Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The $\mathcal L_B$-cohomology on compact torsion-free $\mathrm{G}_2$
  manifolds and an application to 'almost' formality   We study a cohomology theory $H^{\bullet}_{\varphi}$, called the $\mathcal
L_B$-cohomology, on compact torsion-free $\mathrm{G}_2$-manifolds. We show that
$H^k_{\varphi} \cong H^k_{\mathrm{dR}}$ for $k \neq 3, 4$, but that
$H^k_{\varphi}$ is infinite-dimensional for $k = 3,4$. Nevertheless there is a
canonical injection $H^k_{\mathrm{dR}} \to H^k_{\varphi}$. The $\mathcal
L_B$-cohomology also satisfies a Poincar\'e duality induced by the Hodge star.
The establishment of these results requires a delicate analysis of the
interplay between the exterior derivative $\mathrm{d}$ and the derivation
$\mathcal L_B$, and uses both Hodge theory and the special properties of
$\mathrm{G}_2$-structures in an essential way. As an application of our
results, we prove that compact torsion-free $\mathrm{G}_2$-manifolds are
'almost formal' in the sense that most of the Massey triple products
necessarily must vanish.

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The categories of this paper are: eess
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The categories of this paper are: hep-ph
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The categories of this paper are: cs
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The categories of this paper are: supr-con
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The categories of this paper are: patt-sol
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The categories of this paper are: chem-ph
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The categories of this paper are: q-bio
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The categories of this paper are: econ
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The categories of this paper are: q-alg
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The categories of this paper are: mtrl-th
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The categories of this paper are: nucl-th
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The categories of this paper are: astro-ph
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The categories of this paper are: q-fin
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The categories of this paper are: hep-lat
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The categories of this paper are: gr-qc
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The categories of this paper are: adap-org