Superconformal indices of generalized ArgyresDouglas theories from 2d TQFT
JHE
Superconformal indices of generalized ArgyresDouglas theories from 2d TQFT
Jaewon Song 0 1
0 9500 Gilman Dr , La Jolla, CA 92093 , U.S.A
1 Department of Physics, University of California , San Diego
We study superconformal indices of 4d N = 2 class S theories with certain irregular punctures called type Ik;N . This class of theories include generalized ArgyresDouglas theories of type (Ak 1; AN 1) and more. We conjecture the superconformal indices in certain simpli ed limits based on the TQFT structure of the class S theories by writing an expression for the wave function corresponding to the puncture Ik;N . We write the Schur limit of the wave function when k and N are coprime. When k = 2, we also conjecture a closedform expression for the HallLittlewood index and the Macdonald index for odd N . From the index, we argue that certain shortmultiplet which can appear in the OPE of the stressenergy tensor is absent in the (A1; A2n) theory. We also discuss the mixed Schur indices for the N = 1 class S theories with irregular punctures.
Supersymmetric gauge theory; Supersymmetry and Duality; Topological Field

HJEP02(16)45
Theories
1 Introduction
2 Schur index
2.1
2.2
3.1
3.2
4.1
4.2
3 HallLittlewood index
Index from the 3d mirror
Wave function for I2;N
4
Macdonald index
Examples 4.2.1 4.2.2
Wave function for the puncture of type Ik;N
Examples
2.2.1
2.2.2
Lagrangian theories
ArgyresDouglas theories
Wave function for the puncture of type I2;n
Consistency checks
Conjecture for the Macdonald indices of ArgyresDouglas theories
can be realized by wrapping 6d N = (2; 0) theory on a Riemann surface C [1, 2]. This
description enables us to understand dynamics of the 4d theory in terms of geometry of the
Riemann surface. One of the most interesting connection is between the superconformal
index [3, 4] of the 4d theory and the 2d topological eld theory [5{9]. It says that the
superconformal index of a given theory labelled by C (called the UV curve) is given by a
correlation function of the 2d topological eld theory on C, which is a deformed version of
YangMills theory. Especially, in the Schur limit of the index, the TQFT is identi ed as the
qdeformed YangMills theory [10]. It has been shown via localization of 5d Maximal SYM
on S3, that the 2d theory is indeed given by the qdeformed YangMills theory [11{13].
This relation is also extended to the Lens space index [14{16], to the outerautomorphsim
twisted index [17], and to other gauge groups [18{22].
A class S theory is not just labelled by the UV curve but also its local data on the
punctures. There are regular and irregular punctures depending on the boundary condition
{ 1 {
we impose. The regular punctures are labelled by an SU(
2
) embedding into
that labels
the 6d N = (2; 0) theory. The irregular punctures require more elaborate classi cations,
and generally lead to nonconformal theories. But when the UV curve is a sphere, we
can get a SCFT with one irregular puncture and also with or without one regular
puncture [23, 24]. Theories realized in this way includes ArgyresDouglas theory [25, 26] and
its generalizations.
The (generalized) ArgyresDouglas theories are inherently stronglycoupled and have
no weakcoupling limit. Therefore there has been no direct way of computing the
superconformal indices or S1
S3 partition functions. Recently, a progress is made in [27], where
they obtained an ansatz for the TQFT description of the Schur index for some of the
ArgyresDouglas type theories. They were able to verify their result against Sduality [28]
and also by studying dimensional reduction to 3d [29]. Their result has been recently
extended to the Macdonald index [30]. The result of [27] agrees with the general prediction
made in [31], and studied further in [32, 33], that for any N = 2 d = 4 SCFT, there is a
protected sector with in nite dimensional chiral algebra acting on it. This implies that the
Schur indices have to be given by the vacuum character of the corresponding chiral algebra.
Another progress is made in [34]. They observed that the trace of (inverse) monodromy
operator appears in the BPS degeneracy counting [35] agrees with the Schur index (of a
nonconformal theory). From this observation, they were able to predict the Schur indices
of various generalized ArgyresDouglas theories and conjectured that they are given by the
vacuum character of certain nonunitary W minimal models. This agrees with the result
of [31], once applied to the ArgyresDouglas theories with no avor symmetry, yield chiral
algebra given by Virasoro/Walgebra with central charges of the minimal model series.
