On permutation-twisted free fermions and two conjectures

Abstract

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.

Super Quantum Airy Structures

“…Each term a f a,b (x) is declared to be homogeneous of degree 2a + b, as is consistent with our Z-grading. 7 It is easy to check that the condition 3a + b ≥ 0 combined with b ≥ 0 entail that 2a + b ≥ 0, so the degree of each term is always non-negative. For fixed d = 2a + b we have inequalities 2a ≤ d and a ≥ −d, so f may be rewritten as…”

“…We do not review the well known definitions for vertex operator algebras (VOAs), vertex operator super algebras (VOSAs) and their representations. See [11] for relevant definitions in the context of the construction presented here, and for instance [6,7,9,42,49,53] for more details on VOAs and VOSAs.…”

On the Correspondence Between Mirror-Twisted Sectors for N = 2 Supersymmetric Vertex Operator Superalgebras of the Form V ⊗ V and N = 1 Ramond Sectors of V

“…As noted in [B4], the VOSA V = V L ⊗ V d f er , is naturally an N=1 VOSA, and V ⊗ V is naturally an N=2 VOSA. This uses the construction of a VOSA from a positive definite integral lattice, following for instance [DL], [X], [BV1]. Such N=2 VOSAs have more than one mirror map as was shown in [B4], where the author constructed mirror-twisted modules for these VOSAs for the other mirror map.…”