and
This is a sub-entry of geometric models for elliptic cohomology and A Survey of Elliptic Cohomology
See there for background and context.
This entry here indicates how 1-dimensional FQFTs (the superparticle) may be related to topological K-theory.
raw material: this are notes taken more or less verbatim in a seminar – needs polishing
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recall the commercial for supergeometry with which we ended last time: the grading introduced by supergeometry makes it possible to have push-forward diagrams of the kind:
Example of 1-EFT
Example of $(1|1)-EFT$ associated to a spin manifold, there is the spinor bundle
a $\mathbb{Z}/2$-graded vector bundle and on this there is the Dirac operator
where $\Gamma(S) = \Gamma(S^+) \oplus \Gamma(S^-)$. So we can write
there is an involution $invol : \mathbb{R}^{0|1} \to \mathbb{R}^{0|1}$. It maps to
we have the following moduli space of super intervals (super 1d-bordisms)
and these are mapped by the EFT as
(here we are implicitly working in the topos of sheaves on the category of supermanifolds and these equations have to be interpreted in that topos-logic, mapping generalized elements to generalized elements).
So we have for $E$ a $1|1$ EFT a reduced non-susy field theory
Definition $E \in (1|1)EFT$, the partition function $Z_E$ of $E$ is the function
that sends a length to the value of the EFT on the circle of that circumferene.
Example Consider from above the EFT
look at its reduced part
notice that by the above this assigns
where on the right we have the super trace.
This evaluates to
where the super dimension? of the eigenspace $E_\lambda$ is
and this vanishes for $\lambda \neq 0$ since there $D : E_\lambda^+ \stackrel{\simeq}{\to} E_\lambda^-$
is an isomorphism.
So further in the computation we have
where the last step is the Atiyah-Singer index theorem.
So due to supersymmetry , the partition function has two very special properties:
it is constant – in that it does not depend on $t$,
it takes integer values $\in \mathbb{N} \subset \mathbb{R}$.
recall from $V \to X$ a vector bundle with connection $\nabla$ we get a 1d EFT
given by the assignment
a morphism is an interval $[0,t]$ of length $t$ equipped with a map $\gamma : [0,t] \to X$, this is sent to the parallel transport associated with the connection on a bundle
Now refine this example to super-dimension $(1|1)$:
example of a $(1|1)$-EFT over $X$ consider
given by the assignment
so we just forget the super-part and consider the same parallel transport as before.
now to K-theory:
$KO^0(X) =$ Grothendieck group of real vector bundles over $X$
there is a Bott element $\beta \in KO^{-8}(pt)$
such that
now the push-forward in topological K-theory
for $X$ a closed spin structure manifold
then there exists an embedding $X \hookrightarrow S^{n+m}$. Let $\nu$ be the normal bundle to this embedding.
then we define
as follows:
let $D(\nu)$ be the disk bundle? and $S(\nu)$ be the sphere bundle of $\nu$. Then the Thom bundle? is
we get a map
involving the Thom isomorphism
then we set
now start with $X^n$ again a spin manifold
then
theorem (Stolz-Teichner): we have the horizontal isomorphism in the following diagram:
question if we don’t divide out concordance, do we get differential K-theory on the right?
answer presumeably, but not worked out yet
higher category theory and physics: Spectral standard model and gravity
(1,1)-dimensional Euclidean field theories and K-theory
Pokman Cheung, Supersymmetric field theories and cohomology (arXiv:0811.2267)
Stefan Stolz (notes by Arlo Caine), Supersymmetric Euclidean field theories and generalized cohomology Lecture notes (2009) (pdf)