Abstract
In this note we consider smooth elliptic Calabi-Yau four-folds whose fiber ceases to be flat over compact Riemann surfaces of genus g in the base. These non-flat fibers contribute Kähler moduli to the four-fold but also add to the three-form cohomology for g > 0. In F-/M-theory these sectors are to be interpreted as compactifications of six/five dimensional mathcal{N} = (1, 0) superconformal matter theories. The three-form cohomology leads to additional chiral singlets proportional to the dimension of five dimensional Coulomb branch of those sectors. We construct explicit examples for E-string theories as well as higher rank cases. For the E-string theories we further investigate conifold transitions that remove those non-flat fibers. First we show how non-flat fibers can be deformed from curves down to isolated points in the base. This removes the chiral singlet of the three-forms and leads to non-perturbative four-point couplings among matter fields which can be understood as remnants of the former E-string. Alternatively the non-flat fibers can be avoided by performing birational base changes analogous to 6D tensor branches. For compact bases these transitions alternate all Hodge numbers but leave the Euler number invariant.
Highlights
The three-form cohomology leads to additional chiral singlets proportional to the dimension of five dimensional Coulomb branch of those sectors
In the toric setup we demonstrate this in appendix A for geometry Y3,c and show that it admits the same Hodge numbers (3.14) as three-fold Y3,b
First we have shown how non-flat fibers in codimension two contribute to the Hodge numbers and in particular to the three-form cohomology
Summary
One of the main goals of this note is to give a physical explanation to certain Hodge numbers of elliptic Calabi-Yau four-fold X4 with non-flat fibers in the context of F/Mtheory. From a physics perspective this makes sense as the theories might be viewed as compactifications of the 6D/(5D) theories on Riemann-surfaces Cgα to [61, 62].4 This physics interpretation allows us to express the contributions of non-flat fibers to h1,1(X4) and h2,1(X4) in terms of 6D/5D SCFT data from eq (2.11) and eq (2.8) as nc h1n,o1n-flat(X4) = dim(Coulomb5D)α , α nc h2n,o1n-flat(X4) = dim(Coulomb5D)αgα , α (2.12) (2.13). Our main interest though are the chiral singlets that are inherited from h2,1(X4) that do not come from the base and in our case are given by h2,1(Y4) − h2,1(B3) = h2,1(X4)non-flat These are obtained from the expansion of the M-theory C3-form and are denoted by N β with β = 1 . Instead this review should serve as a motivation of why these singlets are interesting and that non-flat fibrations naturally produce them
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