Abstract
In this work, we consider the question of local Hilbert space factorization in 2D conformal field theory. Generalizing previous work on entanglement and open-closed TQFT, we interpret the factorization of CFT states in terms of path integral processes that split and join the Hilbert spaces of circles and intervals. More abstractly, these processes are cobordisms of an extended CFT which are defined purely in terms of the OPE data. In addition to the usual sewing axioms, we impose an entanglement boundary condition that is satisfied by the vacuum Ishibashi state. This choice of entanglement boundary state leads to reduced density matrices that sum over super-selection sectors, which we identify as the CFT edge modes. Finally, we relate our factorization map to the co-product formula for the CFT symmetry algebra, which we show is equivalent to a Boguliubov transformation in the case of a free boson.
Highlights
In its usual path integral formulation, a continuum QFT does not come equipped with a notion of local Hilbert space factorization
This choice of entanglement boundary state leads to reduced density matrices that sum over super-selection sectors, which we identify as the conformal field theory (CFT) edge modes
In this work we proposed an extension of a 2D CFT which gives a factorization of the Hilbert space in terms of operator product expansion (OPE) data
Summary
In its usual path integral formulation, a continuum QFT does not come equipped with a notion of local Hilbert space factorization. In this work we attempt a generalization of the extended TQFT approach to Hilbert space factorization to 2D conformal field theories In this case, the basic cobordisms are specified by the operator product expansion (OPE) coefficients that define a boundary conformal field theory (CFT) (Fig. 3). III we formulate an E brane constraint (3.1) for CFT’s and propose a solution in the form of the vacuum Ishibashi state We explain how this choice is dictated by conformal symmetry, and corresponds to summing over local boundary conditions (3.3) in contrast to the factorization maps previously proposed1 [8]. The vacuum state of the free compact boson to give an explicit example of the E brane, the associated CFT edge modes, and the edge entanglement entropy In this case the sum over boundary conditions restores the Uð1Þ symmetry of the theory and cancels an anomalous term in the entropy that would appear for a fixed boundary condition. Our ideas to tensor network renormalization and “BC bits” formulation of holographic CFT’s [10]
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