Higgs connection

The field theoretic Higgsing connecting the two theories discussed in the previous Section 2.3 is a realization of the geometric extremal transition connecting the two CY manifolds X_F13 andX_F11. This extremal transition is induced by a blow-up/down in the toric ambient space of the fiber PF_i and a complex structure deformation of the hypersurface constraint. Both these transformations are needed in order to preserve that the fiber is a genus-one curve.

On the level of the toric diagram the transition corresponds to deleting/adding a node in the polyhe-dron and adding/deleting a node in the dual polyhedron.

In order to find the node which has to be deleted for a certain Higgsing we definetoric matter. Toric matter is the matter located atsi= sj =0 wheresiandsjneed to be neighboring nodes on the boundary of the dual polyhedron. To make that clearer consider the case ofF₁₃depicted in Figure 2.9. From Table 2.5 we see that all multiplets are at loci where s_i = s_j = 0. But from the dual polyhedron depicted in Figure 2.9 we find that only two of them are toric. In the cases₁= s₃=0 the nodes are not neighboring and in the case s₆ = s₉ = 0 the node s₆ is not on the boundary. Thus there are two possible toric Higgsings associated to the multiplets (2,1,4) and (1,2,4).

To find the node in polyhedron F₁₃ which has to be blown-down in order to induce a Higgsing of the (2,1,4) multiplet we have to make a blow-up in the dual polyhedron F₄ (see Figure 2.9) at the line connecting the two nodes si andsj of the toric matter. In this case between s₁ and s₉. The line connecting the nodes s₁ and s₉ is dual to the nodee₄ which is deleted (blown-down) in the original polyhedronF₁₃by blowing-up between the two nodess₁ands₉in the dual polyhedronF₄.

Figure 2.9: (a) PolyhedronF13and its dual polyhedronF4. Blue denotes the locus of (2,1,4) and red the locus of (1,2,4). The transitions toXF11can be triggered by Higgs fields in the representation (2,1,4) (b) and (1,2,4) (c).

Charge identification

To find a formula for the U(1) charge in the model after the Higgsing we first have to define a torsional Shioda map (2.15) inX_F13²

σ( ˆst)=St−S₀−[K⁻¹_B ]+ 1

2D^SU(2)₁ ¹+ 1

2D^SU(2)₁ ²+ 1 2

D^SU(4)₁ +2D^SU(4)₂ +D^SU(4)₃

. (2.86)

It can be checked from the intersections of the matter nodes with the torsional section ˆst depicted in Table 2.5 that all matter multiplets have charge 0 under the Shioda mapσ( ˆst). This section can now be expressed in terms of the homology classes of the individual divisors

σF₁₃( ˆst)=[e5]−[e4]−[K_B⁻¹]+ 1

2([s1]−[v])+ 1

2[e1]+ 1

2(([s9]−[u]−[e2]−[e3])+2[u]+[e2]). (2.87) On the contrary the Shioda map ofXF₁₁ in terms of the homology classes is given by

σF11( ˆs₁)=[˜e₄]−[v]−[K⁻¹_B ]+ 1

2[e1]+ 1

3([e2]+2[u]). (2.88) Now we consider the Higgsing with the multiplet (2,1,4). To do this we have to blow down the node [e4] inF₁₃as depicted in Figure 2.9. This blow down requires thus [e4] = 0. Then we can express the Shioda map ofX_F11 in terms of the blown down torsional Shioda map and the Cartan divisors ofX_F13.

2This step is not necessary in simpler models without torsion but U(1) symmetries.

This leads to

σF₁₁( ˆs₁)=σF₁₃( ˆst)+1

2D^SU(2)₁ ¹−[s₁]+ 1 6

−3D^SU(4)₁ −2D^SU(4)₂ −D^SU(4)₃

. (2.89)

The base divisor [s1] does not contribute to the charges and as we have seen above all multiplets inX_F13 are uncharged underσF₁₃( ˆs_t). After identifyingT₃^SU(2)1 = ¹₂D^SU(2)₁ ¹ andT₁₅^SU(4) = ¹₆

3D^SU(4)₁ +2D^SU(4)₂ +D^SU(4)₃

we find the charge formula

Q=T₃^SU(2)1 −T₁₅^SU(4). (2.90)

This formula coincides with the field theory expectation³.

