3d superconformal indices and isomorphisms of M2-brane theories

0.24 MB·English

by  Masazumi Honda

PreprinttypesetinJHEPstyle-PAPERVERSION KEK-TH-1576
HRI/ST/1205
3d superconformal indices and isomorphisms
2 of M2-brane theories
1
0
2

t
c
O
9 Masazumi Hondaa,b and Yoshinori Honmab,c

]
h a
Department of Particle and Nuclear Physics,
t
- Graduate University for Advanced Studies (SOKENDAI),
p
e Tsukuba, Ibaraki 305-0801, Japan
h bHigh Energy Accelerator Research Organization (KEK),
[
Tsukuba, Ibaraki 305-0801, Japan

2 cHarish-Chandra Research Institute,
v
Chhatnag Road, Jhusi, Allahabad 211019, India
1
7
3
1
[email protected], [email protected]
.
0
1
2
1 Abstract: We test several expected isomorphisms between the U(N) U(N) ABJM
×
v: theory and (SU(N) SU(N))/ZN theory including the BLG theory by comparing their
×
i
X superconformal indices. From moduli space analysis, it is expected that this equivalence
r can holdif and only ifthe rankN andChern-Simonslevel k arecoprime. We also calculate
a
the index of the ABJ theory and investigate whether some theories with identical moduli
spaces are isomorphic or not .
Keywords: Supersymmetric gauge theory, Chern-Simons Theories, M-theory.
Contents
1. Introduction 1
2. Dual photon and moduli space of vacua 3
3. Magnetic charge and charge quantization condition 5
4. The superconformal index 6
4.1 U(N ) U(N ) ABJ theory 7
1 k 2 −k
×
4.2 SU(N) SU(N) theory 10
k −k
×
5. Test of the conjectured isomorphisms 10
5.1 N = 2 11
5.2 N = 3 13
6. Search for a possibility of extended isomorphism with higher k 14
6.1 U(2+l) U(2) ABJ theory v.s. (SU(2) SU(2) )/Z BLG theory 14
k −k k2 −k2 2
× ×
6.2 U(2+l) U(2) (0 l k ) ABJ theory v.s. SU(2) SU(2)
k −k k2/2 −k2/2
× ≤ ≤ | | ×
BLG theory 15
7. Conclusions 16
A. partition function of SU(2) SU(2) ABJM theory 17
k −k
×
B. Full result 18
B.1 U(2) U(2) ABJM theory 18
×
B.2 (SU(2) SU(2))/Z BLG theory 19
2
×
B.3 U(3) U(3) ABJM theory 20
×
B.4 (SU(3) SU(3))/Z theory 22
3
×
B.5 U(3) U(2) ABJ theory 25
×
B.6 U(4) U(2) ABJ theory 26
×
1. Introduction
Low-energy limit of N coincident M2-branes on the orbifold C4/Z is captured by the
k
3d = 6 superconformal Chern-Simons-matter (ABJM) theory with the gauge group
N
U(N) U(N) [1] (see also [2]). We can take a large N limit of the ABJM theory
k −k
×
by using t’Hooft coupling λ = N/k and this theory still provides fruitful developments in
AdS /CFT correspondence. Meanwhile the BLG theory [3, 4] based on the Lie 3-algebra
4 3
1
[Xa,Xb,Xc] = fabcXd can also lead us to an another description of multiple M2-branes.
d
If we take the structure constant fabcd to be totally anti-symmetric, then the BLG theory
generically has manifest = 8 supersymmetry and SO(8) R-symmetry. In spite of such
R
N
successful structures, it is known that the only nontrivial solution for a generalized Jacobi
identity is the A algebra defined by fabcd = ǫabcd [5, 6] and the resulting A BLG theory
4 4
can be rewritten as the SU(2) SU(2) ABJM theory [7]. Actually moduli space analysis
×
of this theory [8, 9] implies that the interpretation as two indistinguishable M2-branes on
C4/Z can be possible only for k = 1 and k = 2. Therefore the role of the A BLG theory
k 4
with higher k has been somewhat unclear.
