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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.

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