Note: by fixed points, I always mean homotopy fixed points.

As explained in Jacob Lurie's paper on the cobordism hypothesis, we have an action of O(2) on the $\infty $-groupoid $X$ given by considering fully dualizable objects and invertable morphisms in some symmetric monoidal $(\infty ,2)$ category $\mathcal C$. I am interested in the case when $\mathcal C$ is the category where objects are algebras, 1-cells are bimodules, 2-cells maps of bimodules etc... By an algebra, I want to include algebra objects in some $\infty$-category (e.g. chain complexes), so that $\mathcal C$ really has non-trivial 3-cells, 4-cells and so on.

Now, I know (from remark 4.2.7) that the fixed points for the induced action of SO(2) on this space correspond to cyclic Frobenius algebras, i.e. smooth, proper algebras $A$, with a non-degenerate *and $SO(2)$-equivariant* trace

$A \otimes _{A^e} A \to k$.

Here, non-degenerate means that the pairing $A\otimes A \to A\otimes _{A^e} A \to k$ is nondegenerate. Recall also that the Hochschild homology $A \otimes _{A^e} A$ carries and $SO(2)$ action, so it makes sense to talk about $SO(2)$-equivariant map above.

The data of SO(2) equivariance should be equivalent to descending the trace to cyclic homology.

However, the reason why the fixed points are as above is that both can be identified with 2-d oriented TFTs, after theorem 3.1.8 which describes how to extend from a (n-1)-dimensional TFT to an n-dimensional one.

**Question: Can we calculate this action and identify the fixed point space directly?**

Part of this is not too hard to see (I think): naively, the SO(2) action on $X$ gives a canonical (Morita) automorphism of each (f.d.) algebra $A$. This automorphism is given by the bimodule dual $A^!$ (or the other one $A^\vee$...), so being a fixed point means to give an isomorphism $A^! \to A$, which is the same as a non-degenerate map $A\otimes _{A^e} A \to k$. If the base category where our algebra lives has no higher structure, then I think this is enough, but this does not explain the $SO(2)$ equivariance.

**In particular, where does the SO(2)-equivariance data come from?**

To see this, I tried to go a little deeper (with help from Takuo Matsuoka): the action is given by a map $B\mathbb Z =SO(2) \to Aut(X)$, which we transformed into a map $\mathbb Z \to \Omega Aut(X)$. The data described above corresponds to picking an algebra $A$ and composing to get a map $\mathbb Z \to \Omega X$ = Invertable $A-A$-bimodules. However, we haven't used that the original map from SO(2) is a group homomorphism - this should correspond to the map $\mathbb Z \to \Omega Aut(X)$ being an $E_2$-map. So we get braiding data attached to each of the $A^!$, which should be trivialized (because the E_2 -structure on $\mathbb Z$ is trivial (?)). Then to give a fixed point we must take into account all this data...

This has only given me a headache so far, but maybe there is an easier way to think about this? I tried (failed?) to be brief, so let me know if anything here is unclear!