5.3 Elliptic curves as toric hypersurfaces
5.3.1 Points on genus-one curves
In the following we are particularly interested in points on the elliptic curve i.e. a point of the ambient space that solves the given hypersurface constraint (5.34). Two kinds of points will be of main interest:
• A rational point, that solves the hypersurface constraint with rational coefficients. These give rise to well defined divisors in the ambient space that intersects the elliptic curve exactly once.
• A multi-point, that is a multi valued solution of the hypersurface constraint. Hence its divisors intersects the elliptic fiber multiple times.
When a given genus-one curve admits at least one rational point, this point can be used to define a zero-point on the curve and in this case we will call it an elliptic curve. However in the following we also encounter curves that do not have rational points and are thus not an elliptic curve. However these curves can be mapped to itsJacobianthat have the same complex structure modulusτand have a zero-section and are thus elliptic curves. The set of rational points together with the zero-point, form a group under addition7. It can be shown that this group is finitely generated [72] and can therefore be decomposed intor-free andn-discrete parts: Zr⊕Zk1, . . . ,Zkn. When we promote the elliptic curve to a fibration, the rational points become sections that generate U(1)rgauge symmetries in the field theory, whereas thentorsion parts give quotient factors of the total gauge group [73]. We summarize these concepts in a convenient example, that will be also of use later on: The ambient space we consider is given byF11 with the polyhedron depicted in Figure5.3. The Stanley-Reisner ideal is read offto be
Figure 5.3: Polyhedron F11 and the corresponding coordinates. Edges in the dual polyhedron correspond to coordinate monomials, where we have setei=1 for convenience. The choices for the inequivalent rational points are denoted with a dot.
S R={ue1,uw,uv,ue3,e4w,e4v,e4e3,e4e2,e1v,e1e3,e1e2, we3, we2, ve2}. (5.35) The elliptic curve is specified by the vanishing of the hypersurface
pF11 = s1e21e22e3e44u3+s2e1e22e23e24u2v+s3e22e23uv2+s5e21e2e34u2w+s6e1e2e3e4uvw+s9e1vw2. (5.36) Indeed we find that the hyperplane Equation (5.36) is charged as the sum under each scaling or put differently transforms in the anti-canonical bundle
K−1F
11 =O(3H−E1−E2−E3−E4). (5.37)
7For an example of the geometric group law, see appendix A of [71].
5.3 Elliptic curves as toric hypersurfaces
At next we can find the rational points on the curve by considering the coordinates ofF11, [u:v:w:e1 :e2:e3:e4] that admit fiveC∗scalings summed up in the Table 5.1. We can then find the points on the curve, by
Coord. H E1 E2 E3 E4
u 1 -1 -1 0 -1
v 1 0 -1 -1 0
w 1 -1 0 0 0
e1 0 1 0 0 -1
e2 0 0 1 -1 0
e3 0 0 0 1 0
e4 0 0 0 0 1
Table 5.1: Summary of the scalings of the coordinates obtained from linear equivalences and the five divisor classesH,E1,E2,E3,E4.
setting one coordinate to zero and use theC∗scalings to set as many other coordinates to 1 as possible and then solve for the other coordinates8. In the following we give the rational points obtained by that procedure when we use the scalings to set all remaining coordinates that appear with powers greater than one in (5.36).
ˆ
s0 : [1 : 0 :s1: 1 : 1 :−s5: 1], (5.38) ˆ
s1 : [1 :s5 : 1 : 1 :−s9: 0 : 1], (5.39) ˆ
s2 : [s9: 1 : 1 :−s3: 1 : 1 : 0] . (5.40) Using the group law it can be shown that there is a linear relation among the sections and hence there are only two independent rational points. Hence the Mordell-Weil group has rank one [74,71]. Note that we can also find non-rational points by settingw =0 and use the scalings to setv = e1 = e2 = e3 = 1.
The hypersurface (5.36) then becomes
s1e44+s2e24u2+s3u=0, (5.41) that has the solutionse24 = −s3±
√
s2
3−4s1s2u6
2s2u2 that is clearly not a rational expression in the coefficients.
5.3.2 From elliptic curves to Elliptic fibrations
In the next step we have to promote the elliptic curve to an elliptic fibration of an n+1 fold Yn+1
→π Bn. (5.42)
We consider first the Weierstrass form of the elliptic curve, given in Equation (5.24). It is crucial that now the coordinates transform as functions of the base coordinates. More mathematically put they trans-form in powers of the bundle classes LB of the base, that is the anti-canonical bundle K−1B ofBn. As there is one scaling relation among the coordinates, we choose the coordinates (z,x, y) to transform in (L0,L2,L3). But the Weierstrass equation has to transform in a well defined way and hence the coeffi-cients f andghave to transform in the powersL4BandL6B.
