Recent advances and open questions on the susy structure of the chiral de Rham complex

Abstract

We review different constructions of supersymmetry algebras of the chiral de Rham complex of special holonomy manifolds. We describe a mechanism to topologically twist the complex and attach mock modular forms to special holonomy manifolds. We describe the connection of these algebras with quantum Hamiltonian reduction.

(0,2) Mirror Symmetry on homogeneous Hopf surfaces

“…for solutions of (1.6) such that ε is holomorphic (Definition 2.25). We expect similar results for the cases dim V + ∈ {4k, 7, 8}, in relation to other interesting vertex algebras in the literature [34] (see Remark 2.15 and Section 4.5).…”

“…There has been some progress in Borisov's Programme in recent years, mainly due to the work of Heluani, Kac, and their collaborators [34], but this proposal is far from being complete: even for the case of algebraic Calabi-Yau manifolds, the construction of N = 2 structures on the chiral de Rham complex often involves the SU(n)-holonomy metric provided by Yau's theorem, which we do not know explicitly. For the case of pairs (X, E) the replacement of the metric is expected to be a solution of the Hull-Strominger system [39,55], a fully non-linear system of partial differential equations about which we know very little at present.…”

Topological G2 and Spin(7) strings at 1-loop from double complexes

“…where ρ i means the restriction of the polyform ρ to its degree i component, and σ : S → S is the automorphism defined by 35 σ(µ k ) = (−1) k(k+1)/2 µ k . (C.16)…”

“…Hence a choice of G is equivalent to a choice of decomposition (C.19). 35 This convention is different than the one chosen in e.g. [43].…”

“…Note that Fiset and Gaberdiel[30] show that the cohomology of G ↓ −1/2 is not restricted to the chiral ring and so it cannot be the exact BRST operator which captures the geometry of the target space (though the honest BRST operator is likely related to G ↓ −1/2 ). For our purposes, we need only the identification of the complex of special NS states as later we will identify the analogue of the correct BRST operator in the target space.9 Though see, for example,[31][32][33][34][35].…”