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 Unique-Games coboundary expanders. 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，如果定义在其 $k$ 维面（其中 $k\ll d$）上的任意局部一致函数必然来自其顶点上的函数。更精确地说，直接积测试具有一对 $k$ 维面 $(A,A')$ 上的分布 $\mu$，给定对函数 $F\colon X(k)\to\{0,1\}^k$ 的查询访问，它采样 $(A,A')\sim \mu$ 并检查 $F[A]|_{A\cap A'} = F[A']|_{A\cap A'}$。该测试应满足：(1) “完备性”，即任意直接积赋值 $F$ 以概率 $1$ 通过测试；(2) “可靠性”，即如果 $F$ 以概率 $s$ 通过测试，那么 $F$ 必须与某个直接积函数相关。Dinur 和 Kaufman 表明，一个充分好的谱扩张复形 $X$ 在 $s$ 接近 $1$ 的“高可靠性”区间内允许直接积测试。他们询问是否存在支持直接积测试的高维展开复形，在 $s$ 接近 $0$ 的“低可靠性”区间内。我们刻画出在低可靠性区间内支持直接积测试的高维展开复形的特征。我们表明谱扩张是不充分的，复形还必须满足共边界扩展的一个变体，我们称之为唯一博弈共边界扩展复形。这一性质可视为标准共边界扩展概念在非阿贝尔群上二维复形的高维推广。它断言：利用复形低维面获得的任意局部一致唯一博弈实例，必须允许一个良好的全局解。