卡拉比 - 丘轨形上杂化弦自由场构造中的完全无质量单态谱
The complete massless singlet spectrum in the free-field construction of heterotic strings on Calabi--Yau orbifolds
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中文总结 AI 辅助
研究卡拉比 - 丘轨形上杂化弦自由场构造中的无质量单态谱，通过结合最小模型表示与轨形扭曲扇区结构计算，重现已知谱并给出新谱，验证了Gepner点单态数，满足镜像对称检查。
中文摘要 AI 辅助
我们通过计算无质量\(E_6\)单态的全谱，应用在Berglund - Hübsch卡拉比 - 丘轨形上紧致化的杂化弦的自由场构造。通过将不可约\(N = 2\)最小模型表示的精确内容（由沙波瓦洛夫矩阵的秩编码）与轨形的扭曲扇区结构相结合，得到后代顶点的贡献。该方法重现了具有霍奇数\((17,21)\)的五次轨形的已知谱，即\(17\)个世代、\(21\)个反世代和\(234\)个单态。对于五次曲面本身，我们得到\(330\)个单态。我们表明，在Gepner点，这个数字而非从几何描述中频繁引用的\(326\)是正确的，这与Kachru和Witten的朗道 - 金兹堡计算一致。我们还提供了一般顶点的显式构造。然后我们计算了另外两个五次轨形\(Z_5[0,1,2,3,4]\)（具有\((21,1,210)\)）和\(Z_5[0,0,0,1,4]\)（具有\((49,5,258)\)）的新谱，其中特殊霍奇数\(h^{2,1}=49\)来自扭曲扇区。所有四个例子都满足精确的镜像对称检查。
英文摘要
We apply the free-field construction of the heterotic string compactified on Berglund--Hübsch Calabi--Yau orbifolds by computing the full spectrum of massless $E_6$ singlets. The contribution of the descendant vertices is obtained by combining the exact content of the irreducible $N=2$ minimal model representations, encoded in the ranks of the Shapovalov matrices, with the twisted sector structure of the orbifold. The method reproduces the known spectrum of the quintic orbifold with Hodge numbers $(17,21)$, namely $17$ generations, $21$ antigenerations and $234$ singlets. For the quintic itself we obtain $330$ singlets. We show that this number, rather than the frequently quoted value $326$, obtained from the geometric description, is the correct one at the Gepner point, in agreement with the Landau--Ginzburg computation of Kachru and Witten. We also provide an explicit construction of the general vertices. We then compute the new spectra of the two remaining quintic orbifolds, $Z_5[0,1,2,3,4]$ with $(21,1,210)$ and $Z_5[0,0,0,1,4]$ with $(49,5,258)$, where the exceptional Hodge number $h^{2,1}=49$ arises from the twisted sectors. All four examples satisfy exact mirror-symmetry checks.