In[10],anilluminatinganswerhasbeenobtainedbyconsidering(SU(2) SU(2) )/Z
k −k 2
×
rather than SU(2) SU(2) as thecorrect gauge group. Theauthorshas concludedthat
k −k
×
there are several isomorphisms even in the quantum level between
U(2) U(2) ABJM and Z quotient of (SU(2) SU(2) )/Z BLG theory
k −k k k −k 2
× ×
(1.1)
where k is odd. For k = 2, isomorphism between
U(2) U(2) ABJM and SU(2) SU(2) BLG theory
2 −2 2 −2
× ×
(1.2)
hasbeenalsoconjectured. Aswewillseeinthenextsection,theadditionalZ identification
k
in (1.1) is coming from the U(1) baryon symmetry of (SU(2) SU(2))/Z theory. More
B 2
×
generally, they also proposed that the conjecture (1.1) can be extended to arbitrary rank
N as
U(N) U(N) ABJM and Z quotient of (SU(N) SU(N) )/Z theory
k −k k k −k N
× ×
(1.3)
where k and N are coprime.
The conjectures (1.1) for k = 1 and (1.2) have been already tested by comparing the
superconformal indices [11, 12, 13, 14] obtained by applying the localization method [15]
(see also [16]) and actually nontrivial coincidences have been observed [17]. Furthermore,
it has been also found in [17] that the superconformal indices of
U(3) U(2) ABJ theory and (SU(2) SU(2) )/Z BLG theory
2 −2 4 −4 2
× ×
(1.4)
agree with each other. Thus, the nontrivial tests1 beyond the moduli space analysis have
been already performed for the isomorphisms between the ABJ(M) theories with the non-
trivial = 8 SUSY enhancements and the corresponding BLG theories2.
N
1In Appendix A, we provide a further evidence for the conjecture (1.2) by calculating the partition
function on S3.
2One might be curious about U(3)1×U(2)−1 and U(4)2×U(2)−2 ABJ theories. However it is widely
believed that these theories are dualto theU(2)1×U(2)−1 and U(2)2×U(2)−2 ABJ theories from parity
duality, respectively [2]. For a proof by considering thepartition functions on S3, see [18, 19].
2
In this paper, we test the conjecture (1.1) for the case without = 8 SUSY enhance-
N
ments by comparing their superconformal indices. We also check the conjecture (1.3) for
N = 3 and investigate whether extensions of the isomorphisms (1.2) and (1.4) to higher
k are possible or not. This paper is organized as follows. In Section 2, we review the
argument of [10] about the isomorphism (1.3). In Section 3, we briefly look at the charge
quantization condition of the (SU(N) SU(N) )/Z theories. In section 4, we describe
k −k N
×
our calculation of the superconformal indices. In Section 5, we show our results for the
indices and test the conjecture (1.3). In Section 6, we investigate a possibility where the
isomorphisms (1.2) and (1.4) are extended to higher k. Section 7 is devoted to conclusions
and discussions.
2. Dual photon and moduli space of vacua
In this section, we review arguments of [10] about the conjectured isomorphism (1.3). This
canbededucedfromintegratingouttheU(1)fieldorcomparingtheclassicalmodulispaces.
The Lagrangian of the U(N) U(N) ABJM theory can be expressed [10] as
k −k
×
Nk
= gauged + ǫµνλB H , (2.1)
Lu(N)⊕u(N) Lsu(N)⊕su(N) 8π µ νλ
where B is the gauge field of the U(1) baryon symmetry. H is the field strength of the
µ B µν
trivial U(1), which does not couple to all the fields in the SU(N) SU(N) theory. The
k −k
×
second term is the so-called BF term, which is required to make the theory invariant under
the = 6 supersymmetry after gauging the U(1) symmetry. Introducing the Lagrange
B
N
multiplier σ leads to
Nk N
= gauged + ǫµνλB H + σǫµνλ∂ H . (2.2)
Lu(N)⊕u(N) Lsu(N)⊕su(N) 8π µ νλ 8π µ νλ
Integrating this by parts, we obtain
Nk N
= gauged + ǫµνλB H ǫµνλ∂ σH . (2.3)
Lu(N)⊕u(N) Lsu(N)⊕su(N) 8π µ νλ− 8π µ νλ
Then, the equation of motion for H is
µν
1
B = ∂ σ. (2.4)
µ µ
k
From this equation, we find
u(N)⊕u(N)(ZA,ψA,Bµ,Hµνλ) = su(N)⊕su(N)(ekiσZA,ekiσψA), (A = 1, ,4) (2.5)
L L ···
where the U(1) gauge transformation B B +∂ θ in the language of σ is given by
B µ µ µ
→
σ σ+kθ. (2.6)
→
The last term of (2.2) implies that the periodicity of σ is determined by the charge
quantization condition of H . Note that the charge quantization condition is different
µν
3
from the usual Dirac quantization condition since U(N) is not just a product of U(1) and
SU(N) but rather it is (U(1) SU(N))/Z . Recall that H is a sum of a field strength of
N
×
each U(1) factor of U(N) U(N) gauge group. Finally the condition is given by
×
1 4π
dH = ǫµνλ∂ H Z, (2.7)
µ νλ
2 ∈ N
Z Z
which leads the periodicity of σ to 2π. Thus, we must impose the following identification
on the fields
ZˆA e2kπiZˆA, ψˆA e2kπiψˆA. (2.8)
∼ ∼
where we define ZˆA and ψˆA as ZˆA = ekiσZA and ψˆA = ekiσψA, respectively. From this
fact, the authors of [10] have concluded that the U(N) U(N) ABJM theory is also
k −k
×
equivalent to a Z identification on the (SU(N) SU(N) )/Z theory. As we will see
k k −k N
×
later, this equivalence can hold if we impose an additional constraint on N and k.
