Rigidity of center Lyapunov exponents and $su$-integrability

Let $f$ be a conservative partially hyperbolic diffeomorphism, which is homotopic to an Anosov automorphism $A$ on $\mathbb{T}^3$. We show that the stable and unstable bundles of $f$ are jointly integrable if and only if every periodic point of $f$ admits the same center Lyapunov exponent with $A$. In particular, $f$ is Anosov. Thus every conservative partially hyperbolic diffeomorphism, which is homotopic to an Anosov automorphism on $\mathbb{T}^3$, is ergodic. This proves the Ergodic Conjecture proposed by Hertz-Hertz-Ures on $\mathbb{T}^3$.