(n) or bounding (b) spin structure.

There are similar generalizations of the higher dimensional theories discussed below.

\[thm:5\] All of the *classical* theories alluded to in this paper are purely topological and are most systematically defined by refining the class in $h^{n+1}(BG)$ to an object in the corresponding *differential* theory and then transgressing; see [@F1] for the case $n=3$. The *quantum* theories, which are our focus, do not depend on the differential refinement.

The cohomology group $H^3(BG)$ and two-dimensional theories {#sec:2}
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$H^3(BG; {\mathbb Z})$ for arbitrary $G$ {#subsec:2.1}
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Analogous to Proposition \[thm:1\] we have the following.

\[thm:6\] Let $G$ be a compact Lie group. The cohomology group $H^3(BG; {\mathbb Z})$ can be identified with the set of isomorphism classes of central extensions ${\mathbb T}\to{G^\tau }\to G$.

In purely topological terms the class in $H^3(BG; {\mathbb Z})$ attached to a central extension ${\mathbb T}\to{G^\tau }\to G$ is the obstruction to lifting the universal $G$-bundle to a ${G^\tau }$-bundle.[^9] A central extension is equivalently a smooth hermitian line bundle $K={K^\tau }\to G$ together with isomorphisms $$\label{eq:28}
     \theta _{x,y}{\colon}K_x\otimes K_y\to K_{xy},\qquad x,y\in G,$$ for each pair in $G$ which satisfy an associativity constraint $$\label{eq:29}
     \theta _{xy,z}\,(\theta _{x,y} \otimes \operatorname{id}_{K_z})= \theta
     _{x,yz}\,(\operatorname{id}_{K_x}\otimes 