Contact terms, unitarity, and F -maximization in three-dimensional superconformal theories

Abstract

We consider three-dimensional N = 2 superconformal field theories on a three-sphere and analyze their free energy F as a function of background gauge and supergravity fields. A crucial role is played by certain local terms in these background fields, including several Chern-Simons terms. The presence of these terms clarifies a number of subtle properties of F . This understanding allows us to prove the F -maximization principle. It also explains why computing F via localization leads to a complex answer, even though we expect it to be real in unitary theories. We discuss several corollaries of our results and comment on the relation to the F -theorem.

The geometry of supersymmetric partition functions

“…The non-existence of such a choice signals the presence of an anomaly. (See [3,4] for an example in three dimensions.) Throughout, we will ignore possible anomalies and assume that the counterterms have been suitably adjusted.…”

“…In the presence of a U(1) R symmetry, the appropriate supercurrent is the R-multiplet. 3 Below, we will review the R-multiplet, the associated background supergravity fields, and the geometric structures that are needed to preserve supersymmetry. We also identify supersymmetric configurations for background gauge fields.…”

“…The R-multiplet in three-dimensional N = 2 theories was studied in [3,4,7]. It contains the following operators:…”

“…The corresponding supergravity theory was recently studied in superspace [19][20][21], as well as in components [3,4,16]. The supergravity background fields that couple to the operators in (1.15) are given by 16 16) 14 Under these conditions the flat-space theory does not admit a Ferrara-Zumino (FZ) supercurrent multiplet [6,18].…”

Exploring curved superspace (II)

“…The study of supersymmetric field theories on curved manifolds leads to new observables, which can often be calculated exactly using localization techniques [1][2][3][4][5][6][7][8]. Most authors have focused on supersymmetric theories with an R-symmetry.…”