Constant Degree Direct Product Testers with Small Soundness

Let <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$X$</tex> be a d-dimensional simplicial complex. A function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$F: X(k)\rightarrow\{0,1\}^{k}$</tex> is said to be a direct product function if there exists a function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$f: x(1)\rightarrow\{0,1\}$</tex> such that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$F(\sigma)=(f(\sigma_{1}),\ \ldots,\ f(\sigma_{k}))$</tex> for each k-face <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\sigma$</tex>, In an effort to simplify components of the PCP theorem, Goldreich and Safra [1] introduced the problem of direct product testing, which asks whether one can test if <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$F: X(k)\rightarrow\{0,1\}^{k}$</tex> - is correlated with a direct product function by querying <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$F$</tex> on only 2 inputs. Dinur and Kaufman [2] conjectured that there exist bounded degree complexes with a direct product test in the small soundness regime. We resolve their conjecture by showing that for all <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\delta &gt; 0$</tex>, there exists a family of high-dimensional expanders with degree <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O_{\delta}(1)$</tex> and a 2-query direct product tester with soundness <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\delta$</tex> We use the characterization given by [3] and independently by [4], who showed that some form of non-Abelian coboundary expansion (which they called “Unique-Games coboundary expansion”) is a necessary and sufficient condition for a complex to admit such direct product testers. Our main technical contribution is a general technique for showing coboundary expansion of complexes with coefficients in a non-Abelian group. This allows us to prove that the high dimensional expanders constructed by [5] satisfy the conditions of [3], thus admitting a 2-query direct product tester with small soundness.