![](https://csdnimg.cn/release/download_crawler_static/12306967/bg11.jpg)
3 Cohomology in SCFTs
Our aim in this section and the next is to describe a procedure that generalizes the coho-
mological truncation of [4, 7, 8] from N = 4 SCFTs to the more general non-conformal
N = 4 theories on S
3
that were described in section 2. The construction of [4, 7, 8] was
based on identifying two supercharges Q
H
1
and Q
H
2
in N = 4 SCFTs on R
3
, such that
the OPE restricted to their cohomology gives a certain quantization of the Higgs branch.
It was also possible to find another pair of supercharges, Q
C
1
and Q
C
2
, whose cohomology
similarly leads to a quantization of the Coulomb branch, though this second possibility was
not explored in detail. We will generalize both cases to non-conformal theories on S
3
, but,
just as in [4, 7, 8], our main focus will also be the Higgs branch.
We will find that local operators in the cohomology, both for Q
H
i
and for Q
C
i
, can
only be inserted along a great circle S
1
⊂ S
3
.
14
The circle is the fixed point locus of the
U(1) isometry that appears in the anti-commutator {Q
H
1
, Q
H
2
} or {Q
C
1
, Q
C
2
}. In the case of
Q
H
i
, the operators that can be inserted on S
1
will be referred to as “twisted Higgs branch
operators”, because, as we will see, they are in 1-to-1 correpsondence with Higgs branch
chiral ring operators. Similarly, operators in Q
C
i
cohomology will be referred to as “twisted
Coulomb branch operators”.
In section 3.1, we start by reviewing the construction of [4, 7, 8] in flat space, and
then in section 3.2 we translate this construction to S
3
. In section 4 we describe the
generalization of this cohomology directly based on the su(2|1)
`
⊕ su(2|1)
r
algebra.
3.1 SCFT in flat space
Consider theories living on a three-dimensional Euclidean space R
3
with the standard
coordinates ~x = (x
1
, x
2
, x
3
). The bosonic subalgebra of the osp(4|4) superconformal alge-
bra is so(4) ⊕ sp(4), where the generators of sp(4) are rotations M
αβ
, translations P
µ
and
special conformal transformations K
µ
, and the generators of the so (4)
∼
=
su(2)
H
⊕su(2)
C
R-
symmetry are denoted by R
ab
and
¯
R
˙a
˙
b
and act on the Higgs and Coulomb branches, respec-
tively. The fermionic generators are Q
αa ˙a
and S
αa ˙a
, denoting Poincar´e and conformal super-
charges, respectively. The detailed description of this algebra can be found in appendix C.1.
Define the two supercharges Q
H
1
and Q
H
2
by
Q
H
1
= Q
11
˙
2
+
1
2r
S
2
2
˙
2
, Q
H
2
= Q
21
˙
1
+
1
2r
S
1
2
˙
1
. (3.1)
In (3.1), r is some arbitrary parameter with dimensions of length.
15
The supercharges Q
H
1,2
are nilpotent, i.e., (Q
H
1
)
2
= (Q
H
2
)
2
= 0, and their anticommutator is given by
Z =
ir
4
{Q
H
1
, Q
H
2
} = −M
12
+
¯
R
˙
1
˙
1
. (3.2)
Whether we consider the cohomology of Q
H
1
or Q
H
2
, the above equation implies that it
can be represented by elements from the Z = 0 subspace. In order to satisfy Z = 0, local
operators with zero
¯
R
˙
1
˙
1
charge can only be inserted at the fixed point locus of the M
12
rotation, i.e., at the line x
1
= x
2
= 0.
14
There is some freedom in choosing Q
H,C
i
, which corresponds precisely to the choice of great circle on S
3
.
15
When we map (3.1) to S
3
, we will interpret r as the radius of the sphere.
– 15 –