# Higher holonomies for higher local systems

In Jacob Lurie's classification of tqfts, one finds a version of the cobordism hypothesis for $(X,\zeta)$-structure, where an $(X,\zeta)$ structure on a manifold $M$ is the datum of a continuous map $f:M\to X$ and of an isomorphism $\alpha$ of $\mathbb{R}$-vector bundles $T_M\oplus \mathbb{R}^{n-\dim(M)}\simeq f^*\zeta$. Here $\zeta$ is a fixed rank $n$ vector bundle on $X$ and $\dim(M)\leq n$. This version of the cobordism hypothesis states that for any symmetric monoidal $(\infty,n)$-category $\mathcal{C}$ with duals there is an equivalence of $(\infty,0)$ categories $Fun^\otimes(Bord_n^{(X,\zeta)},\mathcal{C}) \simeq Hom_{O(n)}(\tilde{X},\mathcal{C}^{\sim})$, where $\tilde{X}$ is the total space of the principal $O(n)$-bundle on $X$ associated with the vector bundle $\zeta$ and $\mathcal{C}^{\sim}$ is the $(\infty,0)$ category obtained from $\mathcal{C}$ by discarding all the noninvertible morphisms.

In particular, this says (if I'm not wrong here) that given an $O(n)$-equivariant $\mathcal{C}$-valued local system $E$ on $\tilde{X}$ (i.e., an element $E$ in $Hom_{O(n)}(\tilde{X},\mathcal{C}^{\sim})$), to any manifold $M$ of dimension less or equal to $M$ equipped with an $(X,\zeta)$-structure is canonically associated an element $Z(E;M;f,\alpha)$ in $\Omega^{\dim M}\mathcal{C}$.

Now, fix an $n$-dimensional closed manifold $M$ and let $(X,\zeta)=(M,T_M)$. Then $M$ has a canonical $(X,\zeta)$-structure: the one given by the identity morphism! So the above general result on the cobordism hypothesis would say that to any $O(n)$-equivariant $\mathcal{C}$-valued local system $E$ on $\tilde{M}$ is canonically associated an element in $\Omega^n\mathcal{C}$, which I guess one should think of as the $n$-dimensional holonomy of the given local system. In other words, one should have a canonical morphism

$hol_M: Hom_{O(n)}(\tilde{M},\mathcal{C}^{\sim}) \to \Omega^n \mathcal{C}$.

More in general, if $\dim M\leq n$, one can take $(X,\zeta)=(M,T_M\oplus\mathbb{R}^{n-\dim M})$ and again the identity morphism provides a canonical $(X,\zeta)$-structure on $M$. Thus one should have a canonical morphism

$hol_M: Hom_{O(n)}(\tilde{M},\mathcal{C}^{\sim}) \to \Omega^{n-\dim M} \mathcal{C}$

for any closed manifold $M$ of dimension less or equal to $n$.

questions:

i) is this argument correct?

ii) can one give a direct description of the morphism $hol_M: Hom_{O(n)}(\tilde{M},\mathcal{C}^{\sim}) \to \Omega^n \mathcal{C}$ (i.e., without invoking the cobordism hypothesis)? note that some form of the cobordism hypothesis, namely the framed version, seems to be necessary even to state that there is a $O(n)$-action on $\mathcal{C}^\sim$.

iii) references?

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You forgot to pass to fully dualizable objects in the formulation of the cobordism hypothesis. –  Dmitri Pavlov Sep 11 '12 at 13:42
right, thanks! I've now added "with duals" to the description of the $(\infty,n)$-category $\mathcal{C}$ –  domenico fiorenza Sep 11 '12 at 13:47
At some point you switched from $\Omega^n\mathcal C$ to $\Omega^nM$... –  DamienC Sep 11 '12 at 14:47
right! I should re-read more carefully before posting! thanks, damien –  domenico fiorenza Sep 11 '12 at 15:41