Abstract
Motivated by the appearance of fractional powers of line bundles in studies of vector-like spectra in 4d F-theory compactifications, we analyze the structure and origin of these bundles. Fractional powers of line bundles are also known as root bundles and can be thought of as generalizations of spin bundles. We explain how these root bundles are linked to inequivalent F-theory gauge potentials of a G4-flux.While this observation is interesting in its own right, it is particularly valuable for F-theory Standard Model constructions. In aiming for MSSMs, it is desired to argue for the absence of vector-like exotics. We work out the root bundle constraints on all matter curves in the largest class of currently-known F-theory Standard Model constructions without chiral exotics and gauge coupling unification. On each matter curve, we conduct a systematic “bottom”-analysis of all solutions to the root bundle constraints and all spin bundles. Thereby, we derive a lower bound for the number of combinations of root bundles and spin bundles whose cohomologies satisfy the physical demand of absence of vector-like pairs.On a technical level, this systematic study is achieved by a well-known diagrammatic description of root bundles on nodal curves. We extend this description by a counting procedure, which determines the cohomologies of so-called limit root bundles on full blow-ups of nodal curves. By use of deformation theory, these results constrain the vector-like spectra on the smooth matter curves in the actual F-theory geometry.
Highlights
String theory elegantly couples gauge dynamics to gravity
In globally consistent F-theory constructions with the exact chiral spectra of the Standard Model and gauge coupling unification [31], the vector-like spectra on the low-genus matter curves are encoded in cohomologies of a line bundle, which are identified with a fractional power of the canonical bundle
This work is motivated by the frequent appearance of fractional powers of line bundles when studying vector-like spectra of globally consistent 4d F-theory Standard Models with three chiral families and gauge coupling unification [31]
Summary
String theory elegantly couples gauge dynamics to gravity. This makes string theory a leading candidate for a unified theory of quantum gravity. Enormous efforts have been undertaken to achieve this goal Many of these models concentrated on perturbative corners of string theory, such as the E8 × E8 heterotic string [1,2,3,4,5,6,7,8] or intersecting branes models in type II [9,10,11,12,13,14,15] (see [16] and references therein). These perturbative models were among the first compactifications from which the Standard Model gauge sector emerged with its chiral or, in the case of [4, 5], even the vector-like spectrum These constructions are limited due to their perturbative nature in the string coupling, and they typically suffer from chiral and vectorlike exotic matter. The first globally consistent MSSM constructions are [4, 5] (see [17, 18] for more details on the subtle global conditions for slope-stability of vector bundles)
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