Path integral for fermions on circle and background gauge fields

This question is continuation of Path integral for fermion on circle.

I'm reading Witten article Anomaly Inflow and the $$\eta$$-Invariant and wanna to understand some technical details.

In section 4.1 authors consider even number $$n$$ of Majorana fermions: $$I = \int dt \frac{i}{2} \sum_{i=1}^{n} \chi_i \frac{d}{dt} \chi_i$$

To calculate path integral authors want use equations (as I understand, they use (74) and (75) from Path integrals for fermions, susy quantum mechanics, etc..): $$Tr_{\mathcal{H}} e^{iH T} = \int_{NS} D\chi_i \;e^I$$ $$STr_{\mathcal{H}} e^{iH T} = \int_{R} D\chi_i \;e^I$$

At this moment, using $$H=0$$, we can easily obtain results, by calculation of Hilbert space dimension: $$\int_{NS} D\chi_i \;e^I = 2^{\frac{n}{2}}$$ $$\int_{R} D\chi_i \;e^I = 0$$

But authors chose another way:

Though we could add additional terms to the action to get a nonzero Hamiltonian, instead we will turn on a background $$SO(n)$$ gauge field on the circle.

As I understand, this mean $$I \to I_A = \int dt \frac{i}{2} \left(\sum_{i=1}^{n} \chi_i \frac{d}{dt}\chi_i - \chi_i A_{ij} \chi_j \right)$$ $$H_A = \frac{i}{2} \chi_i A_{ij} \chi_j$$

After authors use holonomy U, and say that U is block diagonal matrix. Form of block: $$U_k = \begin{pmatrix} \cos\theta_k& \sin\theta_k \\ -\sin\theta_k & \cos\theta_k \end{pmatrix}$$

I don't understand this step. After they immediately obtain results: $$Tr_{\mathcal{H}} U = \prod_{k=1}^{n/2} 2 \cos(\theta_k/2)$$ $$STr_{\mathcal{H}} U = \pm\prod_{k=1}^{n/2} 2i \sin(\theta_k/2)$$

Why did they use background gauge field?

How did they calculate this traces?

How this result correspondence straight calculation of dimension of Hilbert space?

This can be calculated by rotating to the Cartan subalgebra of the gauge symmetry group (which is why the matrices turned out diagonal) and then computing the functional determinants.

See, for example,

which is part of the list of references I gave on your other question on the worldlien formalism of QFT.

• By the way, I assume your $\theta$ are the modular parameters that come from gauge fixing, that should still be integrated over. If not, please say what the $\theta_{k}$ are?
– lux
Jan 4 '20 at 19:59
• I updated question Jan 4 '20 at 20:10
• OK so my comment is correct - these are modular parameters that cannot be gauged away. These distinguish physically distinct configurations of the gauge field. But they must be integrated over eventually
– lux
Jan 4 '20 at 20:31
• It is not clear for me. Did you see 4.1 from the Witten article? Jan 4 '20 at 20:51
• In the paper I reference in my answer the authors show exactly how to get there....
– lux
Jan 4 '20 at 21:21