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

Area of Science:

• String Theory
• Quantum Gravity
• AdS/CFT Correspondence

Background:

• Defining entanglement entropy in string theory is crucial for understanding the bulk microstate interpretation of the Ryu-Takayanagi formula.
• Entanglement and Hilbert space factorization are poorly understood in string theory, hindering progress in AdS/CFT.
• The topological A-model and Gopakumar-Vafa duality offer a tractable framework for studying these concepts.

Purpose of the Study:

• To study the entanglement entropy of closed strings in the topological A-model as a toy model for AdS/CFT.
• To provide a self-consistent factorization of the closed string Hilbert space.
• To develop a string theory analogue of the Hartle-Hawking state and calculate its entanglement entropy.

Main Methods:

• Utilizing extended Topological Quantum Field Theory (TQFT) methods for Hilbert space factorization.
• Incorporating the factorization map into a Frobenius algebra for Calabi-Yau manifold fusion and splitting.
• Defining a string theory Hartle-Hawking state and calculating entanglement entropy via the reduced density matrix.

Main Results:

• A self-consistent factorization of the closed string Hilbert space on the resolved conifold was achieved.
• String edge modes transforming under a q-deformed surface symmetry group were identified.
• The calculated entanglement entropy matches geometrical replica trick and dual Chern-Simons theory calculations.

Conclusions:

• The study provides a realization of the Susskind-Uglum proposal, equating closed string entanglement entropy with open string thermal entropy.
• The findings offer insights into BPS microstate counting for entanglement entropy.
• Nonlocal aspects of the factorization map show analogies with phenomena in JT gravity.