Prepared for submission to JHEP
A 3d-3d appetizer
Du Pei and Ke Ye
Walter Burke Institute for Theoretical Physics, California Institute of Technology,
Pasadena, CA 91125
E-mail:
pei@caltech.edu
,
kye@caltech.edu
Abstract:
We test the 3d-3d correspondence for theories that are labelled by Lens spaces.
We find a full agreement between the index of the 3d
N
= 2 “Lens space theory”
T
[
L
(
p,
1)]
and the partition function of complex Chern-Simons theory on
L
(
p,
1). In particular, for
p
= 1, we show how the familiar
S
3
partition function of Chern-Simons theory arises from
the index of a free theory. For large
p
, we find that the index of
T
[
L
(
p,
1)] becomes a con-
stant independent of
p
. In addition, we study
T
[
L
(
p,
1)] on the squashed three-sphere
S
3
b
.
This enables us to see clearly, at the level of partition function, to what extent
G
C
complex
Chern-Simons theory can be thought of as two copies of Chern-Simons theory with compact
gauge group
G
.
CALT-TH-2015-013
arXiv:1503.04809v1 [hep-th] 16 Mar 2015
Contents
1 Introduction
1
2 Chern-Simons theory on
S
3
and free chiral multiplets
3
3 3d-3d correspondence for Lens spaces
7
3.1
M
SUSY
vs.
M
flat
8
3.2 Superconformal index
10
3.3
T
[
L
(
p,
1)] on
S
3
b
16
A Complex Chern-Simons theory on Lens spaces
21
1 Introduction
The 3d-3d correspondence is an elegant relation between 3-manifolds and three-dimensional
field theories [1–4]. The general spirit is that one can associate a 3-manifold
M
3
with a 3d
N
= 2 superconformal field theory
T
[
M
3
;
G
], obtained by compactifying the 6d (2,0) theory
on
M
3
6d (2,0) theory on
M
3
3d
N
= 2 theory
T
[
M
3
]
.
(1.1)
In this procedure, the 6d theory is topologically twisted along
M
3
to preserve
N
= 2 super-
symmetry. As a consequence, the 3d
N
= 2 theory
T
[
M
3
;
G
] only depends on the topology
of
M
3
and the simply-laced Lie algebra
g
= Lie
G
that labels the 6d theory. Although the
dictionary between the dynamics of
T
[
M
3
] and topological properties of
M
3
is incredibly rich
[1, 3–7] and only partially explored, there are two very fundamental relations between
M
3
and
T
[
M
3
]. Firstly, the moduli space of supersymmetric vacua of
T
[
M
3
;
G
] on
R
2
×
S
1
is
expected to be homeomorphic to the moduli space of flat
G
C
-connections on
M
3
:
M
SUSY
(
T
[
M
3
;
G
])
'M
flat
(
M
3
;
G
C
)
.
(1.2)
Second, the partition function of
T
[
M
3
] on Lens space
L
(
k,
1) should be equal to the partition
function of complex Chern-Simons theory on
M
3
at level
k
[7, 8]:
Z
T
[
M
3
;
G
]
[
L
(
k,
1)
b
] =
Z
(
k,σ
)
CS
[
M
3
;
G
C
]
.
(1.3)
– 1 –
The level of complex Chern-Simons theory has a real part
k
and an “imaginary part”
1
σ
, and
σ
is related to the squashing parameter
b
of Lens space
L
(
k,
1)
b
=
S
3
b
/
Z
k
by
σ
=
k
·
1
−
b
2
1 +
b
2
.
(1.4)
For
k
= 0,
L
(
k,
1) =
S
1
×
S
2
, and the equation (1.3) maps the superconformal index of
T
[
M
3
]
to partition function of complex Chern-Simons theory at level (0
,σ
) [4]
Index
T
[
M
3
;
G
]
(
q
) = Tr (
−
1)
F
q
E
+
j
3
2
=
Z
(0
,σ
)
CS
[
M
3
;
G
C
]
.
(1.5)
Despite its beauty and richness, the 3d-3d correspondence has been haunted by many
problems since its birth. For example, the theories
T
DGG
[
M
3
] originally proposed in [3] miss
many branches of flat connections and therefore fail even the most basic test (1.2). This
problem was revisited and partially corrected in [11]. As for (1.3) and (1.5), there is simply
no known proposal for
T
[
M
3
] associated to
any
M
3
that passes these stronger tests. Even
the very first example — and arguably the simplest example — of partition function in
Chern-Simons theory found in Witten’s seminal paper [12]
Z
CS
[
S
3
;
SU
(2)
,k
] =
√
2
k
+ 2
sin
(
π
k
+ 2
)
(1.6)
has yet to find its home in the world of 3d
N
= 2 theories.
In [13], a candidate for the 3d theory
T
[
L
(
p,
1)] was proposed and studied
2
:
T
[
L
(
p,
1);
G
] =
3d
N
= 2
G
super-Chern-Simons theory at level
p
+ adjoint chiral multiplet Φ
.
(1.7)
This theory was used to produce Verlinde formula, the partition function of Chern-Simons
theory on
S
1
×
Σ, along with its “complexification” — the “equivariant Verlinde formula”.
Therefore, one may wonder whether this theory could also give the correct partition function
of Chern-Simons theory on
S
3
in (1.6) and its complex analog:
Z
CS
[
S
3
;
SL
(2
,
C
)
,τ,
τ
] =
√
4
τ
τ
sin
(
2
π
τ
)
sin
(
2
π
τ
)
.
(1.8)
Here we have used holomorphic and anti-holomorphic coupling constants
τ
=
k
+
σ,
τ
=
k
−
σ.
(1.9)
Indeed, according to the general statement of the 3d-3d correspondence,
T
[
L
(
p,
1)] needs to
satisfy
Z
T
[
L
(
p,
1);
G
]
[
L
(
k,
1)
b
] =
Z
(
k,σ
)
CS
[
L
(
p,
1);
G
C
]
(1.10)
1
We use the quotation mark here because
σ
can be either purely imaginary or purely real as pointed out
in [9].
