# nLab (2,1)-dimensional Euclidean field theories and tmf

superalgebra

and

supergeometry

## Applications

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

$\begin{array}{ccccccccc}1& & \left(2\mid 1\right){\mathrm{EFT}}^{0}\left(X\right)/\sim & & \stackrel{\simeq \mathrm{conjectural}}{←}& & {\mathrm{tmf}}^{0}\left(X\right)& & \ni 1\\ ↓& & {↓}^{\mathrm{quantization}}& & & & {↓}^{{\int }_{X}}& & ↓\\ {\sigma }_{\left(2\mid 1\right)\left(X\right)}& & \left(2\mid 1\right){\mathrm{EFT}}^{-n}\left(X\right)/\sim & & \stackrel{\simeq \mathrm{conjectural}}{←}& & {\mathrm{tmf}}^{-n}\left(\mathrm{pt}\right)& & \\ & ↘& ↘& & & ↙& ↙\\ & & & & {\mathrm{mf}}^{-n}\\ & & & & {\mathrm{index}}^{{S}^{1}}\left({D}_{LX}\right)=W\left(X\right)\end{array}$\array{ 1 && (2|1)EFT^0(X)/\sim && \stackrel{\simeq conjectural}{\leftarrow}&& tmf^0(X) && \ni 1 \\ \downarrow && \downarrow^{quantization} &&&& \downarrow^{\int_X} && \downarrow \\ \sigma_{(2|1)(X)}&& (2|1)EFT^{-n}(X)/\sim &&\stackrel{\simeq conjectural }{\leftarrow}&& tmf^{-n}(pt) && \\ &\searrow & \searrow &&& \swarrow& \swarrow \\ &&&& mf^{-n} \\ &&&& index^{S^1}(D_{L X}) = W(X) }

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 $\left(2\mid 1\right)$EFT local by refining it to an extended FQFTs. This will not be considered here.

we will explain the following items

• the ring ${\mathrm{mf}}^{•}$ of integral modular forms

${\mathrm{mf}}^{•}\simeq ℤ\left[{c}_{4},{c}_{6},\Delta ,{\Delta }^{-1}\right]/\left({c}_{4}^{3}-{c}_{6}^{2}-1728\Delta \right)$mf^\bullet \simeq \mathbb{Z}[c_4, c_6, \Delta, \Delta^{-1}]/(c_4^{3}- c_6^{2} - 1728 \Delta)

one calls $w=-\frac{n}{2}$ the weight . We have degree of $\Delta$ is $\mathrm{deg}\left(\Delta \right)=-24$, hence $w\left(\Delta \right)=12$.

• $W\left(X\right)$ is the Witten genus

$W\left(X\right)=\sum _{k\in ℤ}{a}_{k}\cdot {q}^{k}\phantom{\rule{thinmathspace}{0ex}},{a}_{k}\in ℤ$W(X) = \sum_{k \in \mathbb{Z}} a_k \cdot q^k \,, a_k \in \mathbb{Z}

where ${a}_{k}=\mathrm{index}\left({D}_{X}\otimes {E}_{k}\right)$ where ${E}_{k}$ is some explicit vector bundle over $X$.

## modular forms

definition An (integral) modular form of weight $w$ is a holomorphic function on the upper half plane?

$f:\left({ℝ}^{2}{\right)}_{+}↪ℂ$f : (\mathbb{R}^2)_+ \hookrightarrow \mathbb{C}

(complex numbers with strictly positive imaginary part)

such that

1. if $A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in {\mathrm{SL}}_{2}\left(ℤ\right)$ acting by $A:\tau ↦=\frac{a\tau +b}{c\tau +d}$ we have

$f\left(A\left(\tau \right)\right)=\left(c\tau +d{\right)}^{w}f\left(\tau \right)$f(A(\tau)) = (c \tau + d)^w f(\tau)

note take $A=\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$ then we get that $f\left(\tau +1\right)=f\left(\tau \right)$

2. $f$ has at worst a pole at $\left\{0\right\}$ (for weak modular forms this condition is relaxed)

it follows that $f=f\left(q\right)$ with $q={e}^{2\pi i\tau }$ is a meromorphic funtion on the open disk.

3. integrality $\stackrel{˜}{f}\left(q\right)={\sum }_{k=-N}^{\infty }{a}_{k}\cdot {q}^{k}$ then ${a}_{k}\in ℤ$

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

$\left(2\mid 1\right){\mathrm{EFT}}^{0}\stackrel{S}{\to }2\mathrm{EFT}\ne E$(2|1)EFT^0 \stackrel{S}{\to} 2 EFT \ne E
$E↦{E}_{\mathrm{red}}$E \mapsto E_{red}

then the partition function is the map ${Z}_{E}:ℂ\to ℝ$

${Z}_{E}:\tau ↦{E}_{\mathrm{red}}\left({T}_{\tau }\right)$Z_E : \tau \mapsto E_{red}(T_\tau)

where

${T}_{\tau }:=ℂ/ℤ×ℤ\cdot \tau$T_\tau := \mathbb{C}/{\mathbb{Z} \times \mathbb{Z} \cdot \tau}

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 $\left(2\mid 1\right)$-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.

## References

Revised on June 28, 2011 11:00:26 by Urs Schreiber (131.211.239.85)