With Alessandro Valentino we have now written a short note on anomalous tqfts and projective representations. In case you’d like to have a preview of it before we post it to the arXiv, any suggestion, comment or criticism is welcome.

]]>together ideas from "network theory" and "(higher) category theory" that

I don't know where to start. It would be useful if there was a context

specified (e.g. TQFTs).

Robert

Is there stuff on the nlab or mathoverflow or...

that my search abilities are not up to finding?

jim ]]>

Here is where I’ll blog my redaing of Topological Quantum Field Theories from Compact Lie Groups

day 1. quickly read the intro and gone directly to section1. main statement there is proposition 1.2, identifying $H^2(B G,\mathbb{Z})$ with the group of abelian characters $Hom(G,U(1))$. in nLab we have a nice understanding of this identification as follows: the set of abelian characters is the set of cocycles $c:\mathbf{B}G\to \mathbf{B}U(1)$, and the above identification follows by the long fibration sequence of oo-Lie groupoids

$\mathbf{B}\mathbb{Z}\to \mathbf{B}\mathbb{R}\to \mathbf{B}U(1)\simeq\mathbf{B}(\mathbb{Z}\hookrightarrow\mathbb{R})\to \mathbf{B}^2\mathbb{Z}\to\cdots$The description of the $0-1$ tqft associated with $c:\mathbf{B}G\to \mathbf{B}U(1)$ is given explicitely. Apparently it can be interpreted as follows: the vector space $F(pt)$ associated to a point is the space of sections of the line bundle on the groupoid $\mathbf{B}G$ induced by the cocycle $c$ (this space of sections will be $1$- or $0$-dimensional depending whether $c$ is the trivial cocycle or not). The complex number $F(S^1)$ associated to $S^1$ is the dimension of this vector space. This can be seen as the integral over $[S^1,\mathbf{B}G]$ of the morphism $[S^1,\mathbf{B}G]\to [S^1,\mathbf{B}U(1)]\to U(1)$ induced by $c$.

That $F(S^1)$ is related this way to the space of maps from $S^1$ to $\mathbf{B}G$ and to the cocycle $c$ seems to be what is really crucial here. Yet in section 1 no deep reason for this being “the right answer” is given at this level and it is only remarked that the above described interpretation of $dim F(S^1)$ is possible.

]]>