F-theory and Heterotic Duality, Weierstrass Models from Wilson lines

We present how to construct elliptically fibered K3 surfaces via Weierstrass models which can be parametrized in terms of Wilson lines in the dual heterotic string theory. We work with a subset of reflexive polyhedras that admit two fibers whose moduli spaces contain the ones of the $E_{8}\times E_{8}$ or $\frac{Spin(32)}{\mathbb{Z}_{2}}$ heterotic theory compactified on a two torus without Wilson lines. One can then interpret the additional moduli as a particular Wilson line content in the heterotic strings. A convenient way to find such polytopes is to use graphs of polytopes where links are related to inclusion relations of moduli spaces of different fibers. We are then able to map monomials in the defining equations of particular K3 surfaces to Wilson line moduli in the dual theories. Graphs were constructed developing three different programs which give the gauge group for a generic point in the moduli space, the Weierstrass model as well as basic enhancements of the gauge group obtained by sending coefficients of the hypersurface equation defining the K3 surface to zero.


Introduction
F-theory compactified on elliptically fibered K3 surfaces is believed to be dual at the quantum level to the heterotic string compactified on a two torus with Wilson lines [1][2][3][4]. In particular one should be able to relate the complex parameters of the moduli space on the F-theory side to the ones on the heterotic one as their moduli space are the same, namely the Narain space [5,6]  The study of the full moduli space is however a tedious exercise and one wants to focus on subspaces with fewer complex modular parameters. In heterotic string one can consider for example compactifications on a two torus with Wilson lines parametrized by few moduli.
In F-theory, one can choose an algebraic K3 with a large Picard number p, as its modular space is parametrized by 20 − p complex variables [7]. Now, a particularly interesting way to construct K3 surfaces is to use lattice polarized K3 obtained by considering reflexive polyhedras in 3 dimensions which define hypersurfaces in toric varieties [8,9]. Thanks to Kreuzer and Skarke [10] it is possible to have access to the totality of the 4319 different reflexive polyhedras in 3 dimensions. One can then first construct K3 surfaces, and afterwards elliptically fibered K3s by considering a particular subdivision of the fan in the dual polytope dictated by a choice of a two dimensional reflexive subpolytope which plays the role of the fiber. In particular it is possible to classify the different reflexive polyhedras with respect to their Picard number p, and therefore focus on moduli spaces associated to particular K3 surfaces with a low number of moduli.
The duality between F-theory and heterotic string has been written explicitly for only two of the 4319 different K3 surfaces one can construct via reflexive polytopes. First the duality between the parameter of a Weierstrass model presenting a particular E 8 × E 8 singularity and the complex structure and Kahler moduli of the two torus on which the E 8 × E 8 heterotic string is compactified was constructed in [11]. Later it was found that a particular reflexive polyhedra admitting two fibrations has for gauge groups E 8 × E 8 and Spin(32) Z 2 [12]. In a more general case with three moduli, Malmendier and Morrison showed that a particular polytope with again two fibers with gauge group E 7 ×E 8 and Spin(28)×SU (2) is related to compactifications of heterotic strings with one Wilson line moduli [13]. All of this suggests that compactifying F-theory on elliptically fibered K3 surfaces described by polytopes with two fibers seem to be related in some cases to the compactification of both heterotic strings with Wilson line moduli.
Here we show that if we focus on particular reflexive polyhedras that are linked in some way to the E 8 × E 8 / Spin (32) Z 2 polytope, we can understand the Wilson line structure of the dual heterotic strings. This is due to the fact that we can recover the torus on which we compactify the heterotic strings as a particular subspace of the moduli spaces of the elliptically fibered K3s. To find these polytopes we construct graphs where a link between two polytopes M + and M − is drawn if, for every elliptically fibered K3 surface obtained via M + , there exist a limit in the moduli space where one obtains elliptically fibered K3 surfaces of the other polytope M − . In particular we will consider the limit where we send monomials of the hypersurface equation defining the K3 surface associated to M + to zero, which is equivalent to removing a point in M + . This can be seen as an extension of the notion of chains presented by Kreuzer and Skarke in [10]. Focusing on polytopes which have two fibers, links between polytopes then correspond to inclusion relations between the moduli spaces of elliptically fibered K3s. Considering polytopes which are linked to  [17,18]). A pedagocical introduction to toric geometry can be found in [19].
Let us consider two dual lattices M (Monomials) and N (faN) in Z n with real extension M R and N R and an associated product < * , * >: M ×N → Z. We note ∆ an integral convex polytope whose vertices are in M and which contains only the origin as an interior point.
We then define the dual of ∆ as As usual we consider ∆ to be reflexive, meaning that is also convex, only contains the origin and has its vertices v i in N . The normal fan of the polytope ∆ whose rays are the vertex of then defines a projective toric variety P ∆ (which is Fano if and only if ∆ is reflexive, which it is in this paper). with c m ∈ C.
We can then construct in some cases an elliptically fibered K3 as X ∆ together with a surjective morphism π : X ∆ → P 1 such that generic fiber are genus one elliptic curves.
They can be constructed by considering the K3 surface we just described, as well as finding a subpolytope (2) of in the N lattice which plays the role of the fiber of the elliptic K3 1 . There are 16 reflexive polyhedras for n = 2, and 4319 reflexive polytopes for n = 3 [10]. It is then possible to obtain Weierstrass models of elliptically fibered K3 surfaces upon a choice of a fan which contains as rays points of the fiber (2) . Quite amazingly, and upon a particular choice of a fan which will be described in section 3.2, the gauge groups associated to singularities of the elliptically fibered K3s can be read off directly once one chooses a particular subpolytope (2) [20]. Indeed, Candelas and Font noticed that the points located on both sides of the fiber of the polytope in the N lattice are exactly the extended Dynkin diagrams which correspond to the gauge groups associated to singularities 1 Such subpolytope cannot be found sometimes, in particular for small Picard numbers.
appearing in the Weierstrass model via the Kodaira and Néron classification [21,22] (see Figure 1). This was later explained by Perevalov and Skarke in [23]. Depending on which of the 16 two dimensional reflexive polyhedras is the fiber, additional contribution coming from the Mordell-Weil group of rational sections of the elliptic fibration can occur [24][25][26][27][28].
In particular the fibers F1, F2 or F4 give additional discrete symmetries Z # and fibers F13, or SU (#) factors can appear, depending on how the polytope (2) intersects with .

