Abstract
Given a two-dimensional conformal field theory with a global symmetry, we propose a method to implement an orbifold construction by taking orbits of the modular group. For the case of cyclic symmetries we find that this approach always seems to be consistent, even in asymmetric orbifold cases where the usual construction does not yield a modular invariant theory; our approach keeps modular invariance manifest but may give a result that is equivalent to the original theory. For the case that the symmetry is a subgroup of a continuous flavor symmetry, we can give explicit constructions of the spectrum, with twisted sectors corresponding to a non-standard group projection on an enlarged twisted sector Hilbert space.
Highlights
Orbifolds have provided a large class of examples of twodimensional conformal field theories (CFTs)
The most typical case starts with a free theory, and constructs the quotient theory by simultaneously restricting to states that are invariant under the isometries, and by including new twisted sectors in which strings are closed only up to the quotient
This procedure can be implemented at the level of the path integral, as we review in Sec
Summary
Orbifolds have provided a large class of examples of twodimensional conformal field theories (CFTs). If H2ðG; Uð1ÞÞ ≠ 0, the full orbifold partition function is not in the orbit of the untwisted sector; in this case one must provide extra information (a choice of discrete torsion) to specify the orbifold theory, and, more seriously from our point of view, the disjoint orbits cannot be obtained from the parent theory and genus one modular invariance alone For cyclic groups, this is not an issue [H2ðZN; Uð1ÞÞ 1⁄4 0], and these will be our main focus in this paper. For cases that would have been anomalous, the resulting orbifold partition functions do not typically correspond to new theories, but rather reconstruct either the original parent theory or a different consistent orbifold thereof Another observation that emerges from our analysis is that even in nonanomalous orbifolds, one may need to use a nonstandard projection in the enlarged twisted sector Hilbert spaces (i.e., states are not invariant under the group, but should transform in a particular projective representation).
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