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This page contains technical details to be used at the main page differential string structure . See there for context.
Contents
Factorization of the -cocycle
Proposition
The -algebra cocycle
factors as
given dually on CE-algebras by
The left morphism is a quasi-isomorphism.
Proof
To see that we have a quasi-isomorphism, notice that the dg-algebra is isomorphic to the one with generators and differentials
where the isomorphism is given by the identity on the s and on and by
The primed dg-algebra is the tensor product , where the second factor is manifestly cohomologically trivial.
Factorization of the differential -cocycle
We now give a concrete construction showing
Proposition
The factorization
from above lifts to a factorization of differential -algebraic cocycles
Proof
This is at its heart trivial, but potentially a bit tedious. We proceed in two steps:
-
consider a “modified Weil algebra” of the twisted string Lie 3-algebra
in The modified Weil algebra;
-
construct the desired factorization by factoring itself through two fairly evident morphisms into and out of the modified Weil algebra,
in The differential lift.
The modified Weil algebra
Our factorization below makes use of an isomorphic copy of the Weil algebra .
Proposition
The Weil algebra of is given on the extra shifted generators (where is the shift operator extended as a graded derivation, see Weil algebra) by
with Bianchi identities
Let be the dg-algebra with the same underlying graded algebra as but with the differential modified as follows
-
;
-
;
-
;
-
;
-
.
-
,
where “” is the new name for the generator that used to be called “”
There is an isomorphism
in dgAlg that is the identity on all generators except on , where it is given by
Corollary
The invariant polynomials on are generated from those of together with and :
The differential lift
We now use the isomorphism
from prop. and obtain the desired factorization, as the composite
Here
-
the unlabelled vertical morphisms are defined as the unique ones that make the respective square commute;
-
the notation stands for all the invariant polynomials of and specifically for the Killing form.