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

Context

Functorial quantum field theory

Super-Geometry

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 XX has spin structure. Now we assume String structure.

So we are looking for a diagram of the form

1 (2|1)EFT 0(X)/ conjectural tmf 0(X) 1 quantization X σ (2|1)(X) (2|1)EFT n(X)/ conjectural tmf n(pt) mf n index S 1(D LX)=W(X) \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)(2|1)EFT local by refining it to an extended FQFTs. This will not be considered here.

we will explain the following items

modular forms

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

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

(complex numbers with strictly positive imaginary part)

such that

  1. if A=(a b c d)SL 2()A = \left( \array{a & b \\ c& d}\right) \in SL_2(\mathbb{Z}) acting by A:τ=aτ+bcτ+dA : \tau \mapsto = \frac{a \tau + b }{c \tau + d} we have

    f(A(τ))=(cτ+d) wf(τ) f(A(\tau)) = (c \tau + d)^w f(\tau)

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

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

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

  3. integrality f˜(q)= k=N a kq k\tilde f(q) = \sum_{k = -N}^\infty a_k \cdot q^k then a ka_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 EE be an EFT

(2|1)EFT 0S2EFTE (2|1)EFT^0 \stackrel{S}{\to} 2 EFT \ne E
EE red E \mapsto E_{red}

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

Z E:τE red(T τ) Z_E : \tau \mapsto E_{red}(T_\tau)

where

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

is thee standard torus of modulus τ\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)(2|1)-EFTs EE such that the partition function Z EZ_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.

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

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: