Line Defects in Liouville Conformal Field Theory: Localized Cosmological Constants and Decohered Hyperbolic Geometries

Abstract

The study of quantum impurities has long been a central and inspiring theme in quantum many-body physics. Localized impurities are modeled by line defects in quantum field theory. We describe a line defect in Liouville CFT realized as a ``localized cosmological constant'': a non-topological line insertion into the Liouville path integral that is tractable at both weak and strong defect coupling. At weak coupling, we analyze the defect perturbatively and characterize it through its correlations with local operators, energy and information transport, the Casimir energies associated with fusion, and corrections to the open string channel spectrum. We also study the effect of a cuspidal deformation of the defect locus on these observables and describe novel monotonicity properties as the cusp angle is varied. These results derived using perturbation theory are more generally applicable to pinning defects constructed from scalar primary operators in compact $2d$ CFTs. At strong coupling, in a semiclassical limit, the defect admits a geometric interpretation in terms of a discontinuity in the extrinsic curvature of the $1d$ defect locus embedded in $2d$ hyperbolic geometries. The observables characterizing the defect in this regime are computed by gluing hyperbolic surfaces across the defect, and are compared with the corresponding weak coupling results. The correlations across the defect, both at weak and strong coupling, can also be realized by an effective ``decohered FZZT interface'' constructed by diagonal gluing of two copies of the fixed-length FZZT boundary state. These line defects also have interesting interpretations in other models, in terms of end-of-the-world branes in Jackiw-Teitelboim gravity, dust shells in AdS$_3$ gravity, and interfaces with a proliferation of non-abelian Wilson loops in $4d$ $\mathcal{N}=2$ gauge theories.

Figures (21)

• Figure 1: The figure on the left describes the kinematics used to compute the matrix elements of a circular defect between two primary states. The figure on the right shows that in the strongly coupled semiclassical limit, the matrix element can be computed by allowing the local operators and the defect to backreact on the sphere giving a hyperbolic surface that with a kink along the defect. The circular boundaries on the top and bottom are geodesics with proper lengths $2\pi r_H$ and $2\pi r_H'$ determined by the conformal weights of the corresponding operators. Located at the waist in the right figure, $2\pi r_0$ is the proper length (non-geodesic) of the defect determined dynamically in terms of the defect cosmological constant and the two geodesic lengths.
• Figure 2: These figures describe the kinematics of the setup used to compute correlations of local operators across the defect with two canonical choices for the curve $\Sigma$ on which the defect is placed. Figure (a) shows an infinite line defect placed along the real axis, with a pair of local vertex operator insertions with Liouville momenta $\alpha$ and $\alpha'$ on the imaginary axis. Figure (b) shows a circular defect around the equator of the Riemann sphere or equivalently around the unit circle on the plane, with the pair of local operators inserted at the poles.
• Figure 3: This figure illustrates the 'doubling trick' that relates boundary Liouville correlators to Virasoro blocks. The left figure defines a torus two-point Virasoro identity block by cutting the path integral of a general compact CFT and projecting to the vacuum module along the pair of non-contractible closed curves denoted by dotted lines. The figure on the right is the corresponding Liouville correlator on a cylinder with ZZ boundaries on either end which computes this block. The small white disks are local operator insertions. Only the identity and its Virasoro descendents propagate along the dotted line segments joining the ZZ boundaries.
• Figure 4: This is another illustration of the 'doubling trick' analogous to Figure \ref{['fig:torus_vacuum_block']}. The left figure defines a Virasoro conformal block on the twice-punctured torus and the right figure is the Liouville correlator on the cylinder with ZZ and FZZT boundaries inserted on the two ends that computes this block. In the right figure, the primary $s$ with conformal weight $h_s=\frac{c-1}{24}+s^2$ and its Virasoro descendents propagate along the line segment joining the ZZ and FZZT boundaries, while the identity and its descendents propagate along the line segment joining the ZZ boundary.
• Figure 5: This figure describes the kinematics used to compute the 1-point function of the defect on the upper half-plane with a conformal boundary condition along the real axis. A cusp making angles $\theta_{1,2}$ with the real axis is introduced on the defect and the cusp is at a distance $y$ from the conformal boundary.