2
More precisely, this is the UV CFT that can flow to numerous different IR theories labelled by UV R-
charges of Φ. The IR theory relevant for the 3d-3d relation is given by
R
(Φ) = 2.
– 2 –
and
Index
T
[
L
(
p,
1);
G
]
(
q
) = Tr (
−
1)
F
q
E
+
j
3
2
=
Z
(0
,σ
)
CS
[
L
(
p,
1);
G
C
]
.
(1.11)
And if we take
p
= 1, the above relation states that the index of
T
[
S
3
] should give the
S
3
partition function of complex Chern-Simons theory. Even better, as there is a conjectured
duality [14, 15] relating this theory to free chiral multiplets, one should be able to obtain
(1.6) and (1.8) by simply computing the index of a free theory! This relation, summarized in
diagrammatic form below,
Chern-Simons
theory on
S
3
3d-3d
←→
Index of
T
[
S
3
]
duality
←→
free chiral
multiplets
(1.12)
will be the subject of section 2. We start section 2 by proving the duality (at the level of
superconformal index) in (1.12) for
G
=
U
(
N
) and then “rediscover” the
S
3
partition function
of
U
(
N
) Chern-Simons theory from the index of
N
free chiral multiplets. Then in section 3
we go beyond
p
= 1 and study theories
T
[
L
(
p,
1)] with higher
p
. We check that the index of
T
[
L
(
p,
1)] gives precisely the partition function of complex Chern-Simons theory on
L
(
p,
1) at
level
k
= 0. In addition, we discover that index of
T
[
L
(
p,
1)] has some interesting properties.
For example, when
p
is large,
Index
T
[
L
(
p,
1);
U
(
N
)]
= (2
N
−
1)!!
(1.13)
is a constant that only depends on the choice of the gauge group. In the rest of section 3, we
study
T
[
L
(
p,
1)] on
S
3
b
and use the 3d-3d correspondence to give predictions for the partition
function of complex Chern-Simons theory on
L
(
p,
1) at level
k
= 1.
2 Chern-Simons theory on
S
3
and free chiral multiplets
According to the proposal (1.7), the theory
T
[
S
3
] is
N
= 2 super-Chern-Simons theory at
level
p
= 1 with an adjoint chiral multiplet. If one takes the gauge group to be
SU
(2), this
theory was conjectured by Jafferis and Yin to be dual to a free
N
= 2 chiral multiplet [14].
The Jafferis-Yin duality has been generalized to higher rank groups by Kapustin, Kim and
Park [15]. For
G
=
U
(
N
), the statement of the duality is:
T
[
S
3
] =
U
(
N
)
1
super-Chern-Simons theory
+ adjoint chiral multiplet
duality
←→
N
free chiral
multiplets
.
(2.1)
In [13], a similar duality was discovered
3
:
T
[
L
(
p,
1)] =
U
(
N
)
p
super-Chern-Simons theory
+ adjoint chiral multiplet
duality
←→
sigma model to
vortex moduli space
V
N,p
.
(2.2)
3
In [13], the adjoint chiral is usually assumed to be massive, which introduces an interesting “equivariant
parameter”
β
. Here we are more concerned with the limit where that parameter is zero.
– 3 –
Here,
V
N,p
∼
=
{
(
q,φ
)
∣
∣
ζ
·
Id =
qq
†
+ [
φ,φ
†
]
}
/U
(
N
)
,
(2.3)
with
q
being an
N
×
p
matrix,
φ
an
N
×
N
matrix and
ζ
∈
R
+
the “size parameter”, was
conjectured to be the moduli space of
N
vortices in a
U
(
p
) gauge theory [16]. For
p
= 1, it
is a well known fact that (see, e.g. [17])
V
N,
1
'
Sym
N
(
C
)
'
C
N
.
(2.4)
This is already very close to proving that
T
[
L
(1
,
1);
U
(
N
)] =
T
[
S
3
;
U
(
N
)] is dual to
N
free
chirals, with only one missing step. In order to completely specify the sigma model, one also
needs to determine the metric on this space. A sigma model to
C
N
with the flat metric is
indeed a free theory, but it is not obvious that the metric on
V
N,
1
is flat
4
. However, as the
superconformal index of a sigma model only depends on topological properties of the target
space, one obtains that
index of
T
[
S
3
;
U
(
N
)] = index of
N
free chirals
,
(2.5)
proving the duality in (1.12) at the level of index. Combining (2.5) with the 3d-3d corre-
spondence, one concludes that the index of free chirals equals the
S
3
partition functions of
Chern-Simons theory. This is what we will explicitly demonstrate in this section.
Chern-Simons theory on the three-sphere
The partition function of
U
(
N
) Chern-Simons theory on
S
3
is
Z
CS
(
S
3
;
U
(
N
)
,k
)
=
1
(
k
+
N
)
N/
2
N
−
1
∏
j
=1
[
sin
πj
k
+
N
]
N
−
j
.
(2.6)
For
N
= 2, this gives back (1.6) for
SU
(2) (modulo a factor coming from the additional
U
(1)). It is convenience to introduce
q
=
e
2
πi
k
+
N
,
(2.7)
the variable commonly used for Jones polynomial, and express (2.6) as (mostly) a polynomial
in
q
1
/
2
and
q
−
1
/
2
:
Z
CS
(
S
3
;
U
(
N
)
,k
)
=
C
·
(ln
q
)
N/
2
N
−
1
∏
j
[
q
j/
2
−
q
−
j/
2
]
N
−
j
.
(2.8)
Here
C
is a normalization factor that does not depend on
q
and such factors will be dropped
in many later expressions without comments.
4
V
N,p
can be obtained using K ̈ahler reduction from
C
N
(
N
+
p
)
as in (2.3), and a K ̈ahler metric is also inherited
in this process. However, this metric on
V
N,p
is not protected from quantum corrections. The quantum metric
is yet unknown to the best of our knowledge, but for the JY-KKP duality to be true, it should flow to a flat
metric in the IR for
p
= 1 — a somewhat surprising prediction.
