This work develops the correspondence between orbifolds and free fermion models. A complete classification is obtained for orbifolds X/G with X the product of three elliptic curves and G an abelian extension of a group (Z_2)^2 of twists acting on X. Each such quotient X/G is shown to give a geometric interpretation to an appropriate free fermion model, including the geometric NAHE+ model. However, the semi-realistic NAHE free fermion model is proved to be non-geometric: its Hodge numbers are not reproduced by any orbifold X/G. In particular cases it is shown that X/G can agree with some Borcea-Voisin threefolds, an orbifold limit of the Schoen threefold, and several further orbifolds thereof. This yields free fermion models with geometric interpretations on such special threefolds.
In this paper we study the properties of two-dimensional CFTs defined by cyclic and symmetric orbifolds of free Dirac fermions, especially by focusing on the partition function and entanglement entropy. Via the bosonization, we construct the twist operators which glue two complex planes to calculate the partition function of Z_2 orbifold CFT on a torus. We also find an expression of Z_N cyclic orbifold in terms of Hecke operators, which provides an explicit relation between the partition functions of cyclic orbifolds and those of symmetric ones. We compute the entanglement entropy and Renyi en…
We carry out a systematic study of primary operators in the conformal field theory of a free Weyl fermion. Using SO(4,2) characters we develop counting formulas for primaries constructed using a fixed number of fermion fields. By specializing to particular classes of primaries, we derive very explicit formulas for the generating functions for the number of primaries in these classes. We present a duality map between primary operators in the fermion field theory and polynomial functions. This allows us to construct the primaries that were counted. Next we show that these classes of primary fiel…
On the Number of Chiral Generations in Z2 X Z2 Orbifolds
来自对撞机实验和宇宙观测站的数据表明存在三代轻物质。在某些弦紧化类中，代数与一个拓扑量有关，即欧拉特征。但是，这并不能解释三代人的存在。在一类与Z2 X Z2轨道紧化相关的自由费米子弦模型中，三代的存在与这类紧化中三个扭曲扇区的存在相关。然而，三代模型是在自由费米子公式中构造的，它们的几何对应关系并不容易获得。本文利用三个复环面上的附加对称位移对Z2 × Z2轨道的子分类。我们表明，三代真空不是以这种方式获得的，这表明自由费米子模型的几何结构更深奥。
In orbifold family unification on the basis of $SU(N)$ gauge theory on the six-dimensional space-time $M^4\times T^2/Z_m$ ($m=2, 3, 4, 6$), enormous numbers of models with three families of the standard model matter multiplets are derived from a massless Dirac fermion with a vectorlike representation $[N, 3] + [N, N-3]$ of $SU(N)$ ($N = 8, 9$). They contain models with three or more than three neutrino singlets and without any non-Abelian continuous flavor gauge symmetries. The relationship between flavor numbers from a fermion with $[N, N-k]$ and those from a fermion with $[N, k]$ are studied…
我们考虑杂种E8 X E8弦论，它产生了四维类标准模型，并允许它们的SO（10）嵌入。我们研究了两种不同的紧化方案：自由费米子公式和orbifold结构。在自由费米子公式中给出的例子中，我们直接在弦水平上使用自由费米子边界条件，而不是在有效低能场论中，探讨了额外希格斯表象的去除。此外，通过采用标准的分析平坦的方向，我们提出了一个准现实的三代弦模型，其中严格的F-和D-平坦的解决方案似乎不存在的所有订单的超势。我们表明，通过选择一个不可因式分解的紧化格，定义为倾斜其标准的简单的根，我们减少了总的代数。最后，给出了E8 × E8轨道紧化的模不变配分函数的构造。
We build rigorously the attractive five-dimensional model where bulk fermions propagate along the $ \mathcal{S}^1 / \mathbb{Z}_2$ orbifold and interact with a Higgs boson localised at a fixed point of the extra dimension. The analytical calculation of the fermion mass spectrum and effective Yukawa couplings is shown to require the introduction of either Essential Boundary Conditions (EBC) imposed by the model definition or certain Bilinear Brane Terms (BBT) in the action, instead of the usual brane-Higgs regularisations. The obtained fermion profiles along the extra dimension turn out to under…
We study constructions and classifications of three-generation models based on magnetized $T^4$ and $T^4/{Z}_2$ orbifold as candidates of the compact space. We focus on chiral fermion zero-mode wave functions in the extra dimensions. Freedoms of constant gauge fields, called Scherk-Schwarz phases are taken into account. Infinite number of three-generation models are yielded, corresponding to the ways in which the magnetic flux can be turned on. We classify them in a systematic manner, clarifying the relationship between different models. The Higgs sector is also studied by analyzing possible a…
The free orbifold point of the D1-D5 CFT must be deformed with a scalar marginal operator driving it to the region in moduli space where the holographic supergravity description of fuzzball microstates becomes available. We discuss the effects of the deformation operator on the twisted Ramond ground states of the CFT by computing four-point functions. One can thus extract the OPEs of the deformation operator with these Ramond fields to find the conformal dimensions of intermediate non-BPS states and the relevant structure constants. We also compute the anomalous dimensions at second order in p…
We study of fermion zero-modes on magnetized $T^6/\mathbb{Z}_N$ orbifolds. In particular, we focus on non-factorizable orbifolds, i.e. $T^6/\mathbb{Z}_7$ and $T^6/\mathbb{Z}_{12}$ corresponding to $SU(7)$ and $E_6$ Lie lattices respectively. The number of degenerated zero-modes corresponds to the generation number of low energy effective theory in four dimensional space-time. We find that three-generation models preserving 4D $\mathcal{N}=1$ supersymmetry can be realized by magnetized $T^6/\mathbb{Z}_{12}$, but not by $T^6/\mathbb{Z}_7$. We use $Sp(6,\mathbb{Z})$ modular transformation for the…
Partition functions of NAHE-based free fermionic string models
杂合弦自由费米子公式产生了一大类三代模型，其基础SO（10）GUT对称性直接在弦水平上被威尔逊线破缺。这些模型中边界条件基向量的一个公共子集是NAHE集，它对应于SO（12）Narain格的Z2 X Z2轨道，其中（h11，h21）=（27，3）。另一种方法是，从格点上的一个Z2 X Z2轨道开始，加上一个自由作用的Z2对合，得到一个具有相同数据的流形，其中（h11，h21）=（51，3）。这两种构造的等价性通过检验相关的配分函数得到了证明。发现了在自由费米子点上再现紧化的位移的显式实现，并指出其它密切相关的位移再现了相同的无质量谱，但不同的有质量谱，从而证明了从全配分函数中提取信息的实用性.这里讨论的类型的自由作用对合，使威尔逊线的使用，打破大统一理论的对称性，并可用于非微扰研究的自由费米子模型。
Universality in anomaly flow by an Aharonov-Bohm (AB) phase $θ_H$ is shown in the flat $M^4 \times (S^1/Z_2)$ spacetime and in the Randall-Sundrum (RS) warped space. We analyze $SU(2)$ gauge theory with doublet fermions. With orbifold boundary conditions the $U(1)$ part of gauge symmetry remains unbroken at $θ_H = 0$ and $π$. Chiral anomalies smoothly vary with $θ_H$ in the RS space. It is shown that anomaly coefficients associated with this anomaly flow are expressed in terms of the values of the wave functions of gauge fields at the UV and IR branes in the RS space. The anomaly coefficients …
