nLab (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

$$\array{ 1 && (2|1)EFT^0(X)/\sim && \stackrel{\simeq conjectural}{\leftarrow}&& tmf^0(X) && \ni 1 \\ \downarrow && \downarrow^{quantization} &&&& \downarrow^{\int_X} && \downarrow \\ \sigma_{(2|1)(X)}&& (2|1)EFT^{-n}(X)/\sim &&\stackrel{\simeq conjectural }{\leftarrow}&& tmf^{-n}(pt) && \\ &\searrow & \searrow &&& \swarrow& \swarrow \\ &&&& mf^{-n} \\ &&&& index^{S^1}(D_{L X}) = W(X) }$$

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 $(2|1)$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

  $$mf^\bullet \simeq \mathbb{Z}[c_4, c_6, \Delta, \Delta^{-1}]/(c_4^{3}- c_6^{2} - 1728 \Delta)$$

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

• $W(X)$ is the Witten genus

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

  where $a_k = index(D_X \otimes E_k)$ 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

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

(complex numbers with strictly positive imaginary part)

such that

1. 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

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

   note take $A = \left( \array{1 & 1 \\ 0& 1}\right)$ 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 $\tilde f(q) = \sum_{k = -N}^\infty a_k \cdot q^k$ 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

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

$$E \mapsto E_{red}$$

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

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

where

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

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 $(2|1)$-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.

Related concepts

A concrete relation between 2d SCFT and tmf is the lift of the Witten genus to the string orientation of tmf. See there fore more.

References

• 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 Hisham Sati, Urs Schreiber (eds.), Mathematical Foundations of Quantum Field and Perturbative String Theory

A hint supporting the conjectured relation of 2d SCFT to tmf, vaguely in line with the lift of the Witten genus to the string orientation of tmf:

• Davide Gaiotto, Theo Johnson-Freyd, Holomorphic SCFTs with small index (arXiv:1811.00589)