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In some lecture notes, in the presence of a Dp-brane, translational symmetry is broken in the direction transverse to the brane. However, they identify the coordinates, $$\Phi^a,\quad a=p+1,\cdots,D-1$$ as the corresponding goldstone bosons. Should the Goldstone bosons be fluctuations around the generators of the broken symmetries, say, a momentum, $$p^a$$ along the transverse direction?

Also, for the case of Goldstone boson corresponding to the transverse coordinate, that $$\Phi^a$$ having vanishing potential is due to their shift symmetry realized as just a phase factor in $$e^{ip^a(\Phi^a+\lambda^a)}|0\rangle = e^{ip^a\lambda^a}e^{ip^a\Phi^a}|0\rangle$$ regarding the first factor on the RHS as a phase---which doesn't seem to be legitimate as the phase is accompanied by a broken translation generator? Can I ask you also for the generalization of this argument if it is correct?

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    $\begingroup$ (This is a slightly revised version of my earlier comment.) In the prototypical SSB example of a 2-component scalar model with symmetry generator $G$, the symmetry is $|\psi\rangle\to\exp(i\theta G)|\psi\rangle$ with constant $\theta$. When the symmetry is spontaneously broken, the Goldstone mode corresponds to an oscillating space-dependent parameter $\theta(x)$, not to oscillations in the generator $G$. SSB of spacetime symmetries can require special handling (arXiv:1405.5532), but if this simple analogy holds, then $\theta(x)$ is analogous to the coordinates $\Phi^a$ in the $p$-brane case. $\endgroup$
    – Chiral Anomaly
    Jun 2, 2019 at 3:16
  • $\begingroup$ @ChiralAnomaly Thank you for the comment. I see my mistake. When a symmetry is spontaneously broken, the broken generators send a vacuum to goldstone bosons. The angle $$e^{i\theta(x)} |\psi\rangle $$ seems to be the consequence of the broken generator's action on $$|\psi\rangle $$ which I call fluctuation around the broken generators, but should have been distinguised from the fluctation around the broken generator, indeed. $\endgroup$
    – Liberty
    Jun 2, 2019 at 3:41

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