Abstract
We study the constraints imposed by superconformal symmetry, crossing symmetry, and unitarity for theories with four supercharges in spacetime dimension $2\leq d\leq 4$. We show how superconformal algebras with four Poincar\'{e} supercharges can be treated in a formalism applicable to any, in principle continuous, value of $d$ and use this to construct the superconformal blocks for any $d\leq 4$. We then use numerical bootstrap techniques to derive upper bounds on the conformal dimension of the first unprotected operator appearing in the OPE of a chiral and an anti-chiral superconformal primary. We obtain an intriguing structure of three distinct kinks. We argue that one of the kinks smoothly interpolates between the $d=2$, $\mathcal N=(2,2)$ minimal model with central charge $c=1$ and the theory of a free chiral multiplet in $d=4$, passing through the critical Wess-Zumino model with cubic superpotential in intermediate dimensions.
Highlights
Accuracy [9,10,11,12]
We study the constraints imposed by superconformal symmetry, crossing symmetry, and unitarity for theories with four supercharges in spacetime dimension 2 ≤ d ≤ 4
We argue that one of the kinks smoothly interpolates between the d = 2, N = (2, 2) minimal model with central charge c = 1 and the theory of a free chiral multiplet in d = 4, passing through the critical Wess-Zumino model with cubic superpotential in intermediate dimensions
Summary
We begin by presenting what we would like to call the dimensional continuation of the superconformal algebra with four Poincare supercharges to an arbitrary spacetime dimension d ≤ 4. The only generators that can appear in the anticommutator of a Poincare and a conformal supercharge are D, R, Mij and Mij. Rotation invariance dictates that D comes multiplied with one of the invariant tensors δαβ, δαβ. We illustrate that our interpolation reduces to the expected algebras in integer number of dimensions This is necessary since they are the unique superconformal algebras with four Poincare supercharges and a U(1) R-symmetry in the respective number of spacetime dimensions. Working on the holomorphic (left-moving) side, we have the usual bosonic generators Ln, n = −1, 0, 1, R-symmetry Ω, and fermionic generators G±±1/2 They satisfy the following (anti)commutation relations [Lm, Ln] = (m − n) Lm+n ,. Where in the last line X stands for either Q or S
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
