nLab
Oberwolfach Workshop, June 2009 -- Friday, June 12

Talk notes for the session of

• Friday June 12

of

• Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology.

Corbett Redden: String structures and 3-forms

• notational conventions

  • $M^n$ a smooth closed $n$-manifold

  • $g$ a Riem. metric

  • $\mathrm{Spin}\to P\to M$ a principal $\mathrm{Spin}(n)$-bundle

  • $A$ a connection on $P$

  • $S$ a string class

Outline

• I $\{\mathrm{String}\mathrm{Structures}\}{/}_{\mathrm{homotopy}}$

• II harmonic 4-forms on $P$

• III geometry and tmf?

I. String structures

definition

• $B\mathrm{String}$ is defined to be the homotopy fiber $B\mathrm{String}(n)\to B\mathrm{Spin}(n)\stackrel{p_1/2}{\to }K(\mathbb{Z},4)$

• a String structure on $\pi :P\to M$ is a lift of the classifying map to $B\mathrm{Spin}$ through $B\mathrm{String}\to B\mathrm{Spin}$

  $$\begin{array}{ccc}P& & B\mathrm{String}\\ \downarrow & \nearrow & \downarrow \\ M& \to & B\mathrm{Spin}\end{array}$$\array{ P && B String \\ \downarrow &\nearrow& \downarrow \\ M &\to& B Spin }

Prop

• String structure exists iff $\frac{1}{2}{p}_1(P)=0\in H^4(X,\mathbb{Z})$ (essentially by definition through homotopy fiber)

• let $i_P^{*}$ the pullback to the fiber of $P$, then we have

  $${String structures}/_{homotopy}

  \simeq_{canonical} {String classes}

  {\rho \in H^3(P,\mathbb{Z}) s. t. i_P^* \rho = 1 \in H^3(Spin, \mathbb{Z})}

• $\{\mathrm{string}\mathrm{class}\}$ is a torsor for $H^3(M,\mathbb{Z})$ under $\rho \mapsto \rho +{\pi }^H{}^3(M)$

Proof. universal example

$$\begin{array}{ccccc}K(\mathbb{Z},3)& \simeq & {\pi }^{*}E\mathrm{Spin}& \to & E\mathrm{Spin}\\ & & B\mathrm{String}& \to & B\mathrm{Spin}& \to & K(\mathbb{Z},4)\end{array}$$\array{ K(\mathbb{Z},3) &\simeq& \pi^* E Spin &\to & E Spin \\ && B String &\to& B Spin &\to& K(\mathbb{Z}, 4) }

hm, here is my (Urs) description of the situation:

consider the following pasting diagram of homtopy pullbacks ($P$ is the given $\mathrm{Spin}$-bundle, $\hat{P}$ its String-lift)

$$\begin{array}{ccccc}\mathrm{String}& \to & \hat{P}& \to & *\\ \downarrow & & \downarrow & & \downarrow \\ \mathrm{Spin}& \to & P& \to & B^{2}U(1)& \to & *\\ \downarrow & & \downarrow & & \downarrow & & \downarrow \\ *& \to & X& \to & B\mathrm{String}(n)& \to & B\mathrm{Spin}(n)\end{array}$$\array{ String &\to& \hat P &\to& {*} \\ \downarrow && \downarrow && \downarrow \\ Spin &\to& P &\to& B^2 U(1) &\to& {*} \\ \downarrow && \downarrow && \downarrow && \downarrow \\ {*} &\to& X &\to& B String(n) &\to& B Spin(n) }

Why String structures?

String structure on $P$ $\stackrel{\mathrm{transgresses}}{\mapsto }$ Spin structure on loop space $L\mathrm{Spin}\to LP\to LM$

but all the reps of this loop group are projective, so there is actually a central $S^1$ extension of $L\mathrm{Spin}$ in the game

$$\widehat{L\mathrm{Spin}}\to L\mathrm{Spin}$$\hat {L Spin} \to L Spin

Need:

$$\begin{array}{ccc}\widehat{L\mathrm{Spin}}& \to & \widehat{LP}\\ \downarrow & & \downarrow \\ L\mathrm{Spin}& \to & LP\end{array}$$\array{ \hat {L Spin} &\to & \hat {L P} \\ \downarrow && \downarrow \\ L Spin &\to& L P }

$$\rho \in H^3(P,\mathbb{Z})\mapsto$$\rho \in H^3(P, \mathbb{Z}) \mapsto

$$\begin{array}{ccc}({\pi }_{!}{\mathrm{ev}}^{*})\rho & \in & H^2(LP,\mathbb{Z})\\ \downarrow & & \downarrow \\ \mathrm{iniv}.\mathrm{ext}.& & H^2(LG,\mathbb{Z})\end{array}$$\array{ (\pi_! ev^*) \rho &\in& H^2(L P, \mathbb{Z}) \\ \downarrow && \downarrow \\ iniv. ext. && H^2(L G, \mathbb{Z}) }

