[Paper Review] Boundary States as Holographic Duals of Trivial Spacetimes

Masamichi Miyaji, Shinsei Ryu+2 more|arXiv (Cornell University)|Dec 19, 2014

Black Holes and Theoretical Physics34 references5 citations

TL;DR

This paper proposes that conformally invariant boundary states in conformal field theories (CFTs) are holographic duals of trivial spacetimes—specifically, spacetimes with zero spacetime volume. By showing these boundary states exhibit negligible real-space entanglement, the authors establish a link between vanishing entanglement and trivial geometry, and further demonstrate that such states serve as ideal infrared (IR) fixed points for constructing continuous multiscale entanglement renormalization ansatz (cMERA) states in general CFTs.

ABSTRACT

We study real-space quantum entanglement included in conformally invariant boundary states in conformal field theories (CFTs). First, we argue that boundary states essentially have no real-space entanglement by computing the entanglement entropy when we bipartite the system into two spatial regions. From the viewpoint of holography, this shows that boundary states are dual to trivial spacetimes of zero spactime volume. Next, we point out that a continuous multiscale entanglement renormalization ansatz (cMERA) for any CFTs can be formulated by employing a boundary state as its infrared unentangled state with an appropriate regularization. Exploiting this idea, we propose an approximation scheme of cMERA construction for general CFTs.

Motivation & Objective

• To identify the holographic dual of a trivial spacetime with zero spacetime volume.
• To show that conformally invariant boundary states in CFTs possess negligible real-space entanglement, consistent with a trivial geometry.
• To establish a framework for constructing cMERA states in generic CFTs using boundary states as the IR unentangled reference state.
• To generalize the cMERA construction beyond known examples by leveraging the boundary state's entanglement properties.

Proposed method

• Computing the real-space entanglement entropy of boundary states in CFTs to quantify their entanglement structure.
• Using the fact that boundary states are built from Ishibashi states, which are maximally entangled across holomorphic and antiholomorphic sectors, to analyze their overall entanglement.
• Formulating a continuous multiscale entanglement renormalization ansatz (cMERA) for general CFTs by initializing the IR state as a boundary state with appropriate regularization.
• Employing scale transformations and mode decoupling in momentum space to define the cMERA evolution across energy scales.
• Analyzing the behavior of the cMERA state under UV and IR limits, showing that the boundary state remains invariant under scale transformations in the IR.
• Extending the construction to massive scalar fields and compactified spatial dimensions (e.g., on a cylinder), confirming consistency across different field theory setups.

Experimental results

Research questions

• RQ1 Can conformally invariant boundary states in CFTs be interpreted as holographic duals of trivial spacetimes with zero spacetime volume?
• RQ2 To what extent do boundary states exhibit real-space entanglement, and how does this relate to the geometry of their dual spacetime?
• RQ3 Can boundary states serve as a universal infrared reference state for constructing cMERA in arbitrary CFTs?
• RQ4 How does the cMERA construction based on boundary states behave under UV and IR limits in both massless and massive scalar field theories?
• RQ5 What is the role of scale invariance and regularization in ensuring the cMERA construction remains consistent across different field theory realizations?

Key findings

• Boundary states in CFTs exhibit negligible real-space entanglement, as their entanglement entropy vanishes when the system is bipartitioned spatially.
• This vanishing entanglement implies that boundary states are holographically dual to trivial spacetimes with zero spacetime volume.
• The boundary state serves as a natural infrared (IR) fixed point for cMERA construction in general CFTs, due to its lack of real-space entanglement.
• A general cMERA construction for any CFT can be achieved by initializing the IR state as a boundary state and evolving it via scale transformations with proper regularization.
• In the massive scalar case, the cMERA state converges to the same IR vacuum state regardless of scale, confirming stability of the construction.
• The cMERA framework remains consistent under compactification on a cylinder, with the radius rescaling appropriately under the scale evolution.