For theories with non-zero Chern-Simons level the S-duality is called a
‘Giveon-Kutasov duality (see [3] for the original paper). The electric theory is a (1 + 2)d N = 2 U(Nc)k gauge theory withNf chiral multipletsQi in the Nc
representation and Nf anti-chiral multiplets Q¯i in theNc representation, where i,¯i= 1, ..., Nf. Here the subscript kdenotes the Chern-Simons level.
The global symmetries areSU(Nf)×SU(Nf)×U(1)A×U(1)R×U(1)J [3]. As was
the case in the discussion of Aharony duality, the Higgs branch of the Giveon-Kutasov duality is parameterised by the meson:
Mi¯i=QiQ
¯i
(7.45)
and the Coulomb branch is parameterised by the chiral superfield monopole operators [83]:
V+ ∼eΣ/g 2
(7.46)
V− ∼e−Σ/g2 (7.47)
where, again, Σ =σ+iγ.
The fields of the electric theory transform under the same global symmetries as the electric theory of Aharony duality (table copied from [87]):
Table 9: Global Symmetries of Electric SQCD
Particle U(1)J SU(Nf) SU(Nf) U(1)A U(1)R Q 0 Nf 1 1 0 Q 0 1 Nf 1 0 M 0 Nf Nf 2 0 V+ +1 1 1 −Nf Nf −Nc+ 1 V− −1 1 1 −Nf Nf −Nc+ 1
The magnetic dual theory is (1 + 2)-dimensionalN = 2 U(Nf +|k| −Nc)−k with Nf
flavours of quark qi in the Nf +k−Ncrepresentation of U(Nf +|k| −Nc) and Nf
branch is parameterised by ˜V+ and ˜V−,which are singlets under U(Nf +|k| −Nc)−k.
In the magnetic theoryM is interpreted as a fundamental field, whilst in the electric theory it is a composite ofQ and Q.
The fields of the magnetic theory transform under the same global symmetries as the magnetic theory of Aharony duality (table copied from [87] and using [2]):
Table 10: Global Symmetries of Magnetic SQCD
Particle U(1)J SU(Nf) SU(Nf) U(1)A U(1)R q 0 Nf 1 −1 1 q 0 1 Nf −1 1 M 0 Nf Nf 2 0 ˜ V+ +1 1 1 Nf Nf −Nc+ 1 ˜ V− −1 1 1 Nf Nf −Nc+ 1
Unlike Aharony duality, Giveon-Kutasov duality does not contain monopole operators in the superpotential of the magnetic theory:
W =M qq¯ (7.48)
Tests of Giveon-Kutasov Duality:
The dual of the dual gives the original theory.
The moduli spaces of the electric and magnetic theories match [3].
Matching of the global symmetries.
Matching of the partition functions of electric and magnetic theories with non-zero real masses and non-zero FI-terms [84].
7.5.1 Giveon-Kutasov Duality with Adjoint Matter
Aharony duality was also formulated for the case of (1 + 2)dN = 2 field theory containing adjoint matter [41,42].
As before, the electric theory is a (1 + 2)dN = 2 U(Nc) gauge theory withNf chiral
multiplets Qi in the Nc representation and Nf anti-chiral multipletsQ¯i in theNc
representation, where i,¯i= 1, ..., Nf. There are also the monopole operatorsV+ and
V−, andM is composite field (meson). In addition there is the adjoint chiral multiplet Φ [41,42].
The electric theory contains the superpotential [41,42]: We= n X i=0 ci n+ 1−iΦ n+1−i (7.49)
With the inclusion of adjoint matter, the magnetic dual theory is a (1 + 2)d N = 2 theory with aU(nNf +n|k| −Nc) gauge group [41,42]. Herenis the integer that
appeared in section4.7. The magnetic theory contains chiral multipletsqi in the nNf +n|k| −Nc representation and anti-chiral multipletsqi in thenNf +n|k| −Nc
representation. There are also the monopole operators ¯V+ and ¯V−, and M is a fundamental field. In addition, there is an adjoint multiplet ¯Φ.
The magnetic theory contains a superpotential which is absent of monopole operators [41,42]: Wm=− n X i=0 ¯ ci n+ 1−iTr ¯Φ n+1−i+ n X i=1 ¯ ¯ ciMiq¯Φ¯n−iq (7.50) ¯
ci and ¯c¯i are functions ofci. Alternatively this can be written [41,42]:
Wm =− c0 n+ 1Tr ¯Φ n+1+ n X i=1 Miq¯Φ¯n−iq (7.51)
Part III
8
Theories with Massive Fundamental and Antifundamen-
tal Matter
In [4], configurations of the form of the right hand diagram in 17 are considered, with various numbers of D5-branes and D3-branes displaced along thex3-direction. When a D5-brane is displaced, it becomes part of a (p, q)-NS5-D5-web; one half of the brane is displaced in the positivex3-direction and labelled as D5+, whilst the other half is displaced in the negative x3-direction and lablled as D5−. When a D3-brane is displaced, the whole brane is moved either in the positive x3-direction or in the negativex3-direction. In this section a variety of such configurations are considered. The massive and massless states that arise from these configurations are found. The induced Chern-Simons terms that arise in the low energy theory are determined, and the resulting flows between Aharony and Giveon-Kutasov dualities are stated.
Notation: Throughout this section all iindices (including those with dashes) are flavour indices. Allj,k,l,m,n,p,q indices (including those with dashes) are colour indices.