Invariant Parameters of the Moduli Space
The number of complex moduli for a K3 surface with Picard number p is 20 − p. Previously we defined an algebraic K3 as an hypersurface (2.2) in the toric variety P ∆ whose number of parameters is a priori given by the number of points in ∆ ∩ M if we consider (2.2). However different sets of those parameters correspond to the same point in the moduli space. For example several of the coefficients can be put to 1 by a reparametrization of the coordinates in the projective space. In order to properly define complex parameters on the moduli space of the K3 surface we use the construction developed in [30]. It was shown there that monomials defined by points interior to facets in ∆ ∩ M can be removed by an appropriate change of coordinates for the different reflexive polyhedras they considered. We therefore as well as the origin. The hypersurface equation can then be written as with v k rays of the normal fan P ∆ . Due to the strong link between the period map of K3 surfaces and their moduli spaces [7], one can seek for parameters of the moduli space by considering the fundamental period of the holomorphic two-forms which can be written in our case as [31]w with C a product of cycles that enclose the hypersurface defined by x i = 0 [30]. This can be recast asw Using the notations of [26]. The polytopes are i1,i2,i4 and i9,i7,i6 using notations of [29].
The only non zero terms in (2.5) are the constant terms 3 in the development ofH l . The fundamental period of the holomorphic two-forms can therefore be parametrized by the following invariants such that Taking As an example let us take the polytope M 476, with Picard number equal to 16 i.e. 4 moduli. Its vertices are given by An additional point, (7) M = (−3, −2, 0), is situated on the edges of the polytope. We can thus consider four inequivalent linear relations between these points, a possibility being which leads using (2.9) to the complex parameters of the moduli space

Obtaining Data on Elliptically Fibered K3s
We now present three computer programs that allow to obtain different information about elliptically fibered K3 surfaces automatically. We note M # the polytope ∆ in the M lattice corresponding to Ref lexiveP olytope(3, #) in SageMath 4 . We write schematically what is 3 By the residue theorem. 4 The vertices of each of the polytopes presented in this paper are written in the Appendix D.
done in Sagemath, more accurate descriptions of the computer programs as well as the programs themselves are available on GitHub at https://github.com/lilianChabrol/ReflexivePolyhedras.