Next we consider the moduli space of the (SU(2) SU(2) )/Z theory with the Z
k −k 2 k
×
identification and check the above result. Discussion for generalization to arbitrary rank
N is essentially the same [10]. Setting the scalar potential to be zero, we can take ZA up
to gauge transformation as
1 i
ZA = rA rAσ , (2.9)
√2 1 − √2 2 3
where rA and rA are complex numbers. These can be regarded as the center of mass coor-
1 2
dinate and the relative coordinate of two M2-branes, respectively. For a later convenience,
we take
1 i
rA = (zA +zA), rA = (zA zA). (2.10)
1 2 1 2 2 2 1 − 2
Recall that the moduli space of SU(2) SU(2) theory is shown to be the orbifold (C4
× ×
C4)/D [8, 9]. Here D is the dihedral group of order 2n, which is equivalent to the
2k n
semi-direct product of Z and Z with Z acting on Z by inversion. Except for the
n 2 2 n
modification (2.7) of the charge quantization condition and the Z identification (2.8), the
k
same argument holds also in the present case. Thus, we can show that the moduli space
of the (SU(2) SU(2))/Z theory with the Z identification is given by
2 k
×
ZUk(1) : z1A ∼ e2kπiz1A, z2A ∼ e2kπiz2A,
Zperm : zA zA,
2 1 ∼ 2
Zk(SU(2)×SU(2))/Z2 : z1A ∼ e2kπiz1A, z2A ∼ e−2kπiz2A, (2.11)
where ZU(1) is the Z identification (2.8) imposed to zA,zA and Zperm is a permutation of
k k 1 2 2
two indistinguishable M2-branes. Note that the last one is slightly different from the one
of the SU(2) SU(2) theory given by
×
Z2SkU(2)×SU(2) : z1A ∼ eπkiz1A, z2A ∼ e−πkiz2A. (2.12)
4
The difference is comes from the modified charge quantization condition (2.7). As we
mentioned above, Zperm and Z(SU(2)×SU(2))/Z2 identifications generate the dihedral group
2 k
D .
k
The moduli space of U(2) U(2) ABJM theory is
×
(R8/Z ) (R8/Z )
k k
× , (2.13)
Z
2
which is given by the following quotient
Z(k1) : z1A ∼ e2kπiz1A, z2A ∼ z2A,
Zperm : zA zA,
2 1 ∼ 2
Z(k2) : z1A ∼ z1A, z2A ∼ e−2kπiz2A. (2.14)
Here Z(1) and Z(2) are the Z identifications of each M2-brane. We can easily show that
k k k
(2.11) and (2.14) are equalwitheach other if andonlyif k is odd3. For arbitrary N, similar
discussionleads usto theconclusion thattheU(N) U(N) ABJMtheoryis isomorphic
k −k
×
to the Z quotient of the (SU(N) SU(N) )/Z theory if N and k are coprime [10] .
k k −k N
×
3. Magnetic charge and charge quantization condition
Here we briefly look at the charge quantization condition of the (SU(N) SU(N) )/Z
k −k N
×
theories. ThefullglobalsymmetryoftheU(N ) U(N )ABJ(M)theoryisSO(6) U(1) .
