An Explicit Construction of Commuting Quantum KdV Hamiltonians in the Universal Virasoro Vertex Algebra
Abstract
We give an explicit formula for elements H2k of conformal weight 2k, k≥1, in the universal Virasoro vertex operator algebra V at every central charge, such that H2k(0)H2l∈ TV for all k,l≥1. Modulo translation, the kernel of H4(0) is generated by [H2k] in each positive even weight and is zero in positive odd weights. The formula uses a realization on differential polynomials by possibly infinite-order differential operators in a completion of the Weyl algebra, together with the inverse of the identity plus a nilpotent operator on each weight space. A Koszul-type exactness argument for the classical KdV flows proves conservation; uniqueness and the Jacobi identity give commutativity. The classical limit recovers the normalized Gelfand--Dickey densities. Explicit low-weight elements and exact symbolic verifications illustrate the construction.
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BibTeX
@misc{zhao2026explicit,
title = {{An Explicit Construction of Commuting Quantum KdV Hamiltonians in the Universal Virasoro Vertex Algebra}},
author = {Hongbo Zhao},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06767},
eprint = {2611.06767},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06767},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An Explicit Construction of Commuting Quantum KdV Hamiltonians in the Universal Virasoro Vertex Algebra
AU - Hongbo Zhao
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06767
UR - https://arxiv.org/abs/2611.06767
AB - We give an explicit formula for elements H_2k of conformal weight 2k, k≥1, in the universal Virasoro vertex operator algebra V at every central charge, such that H_2k(0)H_2l∈ TV for all k,l≥1. Modulo translation, the kernel of H_4(0) is generated by [H_2k] in each positive even weight and is zero in positive odd weights. The formula uses a realization on differential polynomials by possibly infinite-order differential operators in a completion of the Weyl algebra, together with the inverse of the identity plus a nilpotent operator on each weight space. A Koszul-type exactness argument for the classical KdV flows proves conservation; uniqueness and the Jacobi identity give commutativity. The classical limit recovers the normalized Gelfand--Dickey densities. Explicit low-weight elements and exact symbolic verifications illustrate the construction.
ER -