Kinematic space and wormholes

101

We consider the computation of volumes contained in a spatial slice of AdS 3 in terms of observables in a dual CFT. Our main tool is kinematic space, defined either from the bulk perspective as the space of oriented bulk geodesics, or from the CFT perspective as the space of entangling intervals. We give an explicit formula for the volume of a general region in a spatial slice of AdS 3 as an integral over kinematic space. For the region lying below a geodesic, we show how to write this volume purely in terms of entangling entropies in the dual CFT. This expression is perhaps most interesting in light of the complexity = volume proposal, which posits that complexity of holographic quantum states is computed by bulk volumes. An extension of this idea proposes that the holographic subregion complexity of an interval, defined as the volume under its Ryu-Takayanagi surface, is a measure of the complexity of the corresponding reduced density matrix. If this is true, our results give an explicit relationship between entanglement and subregion complexity in CFT, at least in the vacuum. We further extend many of our results to conical defect and BTZ black hole geometries.

mentioning

confidence: 99%

Holographic subregion complexity from kinematic space

Abt

Erdmenger

Gerbershagen

et al. 2019

101

“…In this section we focus on static conical defect spacetimes and consider the kinematic space for a constant time slice. We show that the differential entropy approach [8], and the quotient approach [22] produce different fundamental regions of the same kinematic space, but are entirely equivalent.…”

Section: Kinematic Space From the Bulkmentioning

confidence: 94%

“…The first is a simple application of the differential entropy definition applied to geodesics of all lengths. The second follows [22] in noting that the conical defects can be obtained as a quotient of pure AdS 3 . Under this quotient classes of geodesics are identified, producing a quotient on kinematic space, and leading to a result equivalent to the first approach.…”

Section: Contentsmentioning

confidence: 99%

See 1 more Smart Citation

Kinematic space for conical defects

Cresswell

Peet

2017

Kinematic space can be used as an intermediate step in the AdS/CFT dictionary and lends itself naturally to the description of diffeomorphism invariant quantities. From the bulk it has been defined as the space of boundary anchored geodesics, and from the boundary as the space of pairs of CFT points. When the bulk is not globally AdS$_3$ the appearance of non-minimal geodesics leads to ambiguities in these definitions. In this work conical defect spacetimes are considered as an example where non-minimal geodesics are common. From the bulk it is found that the conical defect kinematic space can be obtained from the AdS$_3$ kinematic space by the same quotient under which one obtains the defect from AdS$_3$. The resulting kinematic space is one of many equivalent fundamental regions. From the boundary the conical defect kinematic space can be determined by breaking up OPE blocks into contributions from individual bulk geodesics. A duality is established between partial OPE blocks and bulk fields integrated over individual geodesics, minimal or non-minimal.Comment: 29 pages, 9 figures. As published in JHE

“…While these works on kinematic space were thorough, they mainly focused on pure AdS. Followup papers [18][19][20][21][22][23][24] have worked towards extending kinematic space and the OPE block duality to more general AdS spacetimes. We will continue this line of inquiry for AdS 3 , where all vacuum solutions to the Einstein equation with negative cosmological constant are locally AdS 3 and can be obtained as quotients.…”

Section: Introductionmentioning

confidence: 99%

Holographic relations for OPE blocks in excited states

Cresswell

Jardine

Peet

2019

We study the holographic duality between boundary OPE blocks and geodesic integrated bulk fields in quotients of AdS 3 dual to excited CFT states. The quotient geometries exhibit non-minimal geodesics between pairs of spacelike separated boundary points which modify the OPE block duality. We decompose OPE blocks into quotient invariant operators and propose a duality with bulk fields integrated over individual geodesics, minimal or non-minimal. We provide evidence for this relationship by studying the monodromy of asymptotic maps that implement the quotients.