Constant Degree Direct Product Testers with Small Soundness

Let X be a d-dimensional simplicial complex. A function F: X(k)→0,1^k is said to be a direct product function if there exists a function f: x(1)→0,1 such that F(Σ)=(f(Σ₁), ·, f(Σ_k)) for each k-face Σ, 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 F: X(k)→0,1^k - is correlated with a direct product function by querying f 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 Δ > 0, there exists a family of high-dimensional expanders with degree O_Δ(1) and a 2-query direct product tester with soundness Δ 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.