##
**Breaking GUT groups in \(F\)-theory.**
*(English)*
Zbl 1269.81121

The possibility of breaking the GUT group to the Standard Model gauge group in \(F\)-theory compactification is considered. The authors’ main idea is to turn on certain \(U(1)\) fluxes on the world volume of the 7-brane [the authors, “Model building with \(F\)-theory”, (2008), arXiv:0802.2969], referred to as [1]]. In a 7-brane with a GUT group \(G= SU(5)\) wrapping a four-ccycle \(S\), to turn on fluxes means to specify a \(U(1)\) bundle \(L\) on \(S\). Such a \(U(1)\) flux breaks the GUT group \(SU(5)\) to the subgroup which commutes with the \(U(1)\) which is the Standard Model gauge group \(SU(3)\times SU(2)\times SU(1)\) (§2).

Of course, this rough arguments need to refine. In §3, considering the decomposition of the \(SU(5)\) adjoint under \(SU(3)\times SU(2)\times SU(1)\) and a local model from an \(SU(5)\) GUT obtained from unfolding an \(E_8\) singularity, it is concluded for the total flux to be properly quantized, only \(c_1(L^{5/6})\) and its multiples need to be integer quantized, but not \(c_1(L)\) or \(c_1(L^{1/6})\). The authors remark this is compatible with the requirement of the absence of massless lepto-quarks.

After brief review of GUT monopoles, which are similar to previous researches [B. R. Greene et al., Nucl. Phys., B 481, No. 3, 513–538 (1996; Zbl 0925.32006)] in §3, threshold corrections for the gauge coupling unification are discussed in §4. It is the main body of the paper.

The corrections to the leading term at the GUT scale are organized in a power series in \(\alpha_{\text{GUT}}\); \[ {16\pi^2\over g^2_a(M_{\text{GUT}})}= {4\pi k_a\over\alpha_{\text{GUT}}}+ \delta^{(0)}_a+ \delta^{(1/2)}_a \alpha^{1/2}_{\text{GUT}}+ \delta^{(1)}_a \alpha_{\text{GUT}}+\cdots. \] In the KK (Kaluza-Klein) modes of the gauge theory, computation of the threshold correction \(\delta^{(0)}_a\) is reduced to the computations of determinants of Dolbeault operators coupled to holomorphic bundles ([1]). The authors use the Ray-Singer determinant as determinants of Dolbeaut operators.

Ray-Singer which use zeta-function regularization. As for zeta-function regularization, the authors say it throw out power law divergence, and keep only the logarithmic divergence. Power law divergence are natural in KK theory; they signal the gauge coupling becomes dimensionful above the KK scale and has power law running. Thus the regularization provided by the microscopic theory underlying \(F\)-theory will certainly not be the zeta-function regularization. But a term with power law divergence that could affect the relations between the \(4d\) gauge couplings must be of the schematic form \(\Delta^2 F^3\) on \(S\). Since \(g_{ij} F^{ij}_Y= 0\) for the internal flux, such a term would have no effect on the \(4d\) gauge coupling. So we can use the zeta function regularization in the study of unification.

The rest of §4 is mainly devoted to the study of KK modes. The authors remark computations of this paper suggests if \(M_{\text{GUT}}/M_{\text{KK}}\sim 10^2\), the unification scale gets lowered to \(1.3\times 10^{16}\text{GeV}\), while if \(M_{\text{GUT}}/M_{\text{KK}}\sim 10^{0.5}\), then the unification scale is \(1.7\times 10^{16}\text{GeV}\).

In §5, the last section, constraints on \(F\)-theory due to observational constraints on the proton decay are discussed. In generic models, there is nothing to prevent the fast proton decay, so there must be some extra structures and some option for eliminating significant proton decay through four and five operators is discussed [the authors, “Gluing branes. I”, (2011), arXiv:1104.2610; “Gluing branes. II: Flavour physics and string duality”, (2011), arXiv:1112.4854].

