### del Pezzo

There are several reasons why a lot of recent work on string compactifications has focussed on the case of Type-II orientifold backgrounds. The most obvious is that turning on fluxes gives a mechanism for moduli stabilization.

Equally important, however, is that, in contrast to heterotic backgrounds, many of the phenomena of interest (nonabelian gauge theories coupled to chiral matter) are *localized* in the target space. This localization (along with the warping due to large fluxes) provides a nice mechanism for generating a large hierarchy of scales. It also emboldens one to take a “tinker-toy” approach, putting together pieces of the desired physics, localized at different locations in some (unspecified) Calabi-Yau orientifold (the Standard Model on a stack of branes at this singularity *over here*, supersymmetry-breaking *over there*, …). Whether, in fact, these pieces can be assembled together in arbitrary ways, or whether there are important constraints that one misses in the purely local analysis, is a crucial question.

Another question, to which there is still not a clear-cut answer, is to what extent one can get the Standard Model (without extra junk) out of these local constructions. We know that, with suitable bundles, on the heterotic side, one can get the Standard Model field content, on the nose. (I am thinking, here, of the work by the Penn group in the context of heterotic M-Theory.) Moreover, on the simply-connected covering space (*i.e.*, before Wilson-line breaking), there is a known dictionary between the heterotic compactification, with bundles constructed via the Freedman-Morgan-Witten construction and the F-theory dual. Unfortunately, the same is not true *after* Wilson-line breaking. So, whereas one can get GUT groups, one cannot, currently, get the Standard Model on the F-theory side.

The “state of the art,” in terms of local constructions, seems to be the model of Verlinde and Wijnholt. The complex codimension-3 singularities of a Calabi-Yau look like a complex cone over a del Pezzo surface, and Verlinde and Wijnholt manage to fit a Standard Model-like theory into the biggest and best of the del Pezzo singularities, dP_{8}. Recently, Buican *et al* showed how the above construction can be embedded in a compact Calabi-Yau geometry.

$S=\text{dP}_8$ 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 dP_{6} singularity, without the somewhat baroque contrivance of demanding that we have a dP_{8} 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 mechanism^{1}. 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 dP_{8} 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.

^{1} The Green-Schwarz mechanism, in 4 dimensions, works as follows. Consider some (anomalous) $U(1)$ gauge symmetry. *If* the coefficient of $Tr(F^3)$ in the anomaly polynomial vanishes (*i.e.*, if we’ve managed to cancel the ${U(1)}^3$ anomaly, as well as all the pure nonabelian anomalies), then we can cancel the mixed gravitational-$U(1)$ anomaly (the $Tr(F)Tr(R^2)$ term in the anomaly polynomial) and/or the mixed $U(1)$-${SU(n)}^2$ anomaly (the $Tr(F) Tr(G^2)$ terms in the anomaly polynomial) by picking some pseudoscalar field, $a$, and giving it a nontrivial transformation under $U(1)$ gauge transformations,
$A\to A +d f, \quad a \to a +f$
and adding a term $\int a\, \left(c Tr(R\wedge R) + c_i Tr(G_i\wedge G_i)\right)$ to the action. In the case at hand, there’s no gravitational anomaly, but there are mixed $U(1)$-${(U(1)')}^2$, $U(1)$-${SU(3)}^2$ and $U(1)$-${SU(2)}^2$ anomalies, which affect two of the 8 $U(1)$s.

## Re: del Pezzo

Hi Jacques,

I just noticed your post on Del Pezzo’s, and thought I would comment on a few things.

“Another question, to which there is still not a clear-cut answer, is to what extent one can get the Standard Model (without extra junk) out of these local constructions. We know that, with suitable bundles, on the heterotic side, one can get the Standard Model field content, on the nose.”

It seems you are asking for two things here. First with D-branes you can only get the MSSM with some extra massive junk. For instance the SU(3) comes with an extra gauged U(1)_B, which will eat a Stueckelberg field and become massive. Secondly, there is the question of interactions. Here it was Herman who first emphasized that in the local context, getting the MSSM spectrum right ought to be sufficient; by open/closed intuition, any deformation of the MSSM should then map to a deformation of the local geometry.

” (I am thinking, here, of the work by the Penn group in the context of heterotic M-Theory.)”

I don’t want this to be seen as belittling the work of the Penn group, since they did a lot of impressive work, but if I remember correctly they did have a massless U(1)_B-L that was hard to get rid off.

“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.”

Well, DP6 and DP8 are related by blowing up -1 curves, not -2 curves.

I do agree that it seems somewhat arbitrary from the geometry. I don’t see anything that would make the Standard Model seem “special.”

Finally, the phenomenological problems have to do with the fact that we were restricting ourselves to oriented quivers. I have been looking at unoriented quivers (including orientifolds) recently; then it appears that one can indeed get exactly the MSSM, though I still have to finish checking that indeed the surplus non-chiral matter can be lifted.