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Solutions of super Knizhnik–Zamolodchikov equations

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Abstract

We establish an explicit bijection between the sets of singular solutions of the (super) KZ equations associated with the Lie superalgebra, of infinite rank, of type \(\mathfrak {a}, \mathfrak {b},\mathfrak {c},\mathfrak {d}\) and with the corresponding Lie algebra. As a consequence, the singular solutions of the super KZ equations associated with the classical Lie superalgebra, of finite rank, of type \(\mathfrak {a}, \mathfrak {b},\mathfrak {c},\mathfrak {d}\) for the tensor product of certain parabolic Verma modules (resp., irreducible modules) are obtained from the singular solutions of the KZ equations for the tensor product of the corresponding parabolic Verma modules (resp., irreducible modules) over the corresponding Lie algebra of sufficiently large rank, and vice versa. The analogous results for some special kinds of trigonometric (super) KZ equations are obtained.

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Acknowledgements

The first author was partially supported by NSFC (Grant Nos. 11571374, 11521101 and 11771461). A part of this research was done during the visit of the second author to Sun Yat-sen University. The second author was partially supported by Ministry of Science and Technology Grant 107-2115-M-006-005-MY2 of Taiwan and he thanks Sun Yat-sen University for hospitality and support.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Yunnan University, Kunming, 650500, China

    Bintao Cao

  2. School of Mathematics, Sun Yat-sen University, Guangzhou, 510275, China

    Bintao Cao

  3. Department of Mathematics, National Cheng Kung University, Tainan, 70101, Taiwan

    Ngau Lam

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  1. Bintao Cao
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  2. Ngau Lam
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Correspondence to Bintao Cao.

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Cao, B., Lam, N. Solutions of super Knizhnik–Zamolodchikov equations. Lett Math Phys 110, 1799–1834 (2020). https://doi.org/10.1007/s11005-020-01275-z

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