1 2 R T
× ×
Here U(1) is the topological symmetry of the ABJ(M) theory whose conserved current is
T
given by4
1
Jµ = ǫµνρ TrF +TrF˜ . (3.1)
νρ νρ
16π
(cid:16) (cid:17)
F and F˜ are the field strengths of U(N ) and U(N ) gauge fields, respectively. The
1 2
operators carrying the U(1) charge are called monopole operators [20, 21] and involve a
T
non-zero magnetic flux in the diagonal U(1) gauge group. The monopole operators can be
labeled by the GNO charges n , ,n and n˜ , ,n˜ which are the monopole charges
1 ··· N1 1 ··· N2
for the Cartan part of the gauge group U(N ) U(N ) [22]. The GNO charges label the
1 2
×
magnetic flux on S2 surrounding the insertion point of the operator and their summation
corresponds to the U(1) charge as
T
k N1 N2
Q = n + n˜ . (3.2)
T i a
4
!
i=1 a=1
X X
The equations of motion for the gauge fields set TrF TrF˜ = 0 and therefore n =
− i i
n˜ . Thus, the U(1) charge can be expressed as
a a T P
P k N1 N2
Q = T with T = n = n˜ . (3.3)
T i a
2
i=1 a=1
X X
3If we parametrize k = 2l−1, (cid:0)ZU(1) ×Z(SU(2)×SU(2))/Z2(cid:1)l and (cid:0)(ZU(1))−1 ×Z(SU(2)×SU(2))/Z2(cid:1)l are
k k k k
equal to Z(1) and Z(2), respectively.
k k
4Weuse thesame notation as in [17].
5
Let us denote w (i = 1, ,rank(G)) as the weight vector of the gauge group G in
i
···
an irreducible representation and Λ(G) as the weight lattice. The quantization condition
imposes exp(ie n w )= 1 and this implies that
i i i
P e n w = 2πZ (3.4)
i i
i
X
forallw Λ(G)5. HereweconsidertheSO(3) = SU(2)/Z andSU(2)asasimpleexample
2
∈
of dual groups 6. In this case, we have
Λ(SO(3)) = 0, 1, 2, 3, ,
{ ± ± ± ···}
1 3
Λ(SU(2)) = 0, , 1, , . (3.5)
{ ±2 ± ±2 ···}
As a result, we find that the magnetic charges must satisfy
en = 2πZ for SO(3),
i
en = 4πZ for SU(2). (3.6)
i
In the A BLG theory, we have (SU(2) SU(2))/Z gauge group, where the Z is
4 2 2
×
embedded diagonally in theproductof the centers of thetwo SU(2) factors. Note that this
isindistinguishablefrom(SU(2)/Z)2. Therefore,theGNOchargesfor(SU(2) SU(2))/Z
2
×
gauge group are allowed to be half the value of those of SU(2) SU(2) gauge group. In
×
our notation this implies that
1
n = Z for (SU(2) SU(2))/Z ,
i 2
2 ×
n = Z for SU(2) SU(2). (3.7)
i
×
A similar discussion can be applied to the (SU(3) SU(3))/Z theories. In this case, we
3
×
finally obtain
1
n = Z for (SU(3) SU(3))/Z ,
i 3
3 ×
n = Z for SU(3) SU(3). (3.8)
i
×
Aswewillseeinthenextsection, thesuperconformalindexisgivenbysummationover
contributions from each GNO charge. Therefore we have to take account of the difference
of the charge quantization conditions for calculating the index.
4. The superconformal index
In this section, we derive usefulexpressions for the superconformalindices of the U(N )
1 k
×
U(N ) ABJ(M) and SU(N) SU(N) theories including the BLG theory. The su-
2 −k k −k
×
perconformal index is defined by
I(x,z) = Tr ( 1)Fxǫ+j3zh , (4.1)
−
h i
5This also means that e~n corresponds to the dual lattice Λ∗(G). This is a weight lattice of a magnetic
(or Langlands) dualgroup G∨ and e~nis its weight vector.
6Fora general SU(N) group, thedual relation is given by SU(N)∨ =SU(N)/Z .
N
6
where F, ǫ, j and h are the fermion number, the energy (or equivalently the conformal
3
dimension), the projection of spin and the charge of a flavor symmetry, respectively. This
quantity is a powerful tool for distinguishing theories with a same moduli space.
