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A Combinatorial Formula and a Hilbert Scheme Realization of a Rank-One Virasoro Irregular Conformal Block

  • Hajime Nagoya
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Abstract

For general central charge, we give a combinatorial formula and a geometric realization of the formal expansion of the rank-one Virasoro irregular conformal block of type (0,0,1) at its irregular singular point. We realize the normalized block as a coefficientwise matrix element on the localized equivariant Hilbert--Fock space associated with Hilbert schemes of points on ℂ2. At c=1, we prove the author's conjecture and express the parameter-independent coefficients in terms of skew-tableau numbers and Littlewood--Richardson coefficients, establishing their nonnegativity and integrality. The proof combines sectorial asymptotics of screening integrals with Laguerre--Jack evaluations. We express the screening kernel using the flow of a degree-raising differential operator and compute the Jack coefficients of its exponential from the one-box Pieri rule and the shifted Jack coefficient formula of Okounkov and Olshanski.

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Cite

BibTeXRIS
BibTeX
@misc{nagoya2026combinatorial,
  title = {{A Combinatorial Formula and a Hilbert Scheme Realization of a Rank-One Virasoro Irregular Conformal Block}},
  author = {Hajime Nagoya},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.39786},
  eprint = {2610.39786},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.39786},
}
RIS
TY  - UNPB
TI  - A Combinatorial Formula and a Hilbert Scheme Realization of a Rank-One Virasoro Irregular Conformal Block
AU  - Hajime Nagoya
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.39786
UR  - https://arxiv.org/abs/2610.39786
AB  - For general central charge, we give a combinatorial formula and a geometric realization of the formal expansion of the rank-one Virasoro irregular conformal block of type (0,0,1) at its irregular singular point. We realize the normalized block as a coefficientwise matrix element on the localized equivariant Hilbert--Fock space associated with Hilbert schemes of points on ℂ^2. At c=1, we prove the author's conjecture and express the parameter-independent coefficients in terms of skew-tableau numbers and Littlewood--Richardson coefficients, establishing their nonnegativity and integrality. The proof combines sectorial asymptotics of screening integrals with Laguerre--Jack evaluations. We express the screening kernel using the flow of a degree-raising differential operator and compute the Jack coefficients of its exponential from the one-box Pieri rule and the shifted Jack coefficient formula of Okounkov and Olshanski.
ER  -