To calculate the charges in a Higgsing with the multiplet (1,2,4) we have to take two additional effects into account: First the toric diagram needs to be reflected, see Figure 2.9. This effectively amounts to the interchange of [u] and [e₁] with [e₃] and [v] respectively. The second effect we have to deal with is that we need to change the zero section. This leads to a redefinition of one Cartan divisor of each gauge group in XF₁₃. Taking all the described effets into account we can express the Shioda map ofXF₁₁ in terms of the divisors ofX_F13 in the following way

σF₁₁( ˆs₁)=−σF₁₃( ˆst)+2[K_B⁻¹]+ 1

2[s1]−1

2[s₃]+ 1

2D^SU(2)₁ ² + 1 6

−3D^SU(4)₁ −2D^SU(4)₂ −D^SU(4)₃ . (2.91) Using the same definitions of the generators of the gauge groups as above we find the following formula for the charges in the effective field theory ofX_F11 after the Higgsing

Q=T₁₅^SU(4)−T₃^SU(2)2. (2.92)

SL(2,Z)transformations

Apart from the field theoretic Higgsing obtained by assigning a vev to the (2,1,4) multiplet in XF₁₃

there is another possible toric Higgsing associated to a vev in the (1,2,4) representation. On the level of the toric diagram this corresponds to blowing down the variable e₅ instead of e₄. Both possible Higgsings lead to theories with the same multiplets but with different multiplicities. In order to match the theory obtained after giving a vev to the (1,2,4) representation with the effective field theory ofXF₁₁

we observe that the toric diagrams do only agree up to mirroring along thee₂-waxis (see Figure 2.9).

In general anS L(2,Z) transformation can occur. The identification has to map the structures onto each other. That means that rational points have to be mapped to rational points and internal points on lines have to be mapped to internal points on lines. Concretely in this case we have to identify the nodes given in Table 2.8.

node inF₁₃ e₄ e₃ e₂ u e₁ w v node inF₁₁ e₄ u e₂ e₃ v w e₁

Table 2.8: The nodes which have to be identified for a proper matching of the theories after the Higgsing with a multiplet in the (1,2,4) representation.

3In [32] this charge formula is given with an overall factor of -1 which is irrelevant in 6D.

In order to identify the points the homology classes need to agree. Thus we get a linear system of equations connecting the homology classes ofF₁₁with the homology classes ofF₁₃after the Higgsing.

Solving this system gives a shift in the bundlesS₇andS₉

S₇→2[K⁻¹_B ]− S₇, S₉ → S₉, (2.93)

and correspondingly in the matter multiplicities.

Allowed region

As discussed in Section 2.3 a specific base fixes the divisor classesS₇, S₉ and [K_B⁻¹]. Since we are dealing with CY threefolds in this section we use in the following the baseB = P². This gives the following identifications

S₇ =n₇H_B, S₉=n₉H_B, [K⁻¹_B ]=3HB. (2.94) The requirement that the sections (2.42) in the hypersurface equation are effective divisors leads to constraints onn₇ andn₉. Since smaller polyhedra like F₁₁ compared toF₁₃have more monomials in the hypersurface equation their values for (n₇,n₉) are more constrained than for larger polytopes. This means that the allowed region of possible values (n7,n₉) ofXF₁₁ is contained in the allowed region of X_F13. Thus there are values for (n₇,n₉) which are allowed in X_F13 but not allowed in X_F11. In Figure 2.10 the allowed region ofXF₁₃ is depicted. The values which are not in the allowed region ofXF₁₁ are colored blue.

1 2 3 4 5 6 n9

-2 2 4 6 8 n7

Figure 2.10: Allowed region ofX_F13. The values for (n₇,n₉) which are not in the allowed region ofX_F11are colored blue. At the green dots the SU(4) is not present. At the yellow dots the SU(2)₁ is not present and at the orange dot both SU(2)’s are not present.

Hence geometrically a transition at the values for (n7,n₉) which are not allowed inXF₁₁is not possible.

The field theoretic reason for this is as follows: In order to preserveN = 1 supersymmetry and thus be able to connect two SUGRA theories we need a D-flat potential. In [87] it was observed that this requires two Higgs fields in six dimensions.

As an example take the HiggsingX_F13 →X_F11by giving a vev to the (2,1,4) multiplet. This multiplet

has the multiplicity

(3[K_B⁻¹]− S₇− S₉)S₉=(9−n₇−n₉)n₉. (2.95) Thus outside the allowed region of X_F11 the multiplicity is smaller then two and a Higgsing is not possible. Nevertheless the Higgsing with the other toric multiplet (1,2,4) which requires a redefinition of the bundles as described in Subsection 2.4.1 is possible.

Additionally we see that at some points at the boundary of the allowed region ofX_F11the multiplicity of the (2,1,4) multiplet is also smaller than two. Specifically this happens at the green, yellow and orange dots in Figure 2.10. At the green dots the SU(4) divisor in XF₁₃ and the SU(3) divisor in XF₁₁

are absent. At the yellow dots the SU(2)₁inX_F13 is not present and at the orange dots both SU(2)’s in X_F13 and the SU(2) inX_F11 are not there. In all cases there are no states in the (2,1,4) representation.

However in all cases the rank of the gauge groups ofXF₁₁andXF₁₃ agree and an adjoint Higgsing with Higgses in the (3,1,1) or the (1,1,15) representations is possible.