Multiple SLEs for κ∈ (0,8): Coulomb gas integrals and pure partition functions

In this article, we give an explicit relationship of SLE partition functions with Coulomb gas formalism of conformal field theory. We first construct a family of SLE_κ partition functions as Coulomb gas integrals and derive their various properties. In accordance with an interpretation as probabilistic correlations in loop $O(n)$ models, they are always positive when κ∈ (8/3,8), while they may have zeroes for κ≤ 8/3. They also admit a Fr\"{o}benius series expansion that matches with the algebraic content from CFT. Moreover, we check that at the first level of fusion, they have logarithmic asymptotic behavior when κ=8/3 and κ=8, in accordance with logarithmic minimal models $M(2,1)$ and $M(2,3)$, respectively. Second, we construct SLE_κ pure partition functions and show that they are continuous in κ∈ (0,8) and they decay to zero as a polynomial of (8-κ) when κ→ 8. We explicitly relate the Coulomb gas integrals and pure partition functions together in terms of the meander matrix. As a by-product, our results yield a construction of global non-simple multiple chordal SLE_κ measures (κ ∈ (4,8)) uniquely determined by their re-sampling property.