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

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

Definition

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

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

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

Properties

The objects of $\mathbf{Q}$

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

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

$U \in$ CartSp;

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

$$\alpha_U^g : A \mapsto WZW(g,A)$$

$$CS(A^g) - CS(A) = d WZW(g,A)$$

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