– 4 –
One can easily obtain the partition function for
GL
(
N,
C
) Chern-Simons theory by notic-
ing that it factorizes into two copies of (2.6) at level
k
1
=
τ/
2 and
k
2
=
τ/
2
Z
CS
(
S
3
;
GL
(
N,
C
)
)
= (ln
q
ln
q
)
N/
2
N
−
1
∏
j
=1
[
q
j/
2
−
q
−
j/
2
]
N
−
j
[
q
−
j/
2
−
q
j/
2
]
N
−
j
.
(2.9)
Here, in slight abusive use of notation (
cf.
(2.7))
q
=
e
4
πi
τ
,
q
=
e
4
πi
τ
.
(2.10)
Notice that the quantum shift of the level
k
→
k
+
N
in
U
(
N
) Chern-Simons theory is absent
in the complex theory [9, 18, 19]. Although (2.9) is almost a polynomial, it contains “ln
q
”
factors. So, at this stage, it is still somewhat mysterious how (2.9) can be obtained as the
index of any supersymmetric field theory.
In (2.9) the level is arbitrary and the
k
= 0 case is naturally related to superconformal
index of
T
[
S
3
] (1.11). For
k
= 0,
q
=
e
4
πi
σ
,
q
=
e
−
4
πi
σ
=
q
−
1
,
(2.11)
and
Z
(0
,σ
)
CS
(
S
3
;
GL
(
N,
C
)
)
= (ln
q
)
N
N
−
1
∏
j
=1
[
(1
−
q
j
)(1
−
q
−
j
)
]
N
−
j
.
(2.12)
This is the very expression that we want to reproduce from the index of free chiral multiplets.
Index of a free theory
The superconformal index of a 3d
N
= 2 free chiral multiplet only receives contributions from
the scalar component
X
, the fermionic component
ψ
and their
∂
+
derivatives. If we assume
the R-charge of
X
to be
r
, then the R-charge of
ψ
is 1
−
r
and the superconformal index of
this free chiral is given by
I
r
(
q
) =
∞
∏
j
=0
1
−
q
1
−
r/
2+
j
1
−
q
r/
2+
j
.
(2.13)
In the
j
-th factor of the expression above, the numerator comes from fermionic field
∂
j
ψ
while
the denominator comes from bosonic field
∂
j
X
. Here
q
is a fugacity variable that counts the
charge under
E
+
j
3
2
=
R
+
j
3
/
2 and it is the expectation of the 3d-3d correspondence [4] that
this
q
is mapped to the “
q
” in (2.12), which justifies our usage of the same notation for two
seemingly different variables. Now the only remaining problem is to decide what are the
R-charges for the
N
free chiral multiplets.
The UV description of theory
T
[
L
(
p,
1)] has an adjoint chiral multiplet Φ and in general
one has the freedom of choosing the R-charge of Φ. Different choices give different IR fix points
which form an interesting family of theories. As was argued in [13] using brane construction,
the natural choice — namely the choice that one should use for the 3d-3d correspondence —
– 5 –
is
R
(Φ) = 2. For example, in order to obtain the Verlinde formula, it is necessary to choose
R
(Φ) = 2 while other choices give closely related yet different formulae. As the
N
free chirals
in the dual of
T
[
S
3
;
U
(
N
)] is directly related to Tr Φ, Tr Φ
2
, . . . , Tr Φ
N
, the choice of their
R-charges should be
r
m
=
R
(
X
m
) = 2
m,
for
m
= 1
,
2
,...,N.
(2.14)
The index for this assignment of R-charges — out of the unitarity bound — contains negative
powers of
q
. However, this is not a problem at all because the UV R-charges are mixed with
the
U
(
N
) flavor symmetries, and
q
counts a combination of R- and flavor charges.
One interesting property of the index of a free chiral multiplet (2.13) is that it will vanish
due to numerator of the (
m
−
1)-th factor
1
−
q
m
−
r
m
/
2
= 0
.
(2.15)
However, there is a very natural way of regularizing it and obtaining a finite result. Namely,
we multiply the
q
-independent normalization coefficient (
r
m
/
2
−
m
)
−
1
to the whole expression
and turn the vanishing term above into
lim
r
m
→
2
m
1
−
q
m
−
r
m
/
2
r
m
/
2
−
m
= ln
q.
(2.16)
And this is exactly how the “ln
q
” factors on the Chern-Simons theory side arise. With this
regularization
I
2
m
(
q
) = ln
q
m
−
1
∏
j
=1
[(
1
−
q
−
j
)(
1
−
q
j
)]
,
(2.17)
and the 2
m
−
1 factors come from the fermionic fields
ψ
m
,
∂
ψ
m
,. . . ,
∂
2
m
−
2
ψ
m
. The contri-
bution of
∂
2
m
−
1+
l
ψ
m
will cancel with the bosonic field
∂
l
X
as they have the same quantum
number. The special log term comes from the field
∂
m
−
1
ψ
m
which has exactly
R
+ 2
j
3
= 0.
Then it is obvious that
Index
T
[
S
3
;
U
(
N
)]
=
N
∏
m
=1
I
2
m
(
q
) = (ln
q
)
N
N
−
1
∏
j
=1
[
(1
−
q
j
)(1
−
q
−
j
)
]
N
−
j
(2.18)
is exactly the partition function of complex Chern-Simons theory on
S
3
(2.12). For example,
if
N
= 1,
Index
T
[
S
3
;
U
(1)]
=
I
2
(
q
) = ln
q.
(2.19)
For
N
= 2,
Index
T
[
S
3
;
U
(2)]
=
I
2
(
q
)
·I
4
(
q
) = (ln
q
)
2
(1
−
q
−
1
)(1
−
q
)
.
(2.20)
To get the renowned
S
3
partition function of
SU
(2) Chern-Simons theory, we just need to
divide the
N
= 2 index by
N
= 1 index and take the square root:
√
Index
T
[
S
3
;
U
(2)]
Index
T
[
S
3
;
U
(1)]
=
√
I
4
(
q
) =
−
i
·
(ln
q
)
1
/
2
(
q
1
/
2
−
q
−
1
/
2
)
.
