Conformal blocks from Wilson lines with loop corrections

TL;DR

This work develops and applies a renormalized open Wilson-line framework in sl(N) Chern-Simons theory to compute Virasoro and W_N conformal blocks at large central charge with 1/c corrections. By introducing a regulator and renormalizing Wilson-line couplings, the authors reproduce known CFT results for N=2 and N=3, including identity and general four-point blocks, as well as heavy-light correlators in conical AdS_3 backgrounds. The approach unifies Coulomb-gas, Virasoro, and W_3 block structures, delivering all-order-in-z control at 1/c for several blocks and providing explicit expressions for higher-spin exchanges. The results illuminate quantum gravity aspects of AdS_3 via higher-spin holography and extend the Wilson-line network method beyond leading order, with potential extensions to supersymmetric cases and more intricate geometries.

Abstract

We compute the conformal blocks of the Virasoro minimal model or its W$_N$ extension with large central charge from Wilson line networks in a Chern-Simons theory including loop corrections. In our previous work, we offered a prescription to regularize divergences from loops attached to Wilson lines. In this paper, we generalize our method with the prescription by dealing with more general operators for $N=3$ and apply it to the identity W$_3$ block. We further compute general light-light blocks and heavy-light correlators for $N=2$ with the Wilson line method and compare the results with known ones obtained using a different prescription. We briefly discuss general W$_3$ blocks.

Figures (6)

• Figure 1: Contribution to the identity W$_3$ block from the product of open Wilson lines at the leading order in $1/c$ and its corrections of self-energy type up to the order $1/c^2$. The straight and wavy lines represent open Wilson lines and propagators of current with spin $s=2$ or $s=3$. (a) Leading order contribution. (b)-(f) Corrections associated with the open Wilson line from 0 to $z$. (g) Correction associated with both the open Wilson lines. (h)-(l) Corrections associated with the open Wilson line from 1 to $\infty$.
• Figure 2: (a) Contribution with the exchange of a current with $s=2$ or $s=3$. (b)-(i) Its $1/c$ corrections from loop diagrams.
• Figure 3: (a),(b) Contributions with the exchange of two currents with $s=2$ or $s=3$.
• Figure 4: Contribution to the two point function from the network of open Wilson lines at the $1/c$ order.
• Figure 5: Contributions to the three point function from the network of open Wilson lines at the $1/c$ order. (a),(b) Corrections of self-energy type. (c) Correction from the exchange of a current with $s=2$.