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differential string structure -- proofs

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Factorization of the L L_\infty-cocycle

Proposition

The L L_\infty-algebra cocycle

μ:𝔰𝔬b 2 \mu : \mathfrak{so} \to b^2 \mathbb{R}

factors as

𝔰𝔬(b𝔰𝔱𝔯𝔦𝔫𝔤)b 2 \mathfrak{so} \stackrel{}{\to} (b \mathbb{R} \to \mathfrak{string}) \stackrel{}{\to} b^2 \mathbb{R}

given dually on CE-algebras by

CE(𝔰𝔬)(t at a b0 cμ)CE(b𝔰𝔱𝔯𝔦𝔫𝔤)(cc)CE(b 2). CE(\mathfrak{so}) \stackrel{ \left( \array{ t^a \mapsto t^a \\ b \mapsto 0 \\ c \mapsto \mu } \right) }{\leftarrow} CE(b \mathbb{R} \to \mathfrak{string}) \stackrel{ \left( \array{ c \mapsto c } \right) }{\leftarrow} CE(b^2 \mathbb{R}) \,.

The left morphism is a quasi-isomorphism.

Proof

To see that we have a quasi-isomorphism, notice that the dg-algebra CE(b𝔰𝔱𝔯𝔦𝔫𝔤)CE(b \mathbb{R} \to \mathfrak{string}) is isomorphic to the one with generators {t a,b,c}\{t^a, b, c'\} and differentials

d| 𝔤 * =[,] * db =c dc =0, \begin{aligned} d|_{\mathfrak{g}^*} & = [-,-]^* \\ d b & = c' \\ d c' & = 0 \end{aligned} \,,

where the isomorphism is given by the identity on the t at^as and on bb and by

cc+μ. c \mapsto c' + \mu \,.

The primed dg-algebra is the tensor product CE(𝔤)CE(inn(b))CE(\mathfrak{g}) \otimes CE( inn(b \mathbb{R})), where the second factor is manifestly cohomologically trivial.

Factorization of the differential L L_\infty-cocycle

We now give a concrete construction showing

Proposition

The factorization

CE(𝔰𝔬)(t at a b0 cμ)CE(b𝔰𝔱𝔯𝔦𝔫𝔤)(cc)CE(b 2) CE(\mathfrak{so}) \stackrel{ \left( \array{ t^a \mapsto t^a \\ b \mapsto 0 \\ c \mapsto \mu } \right) }{\leftarrow} CE(b \mathbb{R} \to \mathfrak{string}) \stackrel{ \left( \array{ c \mapsto c } \right) }{\leftarrow} CE(b^2 \mathbb{R})

from above lifts to a factorization of differential L L_\infty-algebraic cocycles

CE(𝔰𝔬) (t at a b0 cμ) CE(b𝔰𝔱𝔯𝔦𝔫𝔤) (cc) CE(b 2) W(𝔰𝔬) W(b𝔰𝔱𝔯𝔦𝔫𝔤) W(b 2) inv(𝔰𝔬) inv˜(b𝔰𝔱𝔯𝔦𝔫𝔤) inv(b 2). \array{ CE(\mathfrak{so}) & \underoverset{\simeq}{ \left( \array{ t^a \mapsto t^a \\ b \mapsto 0 \\ c \mapsto \mu } \right) }{\leftarrow} & CE(b \mathbb{R} \to \mathfrak{string}) & \stackrel{ \left( \array{ c \mapsto c } \right) }{\leftarrow} & CE(b^2 \mathbb{R}) \\ \uparrow^\mathrlap{} && \uparrow^\mathrlap{} && \uparrow^\mathrlap{} \\ W(\mathfrak{so}) & \underoverset{\simeq}{ }{\leftarrow} & W(b \mathbb{R} \to \mathfrak{string}) & \stackrel{ }{\leftarrow} & W(b^2 \mathbb{R}) \\ \uparrow^\mathrlap{} && \uparrow^\mathrlap{} && \uparrow^\mathrlap{} \\ inv(\mathfrak{so}) & \underoverset{\simeq}{ }{\leftarrow} & \tilde inv(b \mathbb{R} \to \mathfrak{string}) & \stackrel{ }{\leftarrow} & inv(b^2 \mathbb{R}) } \,.
Proof

