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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:

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 (21)EFT 0(X)/ conjectural tmf 0(X) 1 quantization X σ (21)(X) (21)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 (21)(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 mf^\bullet of integral modular forms

    mf [c 4,c 6,Δ,Δ 1]/(c 4 3c 6 21728Δ) mf^\bullet \simeq \mathbb{Z}[c_4, c_6, \Delta, \Delta^{-1}]/(c_4^{3}- c_6^{2} - 1728 \Delta)

    one calls w=n2w = -\frac{n}{2} the weight . We have degree of Δ\Delta is deg(Δ)=24deg(\Delta) = -24, hence w(Δ)=12w(\Delta) = 12.

  • W(X)W(X) is the Witten genus

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

    where a k=index(D XE k)a_k = index(D_X \otimes E_k) where E kE_k is some explicit vector bundle over XX.

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

(21)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 (21)(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.

References

Revised on March 21, 2014 09:25:28 by Urs Schreiber (89.204.138.115)