We develop a version of dipolar conformal field theory based on the central charge modification of the Gaussian free field with the Dirichlet boundary condition and prove that correlators of certain family of fields in this theory are martingale-observables for dipolar SLE. We prove the restriction property of dipolar SLE(8/3) and Friedrich-Werner's formula in the dipolar case. φU.A non-random conformal Fock space field M is a [λ , λ * ]-differential if for any two overlapping charts φ , φ , we haveis called degrees or conformal dimensions of M. Non-random conformal Fock space fields M are called pre-pre-Schwarzian forms, pre-Schwarzian forms, and Schwarzian forms of order µ(∈ C) if the following transformation laws hold:is a local martingale on dipolar SLE probability space. (The process M t (z 1 , · · · , z n ) is stopped when any z j exits D t .) For example, we can use the identity chart of D. Then for [h, h * ]-differentials M with boundary conformal dimensions h ± at q ± , we haveIf M is a pre-Schwarzian form of order µ, then.Similarly, for a Schwarzian form M of order µ, we have M t (z) = (w ′ t (z)) 2 M(w t (z)) + µS w t (z). A dipolar CFTAll our fields in this paper are Fock space (correlational) fields constructed from the Gaussian free field Φ (0) with the Dirichlet boundary condition, its derivatives, and Wick's exponentials e ⊙αΦ (0) (α ∈ C) by means of Wick's calculus. (An alternate notation for Wick's exponentials of Φ (0) is : e αΦ (0) : .) For Fock space fields X 1 , · · · , X n and distinct points (nodes) z 1 , · · · , z n in D, a correlation functionis defined by Wick's formula. We will review its definition and basic properties in Subsection A.2. For example, we define E[Φ (0) (z)] = 0, E[Φ (0) (z 1 )Φ (0) (z 2 )] = 2G(z 1 , z 2 )

We consider OPE family of the central charge modifications of the Gaussian free field with excursion-reflected/Dirichlet boundary conditions in a doubly connected domain and show that the correlations of fields in the OPE family under the insertion of one-leg operator with marked boundary points are martingale-observables for variants of annulus SLEs. By means of screening, we find Euler integral type solutions to the parabolic partial differential equations for the annulus SLE partition functions introduced by Lawler and Zhan to construct the class of martingale-observables associated with these solutions.

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