This is at its heart trivial, but potentially a bit tedious. We proceed in two steps:

  1. consider a “modified Weil algebra” of the twisted string Lie 3-algebra (b𝔰𝔱𝔯𝔦𝔫𝔤)(b \mathbb{R} \to \mathfrak{string})

    in The modified Weil algebra;

  2. 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 W(b𝔤 μ)W(b\mathbb{R} \to \mathfrak{g}_\mu).

Proposition

The Weil algebra W(b𝔤 μ)W(b\mathbb{R} \to \mathfrak{g}_\mu) of (b 2𝔤)(b^2 \mathbb{R} \to \mathfrak{g}) is given on the extra shifted generators {r a:=σt a,h:=σb,g:=σc}\{r^a := \sigma t^a, h := \sigma b, g := \sigma c\} (where σ\sigma is the shift operator extended as a graded derivation, see Weil algebra) by

  • dt a=12C a bct bt c+r ad t^a = -\frac{1}{2} C^a{}_{b c} t^b \wedge t^c + r^a;

  • db=μ+c+hd b = - \mu + c + h;

  • dc=gd c = g,

with Bianchi identities

  • dr a=C a bct br cd r^a = - C^a{}_{b c} t^b \wedge r^c

  • dh=σμgd h = \sigma \mu - g;

  • dg=0d g = 0.

Let W˜(b𝔤 μ)\tilde W(b\mathbb{R} \to \mathfrak{g}_\mu) be the dg-algebra with the same underlying graded algebra as W(b𝔤 μ)W(b\mathbb{R} \to \mathfrak{g}_\mu) but with the differential modified as follows

  • dt a=12C a bct bt c+r ad t^a = -\frac{1}{2} C^a{}_{b c} t^b \wedge t^c + r^a;

  • dr a=C a bct br ad r^a = - C^a{}_{b c} t^b \wedge r^a;

  • db=cs+c+h˜d b = - cs + c + \tilde h;

  • dh˜=,gd \tilde h = \langle -,-\rangle - g;

  • dc=gd c = g .

  • dg=0d g = 0,

where “h˜\tilde h” is the new name for the generator that used to be called “hh

There is an isomorphism

W(b𝔤 μ)W˜(b𝔤 μ) W(b\mathbb{R} \to \mathfrak{g}_\mu) \to \tilde W(b\mathbb{R} \to \mathfrak{g}_\mu)

in dgAlg that is the identity on all generators except on hh, where it is given by

hh˜+(μcs). h \mapsto \tilde h + (\mu - cs) \,.
Note

Where the formula for the differential of W(b𝔤 μ)W(b\mathbb{R}\to \mathfrak{g}_\mu) has the 3-cocycle μ\mu that for W˜(b𝔤 μ)\tilde W(b\mathbb{R}\to \mathfrak{g}_\mu) has the Chern-Simons element cscs. The shift by csμcs-\mu is precisely what shifts the curvature characteristic d W(𝔤)μd_{W(\mathfrak{g})}\mu into the shifted copy of 𝔤 *\mathfrak{g}^* in the Weil algebra, thus exhibiting the modified hh as an invariant polynomial.

Corollary

The invariant polynomials on (b𝔤 μ)(b \mathbb{R} \to \mathfrak{g}_\mu) are generated from those of 𝔤 μ\mathfrak{g}_\mu together with h˜\tilde h and gg:

inv˜(b𝔰𝔱𝔯𝔦𝔫𝔤)=(inv(𝔰𝔬)h˜,g)/(dh˜=,g). \tilde inv(b \mathbb{R} \to \mathfrak{string}) = (inv(\mathfrak{so})\otimes \langle \tilde h, g\rangle)/(d \tilde h = \langle -,-\rangle - g) \,.

