Abstract
A large number of examples of compact G2 manifolds, relevant to supersymmetric compactifications of M-Theory to four dimensions, can be constructed by forming a twisted connected sum of two building blocks times a circle. These building blocks, which are appropriate K3-fibred threefolds, are shown to have a natural and elegant construction in terms of tops, which parallels the construction of Calabi-Yau manifolds via reflexive polytopes. In particular, this enables us to prove combinatorial formulas for the Hodge numbers and other relevant topological data.
Highlights
Given the central position of M-Theory in the web of string dualities, a better understanding of M-Theory compactifications is expected to tie together supersymmetric compactifications of all of the weakly coupled string theories
The first constructions [1, 2] were non-compact examples which are asymptotically conical, see [3, 4] for more examples. These can be thought of as smoothed versions of singular cones, which is very interesting from the point of view of physics: in compactifications of M-Theory on manifolds of G2 holonomy, non-abelian gauge groups and matter arise from singularities of the compactification geometry and interesting singularities can be constructed from such conical manifolds
The construction of G2 manifolds as twisted connected sums [6,7,8] starts from K3-fibred threefolds Z which are called building blocks
Summary
Given the central position of M-Theory in the web of string dualities, a better understanding of M-Theory compactifications is expected to tie together supersymmetric compactifications of all of the weakly coupled string theories. (the singular versions of) these manifolds do not support interesting singularities on their own and their toroidal origin renders them locally flat This prevents to cut and paste the interesting singular non-compact examples studied in the literature as these are not asymptotically flat. As first discussed by Kovalev [6] and further elaborated on in [7, 8], one may construct compact manifolds of G2 holonomy by forming a twisted connected sum (TCS) of two appropriate ‘building blocks’ times a circle. These building blocks can be thought of as K3 fibrations over a P1 base. As the relevant theorems and constructions from [6,7,8] are nicely reviewed in the physics literature in [23], the discussion is limited to a minimum
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