Abstract
It is now well known that the moduli space of a vector bundle for heterotic string compactifications to four dimensions is parameterized by a set of sections of a weighted projective space bundle of a particular kind, known as Looijenga’s weighted projective space bundle. We show that the requisite weighted projective spaces and the Weierstrass equations describing the spectral covers for gauge groups E N (N = 4, · · · , 8) and SU(n + 1) (n = 1, 2, 3) can be obtained systematically by a series of blowing-up procedures according to Tate’s algorithm, thereby the sections of correct line bundles claimed to arise by Looijenga’s theorem can be automatically obtained. They are nothing but the four-dimensional analogue of the set of independent polynomials in the six-dimensional F-theory parameterizing the complex structure, which is further confirmed in the constructions of D 4, A 5, D 6, E 3 and SU(2) × SU(2) bundles. We also explain why we can obtain them in this way by using the structure theorem of the Mordell-Weil lattice, which is also useful for understanding the relation between the singularity and the occurrence of chiral matter in F-theory.
Highlights
We will show that the structure theorem of the Mordell-Weil lattice is useful for understanding the relation between the singularity and the occurrence of chiral matter in F-theory. (This new role of the Mordell-Weil lattice in F-theory was already briefly discussed in [6].) In the literature the relation between sections and the appearance of chiral matter is somewhat indirect
They are nothing but the four-dimensional analogue of the set of independent polynomials in the six-dimensional F-theory parameterizing the complex structure, which is further confirmed in the constructions of D4, A5, D6, E3 and SU(2) × SU(2) bundles
Where Γ denotes the space of the sections, L is the anti-canonical line bundle of the base B, and N is the “twisting” line bundle over B,1 characterizing the vector bundle of the dual heterotic string theory compactified on an elliptic Calabi-Yau Z of complex dimension one less, whose complex structure is identical to that of the elliptic fibration at z = ∞
Summary
We start with the construction of E8 bundles, following [1]. As pointed out there, it is well known that E8 bundles have exceptional features but the construction is important because it is the starting point of all the constructions of the vector bundles for other gauge groups of lower ranks. For A4 bundles, we have obtained a third-order equation in WP3(1,1,1,1) (which is singular but can be smooth by a blow up) with a1,0 ∈ Γ(L−5 ⊗ (L6 ⊗ M)1) a2,1 ∈ Γ(L−4 ⊗ (L6 ⊗ M)1) a3,2 ∈ Γ(L−3 ⊗ (L6 ⊗ M)1) a4,3 ∈ Γ(L−2 ⊗ (L6 ⊗ M)1) a6,5 ∈ Γ(L−0 ⊗ (L6 ⊗ M)1) This agrees with the set of Casimirs of A4 with degrees (with 0): 0, 2, 3, 4, 5. One can show that there is an agreement between the powers of the line bundles and the degrees of the independent Casimirs and the expansion coefficients of the highest weight Note that in these cases there is still a singularity at z = 0 to be further blown up
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