In this paper, we propose the Schur, HallLittlewood (HL) and Macdonald limit of the
superconformal index for a class S theory containing certain irregular puncture called (a
subset of) type Ik;N . We conjecture the wave function for irregular puncture in these limits,
which enables us to use the TQFT description to compute the indices of AD theories. Our
strategy is very similar to [27, 30], but we are mainly interested in the theories with no
avor symmetry. It turns out the corresponding wave functions are much simpler than the
ones with avor symmetries. We rst start with the irregular punctures that appear in the
Lagrangian theories and derive their wave function as an integral transformation of regular
punctures. From here, we extrapolate the expression to nd a wellbehaved wave function
corresponding to other irregular punctures. For the Schur index, we are able to
nd the
wave function for arbitrary coprime k; N . For the HL index and Macdonald index, we nd
the wave functions for the k = 2 cases only.
Curiously, we nd that the indices for nonconformal theories can also be written in
terms of the TQFT. This is the index of the theory at the UV
xed point (zero coupling)
with Gauss law constraint. Even though the theory is nonconformal for nonzero gauge
coupling, the UV index is nevertheless wellde ned. The TQFT description for the
\superconformal index" of a nonconformal N = 2 theory enables us to write the index for the
conformal 4d N = 1 class S theory [36{39].
The outline of this paper is as follows. In section 2, we study Schur index for the
theories with Ik;N punctures. We are able to give a expression when k and N are coprime,
{ 2 {
and it agrees with other proposals. In section 3, we study HallLittlewood index with I2;N
punctures. We obtain the wave function for I2;N and compare with the direct computation
of the index from the 3d mirror theory. In section 4, we conjecture a closedform formula
for I2;N with N odd and perform a number of consistency checks. In section 5, we compute
mixed Schur limit of the index for N = 1 class S theories using the result of 2.
2
Schur index
Superconformal index of an N = 2 d = 4 SCFT is de ned as
I(xi; p; q; t) = Tr(
1
)F pj2+j1 rqj2 j1 rtR+r Y xiFi ;
(2.1)
symmetries. The trace is taken over the 18 BPS states that are annihilated by a supercharge
Q. The index can be simpli ed by taking certain limits [7]. When p ! 0, it is called the
Macdonald index and gets contributions from 14 BPS states. The Macdonald index can be
further simpli ed to the Schur limit upon taking q = t limit. The other limit is to take
q ! 0, and this is called the HallLittlewood (HL) index.
For any class S theories coming from 6d (2; 0) theory of type
wrapped on a Riemann
surface Cg;n, the superconformal index can be written in terms of a correlation function of
a topological eld theory
I(ai; p; q; t) = X C2g 2+n Y (i)(ai; p; q; t) ;
n
i=1
each puncture.
constant is xed to be
where the sum is over all irreducible representations of
and g and n is the genus and
the number of punctures respectively. The function C (p; q; t) is sometimes called the
structure constant of the theory, and
(a; p; q; t) is called the wave function we assign to
The Schur index is obtained by specialization p = 0; q = t. In this limit, the structure
C 1 =
?(a = q ; q) = Qir=1(qdi; q)
;
(q )
where
is the Weyl vector of
the shorthand notation za
de ned as (z; q)
wave function is given by
Qi1=0(1 zqi). For the case of a full regular puncture, the corresponding
= ADE and di are the degrees of the Casmirs. We use
Qi ziai and q
Q q i. Here the qPochhammer symbol is
i
(z; q) = PE
1
q
= Ak 1 and denote the local singularity around
the puncture at z = 1 is given as
(2.8)
(2.9)
(2.10)
(2.11)
(2.12)
or equivalently around z0 = 0 via z0 = 1=z as
xk = zN (dz)k ;
xk =
1
z0N+2k (dz0)k ;
to be Ik;N , following the notation of [23, 40]. One can further deform the singularity by
adding less singular terms. They serve as deformation parameters of the theory. The
regular punctures correspond to N =
k. For the case of k = 2 and N = 2n even, the
wave function is written by [27] as
where R is the spin 2 representation of SU(
2
), and j3 =
2
;
2 + 1;
; .
2
Our goal in this section is to nd a wave function for the punctures of type Ik;N . For
example, we conjecture the wave function for the I2;2n 1 puncture to be
I2;2n 1 (q) =
0
(
(
1
) 2 q 2 ( 2 +1)(n+ 12 )
even;
as we will provide evidences in this section.
