Shift orbifolds, decompactification limits, and lattices

The article studies orbifolds of a family of 2-dimensional conformal field theories known as Narain CFTs which describe string worldsheet theories. The type of the orbifold is shift orbifold and its generator is an element of a finite symmetry group of the Narain CFT. The orbifold theory is again a Narain CFT located at a different point in the Narain moduli space.

The authors focus on 10d heterotic string theory. They study shift orbifolds systematically and classify shifts vectors that exchange $E_8+E_8$ and Spin(32)$/\mathbb Z_2$ heterotic theories. To do so, they compactify the theory on a circle, define the shift orbifold of the Narain CFT including its action on the Narain lattice with signature $(17,1)$, and finally take two decompactification limits as the radius of the circle goes to $\infty$ and $0$ to recover the 10d theory.

They develop an algorithm to classify shift vectors that yield inequivalent lattices for abelian orbifold groups. They present the results for the first few low lying values of the order of the cyclic group for the 10d heterotic string theory. Furthermore, they apply their algorithm to the Leech lattice CFT by embedding the Leech lattice in the Narain lattice with signature $(25,1)$. For cyclic groups, they classify shift vectors and determine which of the 24 Niemeier CFTs (including the original Leech CFT) is obtained from the shift orbifold.

Although, as the authors have pointed out, the lattice shifting method and shift orbifold conformal field theories are well-known subjects and have been studied in different contexts, the novel aspect of the work lies in developing a general algorithm to construct conformal field theories with inequivalent lattices and to classify shift orbifolds. The paper presents interesting results and the algorithm can be extended to other families of conformal field theories and to other types of orbifolds which include shift vectors and therefore, will be useful in wider context. The paper is well-written and the lattice construction and choices of the shift vectors are explained clearly. We recommend the paper for publication in SciPost after minor revisions based on the following points.

1 ) Could the authors clarify why the phase the vertex operator acquires under the orbifold action is of the form given in equation (3.2)? How does it change if the lattice was even but not self-dual?

2) Could the authors specify the rank and signature of the invariant lattices in both heterotic and Niemeier theories.

3) Could the authors comment if and how the algorithm can be used in compactifications to lower dimensions with Narain lattices of signature $(16+d,d)$ and decompactifications to dimensions equal or less than 10? What are the obstructions?

4) How can the authors guarantee that the degeneracy of the states of the constructed shift orbifolds are non-negative integers in all untwisted and twisted sectors?

Publish (easily meets expectations and criteria for this Journal; among top 50%)

1- The article is exceptionally clear and careful about all technical points, many of which are usually not explicitly written out in the literature.

2- The article presents in a very explicit way various useful technical connections between lattice theory and conformal field theory.

3- The article contains a thorough set of references including modern work on the topic, making it a particularly useful working reference.

1- The main weakness of the article is that it does not present significantly novel results, but this is clearly acknowledged by the authors.

Publish (easily meets expectations and criteria for this Journal; among top 50%)

• clear general discussion of a class of heterotic orbifolds,
• poses an interesting problem concerning relation between even-self dual lattices and solves it in a insightful way connected to decompactification along an expectator circle.

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Publish (easily meets expectations and criteria for this Journal; among top 50%)