This paper contains three Appendixes; A: A GUT breaking fluxes constructed through the cylinder map, B: Metric anomaly for holomorphic torsion, and C: Roots of \(E_8\). Before these discussions, reason to use \(F\)-theory in the study of unification, is explained in the introduction.

Of course, this rough arguments need to refine. In §3, considering the decomposition of the \(SU(5)\) adjoint under \(SU(3)\times SU(2)\times SU(1)\) and a local model from an \(SU(5)\) GUT obtained from unfolding an \(E_8\) singularity, it is concluded for the total flux to be properly quantized, only \(c_1(L^{5/6})\) and its multiples need to be integer quantized, but not \(c_1(L)\) or \(c_1(L^{1/6})\). The authors remark this is compatible with the requirement of the absence of massless lepto-quarks.

After brief review of GUT monopoles, which are similar to previous researches [B. R. Greene et al., Nucl. Phys., B 481, No. 3, 513–538 (1996; Zbl 0925.32006)] in §3, threshold corrections for the gauge coupling unification are discussed in §4. It is the main body of the paper.

The corrections to the leading term at the GUT scale are organized in a power series in \(\alpha_{\text{GUT}}\); \[ {16\pi^2\over g^2_a(M_{\text{GUT}})}= {4\pi k_a\over\alpha_{\text{GUT}}}+ \delta^{(0)}_a+ \delta^{(1/2)}_a \alpha^{1/2}_{\text{GUT}}+ \delta^{(1)}_a \alpha_{\text{GUT}}+\cdots. \] In the KK (Kaluza-Klein) modes of the gauge theory, computation of the threshold correction \(\delta^{(0)}_a\) is reduced to the computations of determinants of Dolbeault operators coupled to holomorphic bundles ([1]). The authors use the Ray-Singer determinant as determinants of Dolbeaut operators.

Ray-Singer which use zeta-function regularization. As for zeta-function regularization, the authors say it throw out power law divergence, and keep only the logarithmic divergence. Power law divergence are natural in KK theory; they signal the gauge coupling becomes dimensionful above the KK scale and has power law running. Thus the regularization provided by the microscopic theory underlying \(F\)-theory will certainly not be the zeta-function regularization. But a term with power law divergence that could affect the relations between the \(4d\) gauge couplings must be of the schematic form \(\Delta^2 F^3\) on \(S\). Since \(g_{ij} F^{ij}_Y= 0\) for the internal flux, such a term would have no effect on the \(4d\) gauge coupling. So we can use the zeta function regularization in the study of unification.

The rest of §4 is mainly devoted to the study of KK modes. The authors remark computations of this paper suggests if \(M_{\text{GUT}}/M_{\text{KK}}\sim 10^2\), the unification scale gets lowered to \(1.3\times 10^{16}\text{GeV}\), while if \(M_{\text{GUT}}/M_{\text{KK}}\sim 10^{0.5}\), then the unification scale is \(1.7\times 10^{16}\text{GeV}\).

In §5, the last section, constraints on \(F\)-theory due to observational constraints on the proton decay are discussed. In generic models, there is nothing to prevent the fast proton decay, so there must be some extra structures and some option for eliminating significant proton decay through four and five operators is discussed [the authors, “Gluing branes. I”, (2011), arXiv:1104.2610; “Gluing branes. II: Flavour physics and string duality”, (2011), arXiv:1112.4854].

This paper contains three Appendixes; A: A GUT breaking fluxes constructed through the cylinder map, B: Metric anomaly for holomorphic torsion, and C: Roots of \(E_8\). Before these discussions, reason to use \(F\)-theory in the study of unification, is explained in the introduction.

Reviewer: Akira Asada (Takarazuka)

### MSC:

81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |

81V22 | Unified quantum theories |

81V17 | Gravitational interaction in quantum theory |

58J52 | Determinants and determinant bundles, analytic torsion |

17B81 | Applications of Lie (super)algebras to physics, etc. |

81R30 | Coherent states |

83E15 | Kaluza-Klein and other higher-dimensional theories |