2.1
Wave function for the puncture of type Ik;N
Puncture of type IN; N+1. Let us consider SU(N ) theory with N
avors. This theory
can be realized by a 3punctured sphere with one maximal, one minimal and one irregular
of type IN; N+1. If we write the index of this theory in terms of a TQFT, we get
IN; N+1 denote the wave function corresponding to the minimal and
IN; N+1 puncture respectively. Since it is the same as gauging one of the avors of
bifundamental hypermutliplets, it can be also written as
(2.13)
(2.15)
(2.17)
(2.18)
(2.19)
Y (1
2 +
P 12 n (n +1) I
dz
z )zP n
(z) ; (2.16)
where
+ is the set of positive roots of SU(N ) and W is the Weyl group. Here the index
runs from
= 1;
j +j and the integral measure is given by dz = QiN=11 2dziizi . The
last integral can be rewritten upon applying the Weyl character formula as
I
(w)
dzz
+w( + ) P n
=
X
w2W
(w) w(N =1n)
;
where
= 12 P
the determinant of w and w
w(
+ )
is the shifted Weyl re ection.
For the case of N = 2, it is simple to evaluate the above integral. We get
is the Weyl vector, (w) is the signature of w which is the same as
I2; 1 (q) =
1 X(
1
)nq 21 n(n+1) I
2n
z
2n 2
)
1
jWj(q; q) 21 (N 1)(N 2)
X (
1
)P n q
n 2Z
X
w2W
2 +
=
2
(
n2Z
0
(
1
) 2 q 21 2 ( 2 +1)
dz
2 iz
(z
even;
odd:
For N = 3, we nd the expression as below:
(I3;1; 22)(q) = <
>
>8qk(k+1)+`(`+1)+k`
>
>:0
if 1 = 3k; 2 = 3`;
otherwise;
qk2+`2 1+(k 1)(` 1) if 1 = 3k
2; 2 = 3`
2;
1The author would like to thank Yuji Tachikawa for the discussions lead to this observation.
where k; ` 2 Z 0. We do not have an analytic proof of the formula, but we have checked
this expression up to ( 1; 2) = (12; 12).
This wave function is an analog of GaiottoWhittaker vector [41{43] in the AGT
correspondence [44, 45], which realize pure YM theory when we have two punctures of this type.
In our case, we expect the two point function of GaiottoWhittaker state of the qdeformed
YangMills gives us the `Schur index' of the pure YM theory2
IYM(q) =
X
IN; N+1 (q) IN; N+1 (q) :
We will prove this relation for the case of SU(
2
) in section 2.2.1.
Now, we conjecture that the wave function for the irregular
puncture of type Ik;N when k and N are relative primes, is simply given by rescaling the
q parameter of the wave function for the GaiottoWhittaker state (2.14). More precisely,
we nd
Ik;N (q) =
Ik;1 k (qN+k) :
This can be thought of as an analog of the coherent state considered by [46{48] in the
context of AGT correspondence. We give a number of evidences for (2.21) in later sections.
It would be desirable to give a direct proof of this proposal.
Puncture of type Ik;kn.
We can also consider a irregular puncture of type Ik;N with
N being a multiple of k. In this case we have k
1 mass parameters associated to the
U(
1
)k 1 avor symmetry. As before, let us rst consider a Lagrangian example. Consider
SU(N ) gauge theory with 2N
1 fundamentals. It is realized by a sphere with a maximal,
minimal and IN;0 type irregular punctures. Therefore, we write the wave function for the
(2.20)
(2.21)
(2.23)
(2.24)
2For a nonconformal theory, the superconformal index really means that of the free UV xed point with
the Gausslaw constraint.
{ 6 {
IN;0 puncture as
IN;0 (a; q) =
=
=
I
I
I
[dz](q; q)r Y(qzi=zj ; q) Y
q
i6=j
For the case of N = 2, the wave function is given by [27] as
where the trace is over the spin =2 representation R . We nd that the wave function for
the irregular puncture of type I2;2n can be written almost as a simple rescaling of the q as
We have checked this expression indeed gives us the same wave function as (2.23) found
in [27] to high orders in q.
We were not able to nd a prescription for N
3. It may be possible to nd a correct
prescription by using isomorphism of (A2; A2) = (A1; D4), which has the chiral algebra
scu(3) 3 .
