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

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

# Contents

## Definition

Let $\frac{1}{2}{\stackrel{^}{p}}_{1}:B{\mathrm{Spin}}_{\mathrm{conn}}\to {B}^{3}U\left(1{\right)}_{\mathrm{conn}}$ 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

$Q:={\mathrm{Aut}}_{{B}^{3}U\left(1{\right)}_{\mathrm{conn}}}\left(\frac{1}{2}{\stackrel{^}{p}}_{1}\right)\phantom{\rule{thinmathspace}{0ex}}.$\mathbf{Q} := \mathbf{Aut}_{\mathbf{B}^3 U(1)_{conn}}(\frac{1}{2}\hat \mathbf{p}_1) \,.

## Properties

The objects of $Q$

$\begin{array}{ccccc}B{\mathrm{Spin}}_{\mathrm{conn}}& & \stackrel{{\mathrm{Ad}}_{g}}{\to }& & B{\mathrm{Spin}}_{\mathrm{conn}}\\ & {}_{\frac{1}{2}{\stackrel{^}{p}}_{1}}↘& {⇙}_{\simeq }^{\alpha }& {↙}_{\frac{1}{2}{\stackrel{^}{p}}_{1}}\\ & & {B}^{3}U\left(1{\right)}_{\mathrm{conn}}\end{array}$\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;

$A\in {\Omega }^{1}\left(U,\mathrm{𝔰𝔬}\right)$

${\alpha }_{U}^{g}:A↦\mathrm{WZW}\left(g,A\right)$\alpha_U^g : A \mapsto WZW(g,A)
$\mathrm{CS}\left({A}^{g}\right)-\mathrm{CS}\left(A\right)=d\mathrm{WZW}\left(g,A\right)$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)