To start, we will discuss a duality due to Aharony [53] involvingN= 2 gauge theories with unitary gauge groups, but no Chern-Simons terms. The first theory is simply N = 2 U(Nc) Yang-Mills theory withNf fundamental hypermultiplets (Qa,Q˜a) and no superpotential. As discussed in [4], the Coulomb branch of this nonabelian theory is lifted by instanton corrections to two components of one complex dimension each, parametrized by the monopole operators V±, as we saw in the abelian case in Chapter 6. The Higgs branch may be parameterized byMab =QaQ˜b, and is 2Nf−1 dimensional, not receiving any quantum corrections.
The dual theory isN= 2U(Nf−Nc) gauge theory withNffundamental hypermultiplets (ˆqa,q˜ˆa).
Table 8.1: Global Symmetries of the Dual Theories Field SU(Nf)×SU(Nf) U(1)A U(1)J U(1)R−U V
qa (Nf,1) 1 0 12
˜
qa (1,N¯f) 1 0 12
ˆ
qa ( ¯Nf,1) −1 0 12
˜ˆ
qa (1, Nf) −1 0 12
Mˆba (Nf,N¯f) 2 0 1 Vˆ± (1,1) −Nf ±1 N2f −Nc+ 1
In addition, there areNf2 uncharged chiral multiplets ˆMaband two uncharged chiral multiplets ˆV±, which couple via the following superpotential
˜ˆ
qaMˆbaqb+ ˆV+V˜ˆ−+ ˆV−V˜ˆ+, (8.1) whereV˜ˆ±are monopole operators, parameterizing the Coulomb branch of this theory. We emphasize that ˆV± are elementary (i.e., noncomposite) fields, while ˜V± are monopole operators, so can in principle be expressed in terms of the other fields. Since this contains the effective Coulomb branch parameters V˜ˆ±, it does not give a complete microscopic definition of the theory, but it will suffice for our purposes.
As the notation suggests, the conjectured mapping of chiral multiplets identifies Mab with ˆMab andV± with ˆV±. The Coulomb branch of the dual theory is lifted by the superpotential, which sets
˜ˆ
V±= 0, so all that remains is the Higgs branch. Naively ˆV±is also set to zero, but we must remember that V˜ˆ± are only effective variables and do not apply everywhere, and in fact ˆV± parameterize a moduli space which maps to the Coulomb branch of the original theory. As argued in [4], after some somewhat subtle lifting of parts of the Higgs branch, one can show that it maps to Higgs branch of the original theory.
We can also consider the flavor symmetries of these two theories, and how they are mapped under the duality. Both theories haveSU(Nf)×SU(Nf) global symmetry rotating the two sets of chiral multiplets, as well as aU(1)A rotating all the chirals by the same phase. In addition, there is theU(1)J topological symmetry, and a UVU(1)Rsymmetry (the IRR-symmetry will get mixed with the other global symmetries). Note that this symmetry group is the same for both theories, so we can summarize how the duality acts by thinking of a single symmetry group which acts on both theories, and listing the charges of the fields of both theories under this group, as shown in Table 8.1.
8.1.1 Mapping of Partition Functions
We now consider the partition function for these theories, and test if they are equal for dual theories.
The undeformed partition function for aU(Nc) theory withNf fundamental hypermultiplets ofR-
charge ∆ is:
1 Nc!
Z
dNcλY
i
eNf`(1−∆+iλi)+Nf`(1−∆−iλi)Y
i<j
(2 sinhπ(λi−λj)2) (8.2)
In the dual theory, the contribution of the extra fields just give factorseNf2`(1−∆M)ande`(1−∆V±). Note these do not depend on the integration variablesλi since these fields are not charged.
Corresponding to the two SU(Nf) factors, we can add masses for the two chiral multiplets in each flavor, ma and ˜ma, which are each constrained to sum to zero. In addition, forU(1)A there is an total axial mass µ, and forU(1)J there is the FI term η. We can also set theR-charges for the fields to their UV values, since a mixing of the R-charge with some flavor symmetry can be accounted for by shifting the corresponding deformation parameter by an imaginary value.
Including all of these deformations the partition function for the first theory can be written as:
ZN(U)
f,Nc(η;ma; ˜ma;µ) = 1 Nc!
Z Nc Y
j=1
dλj
Nf
Y
a=1
e`(12+iλj+ima+iµ)+`(12−iλj−im˜a+iµ)
Y
i<j
(2 sinhπ(λi−λj))2 (8.3) For the second theory, we see that the representation of SU(Nf)×SU(Nf)×U(1)A in which the quarks lie is replaced by its conjugate, so all mass terms should come in with the opposite sign.
Inspecting the table above, we see that the 1-loop partition function forMab is:
e`(i(ma−m˜b+2µ) (8.4)
while that ofV± is:
e`(Nc−Nf2 −iNfµ±iη) (8.5)
Thus the dual partition function is given by:
ZN(U)
f,Nf−Nc(η;−ma;−m˜a;−µ)e`(Nc−Nf2 −iNfµ+iη)+`(Nc−Nf2 −iNfµ−iη)Y
a,b
e`(2iµ+ima−im˜b) (8.6)
We wish to show that these two expressions are equal for all complex values of the deformations.
Here we can simply quote the work of [57, 58]. Specifically, the integrals considered in those pa- pers involved the hyperbolic gamma function Γh(z;ω1, ω2), a generalization of the ordinary gamma function, but as shown in [54], this is related to the 1-loop determinant by
Γh(z;i, i) =e`(1+iz), (8.7)
and after translating theorem 5.5.11 of [58] using this identity, we find we get precisely the equality of the dual partition functions. It would be interesting to study in more detail the origin of these integral identities, and see if they give new insight into the field theory dualities.
Note that we have not yet determined the IR R-charge of the theory. Nevertheless, as discussed above, the duality holds for any possible trial R-charge, and so in particular for the correct one.
Thus the duality appears to be completely independent of the issue of the superconformalR-charge.
This is not completely true since, as shown in [54], the duality may sometimes imply that a field on one side or the other must have dimension violating the unitarity bound ∆≥ 12. This signals that there must be new symmetries arising in the IR, possibly from some of the fields becoming decoupled, with which theR-symmetry may mix.