2
tures with x2 / (dz0)2=z03, which means I2; 1. From the TQFT, we get
ITQFT( 1; 1)(q) =
X
X
m2Z 0
It agrees with the integral expression obtained from blindly applying the index formula for
the vector multiplets and then integrating over the gauge group
I
dz
2 iz
ISYM(q) = (q; q)2
(z)(qz 2; q)2 =
qm(m+1) ;
where (z) = 12 (1 z 2) is the Haar measure of SU(
2
). This integral can be easily evaluated
by using the Jacobi triple identity (2.15). It can be considered as the index at the UV xed
point with Gauss law constraint.
SU(
2
) with Nf = 1. The SU(
2
) gauge theory with 1 avor can be realized by a sphere
with 2 irregular punctures I2; 1 and I2;0. From the TQFT, we get
ITQFT( 1;0)(q; a) =
X
I2; 1 I2;0 (a)
= 1 + q +
+
2a2
a
2
2
1
a2 + 2 q2 +
a2 + 4 q4 + a4 +
a
2
1
a4
1
a2 + 2 q
3
3
a2
3a2
+5 q5 +O q6 ;
{ 7 {
(2.25)
(2.26)
(2.27)
(2.28)
(2.29)
which agrees with the one computed from the integral
INf =1 = (q; q)2
I
dz
2 iz
(z)
[S3O;0(]4) + [S0O;3(]4) +2 [S2O;0(]4) +2 [S0O;2(]4) +4 [S1O;0(]4) +4 [S0O;1(]4) +6 [S0O;0(]4) +O(q4);
This agrees with the integral formula
INf =3(ai; q) = (q; q)2
I
dz
2 iz
(z)
upon identifying a1 = bc; a2 = b=c; a3 = a.
SU(3) SYM.
The Schur index of the SU(3) pure YM can be written as
ISU(3) =
=
1
3!
(q; q)4
I
1
dz1
X
dz2
2 iz1 2 iz2
6(q; q)2 n1;2;3;m1;2;32Z
Y (1
2 +
z
)(qz
; q)2
(
1
)n1+m1 q 21 Pi(ni(ni+1)+mi(mi+1)) n1+n2;m1+m2 n1+n3;m1+m3 :
which agrees with the computation from the integral formula. Here we used the Dynkin
label to denote the characters.
and one I2; 1 puncture. This gives
We can also realize the same theory via 3punctured sphere with 2 regular punctures
ITQFT(R;R; 1)(x; y; q) =
X C
(x)
(y) I2; 1 ;
which gives the same answer upon matching the fugacities via a !
pxy and b !
px=y.
SU(
2
) with Nf = 3. The SU(
2
) gauge theory with 3 avors can be realized by a sphere
with 2 regular punctures and 1 irregular puncture I2;0. This gives us the index to be
ITQFT(R;R;0) =
X C
(a)
(b) I2;0 (c)
= 1 + q [S0O;1(;61)] + q2( [S0O;2(;62)] + [S0O;1(;61)] + [0;0;0])
SO(6)
+ q3( [S0O;3(;63)] + [S0O;2(;62)] + [S1O;2(;60)] + [S1O;0(;62)] + [S0O;1(;61)] + [S0O;0(;60)])+O(q4) :
(2.30)
(2.31)
(2.32)
(2.33)
(2.35)
(2.36)
We veri ed this to be the same as the one given by (2.20) with N = 3 to high orders in q.
The rst few terms are
ISU(3) = 1 + q2 + 2q4 + 2q8 + q10 + 2q12 + q16 + 2q18 + 2q20 + O(q24) :
(2.34)
2.2.2
ArgyresDouglas theories
(A1; AN 1) theories.
The ArgyresDouglas theories of type (A1; AN 1) can be realized
by a sphere with single irregular puncture of type I2;N [23, 40]. When N = 2n + 1 is odd,
we have
where
IA2n (q) =
X C 1 I2;2n+1 =
(
1
)j qj(j+1)(n+ 12 ) ;
X
model (2; 2n + 3) to high orders in q, which is consistent with the relation given in [31], and
also computation based on the conjectural relation from the BPS degeneracy [34]. We also
nd that when n = 0 or N = 1, the index becomes 1, which agrees with the expectation
that there is no massless degrees of freedom and vanishing central charges [40] for the
(A1; A0) theory.
HJEP02(16)45
When N = 2n is even, we get [27]
IA2n 1 (x; q) =
X C 1 I2;2n (x) =
0
X [ + 1]q TrR
with one I2;N type irregular puncture and a regular puncture. When N = 2n
This expression exactly agrees with the vacuum character of the a ne Lie algebra
4n . When N = 1, we nd the index of (A1; A3) agrees with that of (A1; D3)
I(A1;A3) =
up on identifying x = a2.
