nLab Generalized higher loop spaces

Generalized higher loop spaces are analogs of loop spaces adapted to the setting of overconvergent global analytic geometry.

Overview

Recall that one may define higher Hochschild homology of a derived scheme as functions on its completed formal higher loop space

$$\hat{L}^n X:=\widehat{Hom}_{dSt}(S^n,X).$$

There is a natural action of the symmetry group of $S^n$ on this space. One may try to define analogs of this construction for other symmetry groups, at least on a local field, to try to use these spaces to study representation theory.

The basic idea here is to start with the non-strict analytic space $GL_n$, and look at the mapping space

$$L_{GL_n}X:=Hom_{dSt}(GL_n,X).$$

It may only be definable for usual spaces, not Artin stacks, but one has to check that.

In any case, there is a natural (in the setting of generalized global analytic geometry) subgroup $U(n)\subset GL_n$, and one may also consider

$$L_{U(n)}X:=\widehat{Hom}_{dSt}(U(n),X).$$

Normally, since $U(n)$ is proper, this space is finite dimensional in the differential stacky sense. One may then define linear and unitary higher Hochschild cohomology of dimension $n$ to be given by functions on the moduli spaces

$$M_{GL_n}X:=[L_{GL_n}X/GL_n]$$

and

$$M_{U(n)}X:=[L_{U(n)}X/U(n)].$$

Up to $D^1$-homotopy, both spaces should be identical. Now the unitary moduli space has a strict global definition, which means it is a very rigid object (essentially, a kind of Arakelov stack, given by a scheme-moduli stack together with an archimedean structure).

The determinant map $det:GL_n\to GL_1$ induces a natural morphism

$$det^*:L_{GL_n}X\to L_{GL_1}X,$$

which may give also a morphism

$$det^*:M_{GL_n}X\to M_{GL_1}X.$$

Remark now that there exists an exact sequence of (non-strict) analytic groups

$$1\to U(1)\to GL_1\to N_1\to 1$$

where $N$ is the group of norms of elements in $GL_1$. In dimension $n$, we only have a groupoid $N_n$. However, there is always the possibility to define a new moduli space given by

$$D_{GL_n}X:=[L_{GL_n}X/U(n)].$$

This moduli space is equipped with an action of the groupoid $N_n$. In dimension $1$ over $C$, if $X$ is, say, projective, we get, as functions on $D_{GL_1}(X)$, a (probably infinite dimensional) space over $C$ together with an action of $N_1\cong \mathbb{R}_+^*$. In the $p$-adic situation, we have $N_1=p^\mathbb{Z}$. On a global field, one may work over adeles to get a sensible construction.

One should also not forget that there is (in characteristic $0$, e.g., on adeles of $\mathbb{Q}$) a logarithm map

$$log:\overset{\circ}{D}(1,1)_{GL_1}\to Lie(GL_1)$$

that allows us to relate the above constructions to the notion of generalized lambda-structure.

Relation with usual Hochschild and cyclic homology

Remark that, even over $\mathbb{C}$, the $D^1$-homotopy invariant version of $M_{U(1)}X$ is not equivalent to the (simplicial) loop space $L^1 X$. Indeed, to get a sensible relation between our moduli spaces and loop spaces (i.e., usual Hochschild and negative cyclic homology), one has to be very careful: inverting $D^1$ is not the right thing to do because it just gives usual loops up to homotopy, not the derived ones, which are necessary for differential constructions. A nice way to circumvent this problem is given by the following construction: let $R$ and $S$ be two base ind-Banach ring, and consider the category of derived stacks on $R$ with values in the (say) stable $D^1$-homotopy category of usual stacks over $S$. If we start with $S=\mathbb{C}$ and $R=(\Z,|\cdot|_0)$, we get a setting that is appropriate for explaining the relation between derived loop spaces and generalized higher loop spaces. Indeed, one may see $GL_n$ as a constant derived stack given by the associated group object of the $D^1$-homotopy category, denoted $GL_n^c$. Then the corresponding moduli space $M^c_{GL_n}X$ is related to the derived loop space in the case $n=1$: they are homotopy equivalent because the $D^1$-homotopy category over $C$ is equivalent (following Ayoub) to the usual homotopy category of simplicial sets.

If one wants to avoid getting back the usual loop space, one may work with the category

$$Shv(DAff^\dagger_S\times SAff^\dagger_R,{}^\infty Grpd)$$

of sheaves of sheaves on smooth affinoid spaces over $R$ on derived affinoid algebras over $S$. This may be seen as an extension of scalars of the $\infty$-category of derived analytic stacks over $S$ along the inclusion of infinity-groupoids into usual analytic higher stacks over $R$ (that form a presentable infinity-category, so that the tensor product operation works).

Recall that we may define two types of linear group stack-valued derived stacks by setting, for $(A,B)\in DAff^\dagger_S\times SAff^\dagger_R$, the constant linear stack to be defined associated to the prestack

$$GL_n^c(A,B)=GL_n(B),$$

and the usual derived analytic stack

$$GL_n^d(A,B)=GL_n(A).$$

Having a derived and a non-derived direction seems to be quite important. Indeed, the derived direction is needed for doing differential calculus, and the non-derived direction is needed for doing $D^1$-homotopy theory. Combining the two may give an optimal setting for relating generalized loop spaces with usual derived loop space.

There is a relation between the homotopy and classical generalized higher loop spaces given by the following: if we assume given a morphism $R\to S$, we may define the diagonal (extension of scalars) functor

$$\Delta:SAff^\dagger_R\longrightarrow DAff^\dagger_R\times SAff^\dagger_S,$$

to get a natural morphism of stack-valued derived stacks induced by

$$\Delta^*GL_n^d(A)\cong GL_n(A)\longrightarrow GL_n(A_S)\cong\Delta^*GL_n^c(A).$$

We may now define a version of the generalized higher loop spaces that is directly related to Hochschild cohomology: we may define

$$L^{dc}_{GL_n}(X):= \{(f,g)\in Hom(GL_n^d,X)\times Hom(GL_n^c,X),\; \Delta^*f\cong \Delta^* g\}.$$

This means that we are choosing (for $n=1$), for every classical derived loop $g:GL_n^c\to X$ (up to $D^1$-homotopy, say), a corresponding generalized derived loop $f:GL_n\to X$, that we may call its representative. Forgetting the representative gives back the usual derived loop spaces if $R=(\Z,|\cdot|_0)$ and $S=(\C,|\cdot|_\infty)$, and $X$ is a derived scheme over $\Z$, and forgetting the classical derived loop gives the generalized higher derived loop space defined at the beginning of this page. We thus have two projections

$$L^d_{GL_n}X\leftarrow L^{dc}_{GL_n}(X)\to L^c_{GL_n}(X)$$

that may be used to define a correspondence between modules on the generalized higher derived loop space and on the classical derived loop space.

Related entry

overconvergent derived loop spaces