(2.21)
– 6 –
For compact gauge group
SU
(2), we substitute in
q
=
e
2
πi
k
+2
(2.22)
and up to an unimportant normalization factor, (2.21) is exactly
Z
CS
(
S
3
;
SU
(2)
,k
) =
√
2
k
+ 2
sin
π
k
+ 2
.
(2.23)
As almost anything in a free theory can be easily computed, one can go beyond index
and check the following relation
Z
N
free chirals
(
L
(
k,
1)
b
) =
Z
(
k,σ
)
CS
(
S
3
;
U
(
N
))
.
(2.24)
The left-hand side can be expressed as a product of double sine functions [20] and with
the right choice of R-charges it becomes exactly the right-hand side, given by (2.6). As
this computation is almost identical for what we did with index, we omit it here to avoid
repetition.
Before ending this section, we comment on deforming the relation (1.12). In the formu-
lation of
T
[
L
(
p,
1)] in (1.7), there is a manifest
U
(1) flavor symmetry that can be weakly
gauged to give an “equivariant parameter”
β
. And partition function of
T
[
L
(
p,
1);
β
] should
be related to
β
-deformed complex Chern-Simons theory studied in [13]:
Z
T
[
L
(
p,
1);
β
]
(
L
(
k,
1)) =
Z
β
-CS
(
L
(
p,
1);
k
)
.
(2.25)
When
p
= 1, if JY-KKP duality is true, this
U
(1) flavor symmetry is expected to be enhanced
to a
U
(
N
) flavor symmetry of
T
[
S
3
;
U
(
N
)] that is only visible in the dual description with
N
free chiral multiplets. Then one can deform
T
[
S
3
] by adding
N
equivariant parameters
β
1
,β
2
,...,β
N
. It is interesing to ask whether Chern-Simons theory on
S
3
naturally admits
such an
N
-parameter deformation and whether one can have a more general matching.
Index
T
[
S
3
]
(
q
;
β
1
,β
2
,...,β
N
) =
Z
CS
(
S
3
;
q,β
1
,β
2
,...,β
N
)
.
(2.26)
As Chern-Simons theory on
S
3
is dual to closed string on the resolved conifold [21, 22],
it would also be interesting to understand whether similar deformation of the closed string
amplitudes
F
g
exists.
3 3d-3d correspondence for Lens spaces
In the previous section, we focused on
T
[
S
3
] and found that it fits perfectly inside the 3d-3d
correspondence. This theory is the special
p
= 1 limit of a general class (1.7) of theories
T
[
L
(
p,
1)] proposed in [13]. In this section, we will test this proposal and see whether it
stands well with various predictions of the 3d-3d correspondence. There are several tests to
run on the proposed Lens space theories (1.7). The most basic one is the correspondence
between moduli spaces (1.2) that one can formulate classically without doing path integral:
M
SUSY
(
T
[
L
(
p,
1);
U
(
N
)])
'M
flat
(
L
(
p,
1);
GL
(
N,
C
))
.
(3.1)
And our first task in this section is to verify that this is indeed an equality.
– 7 –
3.1
M
SUSY
vs.
M
flat
The moduli space of flat
H
-connections on a three manifold
M
3
can be identified with the
character variety:
M
flat
(
M
3
;
H
)
'
Hom(
π
1
(
M
3
)
,H
)
/H.
(3.2)
As
π
1
(
L
(
p,
1)) =
Z
p
, this character variety is particularly simple. For example, if we take
H
=
U
(
N
) or
H
=
GL
(
N,
C
) — the choice between
U
(
N
) or
GL
(
N,
C
) does not even
matter — this space is a collection of points labelled by Young tableaux with size smaller
than
N
×
p
. This is in perfect harmony with the other side of the 3d-3d relation where the
supersymmetric vacua of
T
[
L
(
p,
1);
U
(
N
)] on
S
1
×
R
2
are also labelled by Young tableaux
with the same constraint [13]. We will now make this matching more explicit.
If we take the holonomy along the
S
1
Hopf fiber of
L
(
p,
1) to be
A
, then
M
flat
(
L
(
p,
1);
GL
(
N,
C
))
'{
A
∈
GL
(
N,
C
)
|
A
p
= Id
}
/GL
(
N,
C
)
.
(3.3)
First we can use the
GL
(
N,
C
) action to cast
A
into Jordan normal form. But in order to
satisfy
A
k
= Id,
A
has to be diagonal, and each of its diagonal entries
a
l
has to be one of the
p
-th roots of unity:
a
p
l
= 1
,
for all
l
= 1
,
2
,...,N
.
(3.4)
One can readily identify this set of equations with the
t
→
1 limit of the Bethe ansatz
equations that determine the supersymmetric vacua of
T
[
L
(
p,
1);
U
(
N
)] on
S
1
×
R
2
[13]:
e
2
πipσ
l
∏
m
6
=
l
(
e
2
πiσ
l
−
te
2
πiσ
m
te
2
πiσ
l
−
e
2
πiσ
m
)
= 1
,
for all of
l
= 1
,
2
,...,N.
(3.5)
For
t
= 1, this equation is simply
e
2
πipσ
l
= 1
,
for
l
= 1
,
2
,...,N
.
(3.6)
And this is exactly (3.4) if one makes the following identification
a
l
=
e
2
πiσ
l
.
(3.7)
Of course this relation between
a
l
and
σ
l
is more than just a convenient choice. It can be
derived using the brane construction of
T
[
L
(
p,
1)]. In fact, it just comes from the familiar
relation in string theory between holonomy along a circle and positions of D-branes after
T-duality. Indeed, in the above expression, the
a
l
’s on the left-hand side label the
U
(
N
)-
holonomy along the Hopf fiber, while the
σ
l
’s on the right-hand side are coordinates on the
Coulomb branch of
T
[
L
(
p,
1)] after reduction to 2d, which exactly correspond to positions of
N
D2-branes.
