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

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

Definition

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

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

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

Properties

The objects of $Q$

$$\begin{array}{ccccc}B{\mathrm{Spin}}_{\mathrm{conn}}& & \stackrel{{\mathrm{Ad}}_g}{\to }& & B{\mathrm{Spin}}_{\mathrm{conn}}\\ & {}_{\frac{1}{2}{\hat{p}}_1}\searrow & {⇙}_{\simeq }^{\alpha }& {\swarrow }_{\frac{1}{2}{\hat{p}}_1}\\ & & B^{3}U(1{)}_{\mathrm{conn}}\end{array}$$\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 $g\in \mathrm{Spin}$ and a Wess-Zumino 2-form which exhibits the failure of the Chern-Simons form to be $g$-gauge invariant

$U\in$ CartSp;

$A\in {\Omega }^1(U,\mathrm{𝔰𝔬})$

$${\alpha }_U^g:A\mapsto \mathrm{WZW}(g,A)$$\alpha_U^g : A \mapsto WZW(g,A)

$$\mathrm{CS}(A^g)-\mathrm{CS}(A)=d\mathrm{WZW}(g,A)$$CS(A^g) - CS(A) = d WZW(g,A)

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