Musings

$S=\text{dP}_8$ is $\mathbb{P}^2$, blown up at 8 generic points. The homology, $\mathrm{H}_2(S)$ is generated by the hyperplane class, $H$, and the eight exceptional divisors, $E_i$, $$H\cdot H =1,\quad E_i\cdot E_j = -\delta_{i j}, \quad H\cdot E_i = 0$$

The canonical class, $K=-3H+\sum E_i$, and the sublattice of $\mathrm{H}_2(S)$, orthogonal to $K$, spanned by $$\begin{aligned} \alpha_i &= E_i - E_{i+1},\quad i=1,\dots , 7\\ \alpha_8 &= H- E_1-E_2-E_3 \end{aligned}$$ is isomorphic to the $E_8$ root lattice.

At some general point in the Kähler moduli space of $S=\text{dP}_n$, the gauge theory on a D3-brane transverse to this singularity is a quiver gauge theory with $n+3$ nodes. (The number of nodes is equal to the rank of $K^0(S)$.). As one moves about in the Kähler moduli space, one crosses various surfaces of marginal stability, where the identities of the stable fractional branes changes. Correspondingly, the quiver gauge theories undergo transitions, related to Seiberg dualities.

dP_$n$, for $n\geq 5$, also has a moduli space of complex structures, of dimension $2n-8$, corresponding to the freedom to move the blowup points, after fixing the locations of 4 of them via the $PGL(3)$ symmetry. On a suitable locus, $S=\text{dP}_8$ can develop an $A_2$ singularity. In the associated CY geometry, the dP_$8$ singular point lies on a curve of $A_2$ singularities. In this situation, there is a branch of the moduli space, where some of the fractional branes (the ones corresponding to $\alpha_1, \alpha_2$) can be moved off the del Pezzo singularity, onto the curve of $A_2$ singularities, and can be moved far away. This decouples two of the nodes of the quiver, and the bifundamental matter charged under those nodes.

What remain, according to Verlinde and Wijnholt, is a quiver with 9 nodes, depicted at left. The gauge group is $(U(3)\times U(2) \times {U(1)}^7)/{U(1)}_{\text{diag}}$. There’s a bifundamental chiral multiplet corresponding to each arrow in the quiver, and I’ve suggestively named them by the corresponding supersymmetric Standard Model field. Note that there are 3 generations of quarks, leptons, and right-handed neutrinos as well as 6 generations (six!) of Higgses.

The quiver gauge theory has superpotential terms corresponding to directed loops in the quiver. So we can see, e.g., the usual Yukawa couplings with the Higgses which give masses to the quarks and leptons, as well as a $W=\dots + \lambda N H_u L$, which is part of the mechanism for generating neutrino masses.

I’m not quite sure why this quiver could not arise somewhere in the Kähler moduli space of a dP₆ singularity, without the somewhat baroque contrivance of demanding that we have a dP₈ singularity sitting on top of a curve of $A_2$ singularities, and moving off onto a mixed branch of the moduli space.

This quiver gauge theory doesn’t yet look much like the Standard Model. There’s a plethora of extra $U(1)$s. One linear combination of the $U(1)$s, the one corresponding to the node $\alpha_4$ of the $E_8$ Dynkin diagram, is ${U(1)}_Y$. No problem, say Buican et al., two of the $U(1)$s are anomalous, and the corresponding gauge bosons pick up a mass due to the Green-Schwarz mechanism¹. The remaining $U(1)$s are massless in the noncompact geometry, but are given mass via a Stückelberg mechanism, when this geometry is embedded in a compact Calabi-Yau.

To ensure that ${U(1)}_Y$ remains massless, it suffices to ensure that the homology class, $\alpha_4$, which is nontrivial on $S$, becomes trivial upon embedding in the compact Calabi-Yau geometry. This, they go to some lengths to show can be done, while at the same time ensuring that the dP₈ singularity lies on a curve of $A_2$ singularities in the CY.

Despite the beautiful algebraic geometry at play, there are still some formidable obstacles to interpreting the above theory as the supersymmetric Standard Model. First of all, the $\mu$-term is forbidden by the quiver gauge symmetry. This might actually be a good thing, because we know the $\mu$-term is suppressed. The authors discuss mechanisms for generating the $\mu$-term, either as a nonperturbative correction to the superpotential, or via the Giudice-Masiero mechanism. Much more serious, is that a Majorana mass term for $N$ is forbidden by the quiver gauge symmetry. Rather than being suppressed (like the $\mu$-term), we need this mass to be large ($\sim 10^{15}$ GeV), in order to generate the observed neutrino masses via the seesaw mechanism.

That seems to me to be a rather serious obstacle to trying to use this quiver to embed the Standard model in a Type IIB orientifold.