When N = 2n, the index can be written as
that the Schur indices for the ArgyresDouglas theories of type (Ak 1; AN 1) as
I(Ak 1;AN 1)(q) =
1
Qik=2(qi; q)
X
(q ) Ik;N (q) :
It was conjectured by [34] that for the coprime k; N , the Schur index of (Ak 1; AN 1)
theory is given by the vacuum character of the (k; k + N ) Wkminimal model, which is
given by [49]
0
W (k;k+N)(q) =
(qk+N ; qk+N ) k 1 k 1
W (k;k+N)(q) to high orders in q for a number of cases.
{ 9 {
For ArgyresDouglas theories realized in class S with a nontrivial Higgs branch,3 once
dimensionally reduced to 3d, their mirror theories [51] are known [23, 52]. The Higgs
branch of the dimensionally reduced 3d N = 4 theory is the same as the original 4d N = 2
theory, which is the same as the Coulomb branch of the 3d mirror
MHiggs = MHiggs = M3Cdoumloirmrobr :
4d 3d
Since the HallLittlewood (HL) index is the same as the Hilbert series of the Higgs
branch [7], it should be the same as the Hilbert series of the Coulomb branch [53] of
the 3d mirror theory. This can be also computed using the 3d superconformal index by
taking the `Coulomb limit' as discussed in [54]. Therefore we can write
IH4dL(t) = IH3diggs(t) = IC3doumloimrrbor(t) :
In this section, we review the computation of HL indices from the 3d mirrors [55].
The Coulomb branch index for the 3d N = 4 theory is de ned as
IC (t) = Tr(
1
)F tE RH = Tr(
1
)F tE+j ;
(3.1)
(3.2)
(3.3)
(3.4)
(3.5)
(3.6)
Index from the 3d mirror
where the trace is over the states with E = RC in addition to E
RH
EC
j = 0. Here
RH ; RC are the Cartans of SO(4)R = SU(
2
)RH
SU(
2
)RC and E is the scaling dimension
and j is the Cartan of the rotation group SO(3). The Coulomb branch index contribution
for a hypermultiplet is simply
IhCyp(z; m; t) = t 21 jmj ;
which does not depend on any avor fugacities. Here m is the charge for the background
topological U(
1
) associated to the gauge group.
(A1; A2n 1) theories. The 3d mirror theory is given by a quiver gauge theory with k
U(
1
) nodes where all the nodes are connected by n edges. One of the U(
1
) node has to
be ungauged as usual. See the gure 1. For the U(
1
) gauge theory with n fundamental
hypers, the index is given as
C
I(A1;A2n 1)(w; t) =
1
1
t
X
m2Z
wmt n2 jmj ;
where w is the fugacity for the topological U(
1
)J associated to the gauge group. One can
sum up the formula to get
InC (w; t) =
1
1
t
n
t
1
(1
n
t 2 w)(1
n
t 2 w 1)
=
1
1
t
n
t
1
X
k=0
[k]
SU(
2
)(w)tkn=2 ;
where [k] denotes the characters of SU(
2
) with Dynkin label [k]. Even though we see the
index is written in terms of characters SU(
2
), it is wrong to say that the global symmetry
is actually enhanced to SU(
2
). Note that only when n = 2, we have the extra conserved
current coming from the monopoles [56].
3Even though the theory in the UV might not have a Higgs branch, AD points can sometimes have a
quantum Higgs branch [50].
of bifundamentals between a pair of nodes. One of the U(
1
) has to be decoupled.
The Coulomb branch of U(
1
) gauge theory with n electrons or the Higgs branch of its
mirror theory A^n 1 quiver is known to be given by C2=Zn. One can directly see that the
Hilbert series of this space is the same as above.
For n = 2, the 3d theory is wellknown T [SU(
2
)] theory which is selfmirror. Here the
SU(
2
) enhancement of topological symmetry is evident from the mirror perspective. There
SU(
2
) symmetry is nothing but the avor symmetry of two electrons. In this case, the 4d
theory is the same as the AD theory found from SU(
2
) gauge theory with two avors.
(A2; A3n 1) theories.
