A $d$-dimensional simplicial complex $X$ is said to support a direct product tester if any locally consistent function defined on its $k$-faces (where $k\ll d$) necessarily come from a function over its vertices. More precisely, a direct product tester has a distribution $\mu$ over pairs of $k$-faces $(A,A')$, and given query access to $F\colon X(k)\to\{0,1\}^k$ it samples $(A,A')\sim \mu$ and checks that $F[A]|_{A\cap A'} = F[A']|_{A\cap A'}$. The tester should have (1) the ``completeness property'', meaning that any assignment $F$ which is a direct product assignment passes the test with probability $1$, and (2) the ``soundness property'', meaning that if $F$ passes the test with probability $s$, then $F$ must be correlated with a direct product function. Dinur and Kaufman showed that a sufficiently good spectral expanding complex $X$ admits a direct product tester in the ``high soundness'' regime where $s$ is close to $1$. They asked whether there are high dimensional expanders that support direct product tests in the ``low soundness'', when $s$ is close to $0$. We give a characterization of high-dimensional expanders that support a direct product tester in the low soundness regime. We show that spectral expansion is insufficient, and the complex must additionally satisfy a variant of coboundary expansion, which we refer to as \emph{Unique-Games coboundary expanders}. Conversely, we show that this property is also sufficient to get direct product testers. This property can be seen as a high-dimensional generalization of the standard notion of coboundary expansion over non-Abelian groups for 2-dimensional complexes. It asserts that any locally consistent Unique-Games instance obtained using the low-level faces of the complex, must admit a good global solution.

翻译：一个$d$维单纯复形$X$被称为支持直接积测试，如果任何定义在其$k$-面（其中$k\ll d$）上的局部一致函数必然来自其顶点上的函数。更精确地说，直接积测试有一个分布$\mu$，作用于$k$-面对$(A,A')$，给定对$F\colon X(k)\to\{0,1\}^k$的查询访问，它采样$(A,A')\sim \mu$并检查$F[A]|_{A\cap A'} = F[A']|_{A\cap A'}$。测试应具有（1）“完备性”，即任何作为直接积赋值的赋值$F$以概率$1$通过测试，以及（2）“可靠性”，即如果$F$以概率$s$通过测试，则$F$必须与直接积函数相关。Dinur和Kaufman证明，一个足够好的谱扩展复形$X$在$s$接近$1$的“高可靠性”区域中允许直接积测试。他们提出问题：是否存在高维扩展器，在$s$接近$0$的“低可靠性”区域中支持直接积测试。我们给出了在低可靠性区域中支持直接积测试的高维扩展器的特征刻画。我们证明谱扩展是不充分的，复形还必须满足余边界扩展的一个变体，我们称之为**唯一游戏余边界扩展器**。反之，我们证明这一性质也是获得直接积测试的充分条件。该性质可视为标准余边界扩展概念在非交换群上对二维复形的高维推广。它断言，通过复形的低层面得到的任何局部一致唯一游戏实例，必须允许一个良好的全局解。