Four-Brane and Six-Brane Interactions in M(atrix) Theory

We discuss the proposed description of configurations with four-branes and six-branes in m(atrix) theory. Computing the velocity dependent potential between these configurations and gravitons and membranes, we show that they agree with the short distance string results computed in type IIa string theory. Due to the ``closeness'' of these configuration to a supersymmetric configuration the m(atrix) theory reproduces the correct long distance behavior.Comment: 14 pages, late

Emergence of spacetime from the algebra of total modular Hamiltonians

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.Comment: 27 pages. v2: minor changes. v3: coordinates clarified, version to appear in JHE

Local bulk physics from intersecting modular Hamiltonians

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.Comment: 28 pages, LaTeX. v2: additional discussion of extended modular Hamiltonian. v3: version to appear in JHE

A Note on the Transverse Five-Brane in M(atrix) Theory

We describe a way to compute scattering amplitudes in M(atrix) quantum mechanics, that involve the transverse five-brane. We then compute certain scattering processes and show that they have the expected SO(5) invariance, give the correct transverse-five-brane mass, and agree with the supergravity result.Comment: 13 pages, Latex, final version to appear in PL

Locality, bulk equations of motion and the conformal bootstrap

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 $1/N^2$.Comment: 37 pages, LaTeX. v2: added references and comments on OPE convergence, version to appear in JHE