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

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:

• Axiomatic field theories and their motivation from topology.

• (1,1)-dimensional Euclidean field theories and K-theory

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

$$\begin{array}{ccccccccc}1& & (2\mid 1){\mathrm{EFT}}^0(X)/\sim & & \stackrel{\simeq \mathrm{conjectural}}{\leftarrow }& & {\mathrm{tmf}}^0(X)& & \ni 1\\ \downarrow & & {\downarrow }^{\mathrm{quantization}}& & & & {\downarrow }^{{\int }_X}& & \downarrow \\ {\sigma }_{(2\mid 1)(X)}& & (2\mid 1){\mathrm{EFT}}^{-n}(X)/\sim & & \stackrel{\simeq \mathrm{conjectural}}{\leftarrow }& & {\mathrm{tmf}}^{-n}(\mathrm{pt})& & \\ & \searrow & \searrow & & & \swarrow & \swarrow \\ & & & & {\mathrm{mf}}^{-n}\\ & & & & {\mathrm{index}}^{S^1}(D_{LX})=W(X)\end{array}$$\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 $(2\mid 1)$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

  $${\mathrm{mf}}^{•}\simeq \mathbb{Z}[c_4,c_6,\Delta ,{\Delta }^{-1}]/(c_4^3-c_6^2-1728\Delta )$$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 $\mathrm{deg}(\Delta )=-24$, hence $w(\Delta )=12$.

• $W(X)$ is the Witten genus

  $$W(X)=\sum _{k\in \mathbb{Z}}{a}_k\cdot q^k,a_k\in \mathbb{Z}$$W(X) = \sum_{k \in \mathbb{Z}} a_k \cdot q^k \,, a_k \in \mathbb{Z}

  where $a_k=\mathrm{index}(D_X\otimes E_k)$ 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?

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

(complex numbers with strictly positive imaginary part)

such that

1. if $A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in {\mathrm{SL}}_2(\mathbb{Z})$ acting by $A:\tau \mapsto =\frac{a\tau +b}{c\tau +d}$ we have

   $$f(A(\tau ))=(c\tau +d{)}^{w}f(\tau )$$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 $f(\tau +1)=f(\tau )$

2. $f$ 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.

3. integrality $\tilde{f}(q)={\sum }_{k=-N}^{\mathrm{\infty }}{a}_k\cdot q^k$ 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

$$(2\mid 1){\mathrm{EFT}}^0\stackrel{S}{\to }2\mathrm{EFT}\ne E$$(2|1)EFT^0 \stackrel{S}{\to} 2 EFT \ne E

$$E\mapsto E_{\mathrm{red}}$$E \mapsto E_{red}

then the partition function is the map $Z_E:ℂ\to ℝ$

$$Z_E:\tau \mapsto E_{\mathrm{red}}(T_{\tau })$$Z_E : \tau \mapsto E_{red}(T_\tau)

where

$$T_{\tau }:=ℂ/\mathbb{Z}\times \mathbb{Z}\cdot \tau$$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 $(2\mid 1)$-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.

References

• 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