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.
Stephan Stolz, Peter Teichner, What is an elliptic object? in Topology, geometry and quantum field theory , London Math. Soc. LNS 308, Cambridge Univ. Press (2004), 247-343. (pdf)
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