Anomalies and determinant bundle curvature I heard that anomalies and curvature of determinant bundle are related. Namely, curvature of determinant bundle is related to Chern-Simons form (which are involved in description of gauge anomalies). 
I know about Chern-Simons forms and anomalies, but I have no knowledge about determinant bundle and its relationship with anomalies.
Do you know any literature that would be helpful for understanding it (not very mathematical).
 A: A standard reference packed to the brim with other references to everything you ever wanted to know about anomalies is "Anomalies in Quantum Field Theory" by Bertlmann. This particular topic is what comprises part of chapter 11 there. I'll highlight the main points, but this is a technical topic for which you'll have to go to the references and follow all of the numerous steps in the derivations to understand them:
The relation between a gauge anomaly and a determinant bundle arises once you realize that the gauge anomaly term can be computed by a certain determinant, for instance following the method of Fujikawa. This particular construction is due to Alvarez-Gaumé and Ginsparg. One defines a "square root" $\hat{D}$ of the Dirac operator $D$ as
$$ \hat{D} = \gamma^\mu \left(\partial_\mu + A_\mu P_+\right)$$
where $P_+$ projects onto positive chirality fermions. The partition function of a Dirac fermion $\psi$ with action $\int\bar\psi\mathrm{i}\hat{D}\psi$ is given by the functional determinant of $\mathrm{i}D$. Anomalousness of the path integral measure under a gauge transformation means non-invariance of this determinant, and allows one to compute the anomaly due to a positive chirality Weyl fermion.
The relevant determinant bundle lives on a subspace (a two-sphere, to be precise) of the space of connections $\mathcal{A}$ modulo gauge transformations that vanish at infinity $\mathcal{G}_0$. If the determinant of $\hat{D}$, which is a functional of $A$, is gauge-variant, then it cannot be a well-defined global function on $\mathcal{A}/\mathcal{G}_0$ - the presence of an anomaly is thus detected by this determinant bundle being non-trivial, i.e. lacking global sections.
