Computing the DBI action on $S^5$

Question

I've been looking at this paper (arXiv: 1103.4079). On page 7, from the metric of the giant gravtiton on $AdS_5 \times S^5$,

$$ds^2 = -\cosh^2\rho \, dt^2 + d\rho^2 + \sinh^2 \rho \, d\tilde \Omega_3^2 + d\theta^2 + \sin^2\theta \, d\phi^2 + \cos^2\theta \, d\Omega_3^2,$$

the ansatz

$$ \rho=0, \quad \sigma^0=t, \quad \phi=\phi(t) \quad \sigma^i=\chi_i, $$

and the action $$ S_{D3} = \frac{-N}{2\pi^2} \int d^4\sigma \; \left( \sqrt{-g} -P[C_4]\right), $$

(where $g_{ab} = \partial_a X^M \partial_b X_N$ with $X^M$ the embedding coordinates, and $a,b = 0, \ldots, 3$) the following formula is given:

$$ S = \int dt \; L = -N \int dt\; \left(\cos^3\theta \sqrt{1 - \dot \phi^2 \sin^2\phi} - \dot \phi \cos^4\theta \right)$$

Where does the final line come from? I understand that the final term comes from the pullback in the Wess-Zumino part (the $P[C_4]$), and the square root from the determinant of the metric $g_{ab}$, but how do I go about explicitly computing it? The embedding metric given lowers the index such that $g_{ab} = \partial_a X^M \partial_b (G_{MN} X^N)$, but then I get stuck. How do the derivatives act on the omegas, for example?

Answer 1

It should work as follows. By your "ansatz" the relevant metric reduces to $$ds^2 = -(1- \dot{\phi}^2\sin^2 \theta)dt^2 + d\theta^2 + \cos^2 \theta d\Omega_3^2 \ ,$$ where $d\Omega_3^2$ is the line-element of $S^3$. The volume of the unit $S^3$ is $\text{Vol}(S^3)=2\pi^2$.

Now we can calculate the term $\sqrt{-g}$. We have (for fixed $\theta$) $$\int dt d\Omega_3 \sqrt{-g}=\int dt \sqrt{(1- \dot{\phi}^2\sin^2 \theta) \cdot (\cos^2 \theta)^3} \cdot \text{Vol}(S^3) = 2\pi^2 \int dt \cos^3 \theta \sqrt{1- \dot{\phi}^2\sin^2 \theta}$$

Now we need to calculate $\int C$. We are given $$C = C_{\phi \chi_1 \chi_2 \chi_3}d\phi \wedge d\Omega_3 = \dot{\phi}C_{\phi \chi_1 \chi_2 \chi_3}dt \wedge d\Omega_3$$ with $C_{\phi \chi_1 \chi_2 \chi_3}= \cos^4 \theta \cdot \text{Vol}(S^3)$. Hence, $$\int C = 2\pi^2 \int \dot{\phi} \cos^4 \theta dt \ ,$$ where we only integrated over the angles of $S^3$.

Combining both results yields the asserted equation in the paper. Even the $2\pi^2$ cancels out.:-)