Virasoro Crossing Kernels at Rational Central Charge Demonstrate Novel Solutions with Square-Root Singularities

The fundamental properties of two-dimensional conformal field theories and their connections to three-dimensional topological field theories remain a central challenge in theoretical physics, and recent work by Julien Roussillon of Aalto University and Ioannis Tsiares of Université Paris-Saclay, along with colleagues, significantly advances our understanding of these complex systems. The team establishes novel analytic results for the Virasoro crossing kernels, crucial elements in describing interactions within these theories, for all rational central charge values. Their findings demonstrate a broader range of possible solutions to the equations governing these kernels than previously recognised, and importantly, they derive the physical modular and fusion kernels for a wide range of parameters. This achievement not only confirms the crossing symmetry and modular covariance of timelike Liouville theory, but also reveals a surprising connection between these kernels and semiclassical, one-loop calculations, offering new insights for the conformal bootstrap and the study of three-dimensional quantum gravity.

The research establishes these results for all rational values of a parameter, corresponding to rational central charge values within a defined domain. The team demonstrated that, within this domain, both modular and fusion kernels can be expressed as a linear combination of two admissible crossing kernels. These kernels possess square-root branch point singularities in the Liouville momenta and are not reflection-symmetric.

Virasoro Algebra and Liouville Theory Exploration

Conformal field theory, Liouville theory, and knot theory are interconnected areas of mathematical physics actively explored by scientists. Conformal field theory serves as a foundational framework, while Liouville theory provides a simplified model for understanding two-dimensional quantum gravity. Researchers investigate the Virasoro algebra, the symmetry governing two-dimensional conformal field theories, and employ the modular bootstrap, a powerful technique for solving these theories. They calculate correlation functions to determine the probabilities of different field configurations and study the central charge, a key parameter characterizing the theory.

Knot theory, the study of mathematical knots, connects to conformal field theory through quantum invariants, mathematical properties of knots. Scientists explore quantum modular forms, generalizations of modular forms with quantum properties, to bridge these areas. This work also touches on three-manifold topology, the study of the shapes of three-dimensional spaces, and topological quantum field theories, mathematical frameworks for studying these spaces.

Rational Kernels and Non-Symmetric Singularities

This finding expands the known space of solutions to the fundamental shift relations governing these kernels. Furthermore, the team derived, for the first time, physical modular and fusion kernels for generic values of the Liouville momenta, revealing the presence of square-root branch point singularities. These kernels, expressed as a linear combination of admissible kernels, distinguish themselves from solutions for higher central charge values. As a direct consequence, the researchers demonstrated that timelike Liouville theory, at these specific central charge values, is both crossing symmetric and modular covariant.

Surprisingly, the crossing kernels behave as if they were semiclassical and one-loop exact, a result with implications for the 2D conformal bootstrap and 3D topological field theories describing gravity with negative cosmological constant. The measurements reveal that at rational central charge, the complex functions defining these kernels simplify to ratios and products of the Barnes’ G function, which captures the analytic continuation of the superfactorial. This simplification allows for a more complete understanding of the kernels’ behavior and properties.

Square-Root Singularities in Conformal Field Theory Kernels

This research establishes novel analytic results for kernels central to two-dimensional conformal field theories, three-dimensional topological field theories, and representation theory. Scientists have demonstrated that, for a range of parameters, the modular and fusion kernels can be expressed as combinations of previously known admissible kernels. Importantly, these newly identified kernels possess square-root branch point singularities, expanding the known solution space for these complex mathematical problems. The team further derived, for the first time, physical modular and fusion kernels applicable to a wider range of values, again revealing the presence of these square-root singularities.

This work confirms that certain theoretical models, such as timelike Liouville theory, exhibit both crossing symmetry and modular covariance, properties crucial for consistency and predictive power. The authors acknowledge that a detailed understanding of the branch points within these kernels remains a challenge, and they outline future research directions focused on mapping their precise locations. Further investigation into these branch points and reflection properties may reveal new integration contours and deepen the understanding of these fundamental theoretical structures.