The 3d mirror of this theory is given by U(
1
)
U(
1
) gauge
theory with n bifundamentals and n electrons for each U(
1
). The Coulomb branch index
is given by
(3.7)
(3.8)
1
(1
t)2
X
m1;m22Z
C
I(A2;A3n 1)(w1; w2; t) =
w1m1 w2m2 t 21 (jm1j+jm2j+jm1 m2j)n ;
which can be written in terms of SU(3) characters as
C
I(A2;A3n 1)(a; t) =
1
1
t
t
n 2 1
X
k=0
[SkU;k(]3)(a)tnk
where [k; k] is the Dynkin label for the kth powers of adjoint representation of SU(3) and
the fugacities are mapped to w1 = a1=a22; w2 = a2=a12. From above, we see that the global
symmetry is enhanced to SU(3).
This
avor symmetry can be understood quite easily when n = 1. In this case, the
quiver gauge theory we obtain is nothing but A^2 quiver theory with U(
1
) nodes, which is
mirror to the U(
1
) gauge theory with 3 electrons. As in the previous case, we have SU(3)
avor symmetry for n = 1, which gives us extra conserved current from the monopole
operators. For n > 1, even though we get SU(3) character representations, it does not
mean that our theory has extra conserved currents.
Curiously, we observe that the 3d mirror of (A2; A2) theory is simply given by A^2
quiver theory, which is mirror to U(
1
) with 3 electrons, which is the 3d mirror of (A1; A3)
theory. This is not a surprise, in the sense that generally in class S, there can be many
di erent ways to realize the same 4d SCFT. See [57] for a study of such examples among
the AD theories.
(A1; D2n+2) theories.
This theory is obtained by putting extra U(
1
) punctures to
(A1; A2n 1) theory. The 3d mirror is obtained by gluing the 3d mirror theory
corresponding to the minimal puncture to the 3d mirror of (Ak 1; Akn 1) theory. See the gure 2.
The Coulomb index for the mirror theory is given as
C
I(A1;D2n+2) =
1
(1
t)2
X
m1;m2
w1m1 w2m2 t 21 (njm1 m2j+jm1j+jm2j) :
When n = 1, the above sum can be written as
X
k
C
I(A1;D4) =
[SkU;k(]3)(a)tk ;
which is the same as that of (A2; A2) theory as expected.
3.2
Wave function for I2;N
N = 2n even.
We want to write the index (3.5) using the TQFT as
I(A1;A2n 1)(a) =
X C 1 I2;2n (a) ;
C =
Qir=1(1
1
P HL(t 2 )
tdi )
;
with
where di are the degrees of the Casmirs of . For
= SU(N ), they are given by di =
2; 3;
; N . Here the function P HL(a) is the HallLittlewood polynomial labelled by a
Dynkin label in general, which we normalize so that
h
P HL(z); P HL(z)i =
Z
[dz] (z)PE [t adj(z)] P HL(z)P HL(z) =
:
(3.13)
(3.9)
(3.10)
(3.11)
(3.12)
For the case of = SU(2), we get
The wave function for the I2;2n puncture can be read o from the expression (3.5) to be
P HL(z) =
(p
p
1
1
t
2
t ( (z)
t
if
= 0,
Note that there are many di erent ways decompose the sum (3.5) in the form of (3.11). But
we nd our particular form given by (3.15) is the right choice to maintain the consistent
TQFT description of the index. We nd that this is indeed consistent with the accidental
isomorphism of the AD theories.
One can obtain the wave function for the I2;0 puncture as in the previous section by
integrating the wave function for the regular puncture with the integration kernel given by
vector and hypermultiplets
I
I
I2;0 =
where the wave function for the regular puncture is given as
We nd it agrees with the expression (3.15) with n = 0.
written as
From the TQFT structure of the index, the index for the (A1; D2n+2) theory can be
We indeed nd this expression agrees with the result (3.9) from the 3d mirror.
N = 2n
1 odd.
Again, one can obtain the wave function for I2; 1 by integrating the
regular puncture wave function via vector multiplet measure. We get
I2; 1 =
[dz]Ivec(z)
(z) =
[dz]PE [t adj(z)] P HL(z)
>
>
= <
I
8p
>
>:0
1
p
t 1
t
2
t
2
= 0
= 2
otherwise :
(z)
1
I
From this, we obtain the HL index for the pure SU(
2
) YM to be
ISYM(t) =
X
I2; 1 I2; 1 = 1
t3 ;
which agrees with the direct computation.
