AdS/CFT correspondence

Statement

Let $N \ge 2$ be an integer and let $g_s > 0$. Consider type IIB superstring theory on the spacetime $AdS_5 \times S^5$, with string coupling $g_s$. Here $AdS_5$ is five-dimensional anti-de Sitter space and $S^5$ is the five-sphere, both of radius $L$. There are $N$ units of five-form flux through $S^5$. Consider also $\mathcal{N} = 4$ supersymmetric Yang–Mills theory on four-dimensional Minkowski space, with gauge group $SU(N)$ and coupling $g_{YM}$, which is a conformal field theory. The conjecture states that these two theories are equivalent, with $g_{YM}^2 = 4\pi g_s$ and $L^4 = 4\pi g_s N \alpha'^2$ in a common normalization, where $\alpha'$ is the Regge slope. Each bulk field $\phi$ of the string theory is dual to a local operator $\mathcal{O}$ of the gauge theory. The string partition function with boundary values $\phi_0$ of $\phi$ equals the generating function $\left\langle \exp \int \phi_0\, \mathcal{O} \right\rangle$ of correlators of $\mathcal{O}$.