Equivalence of quantizations of the dispersionless KdV hierarchy
Xavier Blot
TL;DR
The paper proves that two independent quantizations of the dispersionless KdV hierarchy—the meromorphic differential hierarchies (MD) associated to a trivial CohFT and Wang's Heisenberg vertex-algebra construction—produce identical quantum Hamiltonians $H_d$ for all $d\ge-1$, hence the same quantum hierarchy. The identification hinges on the common quantum bracket and a uniqueness lemma (Lemma 8.1) that determines higher Hamiltonians from the leading one, together with a precise comparison of their classical limits and variational derivatives. In the MD framework, $H_d^{\mathrm{MD}}$ are defined via Hodge-type integrals over meromorphic differential strata, while in Wang's framework $H_d^{\mathrm{Wang}}$ are given by explicit formulas with a substitution $u_j \mapsto (\sqrt{-i\hbar})^{j}u_j$ and a generating function $S_{(k+1)}$; despite their different constructions, the two coincide, aligning with DR-based quantizations in related setups. This unifies intersection-theoretic and vertex-algebraic approaches to quantization of the dispersionless KdV hierarchy and suggests broader connections across quantization schemes in the CohFT landscape.
Abstract
Wang recently constructed a quantization of the dispersionless KdV hierarchy using the Heisenberg vertex algebra. Independently, in joint work with Rossi, we obtained a quantization of the dispersionless KdV hierarchy as the trivial Cohomological Field Theory case of the meromorphic differential hierarchies. In this note, we prove that the two constructions coincide.