Extended Dynkin Diagram from Polyhedras
As discussed in the first part of this paper it is possible to have access to the gauge structure of an elliptically fibered K3 upon a choice of reflexive polytope (∆, ), and a choice of fiber (2) . We now present a generic way to find the gauge group associated to each fiber of every reflexive polytope in the Kreuzer-Skarke classification of reflexive polyhedra in 3 dimensions.
We first find all two dimensional reflexive polyhedras (2) which are subpolytopes of modulo SL(3, Z) transformations in the N lattice. Then we identify which of the 16 possible two dimensional reflexive polytope corresponds to each of the fibers (2) . This permits in particular to know if the fiber contains product or quotient by discrete symmetry group [26]: F1, F2 and F3 contribute to a product by Z 3 , Z 2 and Z 4 respectively while fibers F13, F15 contribute by 1 Z 2 and F16 by 1 Z 3 . We do not write the additional contributions of U (1) factors coming from the Mordell Weil group as in the end the gauge group must be of rank 18. We however look for additional SU (#) contribution from the fiber: if polytopes (2) and have a common edge with n points, then there appears an additional SU (n − 1) part in the final gauge group 5 . Finally, the fiber (2) dividing into two parts, we look at points "above" and "below" the fiber and read off the extended Dynkin diagrams.
As an example let us consider the polytope M 476. In Figure 1 we represent the dual polytope N 476 of M 476 for the two inequivalent fibrations (2) it contains. On the left one can read off two extended Dynkin diagram of E 7 . On the right there is a SO(24) as well as 1 Z 2 comming from the fiber F 13, and SU (2) × SU (2) contribution due to the intersection of (2) and symbolised by red points.
The results for K3 surfaces with Picard number 19 and 18 i.e. one and two complex parameters respectively, are presented in the Tables 1 and 2. They were compared with results of an unpublished paper [32] presented at a seminar at CERN [33]. The result with Picard 17 and 3 complex parameters is in the Appendix A (Table 7). Tables with complex parameters up to 5 moduli are available on GitHub, and up to 10 moduli for elliptically fibered K3s admitting only two inequivalent fibrations.

Weierstrass Model, Gauge Groups and Basic Enhancements
The computer program introduced in section 3 is particularly interesting to determine the gauge group at a generic point in the moduli space associated to any fiber of any reflexive polytope in three dimensions. It would however be interesting to get the Weierstrass model which correspond to these gauge groups in order to find their enhancements for particular values of the moduli. Some of the enhancements can then be found quite easily by removing points in the polytope ∆ in the M lattice which amounts to sending to zero a coefficient in the hypersurface equation which defines the K3 surface. This is what the second and third program do: find the Weierstrass model, and the enhancements described above 6 .
We first look at the polytope in the N lattice. As explained in the introduction we then find inequivalent subpolytope (2) of dimension 2 in . For each of this 2 dimensional polytope we want to associate homogenous coordinates such that it describes the fiber. For most cases one can just associate one of them to each vertices of the subpolytope and obtain later the gauge groups expected from reading the extended Dynkin diagrams directly on the polytope . However in 3 cases (F13, F15 and F16) out of the 16 possible two dimensional reflexive polytopes, considering the vertices will not lead to these groups. This is due to the fact that for these particular polytopes the fibrations admit more than one section [12]. Using a similar construction to the one of [33] and in an upcoming paper [32], we then consider the homogeneous coordinates x i of the fiber to be associated to the points as described in Figure 2. Figure 2: In order to obtain the groups associated to the extended Dynkin diagrams on the N lattice we consider the following rays when the two dimensional subpolytopes are F13, F15 and F16.
To define coordinates (s, t) on the base space P (1) we seek for two vectors v s and v t , "above" and "below" the fiber. A fast way to obtain the appropriate Weierstrass model with correct ADE singularities, which correspond to the extended Dynkin diagrams seen in , is then to seek for the closest vectors to the fiber in ∩ N .
Finally we write the hypersurface equation by considering the points on the edges of ∆ and using equation (2.2). To each of these points corresponds a monomial in the hypersurface equation to which we associate a parameter c i ∈ C. Using SageMath we can finally recast this equation into the Weierstrass form where the homogeneous coordinates of the fiber are now [x, y, z] in P (2,3,1) , f and g are respectively polynomials of degree 8 and 12 in (s, t). The discriminant of (3.1) is then ∆ (f,g) = 4f 3 + 27g 2 and vanishes at 24 points which are the locations of 7-branes.
Once one has the Weierstrass form of the elliptically fibered K3, one finds the ADE groups associated to the various singularities using Kodaira and Neron classification [21,22].
Moreover, the moduli can be expressed via the parameters c i as shown in section 2.2.
Sending those parameters to zero, we can therefore find possible enhancements of the group associated to a generic point in the moduli space. The third SageMath program then gives all possible enhancements obtained by sending all possible combinations of parameters c i , when the hypersurface still defines a elliptically fibered K3.

