IR dualities in general 3d supersymmetric SU(N) QCD theories
Received: December
IR dualities in general 3d supersymmetric SU(N )
Ofer Aharony 0 1
Daniel Fleischer 0 1
Open Access 0 1
c The Authors. 0 1
Field Theories, Chern-Simons Theories
0 Rehovot 7610001 , Israel
1 Department of Particle Physics and Astrophysics, Weizmann Institute of Science
In the last twenty years, low-energy (IR) dualities have been found for many pairs of supersymmetric gauge theories with four supercharges, both in four space-time dimensions and in three space-time dimensions. In particular, duals have been found for 3d N = 2 supersymmetric QCD theories with gauge group U(N ), with F chiral multiplets in the fundamental representation, with F chiral multiplets in the anti-fundamental representation, and with Chern-Simons level k, for all values of N, F, F and k for which the theory preserves supersymmetry. For SU(N ) theories the duals have been found in some cases, such as F = F and F = 0, but not in the general case. In this note we find the IR dual for SU(N ) SQCD theories with general values of N, F, F and k 6= 0 which preserve supersymmetry.
Supersymmetric gauge theory; Supersymmetry and Duality; Duality in Gauge
1 Introduction 2 3 4
The flow in theory A
The flow in theory B
A test of the duality
SU(N ) gauge theories exhibit an IR duality, namely they flow to the same theory at low
energies (and, in particular, to the same superconformal field theory at the origin of moduli
space). In the following years this was generalized to many other examples of gauge theories
with four supercharges in four, three and two space-time dimensions. In three dimensions
dualities were first found for U(N ) gauge groups rather than SU(N ) groups. These dualities
were first discovered for theories with F chiral multiplets in the fundamental representation
case where there is also a Chern-Simons (CS) level k [3], and then to arbitrary values of
N, F, F and k for which the theory preserves supersymmetry [4], subject to the Z2-anomaly
constraint [57]
k + (F F)/2 Z .
All of these dualities do not have any rigorous derivation so far, but they pass many
consistency checks, and they are all related to each other by various flows.
The 3d dualities for SU(N ) gauge theories were only discovered relatively recently [8, 9],
theory, we will find the desired duals.
corresponding 4d duality on a circle and carefully flowing to low energies [8], but this
Presumably they can be found by similar methods, or by flowing from the finite k dualities
that we describe here, along the lines of [1012].
We begin in section 2 by describing the flow in theory A, and then in section 3 we
describe the same flow in the dual theory B, leading to our desired dual. Finally, in
section 4 we perform a simple test of the duality by comparing the baryonic flat directions
on both sides.
This note is based on [13], which contains additional details.
The flow in theory A
known dualities; all the necessary background may be found in [8, 10, 14] and references
Chern-Simons theory with F chiral superfields Qi in the fundamental representation F
from now on we take k > 0; results for k < 0 can be obtained by a parity transformation.
theory A. The duality can be written in short as:1
with F F singlets Mi, coupled by the superpotential W = Miiqiq . Mii map to QiQi in
i
i
SU(N )k U(1)N k
The symmetries of theory A are summarized in the following table:
m Q = 0 .
0 1F Fm
1
We want to turn on a real mass for Fm chiral flavors, in order to flow to a theory with
and then integrate them out. Equivalent results for mass flows involving anti-fundamental
flavors can be achieved via a charge conjugation symmetry transformation.
Turning on real masses is equivalent to turning on background scalars in vector
superfields corresponding to the global symmetry currents of the global symmetry group
SU(F )L SU(F )R U(1)B U(1)A. For simplicity, we turn on an equal mass for Fm
flavors in the fundamental representation. The mass matrices then have the form:
1We ignore here the global structure of the gauge group on the right-hand side, which is that of U(N ).
This is important for the consistency of the duality but will not play any role in this paper.
this to theory B it is useful to write it in terms of the background global symmetries: using
we can write (2.2) as:
m Q = mSU(F )L + mU(1)A 1F + mU(1)B 1F ,
m Q = mSU(F )R
+ mU(1)A 1F mU(1)B 1F ,
mU(1)A = mU(1)B =
mSU(F )L =
mSU(F )R = 0 ,
F Fm
with Tr(mSU(F )L ) = 0 as required.
with a vanishing scalar in the vector multiplet. In this vacuum Fm chiral superfields in
the fundamental representation are massive and may be integrated out. We expect that
for many values of N, F, Fm and k this supersymmetric vacuum will survive also in the
quantum theory; as usual in IR dualities this is expected to happen whenever the rank of
the dual group is non-negative (while otherwise we expect no supersymmetric vacuum).
Fm sign (m)
Note that the low-energy theory still satisfies (1.1), as required for co (...truncated)