3,424 research outputs found
Bulk locality from modular flow
We study the reconstruction of bulk operators in the entanglement wedge in
terms of low energy operators localized in the respective boundary region. To
leading order in , the dual boundary operators are constructed from the
modular flow of single trace operators in the boundary subregion. The
appearance of modular evolved boundary operators can be understood due to the
equality between bulk and boundary modular flows and explicit formulas for bulk
operators can be found with a complete understanding of the action of bulk
modular flow, a difficult but in principle solvable task. We also obtain an
expression when the bulk operator is located on the Ryu-Takayanagi surface
which only depends on the bulk to boundary correlator and does not require the
explicit use of bulk modular flow. This expression generalizes the geodesic
operator/OPE block dictionary to general states and boundary regions.Comment: 36 pages, 2 figure
Probing beyond ETH at large
We study probe corrections to the Eigenstate Thermalization Hypothesis (ETH)
in the context of 2D CFTs with large central charge and a sparse spectrum of
low dimension operators. In particular, we focus on observables in the form of
non-local composite operators
with . As a
light probe, is constrained by ETH and satisfies
for a high energy energy eigenstate
. In the CFTs of interests, is related to a Heavy-Heavy-Light-Light (HL)
correlator, and can be approximated by the vacuum Virasoro block, which we
focus on computing. A sharp consequence of ETH for is
the so called "forbidden singularities", arising from the emergent thermal
periodicity in imaginary time. Using the monodromy method, we show that finite
probe corrections of the form drastically alter both sides
of the ETH equality, replacing each thermal singularity with a pair of
branch-cuts. Via the branch-cuts, the vacuum blocks are connected to infinitely
many additional "saddles". We discuss and verify how such violent modification
in analytic structure leads to a natural guess for the blocks at finite : a
series of zeros that condense into branch cuts as . We also discuss
some interesting evidences connecting these to the Stoke's phenomena, which are
non-perturbative effects. As a related aspect of these probe
modifications, we also compute the Renyi-entropy in high energy
eigenstates on a circle. For subsystems much larger than the thermal length, we
obtain a WKB solution to the monodromy problem, and deduce from this the
entanglement spectrum.Comment: 35 pages, 40 figures, additional results and comments adde
A modular toolkit for bulk reconstruction
We introduce new tools for studying modular flow in AdS/CFT. These tools
allow us to efficiently extract bulk information related to causality and
locality. For example, we discuss the relation between analyticity in modular
time and entanglement wedge nesting which can then be used to extract the
location of the Ryu-Takayanagi (RT) surface directly from the boundary theory.
Probing the RT surface close to the boundary our results reduce to the recent
proof of the Quantum Null Energy Condition. We focus on heavy probe operators
whose correlation functions are determined by spacelike geodesics. These
geodesics interplay with the RT surface via a set of rules that we conjecture
and give evidence for using the replica trick.Comment: 31 pages, 10 figures, v2: typos fixed and references adde
Research note: Measuring cultural values - The case of residents' attitudes to the Saltaire Festival
This research note discusses the findings and methodological issues arising from a cultural values survey of residents, particularly in relation to civic pride. In addition to offering insights to the particular case - the Saltaire Festival in the north of England - it provides an approach that might be used elsewhere
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