See there for background and context.
raw material: this are notes taken more or less verbatim in a seminar – needs polishing
The goal here is to define a category of cobordisms that carry the structure of Riemannian manifolds. Where a functor on an ordinary cobordism category defines a TQFT, the assignments of a functor on a category of Riemannian cobordisms do not only depend on the topology of a given cobordism, but also on its Riemannian structure. In physics terms such a functor is a Euclidean quantum field theory .
Notice however that the physicist’s use of the word “Euclidean” is different from the way Stolz-Teichner use it: for a physicist it means that the Riemannian structure is not pseudo-Riemannian. For Stolz-Teichner it means (later on) that the Riemannian metric is flat .
One central technical difference between plain topological cobordisms and those with Riemannian structure is that we want the functors on these to smoothly depend on variations of the Riemannian structure. This requires refining the bordism category to a smooth category. By the logic of space and quantity, one way to do this is to realize it as a stack on Diff with values in categories. This realization will be described here.
the category has
The composition of morphisms is given by gluing of manifolds along their boundary
in all of the following
the symbol denotes -dimensional a Riemannian manifold
note on boundaries technically it is convenient to never ever work with manifolds with Riemannian or other structure with boundary. Instead, we always just mention manifolds without boundary and encoded the way in which they are still to be thouhgt of as cobordisms by injecting collars into them. The manifolds with boundary could be obtained by cutting of at the core of these collars (see the definition below) but, while this is morally the idea, in the construction this is never explicitly considered.
is defined using bicollars from the beginning
an object in is a quintuple
a -dimensional Riemannian manifold ;
a core -manifold sitting in a thickening – being a -manifold –
two disjointly embedded open -dimensional manifolds such that
is in the closure of both and
so the picture of an object, which is missing in this writeup here for the moment, is a -dimensional Riemannian manifold that is thickened a bit in one further othogonal direction
definition A Riemannian bordism from to is a triple where
is a -dimensional Riemannian manifold without boundary
for an open neighbourhood of the core
this defines the intersections with the two collars for each .
a smooth map
is a proper map;
(+) for are isometric embeddings into
i.e. restricted to the (+)-collar the embedding of the thickened object into the would-be cobordisms is isomertric
the core is compact
i.e. cutting of the (+)-collar of the incoming object and the (-)-collar of the outgoing object yields a compact manifold
Remark. Notice that this builds in an asymmetry: the (+)-side is preferred. This is intentionally: also the category of topological vector spaces will have a similar asymmetry (from the fact that for -dimensional vector spaces there is an evaluation map but not necessarily a coevaluation/unit for ), similarly, with the above asymmetric definition we have a cobordims (where is obtained from by reversing orientation) but not one going the other way round.
A big difference between TQFTs and the Riemannian QFTs is that for TQFTs the vector spaces assigned to objects are necessarily finite-dimensional. So this issue here with infinite-dimensional vector spaces and the asymmetry that this introduces is crucial for Riemannian QFTs.
example Given any isometry
such that preserves the decomposition we get a Riemannian cobordism using
We require the commutativity of the following diagram
The isometry is “rel. boundary” if and
so an isomorphism “rel boundary” in the sense here (more “rel collars”, really) is an isometry sitting in a diagram
we decribe explicitly
it has at least the object
which is a point with collar all of .
Lemma every object in which is connected and not the empty set is isomorphic to this
now for consider the morphism
defined as the triple where is the identity map, and where is translation by .
This means that takes the core of in the incoming point to while takes the core of the outgoing point to . Everything in outside of is hence “collar” and this describes what naively one would think of as just the interval regarded as a Riemannian cobordism.
Lemma The composition of these cobordisms is given by
There are also morphisms
which describe morally the same cobordisms as does, but where both boundary components are regarded as incoming or noth as outgoing, respectively.
Here is formall given exactly as only that the map is not the identity, but reflection at the origin. This encodes the orientation reversal at that end.
This is defined for . For the morphism is still defined, but is not!! Exercise: check carefully with the above definition, keeping the asymmetry mentioned there in mind, to show that the obvious definition of does not satisfy the axioms above.
So this means that we have a cobordism of length 0 going , but all cobordisms going the other way round will have to have non-vanishing length.
Another morphism in is the morphism
which just interchanges the two points, without having any length.
Lemma We have the following composition laws:
where in the last line we have the composition that is obvious once you draw the corresponding picture, which in full beuaty is
where the tensor product is given by disjoint union.
theorem the symmetric monoidal category is generated as a symmetric monoidal category by
the morphisms ,
subject to the relations
corollary symmetric monoidal functors
to the category of topological vector spaces are specified by their imagges of these generators. We have
The map is necessarily a nondegenerate and symmetric bilinear form and thus may be used to produce and fix an isomorphism .
This isomorphism is used to get an embedding
The image of this embedding is the set of what in this context will be called “trace class” operators.
With respect to this identification the map is to be understood. For varying the form a semigroup (for instance a typical example would be a space of sections of a vector bundle and for a Laplace operator on ).
note for to be continuous, one cannot use the Hilbert tensor product
the reason is that we have the folloing possible mpas out of the following possible tensor products
(so here the middle is the projective tensor product, the one that we are actually using)
We now refine the definition of the categories and such that they remember smooth stucture.
Effectively, what the following implicitly does is to refine these categories to stacks with values in categories over Diff. The fibred categories that appear in the following, and are the Grothendieck construction of these stacks.
recall that denotes the category of locally convex Hausdorff topological vector space
now let be the fibred category over Diff whose fiber over is the category of topological vector bundles over . This has as objects vector bundles of topological vector spaces, and the morphisms are fiberwise linear -morphisms of bundles in the following sense:
Then a linear map is – for any inclusion
– called at in the direction if
exists in and
Iteratively one defines and then . The morphsims of -bundles are supposed to be maps in this sense (linear in the fibers, of course)
Similarly has as objects submersions and (not necessarily surjective) with a smooth rank-2 tensor on that fiberwise induces the structure of a Riemannian manifold (so these are -families of Riemannian manifolds) such that
recall that a map is a proper map if inverse images of compact sets are compact.
remark Notice that if we fix the topology of the fibers in , then what varies as we vary the fibers is the Riemannian metric on the fibers, so here each can be thought of as a (subspace of a) moduli space of Riemannian metrics on a given topological space. Don’t confuse this with the role the space always called here will play as a kind of “moduli space of field theories”.
a morphism in in
are isometric rel boundary classes of submersions such that
so here is an -family of cobordisms.
A -dimensional Riemannian quantum field theory is a symmetric monoidal functor
and such that it preserves pullback