Conformal covariance for cusps on generic smooth contours

Determine whether a renormalized correlation function for cusped contours formed by generic smooth lines can be defined by taking the ground state of the cutoff-dependent Hamiltonian on each quantization surface, while preserving conformal covariance of the resulting correlator.

Background

For contours made from circular arcs, the authors construct a canonical North–South quantization scheme near each cusp and use it to define renormalized correlation functions that transform covariantly under conformal transformations. For generic smooth lines, however, there is no canonical family of quantization surfaces or associated Hamiltonians; both may depend on the cutoff scale, although they approach the structures determined by the local tangent and curvature near the cusp.

The unresolved issue is whether one can select the ground state of the cutoff-dependent Hamiltonian at every quantization surface so that the resulting renormalized correlator has a well-defined, scheme-independent finite part and retains the primary-like conformal transformation law. The authors express doubt that this is possible because the finite part may depend on the detailed regularization and on the chosen sequence of Hamiltonians, even if the leading divergence is determined solely by the cusp angle.

References

A question that arises given these conditions is whether we may still define a renormalized correlation function by demanding, for example, that the state defined on the quantization surface at scale $\epsilon$ is the ground state of the corresponding Hamiltonian. While we leave a definitive answer to future work, we are doubtful that this can be done in a way that the resulting correlator would still be conformally covariant.

— Cusped Defects: Cusp Operator Expansion, Conformal Properties and Bootstrap Applications  (2609.04035 - Bianchi et al., 3 Sep 2026) in Section 3, subsection “More General Contours”