Abstract
Entanglement entropy is an important quantity in field theory, but its definition poses some challenges. The naive definition involves an extension of quantum field theory in which one assigns Hilbert spaces to spatial sub-regions. For two-dimensional topological quantum field theory we show that the appropriate extension is the open-closed topological quantum field theory of Moore and Segal. With the addition of one additional axiom characterizing the “entanglement brane” we show how entanglement calculations can be cast in this framework. We use this formalism to calculate modular Hamiltonians, entanglement entropy and negativity in two-dimensional Yang-Mills theory and relate these to singularities in the modular ow. As a byproduct we find that the negativity distinguishes between the “log dim R” edge term and the “Shannon” edge term. We comment on the possible application to understanding the Bekenstein-Hawking entropy in two-dimensional gravity.
Highlights
One first issue is that in the algebraic approach to quantum field theory, regions of space are associated with von Neumann algebras, rather than Hilbert space factors
For two-dimensional topological quantum field theory we show that the appropriate extension is the open-closed topological quantum field theory of Moore and Segal
The physical Hilbert space consists of wavefunctionals satisfying local constraints
Summary
The diagrammatic structure of an open-closed TQFT originated from the factorization of string worldsheet amplitudes that describe interactions of open and closed strings [20]. We will review the subject as formulated by [18, 21, 22]. A nice informal treatment is given by [23]
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