– 8 –
G
C
Chern-Simons theory from
G
Chern-Simons theory
The fact that
M
flat
is a collection of points is important for us to compute the partition
function of complex Chern-Simons theory. Although there have been many works on complex
Chern-Simons theory and its partition functions, starting from [9, 23] to perturbative invariant
in [18, 24], state integral models in [7, 25, 26] and mathematically rigorous treatment in [27–
29], what usually appear are certain subsectors of complex Chern-Simons theory, obtained
from some consistent truncation of the full theory. In general, the
full
partition function
of complex Chern-Simons theory is difficult to obtain, and requires proper normalization
to make sense of. Some progress has been made toward understanding the full theory on
Seifert manifolds in [13] using topologically twisted supersymmetric theories. However, if
M
flat
(
M
3
;
G
C
) is discrete and happen to be the same as
M
flat
(
M
3
;
G
), then one can attempt
to construct the full partition function of
G
C
Chern-Simons theory on
M
3
from
G
Chern-
Simons theory. The procedure is the following. One first writes the partition function of
G
Chern-Simons theory as a sum over flat connections:
Z
full
=
∑
α
∈M
Z
α
.
(3.8)
And because the action of
G
C
Chern-Simons theory
S
=
τ
8
π
∫
Tr
(
A∧
d
A
+
2
3
A∧A∧A
)
+
τ
8
π
∫
Tr
(
A∧
d
A
+
2
3
A∧
A∧
A
)
(3.9)
is simply two copies of
G
Chern-Simons theory action at level
k
1
=
τ/
2 and
k
2
=
τ/
2, one
would have
Z
α
(
G
C
;
τ,
τ
) =
Z
α
(
G
;
τ
2
)
Z
α
(
G
;
τ
2
)
,
(3.10)
if
A
and
A
were independent fields. So, one would naively expect
Z
full
(
G
C
;
τ,
τ
) =
∑
α
∈M
Z
α
(
G
;
τ
2
)
Z
α
(
G
;
τ
2
)
.
(3.11)
But as
A
and
A
are not truly independent, (3.11) is in general incorrect and one needs to
modify it in a number of ways. For example, as mentioned before, the quantum shift of the
level
τ
and
τ
in
G
C
Chern-Simons theory is zero, so for
Z
α
(
G
) on the right-hand side, one
needs to at least remove the quantum shift
k
→
k
+
ˇ
h
in
G
Chern-Simons theory, where
ˇ
h
is
the dual Coxeter number of
g
. There may be other effects that lead to relative coefficients
between contributions from different flat connections
α
and the best one could hope for is
Z
full
(
G
C
;
τ,
τ
) =
∑
α
∈M
e
iC
α
Z
′
α
(
G
;
τ
2
)
Z
′
α
(
G
;
τ
2
)
,
(3.12)
– 9 –
where
Z
′
α
(
G
;
τ
2
)
=
Z
α
(
G
;
τ
2
−
ˇ
h
)
.
(3.13)
One way to see that (3.11) is very tenuous, even after taking care of the level shift, is by
noticing that the left-hand side and the right-hand side behaves differently under change of
framing. If the framing of the three-manifold is changed by
s
unit, the left-hand side will pick
up a phase factor
exp
[
φ
fr.
C
·
s
]
= exp
[
πi
(
c
L
−
c
R
)
12
·
s
]
.
(3.14)
Here
c
L
and
c
R
are the left- and right-moving central charges of the hypothetical conformal
field theory that lives on the boundary of complex Chern-Simons theory [9]:
(
c
L
,c
R
) = dim
G
·
(
1
−
2
ˇ
h
τ
,
1 +
2
ˇ
h
τ
)
.
(3.15)
The right-hand side of (3.11) consists of two copies of Chern-Simons theory with compact
gauge group
G
, so the phase from change of framing is
exp
[
φ
fr.
·
s
]
= exp
[
πi
12
(
τ/
2
−
ˇ
h
τ/
2
+
τ/
2
−
ˇ
h
τ/
2
)
dim
G
·
s
]
.
(3.16)
The two phases are in general different
φ
fr.
C
−
φ
fr.
=
2
πi
dim
G
12
.
(3.17)
So (3.11) has no chance of being correct at all and the minimal way of improving it is to add
the phases
C
α
’s as in (3.12) that also transform under change of framing.
It may appears that the expression (3.12) is not useful unless one can find the value of
C
α
’s. However, as it turns out, for
k
= 0 (or equivalently
τ
=
−
τ
), all of the
C
α
’s are constant,
and (3.12) without the
C
α
’s gives the correct partition function
5
. This may be closely related
to the fact that for
k
= 0,
c
L
−
c
R
=
−
2
ˇ
h
dim
G
(
1
τ
+
1
τ
)
= 0
.
(3.18)
3.2 Superconformal index
We have shown that the proposal (1.7) for
T
[
L
(
p,
1)] gives the right supersymmetric vacua and
we shall now move to the quantum level and check the relation between partition functions:
Index
T
[
L
(
p,
1);
U
(
N
)]
(
q
) =
Z
CS
(
L
(
p,
1);
GL
(
N,
C
)
,q
)
.
(3.19)
We have already verified this for
p
= 1 in the previous section. Now we consider the more
general case with
p
≥
1.
5
“Correct” in the sense that it matches the index of
T
[
L
(
p,
1)].
– 10 –
The superconformal index of a 3d
N
= 2 SCFT is given by [10]
I
(
q,t
i
) = Tr
[
(
−
1)
F
e
−
γ
(
E
−
R
−
j
3
)
q
E
+
j
3
2
t
f
i
]
.
