Open Verlinde line operators

TL;DR

This work develops a 3d TFT-inspired framework for Verlinde line operators acting on Virasoro conformal blocks and extends the construction to open lines anchored at punctures. It provides explicit computational rules for generalized conformal blocks, closed and open Verlinde operators, and detailed examples on the four-punctured sphere and one-punctured torus. The paper then links these operators to quantum Teichmüller theory, refined framed BPS degeneracies, and quantum groups, offering a unified approach to connect CFT data with geometric quantization and algebraic structures. The results pave new routes from BPZ blocks to quantum Teichmüller theory, q-deformed traffic rules, and modular double representations, with potential impact on both mathematical physics and gauge-theory applications.

Abstract

We reformulate the action of Verlinde line operators on conformal blocks in a 3d TFT language and extend it to line operators labelled by open paths joining punctures on the Riemann surface. We discuss the possible applications of open Verlinde line operators to quantum Teichmüller theory, supersymmetric gauge theory and quantum groups

Figures (24)

• Figure 1: Left: the pictorial representation of a four-point conformal block, as a trivalent graph inside a ball. We omit for clarity the Liouville momentum labels on the edges. Right: The same conformal block, with the 3d ambient space omitted, and Liouville momenta labels on the edges.
• Figure 2: The pictorial representation of a fusion operation on the four-point conformal block. The fusion "matrix" $F$ acts on the Liouville momentum label on the internal edge. It will be an integral kernel if the external edges carry a generic Liouville momentum, and a finite matrix if at least one external legs is degenerate
• Figure 3: Left: a graph which refines the basic four-point graph $\Gamma_0$ in figure \ref{['fig:4pt']} by an extra degenerate loop linking the middle edge of $\Gamma_0$. We denote the degenerate loop by a thinner line. Right: a graph which refines the basic four-point graph $\Gamma_0$ by an extra rung. We omit the ambient three-ball and indicate the Liouville momenta. The $(2,1)$ label indicates the $\i b + \i b^{-1}/2$ degenerate momentum. The shifts $s_2$, $s_3$ take values $\pm1$
• Figure 4: The pictorial representation of an "A-cycle" Verlinde line operator acting on a four-point conformal block. We omit for clarity the Liouville momentum labels on the edges
• Figure 5: A graphical derivation of the sequence of fusion and braiding operations which realizes an A-cycle Verlinde line operator.