(3.14)
(3.15)
(3.16)
(3.17)
(3.18)
(3.19)
We can obtain the wave function for I2;1 from using the isomorphism (A1; A3) =
(A1; D3). Note that it implies
where we identify x = z2 as we learned from the Schur index. Now, we can use the
orthonormality of the regular puncture wave functions. Multiply
(z) on both sides and
then integrate with the vector multiplet measure to obtain
I2;1 =
X I
[dz]Ivec(z)C 1 I2;4 (z2) (z) = p
1
t
We conjecture the HallLittlewood wave function for I2;2n 1 is the same as that of I2;1,
so that
for n > 0. This gives the HL index for the (A1; A2n) theory to be 1, which is consistent
with the absence of the Higgs branch.
(3.21)
In this section, we discuss the Macdonald index, carrying two fugacities (q; t). The
Macdonald index reduces to the Schur index when q = t and to the HallLittlewood index when
q = 0. We rst construct the wave function for the puncture of type I2;n and then use it
to write the Macdonald index for the (A1; A2n) theory and discuss its implications.
4.1
Wave function for the puncture of type I2;n
Odd n.
As in the previous sections, we start with the I2; 1 puncture. It can be obtained
by a `integral transformation' of a regular puncture via vector multiplet measure. The
wave function for the regular puncture is given by
q adj(z) P (z; q; t) =
1
(t; q)r
Y
2
1
(tz ; q)
P (z; q; t) ;
(4.1)
where P (z; q; t) = N P (z; q; t) is the normalized Macdonald polynomial4 such that
(4.2)
(4.3)
t
h
;
i =
I
N (q; t) =
[dz]Ivec(z)
(z)
(z) =
;
(q; q)(t2q ; q)
(t; q)(q +1; q)
(1
tq )
1
2
:
where the normalization factor for the
= A1 is given by
4Our normalization is slightly di erent from the usual one. We include the `Cartan piece' ((qt;;qq))rr to the
measure.
From now on we suppress the dependence on q; t. When
= A1, we can explicitly write
down the polynomial as
P (a) = N
i=0
X (t; q)i (t; q) i a2i
(q; q)i (q; q) i
;
where we de ne the qPochhammer symbol as (x; q)n = Qin=01(1
xqi).
Now, let us compute the wave function for the GaiottoWhittaker state for the
theory. We evaluate the integral to get
I2; 1 (q; t) =
[dz]Ivec(z)
(z; q; t) =
[dz]PE
adj(z) PE
t
1 q adj(z) P (z)
HJEP02(16)45
I
8
:0
= N
i=0
X (t; q)i(t; q) i I
(q; q)i(q; q) i
= <(
1
) 2 t 2 q 21 2 ( 2 1)N ((qt;;qq)) ==22
I
dz (1
2 iz
q t
1 q
z 2)
2
even;
odd:
(qz 2;0; q)z2i
wave functions.
odd is given by
We see that Schur and HallLittlewood limits of (4.5) indeed reproduces the corresponding
From (4.5), we want to nd a similar rescaling of the fugacities (q; t) as done in the
Schur and HallLittlewood case. Now, we conjecture that the wave function for I2;n for n
8
:0
I2;n (q; t) = <(
1
) 2 t 2
(n+2)
We will provide some evidences of this proposal in the following subsection.
Even n. The wave function for the I2;2n is proposed by [30] as (up to normalization)
I2;2n 2 (x) = N
tn =2qn 2=4
(t; q)
X (t; q)m(t; q)
m=0 (q; q)m(q; q)
m q n( 2 m)2 x2m
:
m
As in the previous sections, we can obtain I2;0 by integrating the wave function for the
regular puncture with vector and hypermultiplet kernel
I2;0 (a; q; t) =
I
(z) =
I
dz (1 z 2) (qz 2;0; q)
2 iz
2
1
(t 2 z a ; q)
P (z; q; t): (4.8)
We have veri ed that this expression agrees with (4.7) for n = 1 up to high orders in q.
4.2
and provide consistency checks.
In this section, we use (4.6) to compute the Macdonald index for a number of examples
(4.4)
= A1
(4.5)
(4.6)
(4.7)
Let us rst consider a number of examples where we can crosscheck the conjectured
formula (4.6) against independent computations.
SU(
2
) SYM. The pure YM theory can be realized by a pair of I2; 1 punctures on a
sphere. From the TQFT structure of the index, we write
ISYM =
X
where t = qT . This result indeed agrees with the direct computation.
(A1; A0) theory. This should describe a theory with no massless degrees of freedom.
We indeed nd
I(A1;A0) =
X C 1 I2;1 (q; t) = 1 ;
(4.10)
to high orders in q. This is rather a nontrivial check of the proposal (4.6), since each term
in the sum has to cancel exactly up on summing over all terms.
