Topological string entanglement

Abstract

We investigate how topological entanglement of Chern-Simons theory is captured in a string theoretic realization. Our explorations are motivated by a desire to understand how quantum entanglement of low energy open string degrees of freedom is encoded in string theory (beyond the oft discussed classical gravity limit). Concretely, we realize the Chern-Simons theory as the worldvolume dynamics of topological D-branes in the topological A-model string theory on a Calabi-Yau target. Via the open/closed topological string duality one can map this theory onto a pure closed topological A-model string on a different target space, one which is related to the original Calabi-Yau geometry by a geometric/conifold transition. We demonstrate how to uplift the replica construction of Chern-Simons theory directly onto the closed string and show that it provides a meaningful definition of reduced density matrices in topological string theory. Furthermore, we argue that the replica construction commutes with the geometric transition, thereby providing an explicit closed string dual for computing reduced states, and Rényi and von Neumann entropies thereof. While most of our analysis is carried out for Chern-Simons on S 3 , the emergent picture is rather general. Specifically, we argue that quantum entanglement on the open string side is mapped onto quantum entanglement on the closed string side and briefly comment on the implications of our result for physical holographic theories where entanglement has been argued to be crucial ingredient for the emergence of classical geometry. A The Topology of the Conifold 50 B Normalized Chern-Simons density matrices 53 C Chern-Simons and topological string observables 57 C.1 Chern-Simons partition sums 57 C.2 Wilson loop generating functions 58 D Brief Review of Topological String Theory 59 -1 -Combining the equations (4.3) and (4.4) and accounting for our normalization choice we learn

Entanglement entropy and edge modes in topological string theory. Part I. Generalized entropy for closed strings

“…Finally, we comment on how our work differs from the recent work [43] which also studied entanglement in the A-model string theory. The essential difference is twofold: first our choice of state cuts through the base manifold S 2 where the closed string worldsheets wrap, whereas the state defined in [43] does not. As a result, our entanglement cut will directly probe the string edge modes that were not revealed by their computation.…”

Entanglement entropy and the large N expansion of two-dimensional Yang-Mills theory

“…The constants a and b can now be determined in terms of α = λA/χ by applying two conditions. The first condition is that there is no term linear in z in the resolvent, 17) and the second is the normalization condition,…”

“…In string theory the sphere diagram contributes to the entropy at order g −2 s ∼ 1/G, and its contribution can be understood as counting open string states with endpoints anchored to the entangling surface. To calculate entanglement entropy directly within string theory remains fraught with challenging conceptual and technical issues [12][13][14][15], though in some cases it is feasible [16,17].…”

Entanglement on multiple S2 boundaries in Chern-Simons theory

“…Recently, the entanglement structure in Chern-Simons theory is discussed within topological string context [31] exploiting Gopakumar-Vafa duality. The implication of our results on the dual closed topological strings is another plausible direction to pursue.…”