Quantum field theory
raw material: this are notes taken more or less verbatim in a seminar – needs polishing
All of the following assumes that the reader is well familiar with the basic ideas indicated at FQFT.
In the following or and the like denotes a category of -dimensional Riemannian cobordisms that are equipped with Riemannian structure, i.e. with Riemannian metric. Similarly or or the like denotes a category of cobordisms with Euclidean structure, by which is meant a flat Riemannian metric.
The definition of this is discussed in detail in
-dimensional Riemannian field theories are symmetric monoidal functors from -dimensional Riemannian bordisms to topological vector spaces.
A field theory is very similar to a representation of a group. Only where a representation of a group is a functor from the delooping of to Vect, an FQFT is a representation of a more complicated domain category.
how does topology enter?
for some topological space there is also a symmetric monoidal category
of Riemannian bordisms equipped with a continuous map to .
Notice that does depend covariantly on . This means that is contravariant in .
When special structure is around, however, we also have a push-forward of such functors along morphisms.
Example: push-forward to the point: for as above and the unique map to the point heuristically we want a map
notice that this push-forward is not an adjoint functor. Instead, it is a map that comes from integration over fibers. In particular it will change the degree of cohomology theories.
heuristically the pushforward
acts on field theories over
by the assignment
for instance when then .
For instance take to be the pair of pants with input and output and let be a map.
Then will take some section to
Here the expression denotes a would-be measure which is still to be defined.
Here is the restriction of to .
Contravariant functors with push-forward also arise as part of a cohomology theory
from the category Diff of smooth manifolds to the category Ab of abelian groups that satisfies the axioms of generalized (Eilenberg-Steenrod) cohomology theory.
These Eilenberg-Steenrod axioms are
homotopy axiom for
Mayer-Vietoris axiom for
here in the last axiom for topological spaces we’d simply have the suspension . Here in order to stay within manifolds we instead do as indicated, where means “compact vertical support”.
Concerning the homotopy axiom: given any functor that sends manifolds to a category of FQFTs over
in s we can make it a homotopy functor by defining
to be concordant if such that in ,
then under this relation
is a homotopy invariant functor
we have that concordant to exactly when for some
Concerning the Mayer-Vietoris axiom: if then the functor should respect this gluing.
suppose is a 2d Riemmanian field theory. let be a loop in . then is a vector space.
If sits neither entirely in or in , then there is no way that the vector space can be reconstructed by knowing just the restriction of to and .
So this is a problem for the definition of field theories so far.
The proposed solution (from What is an elliptic object?) is to use extended FQFTs instead. This introduces locality into FQFTs, at the expense of working with n-categories.
This will however not be studied here for the moment.
Concerning the suspension isomorphism: for that first we need for the notion of a field theory over of degree , i.e.
such that for this is an ordinary Riemannian field theory, in .
This requires to replace manifolds by supermanifolds.
Example let and consider 0-dimensional TFTs over .
in there is only a single object: the empty set which has a unique map
the collecton of morphisms is in there is .
here composition of morphisms is the same as tensor product of objects: both comes from the disjoint union of these finite set domains.
so a priori we have
where on the right we have all maps.
This is not quite what is intended. We want to see smooth maps on the right. To get that, we need to talk about smooth functors on the left. This is the topic of later discussion, which will yield
But one other ting goes wrong the: the corresponding homotopy functor is . So turning this into an Eilenberg-Steenrod theory yields the trivial theory.
The way out to that will be to go to supermanifolds.
Punchline of this session here:
and then it is a that
where is the supergroup .
The degree decomposition on forms then turns out to be the eigenvalue decompositon of the -part, while the deRham differential is the -action (cite Kontsevich, Severa here …)
so then from that one finds that the -invariant bit is the closed 0-forms
So we get -Lemma :
Now with the super-directions included, there is a notion of push-forward of the TFTs that does shift the degree, and we get the following
when is oriented
and similarly after dividing out concordance, the push-forward becomes the push-forward in deRham cohomology.
- Stefan Stolz (notes by Arlo Caine), Supersymmetric Euclidean field theories and generalized cohomology Lecture notes (2009) (pdf)