(3.20)
Here, the trace is taken over the Hilbert space of the theory on
R
×
S
2
. Because of super-
symmetry, only BPS states with
E
−
R
−
j
3
= 0
(3.21)
will contribute. As a consequence, the index is independent of
γ
and only depends on
q
and
the flavor fugacities
t
i
’s. For
T
[
L
(
p,
1)], there is always a
U
(1) flavor symmetry and we can
introduce at least one parameter
t
. When this parameter is turned on, on the other side of the
3d-3d correspondence, complex Chern-Simons theory will become “deformed complex Chern-
Simons theory”. This deformed version of Chern-Simons theory was studied on geometry
Σ
×
S
1
in [13] and will be studied on more general Seifert manifolds in [30]. However, because
in this paper, our goal is to
test
the 3d-3d relation (as opposed to using it to study deformed
Chern-Simons theory), we will usually turn off this parameter by setting
t
= 1, and compare
the index
I
(
q
) with the partition function of the
undeformed
Chern-Simons theory, which is
only a function of
q
, as in (2.12).
Viewing the index as the partition function on
S
1
×
q
S
2
and using localization, (3.20)
can be expressed as an integral over the Cartan
T
of the gauge group
G
[31]:
I
=
1
|W|
∑
m
∫
∏
j
dz
j
2
πiz
j
e
−
S
CS
(
m
)
q
0
/
2
e
ib
0
(
h
)
t
f
0
exp
[
+
∞
∑
n
=1
1
n
Ind(
z
n
j
,m
j
;
t
n
,q
n
)
]
.
(3.22)
Here
h,m
∈
t
are valued in the Cartan subalgebra. Physically,
e
ih
is the holonomy along
S
1
and is parametrized by
z
i
’s, which are coordinates on
T
.
m
=
i
2
π
∫
S
2
F
(3.23)
is the monopole number on
S
2
and take values in the weight lattice of the Langlands dual
group
L
G
.
|W|
is the order of the Weyl group and the other quantities are:
b
0
(
h
) =
−
1
2
∑
ρ
∈
R
Φ
|
ρ
(
m
)
|
ρ
(
h
)
,
f
0
=
−
1
2
∑
ρ
∈
R
Φ
|
ρ
(
m
)
|
f,
0
=
1
2
∑
ρ
∈
R
Φ
(1
−
r
)
|
ρ
(
m
)
|−
1
2
∑
α
∈
ad(
G
)
|
α
(
m
)
|
,
S
CS
=
ip
tr(
mh
)
,
(3.24)
– 11 –
and
Ind(
e
ih
j
=
z
j
,m
j
;
t
;
q
) =
−
∑
α
∈
ad(
G
)
e
iα
(
h
)
q
|
α
(
m
)
|
+
∑
ρ
∈
R
Φ
[
e
iρ
(
h
)
t
q
|
ρ
(
m
)
|
/
2+
r/
2
1
−
q
−
e
−
iρ
(
h
)
t
−
1
q
|
ρ
(
m
)
|
/
2+1
−
r/
2
1
−
q
]
(3.25)
is the “single particle” index.
R
Φ
is the gauge group representation for all matter fields.
Using this general expression, the index of
T
[
L
(
p,
1);
U
(
N
)] can be expressed in the following
form
I
(
q,t
) =
∑
m
1
>
···
>
m
N
∈
Z
1
|W
m
|
∫
∏
j
dz
j
2
πiz
j
N
∏
i
(
z
i
)
pm
i
N
∏
i
6
=
j
t
−|
m
i
−
m
j
|
/
2
q
−
R
|
m
i
−
m
j
|
/
4
(
1
−
q
|
m
i
−
m
j
|
/
4
z
i
z
j
)
N
∏
i
6
=
j
(
z
j
z
i
t
−
1
q
|
m
i
−
m
j
|
/
2+1
−
R/
2
;
q
)
∞
(
z
i
z
j
tq
|
m
i
−
m
j
|
/
2+
R/
2
;
q
)
∞
×
[
(
t
−
1
q
1
−
R/
2
;
q
)
∞
(
tq
R/
2
;
q
)
∞
]
N
.
(3.26)
Here we used the
q
-Pochhammer symbol (
z
;
q
)
n
=
∏
n
−
1
j
=0
(1
−
zq
j
),
W
m
⊂W
is the stabilizer
subgroup of the Weyl group that fix
m
∈
t
and
R
stands for the R-charge of the adjoint chiral
multiplet and will be set to
R
= 2 — the choice that gives the correct IR theory.
In the previous section, we have found the index for
T
[
S
3
] to be exactly equal to the
S
3
partition function of Chern-Simons theory. There, we used an entirely different method by
working with the dual description of
T
[
L
(
p,
1);
U
(
N
)], which is a sigma model to the vortex
moduli space
V
N,p
. For
p
= 1, this moduli space is topologically
C
N
and the index of the sigma
model is just that of a free theory. For
p
≥
2, such simplification will not occur and the index
of the sigma model is much harder to compute
6
. In contrast, the integral expression (3.26)
is
easier
to compute with larger
p
than with
p
= 1, because fewer topological sectors labelled
by the monopole number
m
contribute. As we will see later, when
p
is sufficiently large,
only the sector
m
= (0
,
0
,...,
0) gives non-vanishing contribution. So the two approaches of
computing the index have their individual strengths and are complementary to each other.
Now, one can readily compute the index for any
T
[
L
(
p,
1);
G
] and then compare
I
(
q,t
= 1)
with the partition function of complex Chern-Simons theory on
L
(
p,
1). We will first do a
simple example with
G
=
SU
(2), to illustrate some general features of the index computation.
6
In general, it can be written as an integral of a characteristic class over
V
N,p
that one can evaluate using
Atiyah-Bott localization formula. Similar computations were done in two dimensions in,
e.g.
[1] and [32].
– 12 –
Index of
T
[
L
(
p,
1);
SU
(2)]
We will start with
p
= 1 and see how the answer from section 2 arises from the integral
expression (3.26). In this case, (3.26) becomes:
I
=
∑
m
∈
Z
∫
dz
4
πiz
e
ihm
q
−
2
|
m
|
(
1
−
q
|
m
|
e
ih
)
2
(
1
−
q
|
m
|
e
−
ih
)
2
+
∞
∏
k
=0
1
−
q
k
+1
−
R/
2
1
−
q
k
+
R/
2
=
∑
m
∈
Z
∫
dz
4
πiz
z
m
q
−
2
|
m
|
(
1 +
q
2
|
m
|
−
zq
|
m
|
−
z
−
1
q
|
m
|
)
2
[(
R
−
2) ln
q
]
=
∑
m
∈
Z
∫
dz
4
πiz
z
m
(
q
2
|
m
|
+
q
−
2
|
m
|
+ 4
−
2
(
z
+
1
z
)(
q
|
m
|
+
1
q
|
m
|
)
+
(
z
2
+
1
z
2
))
×
[(
R/
2
−
1) ln
q
]
.
