Abstract
We compute the conformal blocks associated with scalar-scalar-fermion-fermion 4-point functions in 3D CFTs. Together with the known scalar conformal blocks, our result completes the task of determining the so-called `seed blocks' in three dimensions. Conformal blocks associated with 4-point functions of operators with arbitrary spins can now be determined from these seed blocks by using known differential operators.
Highlights
Note that the term denoted by O(1/∆, r) in (A.12) decays at large ∆ and at small r. (Recall that nA ≥ 1 in (2.24).) Plugging (A.12) into (A.10) and expanding up to order ∆1 at large ∆, we find h1∞,l = 0, h2∞,l h3∞,l = 0, h4∞,l with
To determine ci(η), one has to expand the Casimir equations (A.10) to next-to-leading order in 1/∆, and the result to order r0. Such an expansion makes sense because the terms denoted by O(∆0, r) in (A.12) do not contribute in this limit
We find that the ci’s must obey the second order differential equations
Summary
We set up our conventions for embedding 3D spinor fields into a 5D space by following [40]. Space with metric ηAB = ηAB = diag(−1, 1, 1, 1, −1) and coordinates XA To suppress spinor indices on operators we introduce commuting polarizations sα and define Ol(x, s) ≡ sα1 · · · sα2lOα1···α2l(x). Where SI are embedding space spinor polarizations transforming in the fundamental of Sp(4, R) ≃ Spin(3, 2). The form of correlation functions in embedding space is constrained by the SO(3, 2) symmetry (which acts linearly on the embedding coordinates), homogeneity (2.3), and transversality (2.6). Embedding vectors XA are written as embedding bi-spinors defined by XI J ≡ XA(ΓA)I J , where ΓA ≡ (γ2 ⊗γ0 , ⊗γ1 , ⊗γ2 , γ0 ⊗γ0 , γ1 ⊗γ0)
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