Faddeev-Popov determinant and topology of the worldline

Question

I am studying the path integral quantization of relativistic particles, using the BRST quantization method. I have to compute the integral \begin{equation} Z\sim \int Dx \det(\partial_\tau)e^{-\int_0^1d\tau(\frac{1}{4T}\dot{x}^2+m^2T)} \end{equation} where $\det(\partial_\tau)$ is the result of a path integration over the ghost and antighost $b$ and $c$, namely \begin{equation} \det(\partial_\tau)=\int Dc Dbe^{-\int_0^1d\tau b\dot{c}} \end{equation} At this point the professor claims that we need to compute the determinant to resolve the path integral, and that "for the wordline topology of the interval $I$ one obtains the QFT propagator of the scalar particle. The FP determinant is just a constant and it can be absorbed in the normalization and the path integral becomes: \begin{equation} Z_I= \int_0^\infty dTe^{-m^2T}\int_I Dx e^{-\int_0^1d\tau\frac{1}{4T}\dot{x}^2} \end{equation} while for the topology of the circle $S^1$ one obtains the QFT one-loop effective action \begin{equation} Z_{S^1}= -\int_0^\infty \frac{dT}{T}e^{-m^2T}\int_I Dx e^{-\int_0^1d\tau\frac{1}{4T}\dot{x}^2} \end{equation} which contains the extra factor $T^{-1}$ due to the fact that there is a zero-mode in the ghost determinant that signals a traslational symmetry of the circle."

My question: I don't understand the connection with the FD and the topology of the wordline. Why should the determinant be constant, what is the starting point of the reasoning behind this? I can't find suitable references on this particular issue.

Answer 1

The determinant was computed long time ago by Andrew Cohen, Gregory Moore, Philip Nelson, and Joseph Polchinski in An off-shell propagator for string theory. The parameter $T$ in your path integral is the so-called Teichmüller parameter of $\mathfrak{Diff}(I)$ or $\mathfrak{Diff}(S^{1})$, depending on the boundary condition. It is a one-dimensional analogue of the $\tau$ parameter in upper-half plane in string theory. Here, $T$ measures the length of the path in your path integral.

To be more specific, I meant the following procedure.

The path integral of a particle is

$$\int\mathcal{D}e\int\mathcal{D}xe^{-\int dt\left(\frac{1}{4e}\dot{x}^{2}+m^{2}e\right)}$$

where $e(t)$ is called einbein field. In the reference, the author performed a gauge fixing of $e$, i.e.

$$\mathcal{D}e\equiv\det(J) dT\mathcal{D}\xi,$$

where $\xi$ is the gauge redundancy, i.e. diffeomorphisms on an interval, and $J$ is the Jacobian. Using Riemann zeta regularization, one finds that the Jacobian is a constant. Thus, you can replace the functional integral over $e$ by a finite dimensional integral over the Teichmüller parameter.