Half-hypermultiplets and incomplete/complete resolutions in F-theory

Abstract

We consider resolutions of codimension-two enhanced singularities from SO(12) to E 7 and from E 7 to E 8 in six-dimensional F-theory, where a half-hypermultiplet arises for generic complex structures achieving them. The exceptional fibers at the enhanced point exhibit different structures depending on how the colliding 7-brane approaches the stack of gauge 7-branes, as previously observed by Morrison and Taylor in the case of the enhancement from SU(6) to E 6. When the colliding brane approaches them as O(s), where s is the coordinate of the base space along the gauge 7-branes, the resolution process ends up with fewer exceptional fibers than naively expected from the Kodaira classification, with a non-Dynkin intersection matrix including half-integral intersection numbers. We confirm that the exceptional fibers at the enhanced point form extremal rays of the cone of the positive weights of the relevant pseudo-real representation, explaining why a halfhypermultiplet arises there. By altering the ordering of the singularities blown up in the process, we obtain, for both SO(12) → E 7 and E 7 → E 8 , the intersection diagram on every other row of the corresponding box graphs. We present detailed derivations of the intersection diagrams of the exceptional fibers at the singularity enhanced points by examining how an exceptional curve is lifted up on the chart arising due to the subsequent blowing-up process. When the colliding brane approaches the stack of branes as O(s 2), we obtain additional conifold singularity at the enhanced point, which completes the full Dynkin diagram of the enhanced group as was found previously.

Orders of vanishing and U(1) charges in F-theory

“…The two spinor representations of so( 12) both occur at codimension-two loci where the singularity type enhances from I * s 2 (Spin( 12)) to III * (E 7 ). After resolution, the codimension-two singularities can either be incompletely or completely resolved [71,93]. In 6D models, an incomplete resolution locus supports a half-hypermultiplet of spinor matter.…”

“…The 56 representation is pseudoreal, and if the supported 56 matter is uncharged under the u(1), it can be realized as half-hypermultiplets in 6D models. The fibers at such loci would take the shape of an incompletely resolved E 8 diagram [71,93]. However, if the 56 matter has a nonzero u(1) charge, the total representation, which includes the u(1) charge, is no longer pseudoreal.…”

Non-split singularities and conifold transitions in F-theory

“…Let us explain the general procedure of how this is done by taking the present case as an example. Our notation is similar to the one used in our previous paper [26].…”

“…This means that the section f vanishes on the divisor Dv, therefore the fiber type cannot be I5 in this case [19]. 4 In the following, we will refer to a point on the base of the dP9 fibration as a "G point" if Kodaira's classification of the singular fiber just over that point corresponds to a Lie algebra G. 5 We note that analogous phenomena occur on codimension-two loci in 6D split models, where the gauge symmetries are SU(6), SO (12) and E7, and the singularities are enhanced from those to E6, E7 and E8, respectively [17,26,27]. The matter multiplets generated there are half-hypermultiplets in some pseudo-…”