(3.27)
As in section 2, the index will be zero if we naively take
R
= 2 because of the 1
−
q
1
−
r/
2
factor in the infinite product. When
R
→
2, the zero factor becomes
1
−
q
1
−
R/
2
= 1
−
exp [(1
−
R/
2) ln
q
]
≈
(
R/
2
−
1) ln
q.
(3.28)
As in section 2, we can introduce a normalization factor (
R/
2
−
1)
−
1
in the index to cancel
the zero, making the index expression finite.
The integral in (3.27) is very easy to do and the index receives contributions from three
different monopole number sectors
I
=
1
2
ln
q
(
I
m
=0
+
I
m
=
±
1
+
I
m
=
±
2
)
,
(3.29)
with
I
m
=0
=
∫
dz
2
πiz
(
q
0
+
q
−
0
+ 4
)
= 6
,
(3.30)
I
m
=
±
1
=
−
2
∑
m
=
±
1
∫
dz
2
πiz
z
m
(
q
|
m
|
+
q
−|
m
|
)
(
z
+
1
z
)
=
−
4(
q
+
q
−
1
)
,
(3.31)
and
I
m
=
±
2
=
∑
m
=
±
2
∫
dz
2
πiz
z
m
(
z
2
+
1
z
2
)
= 2
.
(3.32)
So the index is
I
=
1
2
ln
q
(
6
−
4(
q
+
q
−
1
) + 2
)
=
−
2 ln
q
(
q
1
/
2
−
q
−
1
/
2
)
2
.
(3.33)
Modulo a normalization constant, this is in perfect agreement with results in section 2. Indeed,
the square root of (3.33) is identical to (2.21) and reproduces the
S
3
partition function of
SU
(2) Chern-Simons theory
Z
CS
(
S
3
;
SU
(2)
,k
) =
√
2
k
+ 2
sin
π
k
+ 2
,
(3.34)
– 13 –
once we set
q
=
e
2
πi
k
+2
.
(3.35)
It is very easy to generalize the result (3.33) to arbitrary
p
. For general
p
, the index is
given by
I
=
1
2
ln
q
∑
m
∈
Z
∫
dz
2
πiz
z
pm
×
(
q
2
|
m
|
+
q
−
2
|
m
|
+ 4
−
2
(
q
|
m
|
+
q
−|
m
|
)
(
z
+
1
z
)
+
(
z
2
+
1
z
2
))
.
(3.36)
And the only effect of
p
is to select monopole numbers that contribute. For example, if
p
= 2,
only
m
= 0 and
m
=
±
1 contribute to the index and we have
I
p
=2
=
1
2
ln
q
(
I
m
=0
+
I
p
=2
m
=
±
1
) =
1
2
ln
q
(6 + 2) = 4 ln
q.
(3.37)
If
p >
2, only the trivial sector is selected, and
I
(
p >
2) =
1
2
ln
q
I
m
=0
= 3 ln
q.
(3.38)
This is a general feature of indices of the “Lens space theory” and we will soon encounter this
phenomenon with higher rank gauge groups.
The test for 3d-3d correspondence
We list the index of
T
[
L
(
p,
1);
U
(
N
)], obtained using
Mathematica
, in table 1. Due to lim-
itation of space and computational power, it contains results up to
N
= 5 and
p
= 6. The
omnipresent (ln
q
)
N
factors are dropped to avoid clutter, and after this every entry in table 1
is a Laurent polynomial in
q
with integer coefficients. Also, when the gauge group is
U
(
N
),
monopole number sectors are labeled by
N
-tuple of integers
m
= (
m
1
,m
2
,...,m
N
) and a
given sector can only contribute to the index if
∑
m
i
= 0.
From the table, one may be able to recognize the large
p
behavior for
U
(3) and
U
(4)
similar to (3.37) and (3.38). Indeed, it is a general feature of the index
I
T
[
L
(
p,
1);
U
(
N
)]
that
fewer monopole number sectors contribute when
p
increases. In order for a monopole number
m
= (
m
1
,...,m
N
) to contribute,
|
pm
i
|≤
2
N
−
2
(3.39)
needs to be satisfied for all
m
i
. For large
p >
2
N
−
2,
I
only receives contribution from the
m
= 0 sector and becomes a constant:
I
(
U
(
N
)
,p >
2
N
−
2) =
I
m
=(0
,
0
,
0
,...,
0)
= (2
N
−
1)!!
.
(3.40)
For
p
= 2
N
−
2, the index receives contributions from two sectors
7
:
I
(
U
(
N
)
,p
= 2
N
−
2) =
I
m
=(0
,
0
,
0
,...,
0)
+
I
m
=(1
,
0
,...,
0
,
−
1)
= [(2
N
−
1)!! + (2
N
−
5)!!]
.
(3.41)
7
Here, double factorial of a negative number is taken to be 1.
