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On a spectral sequence for twisted cohomologies. (English) Zbl 1306.55013

This is an effort of the authors to generalize the works in [R. Rohm and E. Witten, “The antisymmetric tensor field in superstring theory”, Ann. Phys. 170, No. 2, 454–489 (1986)] and [M. Atiyah and G. Segal, Nankai Tracts in Mathematics 11, 5–43 (2006; Zbl 1138.19003)] by working on the details of the claims on page 302 of [V. Mathai and S. Wu, J. Differ. Geom. 88, No. 2, 297–332 (2011; Zbl 1238.58023)].
Let \(H\) be a closed odd degree differential form. Then we define the twist differential \(D=d+H\wedge\) which sends \(\Omega ^{\mathrm{odd}}\) to \(\Omega ^{\mathrm{even}}\) and the other way around. We have \(D^2 =H\wedge d+d (H\wedge )+H\wedge H=0\) since \(H\wedge d+d (H\wedge )=H\wedge H=0\) by the degree of \(H\) being odd. When the degree of \(H\) is \(3\), the authors obtain Theorem 1.1 giving a detailed description of the differential and for the higher degrees, the authors obtain the analogous Theorem 1.2. (both too detailed to be quoted here).
The expression for the Massey product works partially because if \(D\alpha =0\), then \(d\alpha +H\wedge \alpha =0\) and therefore, \(d\alpha =0\) and \(H\wedge \alpha =0\). We notice that we also have \(H\wedge H=0\). And so on.
The definition of the twist differential still works if \(H\) is a sum of closed differential forms of odd degrees. The same arguments apply to the twist differential and we obtain the twisted cohomologies. In the same way, one has the related Massey products.
Now, Let \(K_p =\sum _{i\geq p} \Omega ^i\), then \(D\) sends \(K_p\) into \(K_{p+1}\). Let \(E_1 ^{p,q} =H_D ^{p+q} (K_p /K_{p+1} )\) and so on, one started a spectral sequence, which eventually converges to \(H^* _D\).
The authors obtain Theorem 1.1 and 1.2. It is amazing that Theorem 1.1 implies that the class in \(H^* _D\) only depends on the component of order \(3\) on the spectral sequence level. If this is true, the author should clearly mention it as a Corollary. Then Theorem 1.2 says that one could simply let \(H=H_{2s+1}=0\) once \(H\wedge x_p =0\). This is at least very striking for the reviewer.
In page 636 to 637, the authors discuss the possible generalization to the twisted cohomologies with flat vector bundles.
In Section 6, the authors discuss the indeterminacy of the closed differentials in the spectral sequence.

MSC:

55T25 Generalized cohomology and spectral sequences in algebraic topology
55S20 Secondary and higher cohomology operations in algebraic topology
55S30 Massey products
58J52 Determinants and determinant bundles, analytic torsion
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
55T99 Spectral sequences in algebraic topology
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References:

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