, imposing Dirichlet boundary condition i.e. SDbdy =0 and the form
of bulk on shell action,
where
.
Up to cubic order in action, gauge fields in the 3-vertax is effectively transverse, since we
choose a gauge
.
is 3-point vertex coupled to
transverse gauge fields only.(The boundary on-shell action respects residual gauge
invariance under radial gauge.)
Since boundary on-shell action is generating functional with source J= Aai coupled to
boundary one point expectation value, one can obtain its classical effective action via
Legendre transform using Aai and Πai as
. It is given by
Neumann and Massive and self-dual
deformation
• To impose Neumann boundary condition, we add
a boundary deformation, then
as
therefore,
(Neumann B.C.). In such case, Πai becomes a source couples to
a boundary operator, then boundary on-shell action(generating functional) will be in
terms of Πai . They are given as
since adding SNbdy is effectively Legendre transform from Dirichlet to Neumann.
• To get more an interesting boundary action, we deform the bulk action, of which
on-shell action satisfies massive and self-dual boundary conditions. Massive
deformation and self-dual boundary condition are given by
which is a non-ableian version of self-duality in odd dimension[Townsend et.al.]. To
impose such boundary condition, we deform the bulk action by the double and the
triple trace operators as
where
,
• From such deformations, one obtains the massive self-dual boundary
condition. The on-shell for this case are given by
• The on-shell action is proportional to the Non-abelian massive ChernSimons theory action.
•
“Approximate” electric magnetic
duality
in
SU(2)
Yang-Mills.
It is well-known fact that electric-magnetic duality cannot be demonstrated for
non-Abelian gauge fields theories. Exchanging E and B fields is still possible in
these theories but the real problem is if such transformations are canonical or
not. It is shown that these are not canonical transformations for the non-Ableian
cases by Deser et. al. long ago.
• Recently, however, it is suggested that electric magnetic duality is possible to be
embodied for SU(2) gauge theory when Yang-Mills action is truncated up to cubic
order interactions, at least for a particular gauge(transverse gauge). This duality is
somewhat “approximate”. The meaning of “approximate” is the following: Deser
et. al. have constructed an infinitesimal canonical transformation which is a
natural extension of U(1) duality by retaining Yang-Mills cubic interactions only.
• However, this is not precisely EM duality transform since infinitesimal changes of
electric fields are not proportional to magnetic fields. Therefore, “approximate”
duality only states that there exist a canonical transformation which is a natural
extension of infinitesimal U(1) duality transform.
• The difference between 4-d flat and AdS4 space is their boundaries. EM duality is
not a symmetry of Lagrangian and there will be total derivative terms in the
transformed action. In flat space, one can remove these terms since he/she can
assume that the gauge fields sufficiently die off at the space time infinity.
However, in AdS space, there is AdS boundary at r=0. Therefore, we need to
count boundary terms at r=0 properly.
• Now, let us get into the details. Start with Hamiltonian form of SU(2) YangMills action as
where
and
Which is Hamiltonian density. We have set gauge coupling g=1, and
The negative sign in front of B2 is due to that we are working in Euclidean
space time. Legendre transform is taken with “r” to the action from H.
• The last term in H will be integrated out by imposing Gauss constraint
, since
is a Lagrange multiplier.
• Under such manipulation and choosing transverse gauge, qiAai=0, the
action becomes
where
• It turns out that the bulk action is invariant up to boundary terms and
cubic order in weak fields expansion under the following infinitesimal
transform:
where “higher order” denotes the higher than quadratic orders in weak fields
A or E, η is infinitesimal duality rotation angle and the action varies as
• We interpret the boundary terms as a new deformation terms, which will
map one to another CFT with different boundary condition.
• The variation of the longitudinal part of electric fields does not contribute
to the variation of the action. However, it must be used to show that
Gauss constraint is invariant under such transform.
Bulk Instanton and its boundary
condition
The previous analysis is depending on perturbative analysis in small
amplitude of bulk Yang-Mills fields. Here, we concentrate on nonperturbavtive solutions in Yang-Mills, namely instanton solutions. Since,
Yang-Mills action in AdS4 is exactly mapping to that in flat space by Weyl
transform(GMN -> r2GMN ) of the background metric:
The flat space that we get by such transform is not precisely R4 , actually a half
of it, denoting R4+ since the radial coordinate of AdS space is semi-infinite.
Under the transformation, the equations of motion are invariant but the
gauge condition may change. For example, Lorentz gauge in flat space is
not the same with that in AdS space. However, using the fact that all the
metric factor depends only on radial coordinate “r”, it turns out that the
radial gauge will be the same in both sides.
Since equations of motion are the same in both flat and AdS4 space,
instanton solution in flat space in radial gauge is also solutions in AdS up
to boundary condition. What only matters is boundary condition.
• The instanton with winding number “1” solution in flat 4-d space is given
by
where
,
is the location of
the instanton solution, ρ is size of that and we set r=
for simplicity.
Then the instanton is at AdS boundary.
• The field strength is
,
• and the Yang-Mills action has finite action value as
This solution may satisfy Lorentz gauge condition, so we perform a gauge
transform for the solution to be in radial gauge.
• At the Poincare horizon, at x4=r=∞, the instanton solution approaches to
pure gauge solution(A=0 upto gauge transform), then which does not
change boundary condition at the horizon. However, on the AdS boundary,
r=0, it does not become pure gauge solution, and definitely change its
boundary condition.
• Under radial gauge, we expand the instanton solution in small r near AdS
boundary to figure our its boundary condition as,
where
• yi is the coordinate to the boundary directions(xM=(r,yi)). Then, in our case,
the instanton is located at (r=0, yi0).
• The bulk self-duality condition in the limit of r=0 is
and it is easy to see that the above solution satisfies this condition. What we
want is that express this boundary condition by the boundary field
only.
• One possible way to do this is the following: we re-write
and
by
using a function λ as
We obtain the function λ to be a non-local function of
above relation as
and also
by inverting
, finally with all these, we have
• This is one possible boundary condition that we suggest. It looks like
magnetic field is given by non-local Wilson line type interaction from a
point like source.
• In the action level, the left hand side will come from Chern-Simons action
but we did not determined the precise form of the right hand side in the
action. In sum, it is has a form of
where
will provide non-local interaction term in the boundary condition.
Summary
• We have obtained perturbative solution in the Einstein gravity limit of bulk
SU(2) Yang-Mills system, and its exotic momentum dependent 3-point
correlations.
• By some boundary deformation, we obtained interesting boundary onshell actions, e.g. massive Chern-Simons by massive self-dual deformation.
• SU(2) Yang-Mills in AdS4 enjoys approximate electric magnetic duality and
which provides deformation terms to AdS boundary.
• We also suggested one possible boundary condition for bulk instanton
solution.
Out looks
• Construction of finite EM duality up to cubic interactions and find out a
mapping to all possible boundary CFTs.
• What is the most general boundary conditions for bulk multi-instantons in
arbitrary positions and sizes?