Schreiber
quantomorphism 3-group of 3d Spin-Chern-Simons theory

This is a sub-entry of ∞-geometric prequantization.

Contents

Definition

Let 12p^ 1:BSpin connB 3U(1) conn\frac{1}{2} \hat \mathbf{p}_1 : \mathbf{B}Spin_{conn} \to \mathbf{B}^3 U(1)_{conn} the universal Chern-Simons circle 3-bundle with 3-connection, in H=\mathbf{H} = Smooth∞Grpd.

The quantomorphism 3-group of 3d Spin-Chern-Simons theory is

Q:=Aut B 3U(1) conn(12p^ 1). \mathbf{Q} := \mathbf{Aut}_{\mathbf{B}^3 U(1)_{conn}}(\frac{1}{2}\hat \mathbf{p}_1) \,.

Properties

The objects of Q\mathbf{Q}

BSpin conn Ad g BSpin conn 12p^ 1 α 12p^ 1 B 3U(1) conn \array{ \mathbf{B}Spin_{conn} &&\stackrel{Ad_g}{\to}&& \mathbf{B}Spin_{conn} \\ & {}_{\mathllap{\frac{1}{2}\hat \mathbf{p}_1}}\searrow &\swArrow_{\simeq}^{\alpha}& \swarrow_{{\mathrlap{\frac{1}{2}\hat \mathbf{p}_1}}} \\ && \mathbf{B}^3 U(1)_{conn} }

are pairs consisting of elements gSping \in Spin and a Wess-Zumino 2-form which exhibits the failure of the Chern-Simons form to be gg-gauge invariant

UU \in CartSp;

AΩ 1(U,𝔰𝔬)A \in \Omega^1(U,\mathfrak{so})

α U g:AWZW(g,A) \alpha_U^g : A \mapsto WZW(g,A)
CS(A g)CS(A)=dWZW(g,A) CS(A^g) - CS(A) = d WZW(g,A)

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Revised on June 28, 2012 00:47:35 by Urs Schreiber (82.169.65.155)