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

Talk notes for the session of

• Friday June 12

of

Corbett Redden: String structures and 3-forms

• notational conventions

• $M^n$ a smooth closed $n$-manifold

• $g$ a Riem. metric

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

• $A$ a connection on $P$

• $S$ a string class

Outline

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

• II harmonic 4-forms on $P$

• III geometry and tmf?

I. String structures

definition

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

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

$\array{ P && B String \\ \downarrow &\nearrow& \downarrow \\ M &\to& B Spin }$

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

Proof. universal example

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

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

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

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

Why String structures?

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

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

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

Need:

$\array{ \hat {L Spin} &\to & \hat {L P} \\ \downarrow && \downarrow \\ L Spin &\to& L P }$
$\rho \in H^3(P, \mathbb{Z}) \mapsto$
$\array{ (\pi_! ev^*) \rho &\in& H^2(L P, \mathbb{Z}) \\ \downarrow && \downarrow \\ iniv. ext. && H^2(L G, \mathbb{Z}) }$

(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) 

so give a String manifold with a String class on $M O\langle 8\rangle^{-n} = M String &Spin(T M)$

$M , \rho \mapsto [M, \rho] \mapsto \sigma(M,\rho)$
$\array{ && tmf \\ & {}^{\sigma}\nearrow & \downarrow \\ M String &\stackrel{Witten genus}{\to}& Mod Forms }$

$Witten genus(M) =$$index^{S^1} D_{L M}$

(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

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

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

construction

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

choose a bininvariant metric

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

where the direct sum comes from the splitting of tangent spaces $T_p P$ using the connection that we have

Introduce scaling factor $\delta \gt 0$

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

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 $Ker \Delta_{g_\rho}$

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

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

this means that

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

Theorem (Redden)

Given $P \stackrel{\pi}{\to} g, A$ and $\frac{1}{2}p_1(P) = 0$

then

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

here $CS_3(A)$ 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

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

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 $(P \to M , A) \mapsto \hat {frac{1}{2} p_1}(A) \hat H^4(M)$

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

in particular

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

and secondly $d^* H = 0$

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

Equivariance

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

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

over all what this says is that if we go from

$\array{ Metr(M)\times A(P) \times \{String Class\} & (g,A,S) \\ \downarrow & \downarrow \\ \Omega^3(P) & CS_3(A) - \pi^* H_{q,A,S} \\ \downarrow & \downarrow \\ \Omega^3(M) & H_{g, A \rho} }$
$\array{ && tmf^{-n} \\ & \nearrow & \downarrow \\ M String &\stackrel{}{\to}& MF }$

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

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

• $Ric(g) \gt 0$

• $H_{g,s} = 0 \in \Omega^3(M)$

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

example $M = S^3 = SU(2)$

$p_1 \in H^4(S^3) = 0$
$H^3(S^3, \mathbb{Z}) = \mathbb{Z} = number of string classes$
$d H = d^* H = 0 \Rightarrow H \in H^3(S^3, \mathbb{R}) \simeq \mathbb{R}$
$\array{ String Classes \\ \downarrow \\ M String^{-3} = \pi_3^S = tmf^{-3} = \mathbb{Z}/{24} }$

(can’t type the full diagram…)

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

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