## Minimal distances between SCFTs

##### Abstract

We study lower bounds on the minimal distance in theory space between
four-dimensional superconformal field theories (SCFTs) connected via broad classes of
renormalization group (RG) flows preserving various amounts of supersymmetry (SUSY).
For N = 1 RG flows, the ultraviolet (UV) and infrared (IR) endpoints of the flow can
be parametrically close. On the other hand, for RG flows emanating from a maximally
supersymmetric SCFT, the distance to the IR theory cannot be arbitrarily small regardless
of the amount of (non-trivial) SUSY preserved along the flow. The case of RG flows from
N = 2 UV SCFTs is more subtle. We argue that for RG flows preserving the full N = 2
SUSY, there are various obstructions to finding examples with parametrically close UV
and IR endpoints. Under reasonable assumptions, these obstructions include: unitarity,
known bounds on the c central charge derived from associativity of the operator product
expansion, and the central charge bounds of Hofman and Maldacena. On the other hand,
for RG flows that break N = 2 → N = 1, it is possible to find IR fixed points that are
parametrically close to the UV ones. In this case, we argue that if the UV SCFT possesses
a single stress tensor, then such RG flows excite of order all the degrees of freedom of the
UV theory. Furthermore, if the UV theory has some flavor symmetry, we argue that the
UV central charges should not be too large relative to certain parameters in the theory