(on the right: $S^1$-bundles)

** String orientation ** of tmf = topological modular forms

M O\langle 8\rangle^{-n} = M String &\stackrel{\sigma}{\to}& tmf^{-n}(pt) M O\langle 8\rangle^{-n} = M String &\stackrel{\sigma}{\to}& tmf^{-n}(pt)

so give a String manifold with a String class on $\mathrm{Spin}(TM)$

$$M,\rho \mapsto [M,\rho ]\mapsto \sigma (M,\rho )$$M , \rho \mapsto [M, \rho] \mapsto \sigma(M,\rho)

$$\begin{array}{ccc}& & \mathrm{tmf}\\ & {}^{\sigma }\nearrow & \downarrow \\ M\mathrm{String}& \stackrel{\mathrm{Witten}\mathrm{genus}}{\to }& \mathrm{Mod}\mathrm{Forms}\end{array}$$\array{ && tmf \\ & {}^{\sigma}\nearrow & \downarrow \\ M String &\stackrel{Witten genus}{\to}& Mod Forms }

$\mathrm{Witten}\mathrm{genus}(M)=$ ”${\mathrm{index}}^{S^1}{D}_{LM}$”

(by the way, $\sigma$ is surjective on homotopy classes)

warning: I think above my $\rho$ should really be an $S$

II Harmonic representative of $S$

reminder

$(M,g)\mapsto \Delta =d{d}^{*}+d^{*}d$

$$H^k(M,ℝ){\simeq }_{\mathrm{Hodge}}{\Delta }_g^{*}\subset {\Omega }^k(M)$$H^k(M,\mathbb{R}) \simeq_{Hodge} \Delta^*_g \subset \Omega^k(M)

construction

start with $(\pi :P\to M,g_m,A)$

choose a bininvariant metric

$$g_{\rho }:={\pi }^{*}{g}_m\oplus g_{\mathrm{spin}}$$g_\rho := \pi^* g_m \oplus g_{spin}

where the direct sum comes from the splitting of tangent spaces $T_{p}P$ using the connection that we have

Introduce scaling factor $\delta >0$

$$g_{\rho }:={\pi }^{*}{g}_m\oplus {\delta }^2{g}_{\mathrm{spin}}$$g_\rho := \pi^* g_m \oplus \delta^2 g_{spin}

take the “adiabatic limit” ${\lim }_{\delta \to 0}$

so now there is a 1-parameter family of metric on the bundle, and for each one can look at its Laplacian, so as $\delta$ tends to 0 something becomes singular and one has to be careful, but fortunately others already did that for us…

theorem (Mazzeo-Melrose, Dai, Forman)

the kernel $\mathrm{Ker}{\Delta }_{g_{\rho }}$

• extends smoothly to $\delta =0$ (there is a smooth path in some Grassmannian space)

• and comes from a filtration isomorphic to Serre SS for $(\mathrm{Spin}\to P\to M)$

this means that

$$\Rightarrow H^k(P,\mathbb{Z})\stackrel{\simeq }{\to }\underset{\delta \to 0}{\lim }\mathrm{Ker}{\Delta }_{g_{\rho }}=:H^k(P)$$\Rightarrow H^k(P, \mathbb{Z}) \stackrel{\simeq}{\to} \lim_{\delta \to 0} Ker \Delta_{g_\rho} =: H^k(P)

Theorem (Redden)

Given $P\stackrel{\pi }{\to }g,A$ and $\frac{1}{2}{p}_1(P)=0$

then

$$\begin{array}{ccccc}{H}^3(P,\mathbb{Z})& \to & H^3(P,ℝ)& \to & H^3(P)\\ S& & \mapsto & {\mathrm{CS}}_3(A)& & {\mathrm{CS}}_3(A)-{\pi }^{*}H\end{array}$$\array{ H^3(P, \mathbb{Z}) &\to& H^3(P,\mathbb{R}) &\to& H^3(P) \\ S &&\mapsto& CS_3(A)&& CS_3(A) - \pi^* H }

here ${\mathrm{CS}}_3(A)$ is the Chern-Simons 3-form of the spin-connection

and recall $H^3(P)$ here denotes harmonic forms on $P$ (should really be script font

remark

in genral

$$[\rho {]}_{g_{\delta }}={\mathrm{CS}}_3(A)-{\pi }^{*}H+O(\delta )\neg \in {\pi }^{*}{\Omega }^3(M)$$[\rho]_{g_\delta} = CS_3(A) - \pi^* H + O(\delta) \not\in \pi^* \Omega^3(M)

if we have a product of two groups we accordingly would get CS of one connection minus CS of the other.

What is $H$?

