Dressing bulk fields in AdS3 Kabat, Daniel; Lifschytz, Gilad
The journal of high energy physics,
10/2020, Letnik:
2020, Številka:
10
Journal Article
Recenzirano
Odprti dostop
A
bstract
We study a set of CFT operators suitable for reconstructing a charged bulk scalar field
ϕ
in AdS
3
(dual to an operator
O
of dimension ∆ in the CFT) in the presence of a conserved spin-
n
...current in the CFT. One has to sum a tower of smeared non-primary scalars
∂
+
m
J
m
, where
J
(
m
)
are primaries with twist ∆ and spin
m
built from
O
and the current. The coefficients of these operators can be fixed by demanding that bulk correlators are well-defined: with a simple ansatz this requirement allows us to calculate bulk correlators directly from the CFT. They are built from specific polynomials of the kinematic invariants up to a freedom to make field redefinitions. To order 1/
N
this procedure captures the dressing of the bulk scalar field by a radial generalized Wilson line.
A
bstract
We show that bulk quantities localized on a minimal surface homologous to a boundary region correspond in the CFT to operators that commute with the modular Hamiltonian associated with the ...boundary region. If two such minimal surfaces intersect at a point in the bulk then CFT operators which commute with both extended modular Hamiltonians must be localized at the intersection point. We use this to construct local bulk operators purely from CFT considerations, without knowing the bulk metric, using intersecting modular Hamiltonians. For conformal field theories at zero and finite temperature the appropriate modular Hamiltonians are known explicitly and we recover known expressions for local bulk observables.
A
bstract
In the framework of bulk reconstruction, we elucidate the relationship between the action of CFT modular Hamiltonians on bulk operators, the possible equation of motion for the bulk ...operators, and the charge distribution at infinity corresponding to such bulk fields. In particular for scalar fields interacting with gravity or with gauge fields, we show how CFT considerations of the action of the modular Hamiltonian constrain the possible bulk equation of motion to be consistent with bulk gauge invariance and diffeomorphism invariance. In fact we show that requiring that the action of the modular Hamiltonian on a dressed bulk scalar operator be compatible with some unknown simple equation of motion, fixes, under reasonable assumptions, both the equation of motion and the action of the modular Hamiltonian once the first order
1
N
terms are known.
Dressing bulk spinor fields in AdS3 Lifschytz, Gilad; Patra, Milan
The journal of high energy physics,
04/2023, Letnik:
2023, Številka:
3
Journal Article
Recenzirano
Odprti dostop
A
bstract
We continue the program of bulk reconstruction for fermionic fields. We reconstruct, from the CFT, the Dirac fermion field in AdS
3
coupled to a Chern-Simons gauge field. We show that the ...three conditions; solving the equation of motion, satisfying expected transformation under modular flow and a simple charge distribution at infinity are all compatible and all produce the same bulk operator. We also compute the bulk-boundary tree level three point function from the CFT construction.
A
bstract
We develop the representation of free spinor fields in the bulk of Lorentzian anti-de Sitter space in terms of smeared operators in the dual conformal field theory. To do this we expand the ...bulk field in a complete set of normalizable modes, work out the extrapolate dictionary for spinor fields, and show that the bulk field can be reconstructed from its near-boundary behavior. In some cases chirality and reality conditions can be imposed in the bulk. We study the action of the CFT modular Hamiltonian on bulk fermions to show that they transform with the expected spinor Lie derivative, and we calculate bulk-boundary two-point functions starting from CFT correlators.
