Abstract
We develop techniques for computing superconformal blocks in 4d superconformal field theories. First we study the super-Casimir differential equation, deriving simple new expressions for superconformal blocks for 4-point functions containing chiral operators in theories with N-extended supersymmetry. We also reproduce these results by extending the "shadow formalism" of Ferrara, Gatto, Grillo, and Parisi to supersymmetric theories, where superconformal blocks can be represented as superspace integrals of three-point functions multiplied by shadow three-point functions.
Highlights
Unitarity constraints, we typically have a greater handle on the space of such theories as well as knowledge of protected aspects of the spectrum
By constructing a superconformally-invariant projector we can project 4-point functions onto simple integral expressions for superconformal blocks. We apply this method to 4-point functions containing two chiral and two antichiral operators in theories with N -extended supersymmetry, reproducing the results obtained from the super-Casimir approach
While in general one needs more complicated superspaces to describe CFTs with extended supersymmetry, the superspace defined in section 2 suffices to describe operators which are annihilated by all supersymmetries of one chirality
Summary
We review the construction of superspace in terms of objects which transform linearly under superconformal transformations. We choose a matrix g ∈ GL(2, C) such that MABZBb gba returns back to the Poincare slice The composition of these two transformations defines a map (x+, θ) → (x+, θ ) representing the action of SU(2, 2|N ): δαb ixα+ ̇ a = ZAa → MABZBa ∼ MABZBb gba = ix+αa. This precisely reproduces the usual action of the superconformal group on chiral superspace. Which can be solved by writing xα± ̇ α = xαα ± 2iθαiθiα, x− = x†+, with x real In this way, we recover the usual relation between superspace coordiantes (x, θ, θ) and chiral coordinates (x+, θ). The independent supertwistor Z and dual supertwistor Z transform such that the pairing ZAZA is invariant under the complexified superconformal group SL(4|N )
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