8Note that this procedure only guarantees to find thetoric pointsand do not have to be all rational points.
Yn+1
T2
σ0 σ0
b
b0 Bn
Figure 5.4: Picture of an elliptically fibered Calabi-Yau n+1-fold. The zero-sectionσ0 identifies a point on the elliptic fiber over every point in the baseBn.
Another important fact is that a fibration needs at least one exceptional pointOon the fiber, that dis-tinguishes the fiber part from the base part. This point is usually referred to as the zero point, that gets promoted to thezero sectionσ0of the elliptic curve.
The zero sectionσ0gives an embedding of the base into the fibrationBn,→Yn+1that acts trivial on the base when composed with the projection:π◦σ0 =1B.σ0can then be used to globally keep track of the torus over a generic point in the base, see Figure5.4. We have seen, that this point in the Weierstrass form given as (z,x, y) = (0, λ2, λ3) is independent of f andg and is well defined everywhere over the base. In this case, the zero-section is holomorphic however it happens that even the zero section might not be well defined over certain codimension two loci, where it can wrap whole fiber components and is not just a point [75]. In this case, the zero-section is only rational but it is still sufficient for an F-theory compactification. In the following we generalize this procedure to all the other 16 polyhedra that we have discussed before. When we have at least one rational point, we will use this as the zero-section and if not, we will use the Jacobian fibration that possess one.
As the first step we promote the coordinatesxito be sections of the base bundlesLB. As we havem scal-ing relations form+2 coordinates we can use them to let only two coordinates transform non-trivially under the base bundlesLB. In general the coordinates transform only in two different base bundles that we will take to beS7andS9. When we choose the transformation of the coordinates properly then the monomial coefficientss7ands9in the hyperplane equation transforms exactly in the two bundles base bundlesS7 andS9motivating their names. We can then use adjunction again to fix the anti-canonical class of the base space in order to make the whole fibration a Calabi-Yau n+1 fold.
Lets get concrete by going on with the example ofF11and turn it to a whole fibration over a general base. We now choose the coordinatesu∈ O(H−E1−E2−E4+S9+[Kb]) andv∈ O(H−E2−E3+S9−S7) whereas all the other ones have not changed compared to Table5.1. Again the adjunction formula tells us that the sum of all divisors has to be trivial, which on the other hand fixes the anti-canonical class of the fiber in i.e.
3H−E1−E2−E3−E4+2S9− S7+[KB]=0, (5.43)
→[KB]−1 =3H−E1−E2−E3−E4+2S9− S7. (5.44)
5.3 Elliptic curves as toric hypersurfaces
Remember that the hypersurface constraint has to transform in the anti-canonical bundle given above.
This we can use to read offthe appropriate transformations for the si coefficients summed up in Table 5.2. Note that especially the coefficient of the middle point in the dual polyhedron has to scale with the
section divisor class s1 3[KB−1]− S7− S9 s2 2[KB−1]− S9 s3 [KB−1]+S7− S9 s5 2[KB−1]− S7 s6 [K−1B ]
s9 S9
Table 5.2: Line bundle assignments of the monomial coefficients for the whole space to become a Calabi-Yau n+1-fold.
anti-canonical class. These assignments will be of particular interest, as they will give the multiplicity of the matter curves that are found at loci such as si = sj=0.
5.3.3 Jacobian Fibrations
In the very beginning we have stressed the point that an F-theory compactification needs a zero-section in order to identify the elliptic fibration. As it was first believed that this section should be holomorphic, this paradigm was relaxed to only allow for rational sections. However we can relax this condition even more i.e. the fibration does not need to have a section at all [76, 77]. In that case, we call the fibration agenus-onefibration as opposed to anelliptic fibration that always possesses a zero section.
However we can use the mathematical fact that we can use for every genus-one curveC its Jacobian J(C) constructed by
J(C)=C/(Z+τZ) , (5.45)
with the same τas the genus-one curveC. Furthermore, the invariant Jacobi function J, see (5.27) coincide. The main point is that the Jacobian always has a Weierstrass [78] and in particular a zero point that becomes the zero-section.
It is naturally to expect that F-theory on the genus-one fibrationCas well as its JacobianJ(C) should give the same compactification given thatτandJare the same. The group of curves with the isomorphic Jacobians is defined to be the Tate-Shafarevich groupof a curve and in general is a subgroup ofZ/Zk
wherekis the smallestk-section of the curveC. Hence when there exists a section onC,k =1 and the Tate-Shafarevich group is trivial. In [79,80] it was shown that in that the five dimensional M-theory compactification of genus-one curves differ from each other and correspond to different elements of the Tate-Shafarevich group. However, in the F-theory limit all compactifications coincide. In Chapter6we also encounter non-trivial examples of genus-one curves.