Graphs of Polytopes
Using this we construct graphs of K3 surfaces, generalising the "chains" defined by Kreuzer and Skarke in [10]. Nodes on a graph correspond to polytopes, or equivalently their associated K3 surface. We then link two polytopes if, by sending the same coefficient c i of

F-theory/Heterotic string duality in 8 dimensions: Wilson lines from
Reflexive Polyhedras?

Basic Aspects of Heterotic String
Here we introduce basic notions concerning the compactifications of heterotic strings on a two torus, based on [34,35]. As was presented in the beginning of the introduction, the moduli space of the heterotic string compactified on a two torus is given by the Narain space (1.1). This is due to the fact that the twenty dimensional internal momentum with a = 1, 2 an index corresponding to directions along the two torus and A = 1, ···, 16 an index along the 16 dimensional torus of heterotic string transforms as a vector under O(2, 18, R). One then obtains that P · P ∈ 2Z and P forms a lorentzian lattice of signature (2,18 to U (1) 16 due to the quantization condition of the momentum number.
In [35] is presented the compactifications of heterotic strings on a circle, with one Wilson line parametrized by one parameter. This leads to a wide variety of possible gauge groups one can obtain from heterotic strings. As an example one can consider one Wilson line of the form A = (a 8 , 0 8 ) with a ∈ R breaking SO(32) to SO(16) × SO(16) and E 8 × E 8 to E 7 × E 8 due to the quantization condition of the momentum number. Here we will write the Wilson lines A 1 and A 2 in a complex form A = A 1 + iA 2 . We then seek for possible Wilson lines parametrizations which will correspond to the gauge groups we obtain from elliptic fibrations of K3 surfaces.

Graphs of Polytopes: from Monomials to Wilson lines...
The duality map between F-theory on K3 and heterotic strings on a two torus has been explicitly written for two polytopes having each two inequivalent fibrations. The gauge groups associated to these fibers amazingly are E 8 × E 8 and SO(32) for the first polytope (M 88 using our notation) and E 7 × E 8 and SO(28) × SU (2) (M 221). Using [35], we can see that adding a Wilson line of the form A = (a 2 , 0 14 ) with a ∈ C, using the notation of the previous section 4.1 breaks E 8 × E 8 to E 7 × E 8 , and SO(32) to SO(28) × SU (2) for a generic value of a. On the heterotic side one might be able to interpret the polytope M 221 as a compactification on a two torus, together with one Wilson line of the form (a 2 , 0 14 ). In fact considering this particular parametrization of Wilson line, the enhancements one finds on both heterotic strings and F-theory exactly match, as was presented by Anamaria Font at CERN [33] and studied with more details in an upcomming paper [32].
Now we want to see if we can make similar interpretations by considering polytopes which admit only two fibrations. Following the construction we presented in section 3.3 we seek a graph of polytopes with two fibers in its dual lattice and which contains M 88. As an example let us consider the polytope M 1328 which has Picard number 14. Its moduli space is parametrized by 6 complex parameters. We write the hypersurfaces equations P G = 0 of its two fibers below, where G is the group associated to its ADE singularities with x i homogeneous coordinates of the fiber, and (s, t) coordinates on the base. These hypersurfaces can then be recast into a Weierstrass form where (s, t) correspond to coordinates on the base P 1 9 . Considering the underlined monomials in the equations (4.2) and 9 We do not write the Weierstrass models due to the size of the parameters f , g and ∆ (f,g) · (4.3) gives the Weierstrass models associated to the polytope M 88 and thus corresponds to the heterotic strings without Wilson lines  Figure 6 11 . Going down in the graph, we define four additional complex parameters as We already know that the polytope M 221 is obtained on the heterotic side by adding a Wilson line a(1 2 , 0 14 ) therefore the monomial "c 7 " is associated to this Wilson line. Looking at all the gauge groups in the graph and using results on the compactification of heterotic strings on a circle [35] we find that a possibility for the Wilson lines associated to each monomial is with a, b, c and d in C parametrizing the moduli on the heterotic side. We can see that  12 In the E8 × E8 heterotic string one can just interchange the E8s.

