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Flux Vacua and Modularity for Z2 Symmetric CalabiYau Manifolds
by Philip Candelas, Xenia de la Ossa, Pyry Kuusela, Joseph McGovern
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Authors (as registered SciPost users):  Pyry Kuusela · Joseph McGovern · Xenia de la Ossa 
Submission information  

Preprint Link:  scipost_202304_00018v1 (pdf) 
Date submitted:  20230417 12:59 
Submitted by:  McGovern, Joseph 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We find continuous families of supersymmetric flux vacua in IIB CalabiYau compactifications for multiparameter manifolds with an appropriate Z2 symmetry. We argue, supported by extensive computational evidence, that the numerators of the local zeta functions of these compactification manifolds have quadratic factors. These factors are associated with weighttwo modular forms, these manifolds being said to be weighttwo modular. Our evidence supports the flux modularity conjecture of Kachru, Nally, and Yang. The modular forms are related to a continuous family of elliptic curves. The flux vacua can be lifted to Ftheory on elliptically fibred CalabiYau fourfolds. If conjectural expressions for Deligne’s periods are true, then these imply that the Ftheory fibre is complexisomorphic to the modular curve. In three examples, we compute the local zeta function of the internal geometry using an extension of known methods, which we discuss here and in more detail in a companion paper. With these techniques, we are able to compare the zeta function coefficients to modular form Fourier coefficients for hundreds of manifolds in three distinct families, finding agreement in all cases. Our techniques enable us to study not only parameters valued in Q but also in algebraic extensions of Q, so exhibiting relations to Hilbert and Bianchi modular forms. We present in appendices the zeta function numerators of these manifolds, together with the corresponding modular forms.
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Report
Please see the attached file.
Anonymous Report 1 on 2023615 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202304_00018v1, delivered 20230615, doi: 10.21468/SciPost.Report.7354
Strengths
1
In the article at hand, the authors provide a large class of families of CalabiYau threefolds that are conjecturally modular in the sense that the Hodge structure on their midde cohomology has a direct summand of type $(2,1) + (1,2)$ and rank $2$. This class is characterized by the property that their complex structure parameter space $\mathcal{M}$ admits an action by a $\mathbb{Z}_2$ which is realized as a permutation of coordinates.
2
This action has two effects: One the one hand, it induces the splitting of the Hodge structure and on the other hand the fixed point loci $\mathcal{F}\subset \mathcal{M}$ are solutions to the vacuum equations of supersymmetric flux compactifications in string theory. Therefore, the authors use this class of families of CalabiYau threefolds to give an extensive verification of the flux modularity conjecture of Kachru et al. which says that CalabiYau threefolds corresponding to such vacua are modular in the above sense.
3
Since this class requires $\dim \mathcal{M} > 1$, the authors extend the method of Dwork on CalabiYau crystals to compute the zeta function of $X_\varphi, \varphi\in \mathcal{F}$, from the case of oneparameter families done in an earlier work by Candelas, de la Ossa, and van Straten, to the multiparameter case. The details are deferred to a future publication. The authors use this to test whether the numerator of the zeta function of $X_\varphi$ has a quadratic factor, corresponding to a twodimensional Galois representation and hence to a direct summand of rank 2 in the Hodge structure.
4
Using this method, the authors test the flux modularity conjecture for various explicit families of CalabiYau threefolds and  for each family  a very large number of moduli for which $X_\varphi$ is defined over a number field $K$ with $[K:\mathbb{Q}] \leq 2$.
5
Furthermore, the authors point out a remarkable relation between an abstract elliptic curve over $\mathbb{Q}$ (and its isogeny class) obtained from the twodimensional irreducible Galois representation corresponding to the quadratic factor in the numerator of the zeta function of $X_\varphi$ and the elliptic curve over $\mathbb{C}$ that appears as the fiber of the CalabiYau fourfold in the Ftheory description of the type IIB flux compactification governed by that representation.
6
All this  the numerators of the zeta functions, the modular forms corresponding to the various isogeneous elliptic curves  is supported by a large amount of tables.
Weaknesses
None
Report
The paper is generally very well and clearly written. It provides very valuable foundations for future investigations of modularity of Calabi–Yau threefolds and the geometry of the Calabi–Yau fourfolds for such flux compactifications. In summary, I strongly recommend it for publication after the following minor points have been taken into account.
Requested changes
1
On p. 13, in the definition of $\mathfrak{A}_{i,j}$ after (25), $\varphi_s$ should have an upper index $s$.
2
There is an issue with terminology and notation for algebraic varieties. The standard in mathematics is that if an algebraic variety $X$ is defined over a field $K$, i.e. the coefficients of the equations for $X$ take values in $K$, then one writes $X/K$. If the solutions to these equations take values in a field $K' \supseteq K$, then one writes for the set of $K'$points $X(K')$, see e.g. the beginning of Chapter V.2 in Silverman, ref. [65].
This issue appears at several places in the paper:
On p. 14, last paragraph, it should read $X/\mathbb{K}$ instead of $X(\mathbb{K})$.
On p. 15, first paragraph in Subsection 4.1, it should read "defining a variety $E/\mathbb{F}_{p^n}$ over" instead of $E(\mathbb{F}_{p^n})$. The following sentence, explaining the symbol $E(\mathbb{F}_{p^n})$ is, however, correct, as this describes the solutions to the equations, not the equations themselves.
On p.15 last paragraph, however, it should read twice $E/\mathbb{F}_{p^n}$ as the singularities of $E$ are determined by the equations for $E$, not their solutions (the vanishing of the discriminant is determined by the coefficients of the equations defining $E$).
On p.16, second paragraph, it should read $X_{\mathbf{\varphi}}/\mathbb{K}$ instead of $X_{\mathbf{\varphi}}(\mathbb{K})$ if this really is referring to the field of definition of $X_{\mathbf{\varphi}}$. If, however, the set of points is referred to, then one should write that "the $\mathbb{K}$points of the variety $X_{\varphi}$ will be denoted by $X_{\varphi}(\mathbb{K})$". I rather suspect that the authors mean the latter alternative.
On p.25, in the third paragraph of Subsection 4.6 it should read: $E/\mathbb{Q}(\sqrt{n})$ defined over $\mathbb{Q}(\sqrt{n})$.
3
On p. 24, second line, it should read $\Lambda$ and $\Lambda'$.
4
On p. 24 two lines before eq. (40) it should read $\wp(z,\Lambda)$ instead of $\wp(z,L)$.
5
On p. 33, in the first line of Subsection 5.2, there is a typo in "Frobenius".
6
On p. 34, in eq. (53) the matrices $\mathbb{Y}_i$ do not seem to be defined. Then, below, in the formula for $\theta_iE$, the (1,3) entry of the matrix should read $\ell^T \mathbf{Y}_{ij}$, the (4,4) entry should be $0$. In this formula, the vectors $\mathbf{Y}_{ij}$ are not defined. Of course, from the context it is clear that $(\mathbb{Y}_i)_{jk}=Y_{ijk}$ and $(\mathbf{Y}_{ij})_k = Y_{ijk}$.
Furthermore, while $\mathbf{\epsilon}$ was introduced in Section 3.2 as a vector of formal parameters in a nilpotent ring, here it is interpreted as a vector of (nilpotent) matrices. This implicit identification should be stated explicitly.