Exact Virasoro Blocks from Wilson Lines and Background-Independent Operators

TL;DR

This work provides an exact, bulk-reconstruction-friendly framework in which sl$(2)$ Chern-Simons Wilson lines reproduce Virasoro blocks and OPE blocks in CFT$_2$, enabling background-independent descriptions of AdS$_3$ physics. By formulating Wilson lines as boundary-anchored objects in an infinite-dimensional sl$(2)$ representation, the authors construct Virasoro OPE blocks from Wilson line networks and derive a path-integral representation that automatically yields uniformizing coordinates in arbitrary backgrounds. They perform concrete checks in the large-$c$ and $1/c$ regimes: the vacuum block is computed up to order $1/c^2$, and general Virasoro blocks are obtained at order $1/c$, with explicit kernels $f_a$, $f_b$ and partial $f_c$ that agree with known results and Zamolodchikov recursion. The approach offers a background-independent, non-perturbative handle on bulk gravity in AdS$_3$, with potential implications for black hole information, bulk reconstruction, and the non-perturbative structure of Virasoro blocks, while also highlighting regulatory and computational challenges for higher-order corrections.

Abstract

Aspects of black hole thermodynamics and information loss can be derived as a consequence of Virasoro symmetry. To bolster the connection between Virasoro conformal blocks and AdS$_3$ quantum gravity, we study sl$(2)$ Chern-Simons Wilson line networks and revisit the idea that they compute a variety of CFT$_2$ observables, including Virasoro OPE blocks, exactly. We verify this in the semiclassical large central charge limit and to low orders in a perturbative $1/c$ expansion. Wilson lines connecting the boundary to points in the bulk play a natural role in bulk reconstruction. Because quantum gravity in AdS$_3$ is rigidly fixed by Virasoro symmetry, we argue that sl$(2)$ Wilson lines provide building blocks for background independent bulk reconstruction. In particular, we show explicitly that they automatically compute the uniformizing coordinates appropriate to any background state.

Figures (6)

• Figure 1: Left: A sketch of Wilson lines computing a Virasoro OPE block. Middle: By putting together two such OPE blocks, one obtains a Wilson line network computing a Virasoro conformal block. The blue line indicates the non-trivial vacuum expectation value of the product of the OPE blocks. Right: The Virasoro conformal blocks can also be computed by putting the OPE blocks in the appropriate background bra and ket states.
• Figure 2: This figure shows a single Wilson line $W_h$ ending at two points on the boundary, and a network of three Wilson lines emanating from three boundary points and meeting at a bulk vertex. The vertex must be invariant under the bulk sl$(2)$ gauge group, and so as a function of the internal sl$(2)$ variables $x_i$ it must take the functional form of a conformally invariant 3-point correlator.
• Figure 3: This figure shows some Wilson lines anchored near the boundary at $y=0$ at various points $z_i$ and pointing into the bulk. The Wilson lines are labelled by an $x_i$ variable transforming in the infinite dimensional representation of sl$(2)$ with primary dimension $h_i$. From the near-boundary behavior we deduce that this obeys the Virasoro Ward identity for a correlator of primary operators with dimensions $h_i$ located at $z_i$.
• Figure 4: The Hilbert space associated with an empty cylinder consists of the vacuum and its Virasoro descendants. When we include a Wilson line $W_h$, the space of states includes all Virasoro descendants of a dimension $h$ primary state.
• Figure 5: This figure indicates the contributions to the Virasoro vacuum block at order $\frac{1}{c}$ and $\frac{1}{c^2}$. The Wilson lines appear in black, while the wavy blue lines indicate contractions of stress energy tensors $\langle TT \rangle = \frac{c}{2 z_{ij}^4}$. "Connected" stress tensor correlators begin to contribute at order $\frac{1}{c^3}$.