... and back to Monomials
We are now able for particular polytopes M # whose dual N # contain two fibers to describe K3s as parametrizations of Wilson lines of its dual theory (both for E 8 × E 8 and SO (32)).

Rather we linked
The gauge groups associated to the singularities of these Weierstrass models are E 6 × E 8 and SO(26) respectively. They are exactly what we expect from heterotic string theories with one Wilson line A c 5 in equation (4.8). This means that if we compactify F-theory on these elliptically fibered K3s, we know that the Wilson line content on the dual heterotic strings should be of a similar kind as A c 5 . Using this it is then possible to restrict the study of the duality map between the two theories to a three dimensional moduli space to verify that the enhancements on both F-theory and heterotic sides match.

Wilson line Interpretation for Polytope with more than Two Fibers
The

Conclusion
Here we showed how to interpret particular elliptically fibered K3 surfaces directly from a Wilson line parametrization of the dual E 8 × E 8 and SO(32) heterotic strings. We constructed a graph of polytopes with two fibers where the links can be considered in this particular case as inclusion relations between the moduli spaces of the elliptically fibered K3s associated to each fiber. Because in some limit of the moduli space we now recover the compactifications of F-theory dual to the ones of heterotic strings on a two torus with no Wilson lines, we can interpret the additional complex parameters in the moduli spaces as being dual to Wilson line moduli. They appear in the compactifications of F-theory as additional monomials in the hypersurface equation defining the K3 surface. Therefore there seem to be a close link between monomials on the F-theory side, and Wilson line moduli in the heterotic theories. Adding only one monomial to the hypersurface equations which correspond to the E 8 × E 8 and SO(32) compactifications, we can restrict the number of complex parameters in the moduli space to three i.e. one Wilson line moduli for the heterotic strings. This makes finding possible enhancements more convenient, and a study similar to the ones presented in [33] and upcoming paper [32] should give more insights on the Wilson line description of elliptically fibered K3 surfaces. Moreover, we also showed that this construction can be adapted to polytopes with more than two fibers. We were able to find heterotic duals to the three fibers of the polytope M 2 with one moduli. Using the various SageMath programs we developed as well as graphs of polytopes, we hope to find more insights on K3s with three fibrations and more. Finally here we focused on compactifications of F-theory and heterotic strings to eight dimensions. We expect that the monomial and Wilson line moduli duality should be valuable in studying compactifications to lower dimensions, both involving K3 surfaces and more generally Calabi-Yau threefolds.

B Program 2: Weierstrass Models
Here we present again the typical output of the second computer program. Again on the first line one just specifies in a list the reflexive polytopes # (associated to ReflexivePolytope(3,#). The output is the hypersurface equation for every fibration of the K3 surface as well as the corresponding Weierstrass models (upon a choice fiber described in Figure 2 for F13, F15 and F16. In another file are saved all the hypersurface equations in Sagemath form.
The following is the typical Latex output when putting as an input "[476]".

Polytope M476
Number of different Fiber is 2