SUSY in silico: Numerical D-brane bound state spectroscopy

We numerically construct the supersymmetric and non-supersymmetric wave functions of an $\mathcal{N}=4$ quiver quantum mechanics with two Abelian nodes and a single arrow. This model captures the dynamics of a pair of wrapped D-branes interacting via a single light string mode. A dimensionless parameter $\nu$, which is inversely proportional to the Fayet-Iliopoulos parameter, controls whether the bulk of the wave functions are supported on the Higgs branch or the Coulomb branch. We demonstrate how the supersymmetric and excited states morph as $\nu$ is tuned. We also numerically compute the energy gap between the ground state and the first excited states as a function of $\nu$. An expression for the gap, computed on the Coulomb branch, matches nicely with our numerics at large $\nu$ but deviates at small $\nu$ where the Higgs branch becomes the relevant description of the physics. In the appendix, we provide the Schrödinger equations fully reduced via symmetries which, in principle, allow for the numerical determination of the entire spectrum at any point in moduli space. For the ground states, this numerical determination of the spectrum can be thought of as the first in silico check of various Witten index calculations.

Figure 1

Top two rows: Probability density for the first three energy eigenstates $P\sim \tilde{r}{r}^2|{\mathrm{\Psi }}_0{|}^2$ (after integrating over the angular variables) in the $R=0$, $j=0$ sector with $\theta =-12.5$, and $\mu =10$, corresponding to $\nu \approx 0.172$. The minimum of the potential is at $|\varphi |\approx 3.53$. The energies of the first three eigenstates are respectively $\mathcal{E}\approx (0,1.538,2.984)$. Bottom row: Components of the ground state wave function.

Figure 2

Top two rows: Probability density for the first three energy eigenstates $P\sim \tilde{r}{r}^2|{\mathrm{\Psi }}_0{|}^2$ (after integrating over the angular variables) in the $R=0$, $j=0$ sector for $\theta =-0.911$, $\mu =5$ corresponding to $\nu \approx 1.876$. The minimum of the potential is at $|\varphi |\approx 0.955$. The minimum of $V^{\mathrm{eff}}$ is at $r\approx 0.55$ and the chiral fields become tachyonic for $r<r_h\equiv \sqrt{-\theta /\mu }\approx 0.427$. Note the difference in scales between the $r$-axis and the $|\varphi |$-axis, particularly for the excited states. The energies of the first three eigenstates are respectively $\mathcal{E}\approx (0,0.073,0.079)$. Bottom row: Components of the ground state wave function.

Figure 3

Top left: Expectation value of ${\mu }^{-1/3}⟨|\varphi {|}^2⟩$ as a function of $\nu$. We conclude that $⟨|\varphi {|}^2⟩=-\theta$ in both the Higgs and Coulomb branch. Top right: Expectation value of ${\mu }^{1/3}⟨r⟩$ as a function of $\nu$. As $\nu$ increases the approximate Coulomb branch formula $⟨r⟩=-1/\theta$ becomes more exact. Bottom row: Gap between the ground state and the first excited state as a function of $\nu$. As $\nu$ increases we go from a regime where the gap is given by $\mathrm{\Delta }\mathcal{E}\approx \sqrt{\frac{-2\theta }{\mu }}$ to a regime where the gap is $\mathrm{\Delta }\mathcal{E}\approx \frac{4{\theta }^2}{9\mu }$.

Figure 4

Fraction of the probability density, in the ground state, which has support for $r>r_h\equiv \sqrt{-\theta /\mu }$.