On the ABJ(M) side, we consider the index with the fixed topological charge T. If
thereexistsanisomorphismastheconjectures(1.1)-(1.4), theindexoftheSU(N) SU(N)
×
theory must have contribution from charges of certain symmetry corresponding to U(1)
T
symmetry. As noted in [23], we can write the BLG theory of the product gauge group
formulation [7] in = 2 superspace. Of the original SO(8) R-symmetry this formulation
R
N
manifestly remains only the subgroupSU(4) U(1) . Because this U(1) is not related to
R R
×
the baryonic symmetry, this has nothing to do with the U(1) symmetry in the ABJ(M)
T
theory. Thus the topological charge T of the ABJ(M) theory should correspond to the
charge of a U(1) subgroup of the SU(4) SO(6) as discussed in [17]. We denote this U(1)
∼
subgroup as U(1) . On the BLG side, we treat this U(1) as the flavor symmetry whose
t t
charge assignments are +1( 1) to the (anti-)bi-fundamental. Therefore we introduce the
−
variable z to distinguish the U(1) symmetry of the SU(N) SU(N) theory and compare
t
×
the index with the one of the ABJ(M) theory.
4.1 U(N ) U(N ) ABJ theory
1 k 2 −k
×
By applying the localization method [15, 16], the (whole) superconfomal index of the
U(N ) U(N ) ABJ theory with z = 1 can be represented as
1 k 2 −k
×
I (x) = xǫ0 π dN1λ dN2λ˜ eS0exp ∞ 1f (xp,eipλ,eipλ˜) , (4.2)
ABJ {nX},{n˜}(sym) Z−π (2π)N1 (2π)N2 hXp=1 p tot i
where n and n˜ are the GNO charges, λ and λ˜ are constant holonomy zero modes and
i a
f = f +f ,
tot vec hyper
N1 N2
f (x,eiλ,eiλ˜) = ei(λi−λj)x|ni−nj| ei(λ˜a−λ˜b)x|n˜a−n˜b| ,
vec
− −
Xi6=j (cid:16) (cid:17) Xa6=b(cid:16) (cid:17)
x1/2 x1/2
f (x,eiλ,eiλ˜) = 2 x|ni−n˜b|ei(λi−λ˜b)+ x|ni−n˜b|e−i(λi−λ˜b) ,
hyper
1+x 1+x
!
i,b
X
S = ik n λ ik n˜ λ˜ ,
0 i i a a
−
i a
X X
1 1
ǫ = n n˜ n n n˜ n˜ . (4.3)
0 i b i j a b
| − |− 2 | − |− 2 | − |
i,b i,j a,b
X X X
The factor “(sym)“ denotes the rank of Weyl group for unbroken gauge group. By using
the formula
∞
1
eipλzp = log(1 zeiλ),
p − −
p=1
X
the contribution from the vector multiplet is rewritten as
∞
exp 1f (xp,eipλ,eipλ˜) = 1 x|ni−nj|ei(λi−λj) 1 x|n˜a−n˜b|ei(λ˜a−λ˜b) .
vec
p − −
hXp=1 i Yi6=j(cid:16) (cid:17)aY6=b(cid:16) (cid:17)
7
(4.4)
The contribution from the hyper multiplet is a bit more complicated. By using
∞ 1 yp ∞ ∞
eipλ = log 1 yx2meiλ + log 1 yx2m+1eiλ , (4.5)
p1+xp − − −
Xp=1 mX=0 (cid:16) (cid:17) mX=0 (cid:16) (cid:17)
we obtain
∞
1
exp f (xp,eipλ,eipλ˜)
hyper
p
hXp=1 i
∞ 1 x2m+3/2+|ni−n˜b|ei(λi−λ˜b)1 x2m+3/2+|ni−n˜b|e−i(λi−λ˜b) 2
= − −
" 1 x2m+1/2+|ni−n˜b|ei(λi−λ˜b)1 x2m+1/2+|ni−n˜b|e−i(λi−λ˜b)#
Yi,b mY=0 − −
(x3/2+|ni−n˜b|ei(λi−λ˜b);x2)∞(x3/2+|ni−n˜b|e−i(λi−λ˜b);x2)∞ 2