The differential lift

We now use the isomorphism

W(b𝔰𝔱𝔯𝔦𝔫𝔤)W˜(b𝔰𝔱𝔯𝔦𝔫𝔤) W(b \mathbb{R} \to \mathfrak{string}) \stackrel{\simeq}{\to} \tilde W(b \mathbb{R} \to \mathfrak{string})

from prop. 3 and obtain the desired factorization, as the composite

CE(𝔰𝔬) (t at a b0 cμ) CE(b𝔰𝔱𝔯𝔦𝔫𝔤) = CE(b𝔰𝔱𝔯𝔦𝔫𝔤) (cc) CE(b 2) i 𝔰𝔬 * i (b𝔰𝔱𝔯𝔦𝔫𝔤) * i b 2 * W(𝔰𝔬) (t at a b0 ccs r ar a h˜0 g,) W(b𝔰𝔱𝔯𝔦𝔫𝔤) (h˜h+(csμ)) W˜(b𝔰𝔱𝔯𝔦𝔫𝔤) (cc gg) W(b 2) p 𝔰𝔬 * p (b𝔰𝔱𝔯𝔦𝔫𝔤) * p b 2 * inv(𝔰𝔬) (h˜0 g, ) inv˜(b𝔰𝔱𝔯𝔦𝔫𝔤) = inv˜(b𝔰𝔱𝔯𝔦𝔫𝔤) (gg) inv(b 2). \array{ CE(\mathfrak{so}) & \underoverset{\simeq}{ \left( \array{ t^a \mapsto t^a \\ b \mapsto 0 \\ c \mapsto \mu } \right) }{\leftarrow} & CE(b \mathbb{R} \to \mathfrak{string}) & = & CE(b \mathbb{R} \to \mathfrak{string}) & \stackrel{ \left( \array{ c \mapsto c } \right) }{\leftarrow} & CE(b^2 \mathbb{R}) \\ \uparrow^\mathrlap{i^*_{\mathfrak{so}}} && \uparrow^\mathrlap{i^*_{(b\mathbb{R} \to \mathfrak{string})}} && \uparrow^\mathrlap{} && \uparrow^\mathrlap{i^*_{b^2 \mathbb{R}}} \\ W(\mathfrak{so}) & \underoverset{\simeq}{ \left( \array{ t^a \mapsto t^a \\ b \mapsto 0 \\ c \mapsto cs \\ r^a \mapsto r^a \\ \tilde h \mapsto 0 \\ g \mapsto \langle-,-\rangle } \right) }{\leftarrow} & W(b \mathbb{R} \to \mathfrak{string}) & \underoverset{\simeq}{ \left( \array{ \tilde h \mapsto h + (cs - \mu) } \right) } {\leftarrow} & \tilde W(b \mathbb{R} \to \mathfrak{string}) & \stackrel{ \left( \array{ c \mapsto c \\ g \mapsto g } \right) }{\leftarrow} & W(b^2 \mathbb{R}) \\ \uparrow^\mathrlap{p^*_{\mathfrak{so}}} && \uparrow^\mathrlap{} && \uparrow^\mathrlap{p^*_{(b \mathbb{R} \to \mathfrak{string})}} && \uparrow^\mathrlap{p^*_{b^2 \mathbb{R}}} \\ inv(\mathfrak{so}) & \underoverset{\simeq}{ \left( \array{ \tilde h \mapsto 0 \\ g \mapsto \langle -,-\rangle \\ \langle \cdots \rangle \mapsto \langle \cdots \rangle } \right) }{\leftarrow} & \tilde inv(b \mathbb{R} \to \mathfrak{string}) & = & \tilde inv(b \mathbb{R} \to \mathfrak{string}) & \stackrel{ \left( \array{ g \mapsto g } \right) }{\leftarrow} & inv(b^2 \mathbb{R}) } \,.

Here

  • the unlabelled vertical morphisms are defined as the unique ones that make the respective square commute;

  • the notation \langle \cdots \rangle stands for all the invariant polynomials of 𝔰𝔬\mathfrak{so} and ,\langle-,-\rangle specifically for the Killing form.

Revised on September 8, 2011 13:52:11 by Urs Schreiber (131.211.239.172)