Tau functions of the charged free bosons
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https://doi.org/10.1007/s11425-019-1735-4
. ARTICLES .
Tau functions of the charged free bosons Naihuan Jing1,2 & Zhijun Li1,∗ 1School
of Mathematics, South China University of Technology, Guangzhou 510640, China; of Mathematics, North Carolina State University, Raleigh, NC 27695, USA
2Department
Email: [email protected], [email protected] Received August 26, 2019; accepted July 8, 2020
Abstract
We study bosonic tau functions in relation with the charged free bosonic fields. It is proved that up
to a constant the only tau function in the Fock space M is the vacuum vector, and some tau functions are given f by using Schur functions. We also give a new proof of Borchardt’s identity and obtain in the completion M several q-series identities by using the boson-boson correspondence. Keywords MSC(2010)
tau functions, boson-boson correspondence, vertex operator algebras, symmetric functions 17B37, 58A17, 15A75, 15B33, 15A15
Citation: Jing N, Li Z J. Tau functions of the charged free bosons. Sci China Math, 2020, 63, https://doi.org/ 10.1007/s11425-019-1735-4
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Introduction
The bc fermionic fields and charged free bosons are important in conformal field theory partly due to their applications in free field realizations of affine Lie algebras and related algebras (see [17] and the references b ∞ -modules [21,22], and a therein). In this regard the bc fermionic fields provide gl∞ -modules, level one gl free field realization of W1+∞ algebra in positive integral central charge [14]. The bc fermionic fields also provide a Lie theoretic approach to the celebrated Kadomtsev-Petviashvili (KP) hierarchy [8,9]. One can formulate the Hirota equation for the KP hierarchy as Resz b(z) ⊗ c(z)(τ ⊗ τ ) = 0,
(1.1)
where b(z) and c(z) are fermionic fields satisfying (2.7). The solutions τ of (1.1) are called the KP tau functions. Under the boson-fermion correspondence, the tau functions can be expressed as certain symmetric functions. In particular, one can get polynomial soliton solutions of the KP hierarchy [6, 21] in the Fock space representations. The charged free bosons φ(z) and φ∗ (z) enjoy similar properties to the bc fermionic fields in conformal b ∞ -modules, and a free field realization of W1+∞ field theory. They also provide gl∞ -modules, level −1 gl algebra with negative integral central charge [22, 25, 29]. By the boson-boson correspondence or FriedanMartinec-Shenker (FMS) bosonization [4, 17, 25], one can also consider the corresponding PDEs and the bosonic tau functions as solutions of the equation (see [27]) Resz φ∗ (z) ⊗ φ(z)(τ ⊗ τ ) = 0.
(1.2)
* Corresponding author c Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020 ⃝
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Jing N et al.
2
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Our first result is to show that the only bosonic tau function in the Fock space M is the vacuum f In fact, we vector |0⟩ up to constant. So the natural tau functions in this case lie in the completion M. will show that ( ∑ ) ∗ τ = exp ci,j φ−i φ−j · |0⟩ i>0,j>0
f are bosonic KP-like tau functio
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