D-Branes at Singularities : A Bottom-Up Approach to the String Embedding of the Standard Model

TL;DR

This work advocates a bottom-up program for realizing the Standard Model within string theory by placing Type IIB D3- and D7-branes at singularities, particularly IC^3/Z_N orbifolds. It derives the local spectra, anomaly cancellation, and hypercharge structure, showing that the Z_3 case naturally yields three generations and a robust hypercharge embedding, and explores both supersymmetric and non-supersymmetric realizations. The authors demonstrate explicit 4D compactifications (orbifold, orientifold, and F-theory) that host these local SM/LR sectors, including scenarios with gravity-mediated SUSY breaking via trapped anti-branes. Phenomenological implications are examined, highlighting gauge coupling unification in left-right models, Yukawa textures tied to moduli, and proton stability within B-L–protected frameworks. Overall, the paper presents a concrete, flexible route to semi-realistic string embeddings of the SM with concrete predictions and broad global completion strategies.

Abstract

We propose a bottom-up approach to the building of particle physics models from string theory. Our building blocks are Type II D-branes which we combine appropriately to reproduce desirable features of a particle theory model: 1) Chirality ; 2) Standard Model group ; 3) N=1 or N=0 supersymmetry ; 4) Three quark-lepton generations. We start such a program by studying configurations of D=10, Type IIB D3-branes located at singularities. We study in detail the case of Z_N, N=1,0 orbifold singularities leading to the SM group or some left-right symmetricextension. In general, tadpole cancellation conditions require the presence of additional branes, e.g. D7-branes. For the N=1 supersymmetric case the unique twist leading to three quark-lepton generations is Z_3, predicting $\sin^2θ_W=3/14=0.21$. The models obtained are the simplest semirealistic string models ever built. In the non-supersymmetric case there is a three-generation model for each Z_N, N>4, but the Weinberg angle is in general too small. One can obtain a large class of D=4 compact models by considering the above structure embedded into a Calabi Yau compactification. We explicitly construct examples of such compact models using Z_3 toroidal orbifolds and orientifolds, and discuss their properties. In these examples, global cancellation of RR charge may be achieved by adding anti-branes stuck at the fixed points, leading to models with hidden sector gravity-induced supersymmetry breaking. More general frameworks, like F-theory compactifications, allow completely $\NN=1$ supersymmetric embeddings of our local structures, as we show in an explicit example.

Figures (8)

• Figure 1: Pictorial representation of the bottom-up approach to the embedding of the standard model in string theory. In step i) the standard model is realized in the world-volume of D3-branes sitting at a singular non-compact space $X$ (in the presence of D7-branes, not depicted in the figure). In step ii) this local configuration is embedded in a global context, like a compact Calabi-Yau threefold. Many interesting phenomenological issues depend essentially only on the local structure of $X$ and are quite insensitive to the details of the compactification in step ii). The global model may contain additional structures (like other branes or antibranes) not shown in the figure.
• Figure 2: A non-compact Type IIB $Z Z_3$ orbifold singularity yielding SM spectrum. Six D3 branes sit on top of a $Z Z_3$ singularity at the origin. Tadpoles are canceled by the presence of intersecting D7-branes with their worldvolumes transverse to different complex planes.
• Figure 3: D-brane configuration of a SM $Z Z_3$ orbifold model. Six D3-branes (with worldvolume spanning Minkowski space) are located on a $Z Z_3$ singularity and the symmetry is broken to $U(3)\times U(2)\times U(1)$. For the sake of visualization the D3-branes are depicted at different locations, even though they are in fact on top of each other. Open strings starting and ending on the same sets of D3-branes give rise to gauge bosons; those starting in one set and ending on different sets originate the left-handed quarks, right-handed U-quarks and one set of Higgs fields. Leptons, and right-handed D-quarks correspond to open strings starting on some D3-branes and ending on the D7-branes (with world-volume filling the whole figure).
• Figure 4: D-brane configuration of a LR $Z Z_3$ orbifold model. Seven D3-branes (with worldvolume spanning Minkowski space) are located on a $Z Z_3$ singularity and the symmetry is broken to $U(3)\times U(2)\times U(2)$. For the sake of visualization the D3-branes are depicted at different locations, even though they are actually coincident. Open strings starting and ending on the same sets of D3-branes give rise to gauge bosons; those starting and ending on different sets originate the quarks and Higgs fields. Leptons correspond to open strings starting on some D3-branes and ending on D7-branes.
• Figure 5: A compact Type IIB $T^6/Z Z_3$ orbifold model with a LR subsector. The points marked $(0)$ correspond to fixed points without D3-branes; those marked $(*)$ contain anti-D3-branes and those marked $(x)$ have D3-branes located on them. Seven D3-branes, leading to a LR model of the type studied in section 3.4, reside at the origin. The overall RR charge cancels.