论文标题
假Z
Fake Z
论文作者
论文摘要
最近引入了量子代码和Narain CFT之间的连接，提供了一个简单的ANSATZ，以表达模块化不变的函数$ z（τ，\ barτ）$在多元多项式满足某些其他属性方面。这些属性包括代数身份，可确保$ z（τ，\τ）$的模块化不变性，以及系数的阳性和积分，这意味着$ \ mathfrak {u}（u}（u}（1）^n n times^nimes^times^times \ mathfrak {u}（1）^u}^n $ z（$）这种多项式自然来自代码，从某种意义上说，某种类型的每个代码都会产生所谓的枚举多项式，该多项式自动满足所有必要的属性，而所得的$ z（τ，\ barτ）$是代码CFT的分区函数 - narain理论从代码构建的narain理论毫无疑问。然而，也有``假''多项式满足所有必要属性，与任何代码无关。它们导致$ z（τ，\ barτ）$满足所有模块化bootstrap约束（模块化不变性，字符扩展的阳性和正成），但是它们是否是任何实际CFT的分区函数，尚不清楚。我们将六个最简单的假多项式中的组中的组视为假货：我们表明它们都不是任何纳兰理论的圆环分区函数。此外，其中四个不是任何统一2D CFT的分区函数。我们对其他两个的分析尚无定论。我们的发现指出了模块化引导方法的明显局限性：并非整个圆环模块化自举约束的解决方案都是由于实际的CFT造成的。在本文中，我们考虑了六个简单的示例，请记住可以构建成千上万的示例。
Recently introduced connections between quantum codes and Narain CFTs provide a simple ansatz to express a modular-invariant function $Z(τ,\bar τ)$ in terms of a multivariate polynomial satisfying certain additional properties. These properties include algebraic identities, which ensure modular invariance of $Z(τ,\bar τ)$, and positivity and integrality of coefficients, which imply positivity and integrality of the $\mathfrak{u}(1)^n \times \mathfrak{u}(1)^n$ character expansion of $Z(τ,\bar τ)$. Such polynomials come naturally from codes, in the sense that each code of a certain type gives rise to the so-called enumerator polynomial, which automatically satisfies all necessary properties, while the resulting $Z(τ,\bar τ)$ is the partition function of the code CFT -- the Narain theory unambiguously constructed from the code. Yet there are also ``fake'' polynomials satisfying all necessary properties, that are not associated with any code. They lead to $Z(τ,\bar τ)$ satisfying all modular bootstrap constraints (modular invariance and positivity and integrality of character expansion), but whether they are partition functions of any actual CFT is unclear. We consider the group of the six simplest fake polynomials and denounce the corresponding $Z$'s as fake: we show that none of them is the torus partition function of any Narain theory. Moreover, four of them are not partition functions of any unitary 2d CFT; our analysis for other two is inconclusive. Our findings point to an obvious limitation of the modular bootstrap approach: not every solution of the full set of torus modular bootstrap constraints is due to an actual CFT. In the paper we consider six simple examples, keeping in mind that thousands more can be constructed.