=
i,b"(x1/2+|ni−n˜b|ei(λi−λ˜b);x2)∞(x1/2+|ni−n˜b|e−i(λi−λ˜b);x2)∞#
Y
= x,n n˜ ,ei(λi−λ˜b) , (4.6)
hyper i b
I −
Yi,b (cid:16) (cid:17)
where (a;q) = ∞ (1 aqm) is the q-Pochhammer symbol and
∞ m=0 −
Q 2
(x3/2+|n|y;x2) (x3/2+|n|y−1;x2)
∞ ∞
(x,n,y) = (4.7)
Ihyper (x1/2+|n|y;x2) (x1/2+|n|y−1;x2)
" ∞ ∞#
Thus, the superconfomal index becomes the following simple form
xǫ0 π dN1λ dN2λ˜
I (x) = eS0
ABJ {n},{n˜}(sym)Z−π (2π)N1 (2π)N2
X
1 x|ni−nj|ei(λi−λj) 1 x|n˜a−n˜b|ei(λ˜a−λ˜b)
× − −
Yi6=j(cid:16) (cid:17)aY6=b(cid:16) (cid:17)
x,n n˜ ,ei(λi−λ˜b) . (4.8)
hyper i b
× I −
Yi,b (cid:16) (cid:17)
Next we perform a change of variables as
µ = λ λ (i = 1, ,N 1), µ = λ , ν = λ˜ λ . (4.9)
i i− N1 ··· 1− N1 N1 a a− N1
Then we can easily integrate over µ and obtain
N1
IABJ(x) = (sxyǫm′0 ) (d2πN)1N−11−µ1(2dπN)2Nν2eS0′
n1,···,nN1X−1,n˜1,···,n˜N2 Z
1 x|ni−nj|ei(µi−µj) 1 x|n˜a−n˜b|ei(νa−νb)
× − −
Yi6=j(cid:16) (cid:17)aY6=b(cid:16) (cid:17)
1 x|ni+Pjnj−Pan˜a|eiµi 1 x|ni+Pjnj−Pan˜a|e−iµi
× − −
Yi (cid:16) (cid:17)(cid:16) (cid:17)
8
x,n n˜ ,ei(µi−νb) x, n˜ n˜ n ,eiνa ,
hyper i b hyper b a i
× I − I − −
!
Yi,b (cid:16) (cid:17)Ya Xb Xi
(4.10)
where
N1−1
S′ = ik n µ ik n˜ ν ,
0 i i − a a
i=1 a
X X
ǫ′ = n n˜ n n n˜ n˜
0 | i− b|− | i − j|− | a− b|
i,b i<j a<b
X X X
+ n + n˜ n˜ n + n n˜ . (4.11)
j b a i j b
(cid:12)− − (cid:12)− (cid:12) − (cid:12)
a (cid:12) j b (cid:12) i (cid:12) j b (cid:12)
X(cid:12) X X (cid:12) X(cid:12) X X (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
Furthermore, we perform a change of the variables as
(cid:12) (cid:12) (cid:12) (cid:12)
y = eiµi, w =eiνa, (4.12)
i a
where each of them run over the unit circle in the complex plane. Then we obtain
I (x) = xǫ′0 dN1−1y dN2w ykni−1 w−kn˜a−1
ABJ (sym) (2πi)N1−1(2πi)N2 i i
n1,···,nN1X−1,n˜1,···,n˜N2 I Yi (cid:16) (cid:17)Ya (cid:16) (cid:17)
1 x|ni−nj|y y−1 1 x|n˜a−n˜b|w w−1
× − i j − a b
Yi6=j(cid:16) (cid:17)aY6=b(cid:16) (cid:17)
× 1−x|ni+Pjnj−Pan˜a|yi 1−x|ni+Pjnj−Pan˜a|yi−1
Yi (cid:16) (cid:17)(cid:16) (cid:17)
x,n n˜ ,y w−1 x, n˜ n˜ n ,w .
× Ihyper i− b i b Ihyper b− a− i a
!
i,b a b i
Y (cid:0) (cid:1)Y X X
(4.13)
Thus we can write the index with the fixed topological charge T as
(T) xǫ′0
I (x) = δ
ABJ Pan˜a,T(sym)
n1,···,nN1X−1,n˜1,···,n˜N2
dN1−1y dN2w
ykni−1 w−kn˜a−1
(2πi)N1−1(2πi)N2 i i
I Yi (cid:16) (cid:17)Ya (cid:16) (cid:17)
1 x|ni−nj|y y−1 1 x|n˜a−n˜b|w w−1
× − i j − a b
Yi6=j(cid:16) (cid:17)aY6=b(cid:16) (cid:17)
× 1−x|ni+Pjnj−Pan˜a|yi 1−x|ni+Pjnj−Pan˜a|yi−1
Yi (cid:16) (cid:17)(cid:16) (cid:17)
x,n n˜ ,y w−1 x, n˜ n˜ n ,w .
× Ihyper i− b i b Ihyper b− a− i a
!
i,b a b i
Y (cid:0) (cid:1)Y X X
9