On the Virasoro Crossing Kernels at Rational Central Charge
Julien Roussillon, Ioannis Tsiares
TL;DR
The work advances the analytic understanding of Virasoro crossing kernels by constructing two independent kernel families at rational central charges: for c≥25, two non-meromorphic, square-root–branch kernels split from the known meromorphic kernels; for c≤1, physical kernels are derived via Virasoro-Wick Rotation as averages of rotated components, preserving crossing symmetry and modular covariance in timelike Liouville. Central to the construction are the Garoufalidis-Kashaev state-integral technique and the emergence of quantum modular/fusion polynomials whose discriminants encode geometric data reminiscent of hyperbolic tetrahedra, yielding a rich, non-meromorphic analytic structure. The results imply semiclassical, one-loop–exact behavior of these kernels and connect to broader frameworks including 2d CFT bootstrap, 3d gravity TQFT, and quantum group representations, suggesting a deep, possibly modular-form–like structure in b^2. The paper also demonstrates explicit Liouville crossing and modular covariance at rational c, strengthening the bridge between analytic bootstrap and topological quantum field theory, with future directions exploring the full implications of the quantum modular form conjecture.
Abstract
We report novel analytic results for the Virasoro modular and fusion kernels relevant to 2d conformal field theories (CFTs), 3d topological field theories (TQFTs), and the representation theory of certain quantum groups. For all rational values of the parameter $b^2\in\mathbb{Q}^{\times}$ -- corresponding in 2d CFT to all rational central charge values in the domain $(-\infty,1]\cup[25,\infty)$ -- we establish two main results. First, in the domain $c\in\mathbb{Q}_{[25,\infty)}$ we show that the modular and fusion kernels derived by Teschner and Teschner-Vartanov respectively can be expressed as a linear combination of two functions, which (i) are themselves admissible crossing kernels, (ii) have square-root branch point singularities in the Liouville momenta, (iii) are not reflection-symmetric in the Liouville momenta. These features illustrate that the space of solutions to the basic shift relations determining these kernels is broader than previously assumed. Second, in the domain $c\in\mathbb{Q}_{(-\infty,1]}$ we derive for the first time the physical modular and fusion kernels for generic values of the Liouville momenta. These can again be written as a linear combination of two other admissible kernels but overall, and unlike the Teschner and Teschner-Vartanov solutions for $c\geq 25$, they possess square-root branch point singularities. As a corollary, we demonstrate that timelike Liouville theory at $c\in\mathbb{Q}_{(-\infty,1]}$ is crossing symmetric and modular covariant. Surprisingly, the crossing kernels at any $b^2\in\mathbb{Q}^{\times}$ behave as if they were semiclassical and one-loop exact, and we discuss the interpretation of this fact in the context of the 2d conformal bootstrap and the 3d TQFT that captures pure 3d gravity with negative cosmological constant.