K-theoretic boson-fermion correspondence and melting crystals
K-theoretic boson-fermion correspondence and melting crystals
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K 理论玻色子-费米子对应和熔化晶体
DOI：
10.1088/1751-8113/47/44/445202
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
K. Motegi and K. Sakai
中科院分区：
文献类型：
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作者：
上島裕司;与儀千尋;辻孝祐;小島一男;Tomokatsu Hayakawa;K. Motegi and K. Sakai
我们从Grothendieck多项式的角度研究了非厄米可积费米子和玻色子系统。在这篇文章中考虑的模型是作为一个费米子系统的五顶点模型和作为玻色子系统的非厄米相位模型。这两个模型的特点是不同的解决方案满足相同的杨巴克斯特关系。在我们以前关于五顶点模型的波函数与Grothendieck多项式之间的识别的工作的基础上，我们引入了斜Grothendieck多项式，并导出了它们之间的加法定理。利用这些关系，我们导出了非厄米相位模型的波函数的行列式形式，它也可以表示为格罗滕迪克多项式。也就是说，我们建立了波函数水平上的K理论玻色子-费米子对应。作为一个副产品，一个三维（3D）熔融晶体的统计力学模型的配分函数精确计算使用的相模型的波函数的标量积。所得的表达式可以被看作是描述平面划分的生成函数的MacMahon函数的K理论推广，该MacMahon函数插值二维（2D）和（3D）Young图的生成函数。
We study non-Hermitian integrable fermion and boson systems from the perspectives of Grothendieck polynomials. The models considered in this article are the five-vertex model as a fermion system and the non-Hermitian phase model as a boson system. Both models are characterized by different solutions satisfying the same Yang–Baxter relation. From our previous works on the identification between the wavefunctions of the five-vertex model and Grothendieck polynomials, we introduce skew Grothendieck polynomials and derive the addition theorem among them. Using these relations, we derive the wavefunctions of the non-Hermitian phase model as a determinant form, which can also be expressed as Grothendieck polynomials. Namely, we establish a K-theoretic boson–fermion correspondence at the level of wavefunctions. As a by-product, the partition function of the statistical mechanical model of a three-dimensional (3D) melting crystal is exactly calculated by use of the scalar products of the wavefunctions of the phase model. The resultant expression can be regarded as a K-theoretic generalization of the MacMahon function describing the generating function of the plane partitions, which interpolates the generating functions of two-dimensional (2D) and (3D) Young diagrams.