Quiver Standard Models

Martijn Wijnholt presented his work with Herman Verlinde. It is an interesting and original approach to string phenomenology.

At the beginning, Martijn had to face the very same questions and objections he heard from me during the lunch on Wednesday - except that they were from Andy who essentially argued that the predictions are just predictions of one particular realization of the real world within string theory - and cannot be viewed as general predictions of string theory because we can surely find a googol of other backgrounds that predict different things.

Eventually this dissent has been silenced ;-) and Martijn could continue. Let me try to make a case for their kind of model.

The Planck scale is very far which is why it is phenomenologically justifiable to study the low-energy non-gravitational physics separately from quantum gravity. In the brane realizations of the observed physics we may use some intuition from the gauge/gravity duality. Physical phenomena near the brane encode the low-energy physics. This is what we are interested in. Consequently, we should study physics near the brane. Because we want to get a rather convoluted gauge theory - the MSSM - we need to place the D-brane on a non-trivial kind of singularity.

Physics of D-branes at singularities is described by quiver gauge theories. Quiver is a diagram with nodes representing the U(N) gauge groups and the corresponding vector multiplets and links between these nodes that represent chiral supermultiplets in the bifundamental representations. The overall number of incoming vs. outgoing links from a node must be equal because of non-Abelian gauge anomaly cancellation.

The simplest quiver that contains the MSSM as a proper subset can be constructed from five nodes - U(3), U(2), U(1), U(1), U(1). If you construct an ordinary pentagon with these vertices, all links except for two exactly horizontal links are present in the quiver diagram. Among the five U(1)'s, one of them decouples because nothing is charged under it, and two of them get massive via the Stueckelberg mechanism The remaining two - the hypercharge and B-L - survive.

Once you decide that this is the interesting quiver - the MQSSM quiver i.e. the minimal quiver for the constructions of MSSM in terms of D3-branes on singularities - you may try to analyze what singularity you need to obtain such quivers. After some analysis, you will conclude that you need a del Pezzo 8 singularity, one of the three major classes of singularities found in Calabi-Yau three-folds. (Incidentally, Martijn claimed that even the ordinary quintic has points on its moduli space that contain this dP8 singularity.) Equivalently, you can also work with the Delta_{27} singularity which seems to be a special case. The phenomenologically attractive features of its quivers were noticed already around 2000.

Martijn ended up with a dictionary between various parameters of the geometry on one side and the MSSM couplings on the other side.

Once you understand the physics of the singularity, you may try to embed it into a compact Calabi-Yau which amounts to a UV completion of your low-energy effective field theory. Martijn argued that there could be general predictions from such an attempted UV completion, but the existence of these general predictions remained controversial.

For example, the coordinates on the Calabi-Yau may be non-commutative. Martijn argued that the non-commutativity parameter Theta^{ij} may also be represented as an (n-2,0) closed form on the Calabi-Yau which must vanish. I actually think that the non-commutativity parameter arises from a more fundamental B-field which is a two-form, and closed two-forms (which is probably the natural constraint we get) do exist on Calabi-Yaus. In my opinion, the condition for the (n-2,0) form Martijn that Martijn works with is not saying that the form should be closed.

Alessandro Tomasiello argued that the non-commutativity parameter could still be realized within the Calabi-Yau by the tricks of generalized geometry, and I have no idea how to decide whether it is the case.

Finally, there was an unsolved debate about the question what they mean by the "singularity dual to MSSM" because MSSM itself does not exist at high energies because of the hypercharge Landau pole, and it should therefore be impossible to extend the neighborhood of the singularity arbitrarily far.