(first digression)

theorem (Chern-Simons,…)

given $(P\to M,A)\mapsto \widehat{\mathrm{frac}12p_1}(A){\hat{H}}^4(M)$

$$\begin{array}{ccccccccc}{\Omega }_{\mathbb{Z}}^3& \to & (M){\Omega }^3(M)& \stackrel{a}{\mathrm{to}}& {\hat{H}}^4(M)& \to & H^4(M,\mathbb{Z})& \to & 0\\ & & H& \mapsto & \widehat{\frac{1}{2}{p}_1(A)}& \mapsto & \frac{1}{2}{p}_1(P)=0& & \end{array}$$\array{ \Omega^3_{\mathbb{Z}} &\to& (M)\Omega^3(M) &\stackrel{a}{to}& \hat H^4(M) &\to& H^4(M,\mathbb{Z}) &\to& 0 \\ && H &\mapsto& \hat{\frac{1}{2}p_1(A)} &\mapsto& \frac{1}{2}p_1(P) = 0 && }

in particular

$$\begin{array}{ccc}& & ℝ\\ & {}^{\int H}\nearrow & \downarrow \\ {\mathbb{Z}}_3(M)& \stackrel{\widehat{\frac{1}{2}{p}_1(A)}}{\to }& ℝ/\mathbb{Z}\end{array}$$\array{ && \mathbb{R} \\ & {}^{\int H} \nearrow & \downarrow \\ \mathbb{Z}_3(M) &\stackrel{\hat {\frac{1}{2}p_1(A)}}{\to}& \mathbb{R}/\mathbb{Z} }

and secondly $d^{*}H=0$

these two properties determine $H$ uniquely up to harmonic forms ${\mathscr{H}}_{\mathbb{Z}}^3(M)=\ker {\Delta }_g$

Equivariance

$$H_{\rho +{\pi }^{*}\psi }=H_{\rho }+{\pi }^{*}H4$$H_{\rho + \pi^* \psi} = H_\rho + \pi^* H4

where $\psi \in H^3(M,\mathbb{Z})$ $\mapsto H_4\in {\mathscr{H}}^3(M)$

over all what this says is that if we go from

$$\begin{array}{cc}\mathrm{Metr}(M)\times A(P)\times \{\mathrm{String}\mathrm{Class}\}& (g,A,S)\\ \downarrow & \downarrow \\ {\Omega }^3(P)& {\mathrm{CS}}_3(A)-{\pi }^{*}{H}_{q,A,S}\\ \downarrow & \downarrow \\ {\Omega }^3(M)& H_{g,A\rho }\end{array}$$\array{ Metr(M)\times A(P) \times \{String Class\} & (g,A,S) \\ \downarrow & \downarrow \\ \Omega^3(P) & CS_3(A) - \pi^* H_{q,A,S} \\ \downarrow & \downarrow \\ \Omega^3(M) & H_{g, A \rho} }

$$\begin{array}{ccc}& & {\mathrm{tmf}}^{-n}\\ & \nearrow & \downarrow \\ M\mathrm{String}& \stackrel{}{\to }& \mathrm{MF}\end{array}$$\array{ && tmf^{-n} \\ & \nearrow & \downarrow \\ M String &\stackrel{}{\to}& MF }

conjecture (S. Stolz) if $M$ is String and admits a positive Ricci curvature metric, then $\mathrm{Witten}(M)=0$

question: also $\sigma (M,\rho )=0$? no, no way!

hypothesis

If $M$ is a Spin manifold that admits a metric and String structure $(g,S)$ and $A$ is the Levi-Civita connection

such that

• $\mathrm{Ric}(g)>0$

• $H_{g,s}=0\in {\Omega }^3(M)$

$\Rightarrow \sigma (M,S)=0\in {\mathrm{tmf}}^{-n}(\mathrm{pt})$

example $M=S^3=\mathrm{SU}(2)$

$$p_1\in H^4(S^3)=0$$p_1 \in H^4(S^3) = 0

$$H^3(S^3,\mathbb{Z})=\mathbb{Z}=\mathrm{number}\mathrm{of}\mathrm{string}\mathrm{classes}$$H^3(S^3, \mathbb{Z}) = \mathbb{Z} = number of string classes

$$dH=d^{*}H=0\Rightarrow H\in H^3(S^3,ℝ)\simeq ℝ$$d H = d^* H = 0 \Rightarrow H \in H^3(S^3, \mathbb{R}) \simeq \mathbb{R}

$$\begin{array}{c}\mathrm{String}\mathrm{Classes}\\ \downarrow \\ M{\mathrm{String}}^{-3}={\pi }_3^S={\mathrm{tmf}}^{-3}=\mathbb{Z}/24\end{array}$$\array{ String Classes \\ \downarrow \\ M String^{-3} = \pi_3^S = tmf^{-3} = \mathbb{Z}/{24} }

(can’t type the full diagram…)

consider a 1-parameter family of “berger metrics” on $S^3$

rescaling the fiber in the Hopf fibration $S^1\to S^3\to S^2$

Konrad Waldorf

Urs: I had to miss that and the following two talks, hopefully somebody else has notes. Konrad’s talk is based on his new article

• Konrad Waldorf, String connections and Chern-Simons theory (arXiv)

one more

…

Mike Hopkins : Kervaire invariant one

…

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