Abstract
We show that exactly marginal operators of Supersymmetric Conformal Field Theories (SCFTs) with four supercharges cannot obtain a vacuum expectation value at a generic point on the conformal manifold. Exactly marginal operators are therefore nilpotent in the chiral ring. This allows us to associate an integer to the conformal manifold, which we call the nilpotency index of the conformal manifold. We discuss several examples in diverse dimensions where we demonstrate these facts and compute the nilpotency index.
Highlights
Mc, and an exactly marginal operator O in a generic direction can be expressed as dimMc
Razamat,b Orr Selab and Adar Sharonc aSimons Center for Geometry and Physics, Stony Brook, New York, U.S.A. bDepartment of Physics, Technion, Haifa 32000, Israel cDepartment of Particle Physics and Astrophysics, Weizmann Institute of Science, Rehovot 7610001, Israel E-mail: zkomargodski@scgp.stonybrook.edu, razamat@physics.technion.ac.il, sorrsela@campus.technion.ac.il, adar.sharon@weizmann.ac.il Abstract: We show that exactly marginal operators of Supersymmetric Conformal Field Theories (SCFTs) with four supercharges cannot obtain a vacuum expectation value at a generic point on the conformal manifold
K can only jump along complex co-dimension 1 directions of Mc, and so it should be identical for generic operators of the form (1.1). This endows the conformal manifold with a global integer invariant k(Mc) which is the nilpotency index of an exactly marginal operator corresponding to a generic direction at a generic point of the conformal manifold (away from co-dimension 1 singularities)
Summary
In this subsection we would like to review a few important facts about supersymmetric theories with four supercharges and an R-symmetry. Nelson-Seiberg [23] argued that if F is a “generic” function no SUSY vacua will exist since the critical points of W require F = ∂iF = 0 which gives one more equations than variables in F and in general no solution is expected to exist. Since in supersymmetric theories the superpotential is not renormalized, F does not have to be generic [24, 25] and there are many examples with SUSY vacua which spontaneously break the R-symmetry (the maximally supersymmetric theory being one such example). Let us assume that the equations F = ∂iF = 0 admit solutions (i.e. SUSY vacua) which depend analytically on A. The general result (2.1) applies away from potential complex co-dimension 1 (or higher) sub-manifolds, where the space of solutions may jump.
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