On the Virasoro Crossing Kernels at Rational Central Charge
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 b2∈Q× – corresponding in 2d CFT to rational central charge values in the domain (−∞,1]∪[25,∞) – we establish two main results. First, in the domain c∈Q[25,∞) 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∈Q(−∞,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≥25, they possess square-root branch point singularities. As a corollary, we demonstrate that timelike Liouville theory at c∈Q(−∞,1] is crossing symmetric and modular covariant. Surprisingly, the crossing kernels at any b2∈Q× 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.
Related
- Classification of Strongly Rational Vertex Operator Algebras with Central and Effective Central Charge One
- Wall-crossing formula and genus-one Virasoro conjecture for Fano complete intersections
- On wall-crossing invariant combinations of Welschinger numbers for rational surfaces with a conic bundle
- K-Moduli Wall Crossing and Automorphic Forms for the Moduli Space of Rational Elliptic Surfaces
- Least Variability in a Polynomial-Square Class of Rational Kernels
BibTeX
@article{roussillon2026virasoro,
title = {{On the Virasoro Crossing Kernels at Rational Central Charge}},
author = {Julien Roussillon and Ioannis Tsiares},
journal = {SciPost Physics},
year = {2026},
month = jun,
doi = {10.21468/scipostphys.20.6.167},
eprint = {2512.03172},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05419554v1},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - On the Virasoro Crossing Kernels at Rational Central Charge
AU - Julien Roussillon
AU - Ioannis Tsiares
PY - 2026
DA - 2026/06/17
JO - SciPost Physics
DO - 10.21468/scipostphys.20.6.167
UR - https://hal.science/hal-05419554v1
AB - 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 b2∈Q× – corresponding in 2d CFT to rational central charge values in the domain (−∞,1]∪[25,∞) – we establish two main results. First, in the domain c∈Q[25,∞) 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∈Q(−∞,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≥25, they possess square-root branch point singularities. As a corollary, we demonstrate that timelike Liouville theory at c∈Q(−∞,1] is crossing symmetric and modular covariant. Surprisingly, the crossing kernels at any b2∈Q× 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.
SN - 2542-4653
ER -