Pfaffian identities and Virasoro operators

Kazuya Aokage; Eriko Shinkawa; Hiro Fumi Yamada

doi:10.1007/s11005-020-01265-1

Pfaffian identities and Virasoro operators

Kazuya Aokage, Eriko Shinkawa, Hiro Fumi Yamada

Research output: Contribution to journal › Article › peer-review

Abstract

A formula for Schur Q-functions is presented which describes the action of the Virasoro operators. For a strict partition λ= (λ₁, λ₂, … , λ₂_m) , we show that, for k≥ 1 , LkQλ=∑i=12m(λi-k)Qλ-2kϵi, where L_k is the Virasoro operator given as the quadratic form of free bosons. This main formula follows from the Plücker-like bilinear identity of Q-functions as Pfaffians: ∑i=22m(-1)i∂1Qλ1,λi∂1Qλ2,…,λi^,…,λ2m=0, where ∂₁= ∂/ ∂t₁. This bilinear identity must be explained in geometric words. We conjecture that the Hirota bilinear equations of the KdV hierarchy are derived from this bilinear identity.

Original language	English
Pages (from-to)	1381-1389
Number of pages	9
Journal	Letters in Mathematical Physics
Volume	110
Issue number	6
DOIs	https://doi.org/10.1007/s11005-020-01265-1
Publication status	Published - 2020 Jun 1
Externally published	Yes

Keywords

Bilinear identity
KdV hierarchy
Q-functions
Virasoro operators

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Mathematical Physics

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10.1007/s11005-020-01265-1

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title = "Pfaffian identities and Virasoro operators",

abstract = "A formula for Schur Q-functions is presented which describes the action of the Virasoro operators. For a strict partition λ= (λ1, λ2, … , λ2m) , we show that, for k≥ 1 , LkQλ=∑i=12m(λi-k)Qλ-2kϵi, where Lk is the Virasoro operator given as the quadratic form of free bosons. This main formula follows from the Pl{\"u}cker-like bilinear identity of Q-functions as Pfaffians: ∑i=22m(-1)i∂1Qλ1,λi∂1Qλ2,…,λi^,…,λ2m=0, where ∂1= ∂/ ∂t1. This bilinear identity must be explained in geometric words. We conjecture that the Hirota bilinear equations of the KdV hierarchy are derived from this bilinear identity.",

keywords = "Bilinear identity, KdV hierarchy, Q-functions, Virasoro operators",

author = "Kazuya Aokage and Eriko Shinkawa and Yamada, {Hiro Fumi}",

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year = "2020",

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doi = "10.1007/s11005-020-01265-1",

language = "English",

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T1 - Pfaffian identities and Virasoro operators

AU - Aokage, Kazuya

AU - Shinkawa, Eriko

AU - Yamada, Hiro Fumi

PY - 2020/6/1

Y1 - 2020/6/1

N2 - A formula for Schur Q-functions is presented which describes the action of the Virasoro operators. For a strict partition λ= (λ1, λ2, … , λ2m) , we show that, for k≥ 1 , LkQλ=∑i=12m(λi-k)Qλ-2kϵi, where Lk is the Virasoro operator given as the quadratic form of free bosons. This main formula follows from the Plücker-like bilinear identity of Q-functions as Pfaffians: ∑i=22m(-1)i∂1Qλ1,λi∂1Qλ2,…,λi^,…,λ2m=0, where ∂1= ∂/ ∂t1. This bilinear identity must be explained in geometric words. We conjecture that the Hirota bilinear equations of the KdV hierarchy are derived from this bilinear identity.

AB - A formula for Schur Q-functions is presented which describes the action of the Virasoro operators. For a strict partition λ= (λ1, λ2, … , λ2m) , we show that, for k≥ 1 , LkQλ=∑i=12m(λi-k)Qλ-2kϵi, where Lk is the Virasoro operator given as the quadratic form of free bosons. This main formula follows from the Plücker-like bilinear identity of Q-functions as Pfaffians: ∑i=22m(-1)i∂1Qλ1,λi∂1Qλ2,…,λi^,…,λ2m=0, where ∂1= ∂/ ∂t1. This bilinear identity must be explained in geometric words. We conjecture that the Hirota bilinear equations of the KdV hierarchy are derived from this bilinear identity.

KW - Bilinear identity

KW - KdV hierarchy

KW - Q-functions

KW - Virasoro operators

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JO - Letters in Mathematical Physics

JF - Letters in Mathematical Physics

SN - 0377-9017

IS - 6

ER -

Pfaffian identities and Virasoro operators

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Keywords

ASJC Scopus subject areas

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