Algebraic Loop Spaces and Spherical Derived Traces

Setting - Spherical Derived Geometry

Algebraic Traces

The Tannakian comparison

$$Ma{p}_{Fun(CAlg,S)}(S^1,X)\to Fu{n}^L(Perf(X),Loc(S^1))$$

is responsible for a theory of traces: the dimension of a $1$-categorical sheaf $V\in Perf(X)$ is stored as datum of a local system on a loop.

There are dual ways to view this map

• Fixing $X$, we see the above is the Tannakanization Defn- Tannakianization map $X\to Tnk(X)$ evaluated on the stack $S^1$
• Fixing $S^1$, we get a natural transformation

$$Ma{p}_{Fun(CAlg,S)}(S^1,-)\to Fu{n}^L(Perf(-),Loc(S^1))$$

We’ll adopt the latter, viewing this construction as providing as a “theory of traces”, which encodes monodromy of 1-categorical sheaves around $S^1$-loops as local systems on the anima $S^1$.

We run the story above, but replacing the “Betti-circle” (the constant stack $S^1$) with the “algebraic circle” (the punctured affine plane ${\mathbb{G}}_m$).

Construction (Algebraic traces)

$$Ma{p}_{Fun(CAlg,S)}({\mathbb{G}}_m,-)\to Fu{n}^L(Perf(-),Perf({\mathbb{G}}_m))$$

This is a new “theory of traces”, which encodes monodromy of 1-categorical sheaves around algebraic-loops as local systems on the punctured affine line ${\mathbb{G}}_m$.

remark: These should then be input to Cartier-Transforms.

Comparisons

remark

$\mathbb{S}[\mathbb{Z}]$ is 1-categorically Morita equivalent to $Fre{e}^{(1)}(\mathbb{S})$. Let’s denote these by

• $\mathfrak{auto}(Sp):=LMo{d}_{Fre{e}^{(1)}(S)}$
  • $\mathfrak{auto}(Sp){\times }_{Sp}\{V\}\simeq Au{t}_{Sp}(V)$. An action is an automorphism
• $Auto(Sp):=LMo{d}_{\mathbb{S}[\mathbb{Z}]}$
  • $Auto(Sp){\times }_{Sp}\{V\}\simeq Ma{p}_{Al{g}^{(1)}(S)}({\mathbb{N}}^{gp},En{d}_{Sp}(V))\simeq Ma{p}_{Al{g}^{(1)}(S)}(\mathbb{N},Au{t}_{Sp}(V))$. An action is an automorphism, as N is free E_1 monoid on a point.

Circle and free $E_1$ algebra

looping the above observation, $S^1$ is Morita-equivalent to $Fre{e}^{(1)}(\mathbb{S})$. We can also see this directly via a kind of Riemann-Hilbert argument:

$QCoh(S^1)\simeq {\lim }_{S^1}Sp\simeq Fun(S^1,Sp)$ We see the data of exhibiting a spectrum $V$ as global sections of a sheaf on $S^1$ is precisely data of an automorphism of $V$, which by earlier is equivalent to data of a $Fre{e}^{(1)}(\mathbb{S})$-action.

(Remark: this should have some consequence about spherical cohomology of $S^1$? We should do a sanity check with target $Mo{d}_k$)

Comparison Maps The map of spectra $\mathbb{S}\to \mathbb{S}[\mathbb{Z}]$ classifying the multiplicative unit of the rings structure on the RHS, when remembering the multiplicative $E_1$ structure on the RHS, gives a map of $E_1$ algebras

$$Fre{e}^{(1)}(\mathbb{S})\to \mathbb{S}[\mathbb{Z}]$$

this is an equivalence of E_1 rings, so it was a bit unecessary calling it an equivalence. BTW this shows up in 2-categorical G_m reps.

Base change along this map gives an adjunction

$l:\mathfrak{auto}(Sp)\leftrightarrow Auto(Sp):r$ which is an equivalence (note: this is a silly little redundant sentence back when we thought these spaces could be different)

Trace Maps

Reminder: The Ordinary Chern Character

In the non-strict setting, the trace map is constructed as a special case of the following

$V^{\otimes }$ symmetric monoidal $\infty$-category with duals, there’s a trace map

$Tr:\mathfrak{auto}(V)\to En{d}_V(1)$ Taking $V=Perf(LX:=\underline{Map}(S^1,X))$, this is a map

$$\mathfrak{auto}(Perf(LX))\to {\mathcal{O}}_{LX}$$

The left hand side is

$Fun(S^1,Perf(LX))$ From now on, take $X$ to be a perfect stack. There’s a map $Perf(X)\to Fun(S^1,Perf(LX))$ classified by the map

$S^1\otimes Perf(X)\stackrel{Id}{\to }{S}^1\otimes Perf(X)\simeq Perf(LX)$ Composing these, we get a map $\chi :Perf(X{)}^{\simeq }\to \Gamma {\mathcal{O}}_{LX}$, which we call the Chern Character.

The derived Chern character

We again run the same story but replacing Betti circles with their algebraic counterparts. The map $Perf({\mathbb{G}}_m)\otimes Perf(X)\stackrel{Id}{\to }Perf({\mathbb{G}}_m)\otimes Perf(X)\simeq Perf({\mathbb{G}}_m\times X)$ classifies a map $Perf(X)\to Perf({\mathbb{G}}_m)\otimes Perf({\mathbb{G}}_m\times X)\simeq Auto(Perf({\mathbb{G}}_m\times X))$