– 14 –
p
= 1
p
= 2
p
= 3
p
= 4
p
= 5
p
= 6
U
(2)
2(1
−
q
)(1
−
q
−
1
)
4
3
3
3
3
U
(3)
6(1
−
q
)
2
(1
−
q
2
)
(1
−
q
−
1
)
2
(1
−
q
−
2
)
28
−
6
q
−
2
−
8
q
−
1
−
8
q
−
6
q
2
23 + 2
q
−
1
+ 2
q
16
15
15
U
(4)
24(1
−
q
)
3
(1
−
q
2
)
2
(1
−
q
3
)(1
−
q
−
1
)
3
(1
−
q
−
2
)
2
(1
−
q
−
3
)
504+
84
q
−
4
−
96
q
−
3
−
80
q
−
2
−
160
q
−
1
−
160
q
−
80
q
2
−
96
q
3
+ 84
q
4
204
−
30
q
−
3
−
48
q
−
2
−
24
q
−
1
−
24
q
−
48
q
2
−
30
q
3
188 + 10
q
−
2
+24
q
−
1
+ 24
q
+10
q
2
121+
2
q
−
1
+ 2
q
108
U
(5)
120(1
−
q
)
4
(1
−
q
2
)
3
(1
−
q
3
)
2
(1
−
q
4
)
(1
−
q
−
1
)
4
(1
−
q
−
2
)
3
(1
−
q
−
3
)
2
(1
−
q
−
4
)
12336+
120
q
−
10
+ 192
q
−
9
−
1080
q
−
8
+ 48
q
−
7
+120
q
−
6
+ 3792
q
−
5
−
2016
q
−
4
−
1296
q
−
3
−
3312
q
−
2
−
2736
q
−
1
−
2736
q
−
3312
q
2
−
1296
q
3
−
2016
q
4
+3792
q
5
+ 120
q
6
+48
q
7
−
1080
q
8
+192
q
9
+ 120
q
10
3988+
180
q
−
6
+ 388
q
−
5
−
294
q
−
4
−
932
q
−
3
−
584
q
−
2
−
752
q
−
1
−
752
q
−
584
q
2
−
932
q
3
−
294
q
4
+388
q
5
+ 180
q
6
2144
−
240
q
−
4
−
320
q
−
3
−
320
q
−
2
−
192
q
−
1
−
192
q
−
320
q
2
−
320
q
3
−
240
q
4
1897+
70
q
−
3
+ 192
q
−
2
352
q
−
1
+ 352
q
+192
q
2
+ 70
q
3
1188+
14
q
−
2
+ 40
q
−
1
40
q
+ 14
q
2
Table 1
. The superconformal index of the “Lens space theory”
T
[
L
(
p,
1)
,U
(
N
)], which agrees with the partition
function of
GL
(
N,
C
) Chern-Simons theory at level
k
= 0 on Lens space
L
(
p,
1).
While the ln
q
factors (that we have omitted) are artifact of our scheme of removing zeros in
I
, the constant coefficient (2
N
−
1)!! in (3.40) is counting BPS states. Then one can ask a
series of questions: 1) what are the states or local operators that are being counted? 2) and
why the number of such operators are independent of
p
when
p
is large?
Partition functions
Z
CS
of complex Chern-Simons theory on Lens spaces can also be
computed systematically. Please see appendix A for details of the method we use. For
k
= 0,
G
C
=
GL
(
N,
C
), the partition functions on
L
(
p,
1) only depend on
q
=
e
4
πi/τ
as
q
=
e
4
πi/
τ
=
q
−
1
. After dropping a (ln
q
)
N
factor as in the index case, it is again a polynomial.
We have computed this partition function up to
N
= 5 and
p
= 6 and found a perfect
agreement with the index in table 1.
From the point of view of complex Chern-Simons theory, this large
p
behavior (3.40) seems
– 15 –
to be even more surprising — it predicts that the partition functions of complex Chern-Simons
theory on
L
(
p,
1) at level
k
= 0 are constant when
p
is greater than twice the rank of gauge
group. One can then ask 1) why is this happening? and 2) what is the geometric meaning of
this (2
N
−
1)!! constant?
3.3
T
[
L
(
p,
1)]
on
S
3
b
In previous sections, we have seen that the superconformal index of
T
[
L
(
p,
1)] agrees com-
pletely with the partition function of complex Chern-Simons theory at level
k
= 0 given by
(3.12) with trivial relative phases
C
α
= 0
Z
(
G
C
;
τ,
τ
) =
∑
α
∈M
Z
′
α
(
G
;
τ
2
)
Z
′
α
(
G
;
τ
2
)
,
(3.42)
for
G
=
U
(
N
). But for more general
k
, one can no longer expect this to be true. We will
now consider
S
3
b
partition function of
T
[
L
(
p,
1)], which will give partition function of complex
Chern-Simons theory at level [8]
(
k,σ
) =
(
1
,
1
−
b
2
1 +
b
2
)
.
(3.43)
And we will examine for what choices of
N
and
p
, setting all phases
C
α
= 0 becomes a mistake,
by comparing the
S
3
b
partition function of
T
[
L
(
p,
1)] to the “naive” partition function (3.42)
of complex Chern-Simons theory at level
k
= 1 on
L
(
p,
1).
There are two kinds of squashed three-spheres breaking the
SO
(4) isometry of the round
S
3
: the first one preserves
SU
(2)
×
U
(1) isometry while the second one preserves
U
(1)
×
U
(1)
[33]. However, despite the geometry being different, the partition functions of 3d
N
= 2
theories that one gets are the same [33–36]. In fact, as was shown in [37], three-sphere
partition functions of
N
= 2 theories only admit a one-parameter deformation. We will
choose the “ellipsoid” geometry with the metric
ds
2
3
=
f
(
θ
)
2
dθ
2
+ cos
2
θdφ
2
1
+
1
b
4
sin
2
θdφ
2
2
,
(3.44)
where
f
(
θ
) is arbitrary and does not affect the partition function of the supersymmetric
theory.
Using localization, partition function of a
N
= 2 gauge theory on such ellipsoid can
be written as an integral over the Cartan of the gauge group [33, 35]. Consider an
N
= 2
Chern-Simons-matter theory with gauge group being
U
(
N
). A classical Chern-Simons term
with level
k
contributes
Z
CS
= exp
(
i
b
2
k
4
π
N
∑
i
=1
λ
2
i
)
(3.45)
to the integrand. The one-loop determinant of
U
(
N
) vector multiplet, combined with the
Vandermonde determinant, gives
Z
gauge
=
N
∏
i<j
(
2 sinh
λ
i
−
λ
j
2
)(
2 sinh
λ
i
−
λ
j
2
b
2
)
.
(3.46)
– 16 –