Abstract
We realise the Shatashvili-Vafa superconformal algebra for G2 string compactifications by combining Odake and free conformal algebras following closely the recent mathematical construction of twisted connected sum G2 holonomy manifolds. By considering automorphisms of this realisation, we identify stringy analogues of two mirror maps proposed by Braun and Del Zotto for these manifolds.
Highlights
Another possible approach to constructing G2 manifolds is to start with the product of a Calabi-Yau 3-fold with a circle and take the quotient by an involution [13]
The clear relationship of our result with twisted connected sum (TCS) manifolds calls readily for mirror symmetry applications in type II string theory, in the sense recently suggested by Braun and Del Zotto [8, 9]
Each of these admit automorphisms and some of them are known to be related to T-duality and Calabi-Yau mirror symmetry in type II string theory [43]
Summary
We begin with a short informal account of the twisted connected sum (TCS) construction of 7-dimensional manifolds with holonomy group G2. We will not need much more mathematical rigour for our purposes It suffices to picture a distinguished real direction in X along which the manifold asymptotes to a translationinvariant Calabi-Yau X∞ with structure forms as given above. This cannot be done in the most naive way because this would lead to a manifold with infinite fundamental group It is an important fact in G2 geometry that a compact manifold M with torsion-free G2-structure has holonomy exactly G2 if and only if π1(M ) is finite. This is where the “twist” plays a useful role. This suggests that it remains adequate even in the limit of a short neck, i.e. in the phase of the conformal manifold where geometric control starts to disappear
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