Mirror symmetry for G 2-manifolds: twisted connected sums and dual tops

Abstract

Recently, at least 50 million of novel examples of compact G 2 holonomy manifolds have been constructed as twisted connected sums of asymptotically cylindrical CalabiYau threefolds. The purpose of this paper is to study mirror symmetry for compactifications of Type II superstrings in this context. We focus on G 2 manifolds obtained from building blocks constructed from dual pairs of tops, which are the closest to toric CY hypersurfaces, and formulate the analogue of the Batyrev mirror map for this class of G 2 holonomy manifolds, thus obtaining several millions of novel dual superstring backgrounds. In particular, this leads us to conjecture a plethora of novel exact dualities among the corresponding 2d N = 1 sigma models.

Compact, singular G2-holonomy manifolds and M/heterotic/F-theory duality

“…3 We can compute the Hodge numbers of Z as well the ranks of the lattices N and K in purely combinatorial terms. The result is that h i,0 (Z) = 0 and [11,20] …”

“…The faces Θ • [1] and Θ [1] are the unique pair of faces on ∆ F , ∆ • F obeying Θ [1] , Θ • [1] . Note that exchanging the roles of ♦ and ♦ • exchanges h 2,1 and |K| but keeps b 2 + b 3 invariant, which is relevant for G 2 mirror symmetry [11].…”

“…In the case of * (σ n (Θ [k] )) this refers to the points on ♦ • in the relative interior of σ n (Θ [k] ). Finally, in the expression for |N |, the sum only runs over vertically embedded (ve) faces Θ • [1] of ∆ • F , which are those faces bounding a face Θ • [2] of ♦ • which is perpendicular to F = ν ⊥ e , see [11,20] for a detailed exposition. The faces Θ • [1] and Θ [1] are the unique pair of faces on ∆ F , ∆ • F obeying Θ [1] , Θ • [1] .…”

“…These so-called twisted connected sum (TCS) G 2 -manifolds have a K3-fiberation over a three-manifold. In the physics literature, M-theory compactifications on such geometries were studied in [8][9][10] and in string theory compactifications [11].…”

Marginal deformations of heterotic G2 sigma models

“…Topological twists of the (1, 1) models are also closely related to the generalisation of mirror symmetry to the G 2 setting [39,[63][64][65][66][67]. Since there is a notion of heterotic mirror symmetry in terms of quantum sheaf cohomology, see [68,69] with references therein, one might speculate in analogy that a similar generalisation to the (0, 1) heterotic G 2 setting exists.…”