Symplectic Fermions

TL;DR

This work constructs the maximal local non-chiral CFT SF from the two-component symplectic fermion with central charge $c=-2$, exhibiting logarithmic operators through indecomposable representations. It exploits a global $SL(2,C)$ symmetry to introduce twisted sectors and organizes consistent twisted amplitudes, then forms orbifold models $ ext{SF}[ ext{G}]$ by projecting to $G$-invariant states, obtaining modular-invariant partition functions for abelian $ ext{G}$ and connecting the $C_2$ orbifold to the triplet algebra and the $C_4$ orbifold to dense polymers. The construction relies on comultiplication methods to derive non-chiral amplitudes from fundamental fields and twisted comultiplications to handle twist fields, yielding explicit two-, three-, and four-point functions. The results illuminate the structure of logarithmic CFTs, establish isomorphisms with the triplet model, and provide concrete predictions for polymer-related observables and their lattice realizations.

Abstract

We study two-dimensional conformal field theories generated from a ``symplectic fermion'' - a free two-component fermion field of spin one - and construct the maximal local supersymmetric conformal field theory generated from it. This theory has central charge c=-2 and provides the simplest example of a theory with logarithmic operators. Twisted states with respect to the global SL(2,C)-symmetry of the symplectic fermions are introduced and we describe in detail how one obtains a consistent set of twisted amplitudes. We study orbifold models with respect to finite subgroups of SL(2,C) and obtain their modular invariant partition functions. In the case of the cyclic orbifolds explicit expressions are given for all two-, three- and four-point functions of the fundamental fields. The C_2-orbifold is shown to be isomorphic to the bosonic local logarithmic conformal field theory of the triplet algebra encountered previously. We discuss the relation of the C_4-orbifold to critical dense polymers.