A
bstract
We study the action of the CFT total modular Hamiltonian on the CFT representation of bulk fields with spin. In the vacuum of the CFT the total modular Hamiltonian acts as a bulk Lie ...derivative, reducing on the RT surface to a boost perpendicular to the RT surface. This enables us to reconstruct bulk fields with spin from the CFT. On fields with gauge redundancies the total modular Hamiltonian acts as a bulk Lie derivative together with a compensating bulk gauge (or diffeomorphism) transformation to restore the original gauge. We consider the Lie algebra generated by the total modular Hamiltonians of all spherical CFT subregions and define weakly-maximal Lie subalgebras as proper subalgebras containing a maximal set of total modular Hamiltonians. In a CFT state with a bulk dual, we show that the bulk spacetime parametrizes the space of these weakly-maximal Lie subalgebras. Each such weakly-maximal Lie subalgebra induces Lorentz transformations at a particular point in the bulk manifold. The bulk metric dual to a pure CFT state is invariant at each point under this transformation. This condition fixes the metric up to a conformal factor that can be computed from knowledge of the equation parametrizing extremal surfaces. This gives a holographic notion of the invariance of a pure CFT state under CFT modular flow.
A
bstract
To
O
1
/
N
we derive, purely from CFT data, the bulk equations of motion for interacting scalar fields and for scalars coupled to gauge fields and gravity. We first uplift CFT operators to ...mimic local AdS fields by imposing bulk microcausality. This requires adding an infinite tower of smeared higher-dimension double-trace operators to the CFT definition of a bulk field, with coefficients that we explicitly compute. By summing the contribution of the higher-dimension operators we derive the equations of motion satisfied by these uplifted CFT operators and show that we precisely recover the expected bulk equations of motion. We exhibit the freedom in the CFT construction which corresponds to bulk field redefinitions.
A
bstract
We develop an approach to construct local bulk operators in a CFT to order 1
/N
2
. Since 4-point functions are not fixed by conformal invariance we use the OPE to categorize possible forms ...for a bulk operator. Using previous results on 3-point functions we construct a local bulk operator in each OPE channel. We then impose the condition that the bulk operators constructed in different channels agree, and hence give rise to a well-defined bulk operator. We refer to this condition as the “bulk bootstrap.”
We argue and explicitly show in some examples that the bulk bootstrap leads to some of the same results as the regular conformal bootstrap. In fact the bulk bootstrap provides an easier way to determine some CFT data, since it does not require knowing the form of the conformal blocks. This analysis clarifies previous results on the relation between bulk locality and the bootstrap for theories with a 1
/N
expansion, and it identifies a simple and direct way in which OPE coefficients and anomalous dimensions determine the bulk equations of motion to order 1
/N
2
.
A
bstract
Perturbative bulk reconstruction in AdS/CFT starts by representing a free bulk field
ϕ
(0)
as a smeared operator in the CFT. A series of 1
/N
corrections must be added to
ϕ
(0)
to represent ...an interacting bulk field
ϕ
. These corrections have been determined in the literature from several points of view. Here we develop a new perspective. We show that correlation functions involving
ϕ
(0)
suffer from ambiguities due to analytic continuation. As a result
ϕ
(0)
fails to be a well-defined linear operator in the CFT. This means bulk reconstruction can be understood as a procedure for building up well-defined operators in the CFT which thereby singles out the interacting field
ϕ
. We further propose that the difficulty with defining
ϕ
(0)
as a linear operator can be re-interpreted as a breakdown of associativity. Presumably
ϕ
(0)
can only be corrected to become an associative operator in perturbation theory. This suggests that quantum mechanics in the bulk is only valid in perturbation theory around a semiclassical bulk geometry.
A
bstract
We compute modular Hamiltonians for excited states obtained by perturbing the vacuum with a unitary operator. We use operator methods and work to first order in the strength of the ...perturbation. For the most part we divide space in half and focus on perturbations generated by integrating a local operator
J
over a null plane. Local operators with weight
n
≥ 2 under vacuum modular flow produce an additional endpoint contribution to the modular Hamiltonian. Intuitively this is because operators with weight
n
≥ 2 can move degrees of freedom from a region to its complement. The endpoint contribution is an integral of
J
over a null plane. We show this in detail for stress tensor perturbations in two dimensions, where the result can be verified by a conformal transformation, and for scalar perturbations in a CFT. This lets us conjecture a general form for the endpoint contribution that applies to any field theory divided into half-spaces.
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