SciPost Submission Page
A Generalized Construction of CalabiYau Models and Mirror Symmetry
by Per Berglund, Tristan Hubsch
This Submission thread is now published as
Submission summary
Authors (as registered SciPost users):  Per Berglund 
Submission information  

Preprint Link:  http://arxiv.org/abs/1611.10300v3 (pdf) 
Date accepted:  20180114 
Date submitted:  20171122 01:00 
Submitted by:  Berglund, Per 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We extend the construction of CalabiYau manifolds to hypersurfaces in nonFano toric varieties, requiring the use of certain Laurent defining polynomials, and explore the phases of the corresponding gauged linear sigma models. The associated nonreflexive and nonconvex polytopes provide a generalization of Batyrev's original work, allowing us to construct novel pairs of mirror models. We showcase our proposal for this generalization by examining CalabiYau hypersurfaces in Hirzebruch nfolds, focusing on n=3,4 sequences, and outline the more general class of sodefined geometries.
Author comments upon resubmission
List of changes
To the Referee 1:
1) We have thoroughly rewritten Appendix A, expanding to spell out the motivations and details of the "working definition" that has been used throughout the analysis reported in this article. We also clarify (in several places including the conclusions) that we have verified, wherever possible, that the soobtained results are both selfconsistent, and also consistent with the by now well established results of Refs. [1,5,6].
2) In the text preceding Eq.(16), we have changed the specification of $\nu_\rho$ from "...vertices..." to "...the $N$integral points $\nu_\rho$ of which being the minimal generators..." as this covers also the exceptional $m=2$ case, when $\nu_1=(1,\dots,1,0)$ is in a face, corresponding to the MPCP exceptional set.
3) The sentence containing Eq.(15) now correctly qualifies its use for reflexive (and convex) polytopes.
4) The definition is inserted in footnote 9, at the first mention of the "degree," on p.9. Its effective use in counting is mentioned right after Construction 3.1.
5) We have rewritten Construction 3.1 so as to refer to the full disjoint union of all faces, with the nonconvex ones subdivided into convex ones. In our experience, this is very redundant, but does not introduce undue quandaries of nonuniqueness and their relevance. In turn, the subdivision of nonconvex kfaces into convex kfaces poses no problems, since they are all kcoplanar so that their polars coincide. This has been clarified in the revised document.
6) The transpolar polytope is assembled "from ground up," using the particular "dually" implied relations, which are now spelled out as Eqs. (17) at the end of Step 3 in Construction 3.1. Their use is illustrated in the explicit construction in Section 3.2, and we amplified the discussion of the observed sufficiency of the specifications in Construction 3.1 in the narrative on p.910.
7) As now clarified in several places, we do not have a proof that the specifications of Construction 3.1 always suffice, but have observed that they do in dozens upon dozens of examples (including some infinite sequences) purposefully invented to test it. Indeed, we wholeheartedly welcome any and all attempt to either prove or improve Construction 3.1  and in a twin fashion also provide a conclusive definition of VEX polytopes as a maximal closure of transpolar pairs of polytopes.
8) We have included an explicit reference to stardomains, noting however that they must be understood in a generalized fashion, implicit in the (cited) literature on "multifans."
To Referee 2:
1. We have thoroughly rewritten Appendix A, expanding to spell out the motivations and details of the "working definition" that has been used throughout the analysis reported in this article. We also clarify (in several places including the conclusions) that we have verified, wherever possible, that the soobtained results are both selfconsistent, and also consistent with the by now well established results of Refs. [1,5,6].
2. In the rewritten Appendix A (as well as in Section 2), we highlight the places where and why the related notions of "baselocus" and "transversality" (and Batyrev's $\Delta$regularity) turn up. We now also explicitly state that our article addresses the lowenergy regime of the considered GLSMs, and defer the UV analysis.
3. The related issues of "intrinsic limits," transversality and nonsingularity are detailed in Appendix A, and noted also elsewhere throughout this revision.
Published as SciPost Phys. 4, 009 (2018)
Reports on this Submission
Anonymous Report 2 on 20171130 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1611.10300v3, delivered 20171130, doi: 10.21468/SciPost.Report.286
Strengths
As in my previous report
Weaknesses
none
Report
The authors have carefully revised their paper and significantly clarified the discussion. In particular, all of the shortcomings noted in my previous report have been well addressed.
Anonymous Report 1 on 20171126 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1611.10300v3, delivered 20171125, doi: 10.21468/SciPost.Report.282
Strengths
The strengths remain as stated in my initial report
Weaknesses
The weaknesses in the paper have been largely addressed by the authors in the revised version.
Report
This revision meets my expectations and I recommend the work for publication.