# Born-Infeld term for $D$-brane low energy effective action

I am trying to reproduce the Born-Infeld term for the $$D$$-brane action as explained in Szabo's BUSSTEP Lectures and must admit I am utterly confused by some of the steps.

This is a long and technical question, so I first give some context, then I give my problems. All of this can be answered easily if someone can point me to a detailed derivation of that result.

The context

Szabo starts with the effective bosonic open string action in the conformal gauge with the endpoints coupled to a gauge field, (7.21), $$S[x,A] =\frac{1}{4\pi\alpha'} \int d^2z \, \partial x^\mu \bar \partial x_\mu -\frac{i}{2} \int_0^{2\pi} \partial_\theta x^\mu A_\mu \Big|_{r=1}$$ Here $$\theta$$ is the polar angle describing the unit disc. He then uses the background field method of separating out the zero modes $$x^\mu =x_0^\mu + \xi^\mu$$ and path integrating over the fluctuations $$\xi$$. Integrating out the bulk modes leaves a boundary path integral $$Z = g_s^{-1} \int d\vec{x}_0 \int \mathcal D \xi^\mu(\theta) e^{-S_b[\xi,A]} \tag{1}$$ with the boundary action $$S_b[\xi,A] = \int_0^{2\pi} d\theta \left[ \frac{1}{4\pi\alpha'} \xi^\mu N^{-1} \xi_\mu + iF_{\mu\nu} \xi^\mu\dot\xi^\nu \right] \tag{2}$$ with $$N^{-1}$$ the space inverse of the Green's function on the disc with Neumann boundary conditions. It is given by (7.31) $$N^{-1}(\theta,\theta') = -\frac{1}{\pi} \sum_{n=1}^\infty n \cos [n(\theta-\theta')] \tag{3}$$ He expands the fluctuations in Fourier modes $$\xi^\mu (\theta) = \sum_{n=1}^\infty \left[ a_n^\mu \cos n\theta + b_n^\mu \sin n\theta\right]$$ and then rotates the field tensor in a canonical Jordan form $$F_{\mu\nu} = \begin{pmatrix} 0 & - f_1 & & 0 & 0 \\ f_1 &0 & & 0 & 0 \\ & & \ddots & & \\ 0& 0 & &0&-f_{D/2}\\ 0 & 0 & & f_{D/2} &0 \end{pmatrix}$$ and claims that we end up with Gaussian integrations with quadratic forms (7.35). Ignoring some coefficients these are $$\begin{pmatrix} a_m^{2\ell-1} & a_n^{2\ell} \end{pmatrix} \begin{pmatrix} 1 & -2\pi \alpha' f_\ell \\ 2\pi \alpha' f_\ell & 1 \end{pmatrix} \begin{pmatrix} a_m^{2\ell-1} \\ a_n^{2\ell} \end{pmatrix} \tag{4}$$ and a similar expression with the $$b_n^\ell$$'s.

My questions (4 is the most pressing one)

1. In (1) why integrate over the space coordinate $$d\vec{x}_0$$ only? What happened with the time coordinate
2. How do you derive the kinetic term $$\xi^\mu N^{-1} \xi_\mu$$ in (2)?
3. How do you prove (3)?
4. How do you derive (4)? In particular

$$\quad$$ (4a) The kinetic term has a triple sum (coming from two $$\xi$$'s and one $$N^{-1}$$). It leads to integrations of the form $$f(n\theta) \cos k \theta g(n\theta)$$ where $$f$$ and $$g$$ can be $$\cos$$ or $$\sin$$. This gives one delta function, so we are left with a double sum. (4) has only a single sum over $$n$$? How is that possible?

$$\quad$$ (4b) The gauge field part has a $$\xi^\mu \dot \xi^\nu$$. The $$\dot \xi^\nu$$ essentially switches the $$\cos$$ and the $$\sin$$ in the Fourier expansion. The non-zero $$\theta$$ integrations are then over $$a_m^\mu b_n^\nu \cos (m\theta) \cos (n\theta)$$ and $$b_m^\mu a_n^\nu\sin (m\theta) \sin (n\theta)$$. i.e. the $$a$$'s and the $$b$$ get mixed up, whereas in (4) they are not. What did I miss?

$$\quad$$ (4c) How did the coefficient $$i$$ disappear from (2) to (4)?

Edit Note added: In the meanwhile I have found Non-linear electrodynamics from quantized strings” by E.S. Fradkin and A.A. Tseytlin which may contain the answers to (at least) some of these questions.