Product-Group Unification in Type IIB String Thoery

TL;DR

The paper addresses how to realize a product-group grand unification model based on $SU(5)_{\rm GUT}\times U(N)_{\rm H}$ ($N=2,3$) within Type IIB string theory, preserving the key benefits of the missing-partner mechanism and suppressing dim-5 proton decay via a mod-4 $R$ symmetry. It constructs a local brane setup using a GUT-breaking sector on a D3–D7 system in ${\bf C}^2/{\bf Z}_M\times {\bf C}$, with Higgs multiplets and quarks/leptons arising from open strings at D7-brane intersections, and shows how NMSSM-like interactions and a non-parallel family structure can emerge from the geometry and Chan–Paton data. The work derives local anomaly-cancellation and RR-charge considerations that shape the viable chiral spectra and connect to proposed geometric mechanisms for family structure, while highlighting that an explicit global solution with all RR charges canceled is not yet provided. Overall, the paper offers a local, string-theoretic realization of a phenomenologically appealing GUT framework, with potential implications for Higgs physics, neutrino mixing, and NMSSM signatures, while leaving open questions on Yukawa couplings, moduli stabilization, and explicit global embeddings.

Abstract

The product-group unification is a model of unified theories, in which masslessness of the two Higgs doublets and absence of dimension-five proton decay are guaranteed by a symmetry. It is based on SU(5) x U(N) (N=2,3) gauge group. It is known that various features of the model are explained naturally, when it is embedded in a brane world. This article describes an idea of how to accommodate all the particles of the model in Type IIB brane world. The GUT-breaking sector is realized by a D3--D7 system, and chiral quarks and leptons arise from intersection of D7-branes. The D-brane configuration can be a geometric realization of the non-parallel family structure of quarks and leptons, an idea proposed to explain the large mixing angles observed in the neutrino oscillation. The tri-linear interaction of the next-to-minimal supersymmetric standard model is obtained naturally in some cases.

Figures (5)

• Figure 1: A schematic picture of the local geometry of $CY_3$ and D-brane configuration in it. Only the parts relevant to subsection \ref{['ssec:GUT']} (GUT-breaking sector) and \ref{['ssec:Higgs']} (Higgs particles) are described here.
• Figure 2: A schematic picture of the configuration for the chiral multiplets in the anti-symmetric-tensor representation. A holomorphic 4-cycle $\Sigma_{1'}$ is the image of $\Sigma_1$ under an involution associated with the O7-plane. They intersect at a holomorphic curve $\Sigma_1 \cdot \Sigma_{1'}$ (thick curve in the figure). D7-branes are wrapped on the 4-cycles, and the SU(5)$_{\rm GUT}$ gauge field propagates all over the world volume. The matter in the anti-symmetric-tensor representation arises at the intersection. The GUT-breaking sector is realized by fractional D3-branes located at a ${\bf C}^2/{\bf Z}_2 \times {\bf C}$ singularity on the 4-cycle $\Sigma_1$.
• Figure 3: Toy model explained in subsubsection \ref{['sssec:toy']}. $z^1$ is the coordinate of ${\bf T}^2$, and $(z^2,z^3)$ are those of ${\bf C}^2$. Orientifold projection is associated with the reflection in $z^3$-direction, and hence the O7-plane is the fixed locus of the reflection, i.e., $z^3 = 0$ hypersurface. D7$_-$ is the orientifold mirror image of D7$_+$.
• Figure 4: A schematic picture of the D-brane configuration for quarks and leptons. The SU(5)$_{\rm GUT}$ gauge field propagates on the D7-branes on $\Sigma_1$ and $\Sigma_{1'}$. The chiral multiplets in the 10 representation are localized on $\Sigma_1 \cdot \Sigma_{1'}$, and those in the ${\bf 5}^*$ representation (and right-handed neutrinos) on $\Sigma_1 \cdot \Sigma_2$. The GUT-breaking sector is denoted by a small dot on $\Sigma_1$ (and its orientifold image on $\Sigma_{1'}$).
• Figure 5: Phenomenological models of family structure in WYfamily can be considered as effective theories obtained by projecting onto the complex curve $\Sigma_1 \cdot \Sigma_{1'}$ (A). Multiplets in the 5$^*$ representation and right-handed neutrinos are localized in the internal space, and the large mixing between them may be understood in terms of geometry. Another model in HbMr can be considered as an effective theory obtained by projecting onto the complex curve $\Sigma_1 \cdot \Sigma_2$ (B). Multiplets in the 10 representation are localized in the internal space, and the large hierarchy between them may be explained geometrically.