On permutation-twisted free fermions and two conjectures

21st International Conference on Integrable Systems and Quantum Symmetries

Barron, Katrina^1 ; Werf, Nathan Vander^1

Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, United States^1

关键词: Automorphisms;    Boson fermions;    Cyclic permutations;    Even orders;    Invariant subspace;    Tensor products;    Vertex operators;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/474/1/012009/pdf
DOI  :  10.1088/1742-6596/474/1/012009

来源: IOP

【 摘 要 】

We conjecture that the category of permutation-twisted modules for a multi-fold tensor product vertex operator superalgebra and a cyclic permutation of even order is isomorphic to the category of parity-twisted modules for the underlying vertex operator superalgebra. This conjecture is based on our observations of the cyclic permutation-twisted modules for free fermions as we discuss in this work, as well as previous work of the first author constructing and classifying permutation-twisted modules for tensor product vertex operator superalgebras and a permutation of odd order. In addition, we observe that the transposition isomorphism for two free fermions corresponds to a lift of the -1 isometry of the integral lattice vertex operator superalgebra corresponding to two free fermions under boson-fermion correspondence. We conjecture that all even order cyclic permutation automorphisms of free fermions can be realized as lifts of lattice isometries under boson-fermion correspondence. We discuss the role of parity stability in the construction of these twisted modules and prove that in general, parity-unstable weak twisted modules for a vertex operator superalgebras come in pairs that form orthogonal invariant subspaces of parity-stable weak twisted modules, clarifying their role in many other settings.