Constraining the Kähler Moduli in the Heterotic Standard Model

Abstract

Phenomenological implications of the volume of the Calabi-Yau threefolds on the hidden and observable M-theory boundaries, together with slope stability of their corresponding vector bundles, constrain the set of Kähler moduli which give rise to realistic compactifications of the strongly coupled heterotic string. When vector bundles are constructed using extensions, we provide simple rules to determine lower and upper bounds to the region of the Kähler moduli space where such compactifications can exist. We show how small these regions can be, working out in full detail the case of the recently proposed Heterotic Standard Model. More explicitely, we exhibit Kähler classes in these regions for which the visible vector bundle is stable. On the other hand, there is no polarization for which the hidden bundle is stable. December, 2005 Recently, phenomenologically interesting Calabi-Yau compactifications of the heterotic string have appeared in the literature [2], [5]. Using certain elliptically fibered threefold with fundamental group Z Z 3 × Z Z 3 , and an SU (4) × Z Z 3 × Z Z 3 instanton living on the visible E 8 -bundle, give rise to an effective field theory on IR 4 which has the particle spectrum of the Minimal Supersymmetric Standard Model (MSSM), with no exotic matter but an additional pair of Higgs-Higgs conjugate superfields. In these models, vector bundles are constructed using vector bundle extensions, which correspond to Hermitian Yang-Mills connections when they are slope-stable. We use this specific construction to exemplify how a systematic selection of realistic Kähler moduli can be done. 1 The organization of the paper is as follows: section 2 contains an outline of the natural criteria for selecting Kähler moduli in realistic Calabi-Yau compactifications of the heterotic string. In section 3, we analyze the case of the Heterotic Standard Model, describe the geometry of the elliptic Calabi-Yau and construct its Kähler cone. Section 4 provides lower and upper bounds to the region of the Kähler cone that makes stable the observable vector bundle of the HSM. In such construction we find a destabilizing sub-line bundle for the hidden vector bundle, and exhibit Kähler classes that make stable the visible one. 1 Recently, Donagi and Bouchard [8] have also proposed an independent CY compactification of the heterotic string with the spectrum of the MSSM and no exotic matter, using a different Calabi-Yau with an explicitly slope-stable vector bundle in the observable sector. It would be also interesting to study in detail these questions with the vector bundle which has just appeared in [4], on the same CY [5].3 Being rigorous, we should work with the dimensionfull measure α ′ ω 3 , although this will be irrelevant for our purposes because α ′ factorizes out in the formulae that we use. In the small volume limit this approximation can fail, and we should use conformal field theory to give a more accurate estimation.

Proton stability in grand unified theories, in strings and in branes

“…However, it has been pointed out [334] that the hidden sector bundle of the work of Ref. [333] is not slope-stable which would require changing the hidden sector and will result in different phenomenological properties [333].…”

Schoen manifold with line bundles as resolved magnetized orbifolds

“…Concretely, for the orbifold standard embedding we find for this orbifold 3 + 2 · 8 = 19 chiral 27-plets and 3 + 2 · 8 = 19 chiral 27-plets of E 6 , i.e. the Hodge numbers are (19,19). In fact, because supersymmetry is broken non-locally, any DW(0-2) orbifold is non-chiral, independently of the choice of shifts and Wilson lines.…”

“…Finally, one may consider D 1,00 and D 2,00 as the divisor classes associated to the rational elliptic surfaces B ′ and B, respectively. Indeed, the Euler number of D 1,00 equals χ(D 1,00 ) = 12, and, since D 1,00 E 2 r = −2, D 1,00 contains the same fixed points of g θ as B ′ , see (19). Hence, we may identify B ′ = D 1,00 and similarly B = D 2,00 .…”

“…Consequently, the number of Kähler deformation equals h 11 = 2 · 10 − 1 = 19. In summary the Hodge numbers of X are (19,19).…”

Green-Schwarz mechanism in heterotic (2,0) gauged linear sigma models: torsion and NS5 branes

“…As smooth CY spaces with stable bundles are very complicated to construct, finding the MSSMlike models has proven very difficult, especially because of the issue of bundle stability [31]. Ongoing efforts of refs.…”