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 2-dimensional FQFTs may be related to tmf.
raw material: this are notes taken more or less verbatim in a seminar – needs polishing
Previous:
recall the big diagram from the end of the previous entry.
The goal now is to replace everywhere topological K-theory by tmf.
previously we had assumed that $X$ has spin structure. Now we assume String structure.
So we are looking for a diagram of the form
the vertical maps here are due to various theorems by various people – except for the “physical quantization” on the left, that is used in physics but hasn’t been formalized
the horizontal maps are the conjecture we are after in the Stolz-Teichner program: The top horizontal map will involve making the notion of $(2|1)$EFT local by refining it to an extended FQFTs. This will not be considered here.
we will explain the following items
the ring $mf^\bullet$ of integral modular forms
one calls $w = -\frac{n}{2}$ the weight . We have degree of $\Delta$ is $deg(\Delta) = -24$, hence $w(\Delta) = 12$.
$W(X)$ is the Witten genus
where $a_k = index(D_X \otimes E_k)$ where $E_k$ is some explicit vector bundle over $X$.
definition An (integral) modular form of weight $w$ is a holomorphic function on the upper half plane
(complex numbers with strictly positive imaginary part)
such that
if $A = \left( \array{a & b \\ c& d}\right) \in SL_2(\mathbb{Z})$ acting by $A : \tau \mapsto = \frac{a \tau + b }{c \tau + d}$ we have
note take $A = \left( \array{1 & 1 \\ 0& 1}\right)$ then we get that $f(\tau + 1) = f(\tau)$
$f$ has at worst a pole at $\{0\}$ (for weak modular forms this condition is relaxed)
it follows that $f = f(q)$ with $q = e^{2 \pi i \tau}$ is a meromorphic funtion on the open disk.
integrality $\tilde f(q) = \sum_{k = -N}^\infty a_k \cdot q^k$ then $a_k \in \mathbb{Z}$
by this definition, modular forms are not really functions on the upper half plane, but functions on a moduli space of tori. See the following definition:
if the weight vanishes, we say that modular form is a modular function .
definition (2|1)-dim partition function
Let $E$ be an EFT
then the partition function is the map $Z_E : \mathbb{C} \to \mathbb{R}$
where
is thee standard torus of modulus $\tau$.
then the central theorem that we are after here is
therorem (Stolz-Teichner) (after a suggestion by Edward Witten)
There is a precise definition of $(2|1)$-EFTs $E$ such that the partition function $Z_E$ is an integral modular function
(so this is really four theorems: the function is holomorphic, integral, etc.)
moreover, every integral modular function arises in this way.
Pokman Cheung, Supersymmetric field theories and cohomology (arXiv:0811.2267)
Stefan Stolz, Peter Teichner, Supersymmetric Euclidean field theories and generalized cohomology , in Branislav Jurčo, Hisham Sati, Urs Schreiber (eds.), Mathematical Foundations of Quantum Field and Perturbative String Theory