论文标题
Narain CFT和较高属的量子代码
Narain CFTs and Quantum Codes at Higher Genus
论文作者
论文摘要
代码CFT是由错误校正代码定义的2D共形场理论。最近，Dymarsky和Shapere概括了代码CFT的构建，包括量子错误校正代码。在本文中，我们探讨了高级属的这种联系。我们证明，较高的分区函数采用了高重theta函数的多项式形式，并且高生组模块组充当这些多项式的简单线性变换。我们为如何明确地解释了如何对模块化约束解决，这是我们为属2所做的。结果是，在第1属和属2属的模块化不变性比单独的属1更具限制性。这使我们能够大大降低可能的代码CFT的空间。我们还考虑了许多“同一理论”的示例 - 具有相同属1分区函数的CFT-我们发现它们具有不同的属2分区函数。最后，我们与模块化引导程序已知的一些2D CFT建立连接。 $ n = 4 $理论猜想具有最大的差距，$ so（8）$ WZW模型是代码CFT，使我们能够为其属2分区功能提供表达。我们还发现了其他一些已知的CFT，这些CFT不是代码理论，而是其分区函数满足与代码理论相同的简单多项式ANSATZ。这使我们推测了代码多项式形式的有用性，而不是对CFT的研究。
Code CFTs are 2d conformal field theories defined by error-correcting codes. Recently, Dymarsky and Shapere generalized the construction of code CFTs to include quantum error-correcting codes. In this paper, we explore this connection at higher genus. We prove that the higher-genus partition functions take the form of polynomials of higher-weight theta functions, and that the higher-genus modular group acts as simple linear transformations on these polynomials. We explain how to solve the modular constraints explicitly, which we do for genus 2. The result is that modular invariance at genus 1 and genus 2 is much more constraining than genus 1 alone. This allows us to drastically reduce the space of possible code CFTs. We also consider a number of examples of "isospectral theories" -- CFTs with the same genus 1 partition function -- and we find that they have different genus 2 partition functions. Finally, we make connection to some 2d CFTs known from the modular bootstrap. The $n = 4$ theory conjectured to have the largest possible gap, the $SO(8)$ WZW model, is a code CFT, allowing us to give an expression for its genus 2 partition function. We also find some other known CFTs which are not code theories but whose partition functions satisfy the same simple polynomial ansatz as the code theories. This leads us to speculate about the usefulness of the code polynomial form beyond the study of code CFTs.
