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**Towards the massless spectrum of non-Kähler heterotic compactifications.**
*(English)*
Zbl 1132.81042

The number of particles of a ten-dimensional Universe is thought to be determined by the geometry of a compactified six-dimensional space. According to this argument, the massless spectrum of compactification of heterotic supergravity is studied by using the construction developed by J. A. Fu and S. T. Yau [The theory of superstring with flux on non-Kähler manifolds and line complex Monge-Ampère equation, arXiv:hep-th/0604063, hereafter refered to as [1]; compare also Commun. Anal. Geom. 15, No. 1, 29–76 (2007; Zbl 1122.35145)].

Compactifications of heterotic string theory that preserve \({\mathcal N}= 1\) supersymmetry in four dimensions with non-zero \(H\)-flux was studied in A. Strominger [Supergravity with torsion, Nuclear Phys. B 274, No. 2, 253–284 (1996)], hereafter refered to as [2]. In [2], under suitable assumptions on the metric \(g\) of the internal six-dimensional manifold \(K\), \((K, g)\) is shown to be a complex Hermitian manifold with vanisihing first Chern class and equations on fundamental form \(J\) of the compelx structure, flux \(H\) and the curvature \(F\) (Strominger’s system) are derived. These are reviewed in §2. \(T^2\)-bundles over a Calabi-Yau manifold which satisfy part of Strominger’s system (GP manifolds), are discovered in E. Goldstein and S. Prokushkin [Geometric models for complex non-Kähler manifolds with \(\text{SU}(3)\) structure, Commun. Math. Phys. 251, No. 1, 65–78 (2004; Zbl 1085.32009)]. Solutions of Strominger’s system in the subclass of GP manifolds are obtained in [1]. In this case, base manifold is a \(K3\) surface and the curvature of the \(T^2\)-bundle is a \((1,1)\)-form \({\omega_P\over 2\pi}+{\omega_Q\over 2\pi}\). Studies on these manifolds are called FSY geometry by the authors (§3.1). The rest of §3 is devoted to the study of volumes of total space and fibres. As a conclusion, owing to the topology of \(K3\) surface, we cannot make both circles of the \(T^2\)-bundle arbitrary large.

In §4, from the string frame action ((4.1). cf. Chap. 13 of M. B. Green, J. H. Schwarzand E. Witten [Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology, Cambridge Monographs on Mathematical Physics. Cambridge University Press. (1987; Zbl 0619.53002)], exposed in Appendix A), linearized equations of motion for gaugino, dilation and gravitino ((4.2)–(4.4)) are presented. As for the massless bosonic fields, the authors say they will be superpartner of massless fermionic fields. The authors also remark linearized equations ignore higher-order corrections, but the answer will provide an upper bound to the number of massless fields. Then under suitable absatz (4.8), the gaugino equation of motion is simplified ((4.9). Detailed calculations are given in Appendix C). Under rescaling, this equation takes the form \[ {\mathcal D}C_{(0)}+ 4{\mathcal D}^{†}C_{(2)}= 0,\quad{\mathcal D}C_{(2)}= 0 \] (4.11). Here \(C_a\in\Omega^a st(K; V)\) are \(V\)-valued forms, where \(V\) is a representation of the gauge group, and \({\mathcal D}: \Omega^{p,q}(K; V)\to\Omega^{p+ 1,q}(K; V)\) is a differential operator. Since \({\mathcal D}^2\) is the \((2,0)\) part of the curvature form, the question of counting massless gaugino modes reduces to the question of counting the dimensions of the \({\mathcal D}\)-cohomology groups. By suitable resealing, \({\mathcal D}\) becomes a standard twisted Dolbeault operator. Hence the question reduces to computation of Hodge diamond of GP manifolds, which is reviewed in §5. Product rule with respect to the Levi-Cività connection on non-Kähler manifolds, which are used in §5 are computed in Appendix B.

Compactifications of heterotic string theory that preserve \({\mathcal N}= 1\) supersymmetry in four dimensions with non-zero \(H\)-flux was studied in A. Strominger [Supergravity with torsion, Nuclear Phys. B 274, No. 2, 253–284 (1996)], hereafter refered to as [2]. In [2], under suitable assumptions on the metric \(g\) of the internal six-dimensional manifold \(K\), \((K, g)\) is shown to be a complex Hermitian manifold with vanisihing first Chern class and equations on fundamental form \(J\) of the compelx structure, flux \(H\) and the curvature \(F\) (Strominger’s system) are derived. These are reviewed in §2. \(T^2\)-bundles over a Calabi-Yau manifold which satisfy part of Strominger’s system (GP manifolds), are discovered in E. Goldstein and S. Prokushkin [Geometric models for complex non-Kähler manifolds with \(\text{SU}(3)\) structure, Commun. Math. Phys. 251, No. 1, 65–78 (2004; Zbl 1085.32009)]. Solutions of Strominger’s system in the subclass of GP manifolds are obtained in [1]. In this case, base manifold is a \(K3\) surface and the curvature of the \(T^2\)-bundle is a \((1,1)\)-form \({\omega_P\over 2\pi}+{\omega_Q\over 2\pi}\). Studies on these manifolds are called FSY geometry by the authors (§3.1). The rest of §3 is devoted to the study of volumes of total space and fibres. As a conclusion, owing to the topology of \(K3\) surface, we cannot make both circles of the \(T^2\)-bundle arbitrary large.

In §4, from the string frame action ((4.1). cf. Chap. 13 of M. B. Green, J. H. Schwarzand E. Witten [Superstring theory. Volume 1: Introduction. Volume 2: Loop amplitudes, anomalies and phenomenology, Cambridge Monographs on Mathematical Physics. Cambridge University Press. (1987; Zbl 0619.53002)], exposed in Appendix A), linearized equations of motion for gaugino, dilation and gravitino ((4.2)–(4.4)) are presented. As for the massless bosonic fields, the authors say they will be superpartner of massless fermionic fields. The authors also remark linearized equations ignore higher-order corrections, but the answer will provide an upper bound to the number of massless fields. Then under suitable absatz (4.8), the gaugino equation of motion is simplified ((4.9). Detailed calculations are given in Appendix C). Under rescaling, this equation takes the form \[ {\mathcal D}C_{(0)}+ 4{\mathcal D}^{†}C_{(2)}= 0,\quad{\mathcal D}C_{(2)}= 0 \] (4.11). Here \(C_a\in\Omega^a st(K; V)\) are \(V\)-valued forms, where \(V\) is a representation of the gauge group, and \({\mathcal D}: \Omega^{p,q}(K; V)\to\Omega^{p+ 1,q}(K; V)\) is a differential operator. Since \({\mathcal D}^2\) is the \((2,0)\) part of the curvature form, the question of counting massless gaugino modes reduces to the question of counting the dimensions of the \({\mathcal D}\)-cohomology groups. By suitable resealing, \({\mathcal D}\) becomes a standard twisted Dolbeault operator. Hence the question reduces to computation of Hodge diamond of GP manifolds, which is reviewed in §5. Product rule with respect to the Levi-Cività connection on non-Kähler manifolds, which are used in §5 are computed in Appendix B.

Reviewer: Akira Asada (Takarazuka)

### MSC:

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

81T60 | Supersymmetric field theories in quantum mechanics |

81V22 | Unified quantum theories |

83E30 | String and superstring theories in gravitational theory |

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

83E50 | Supergravity |

32Q25 | Calabi-Yau theory (complex-analytic aspects) |