(A1; A3) = (A1; D3) theory.
The (A1; A3) theory is isomorphic to (A1; D3) theory.
The former description can be obtained from a single I2;4 puncture and the latter
description can be obtained from I2;1 and a regular puncture. We nd that these two descriptions
indeed give us the same index
I(A1;A3)(x) =
upon identifying x = a2. This result provides a consistency check between (4.6) and (4.7).
We nd that the Schur limit of this index can be written in a very simple form
I(A1;D3)(a) = PE
(1
q
q)(1
q
3
q3) adj(a) :
The rst term inside the PE is coming from the conserved current multiplet.
4.2.2
Conjecture for the Macdonald indices of ArgyresDouglas theories
(A1; A2) theory. It can be obtained from I2;3 punctured sphere. We conjecture its
Macdonald index is given as
I(A1;A2) =
X C 1 I2;3 = 1 + q2T + q3T + q4T + q5T + q6(T 2 + T ) + q7(T 2 + T )
(4.12)
(4.13)
+ q8(2T 2 + T ) + q9(2T 2 + T ) + q10(3T 2 + T ) + O(q11) ;
where t = qT . This theory does not have a Higgs branch. This can be seen from triviality
of the HallLittlewood limit of the index q ! 0.
We nd that this expression can be also written as
I(A1;A2) = PE
q2T
q4T 2
(1
q)(1
q5T 2)
+ O(q11) ;
(4.14)
where the O(q11) terms vanish in the limit T
coming from the short multiplet C^0(0;0) (and their powers) using the notation of [58]. See
also appendix B of [7]. This is the multiplet containing the stressenergy tensor. The
Macdonald index for the short multiplet C^R(j1;j2) is given as
! 1. The rst term inside the PE is
We see that the stressenergy tensor multiplet contributes 1q2Tq to the Macdonald index.
Since any SCFT has a stressenergy tensor multiplet, the operator appear in the OPE
of it should be also present in the theory. The OPE of the stressenergy tensor multiplet
which is the value of (A1; A2) theory [61]. See also [62].
other than ` = 0, which contains the stressenergy tensor.
where we have only written short multiplets appear in the OPE that contributes to the
Macdonald index. The C^0( 2` ; 2` ) multiplets are higherspin conserved currents which have to
be absent unless the theory is free or has a decoupled sector [60]. This multiplet contributes
to the index by q`1+2qT . Indeed, we see from the index (4.13) that there is no C^0( 2` ; 2` ) multiplet
Among the terms appear on the r.h.s. of the OPE (4.16), C^1( 2` ; 2` ) multiplet contributes
q`+3T 2
1 q
to the index. Since the index has coe cient 0 for the q4T 2 term, C^1( 12 ; 12 ) cannot
be present. We also see C^1( 2` ; 2` ) with even ` is absent. Our result agrees with the analysis
^
of [59] where they show that C1( 12 ; 1 ) is absent for the theory with central charge c = 1310
2
(A1; A4) theory. It can be obtained from I2;5 punctured sphere. Our conjectured
Mac
X C 1 I2;5 = 1+q2T +q3T +q4(T 2 +T )+q5(T 2 +T )+q6(2T 2 +T )
donald index is
I(A1;A4) =
+ q7(2T 2 + T ) + q8(T 3 + 3T 2 + T ) + O(q9) :
It also reduces to 1 in the HallLittlewood limit q ! 0 as expected.
We nd the index can be written in terms of a Plethystic exponential as
I(A1;A4) = PE
q2T
q6T 3
(1
q)(1
q7T 3)
+ O(q15) ;
where O(q15) term vanishes as T ! 1.
5The author would like to thank Wenbin Yan for discussions on this point.
(4.18)
Here we see that some of the shortmultiplets appear in the OPE of the 3 stressenergy
tensor multiplets
should be absent, because there is no term of the form q6T 3 in the index. Among the
operators appear in the OPE of 3 stressenergy tensors, the multiplet contributing q16Tq3
has to be absent. The natural candidate would be C^2(1;1), but we cannot rule out other
possibilities from the index before working out the selection rule, because any C^R(1;j2) with
R + j2 = 3 for an integer j2 will give the same index.
(A1; A2n) theory.
We put I2;2n+1 puncture on a sphere. We conjecture the Macdonald
q)(1
(q2T )n+1
q2n+3T n+1)
+
;
(4.20)
where omitted piece vanishes in the Schur limit T ! 1.
There is no (q2T )n+1 term in the index. Therefore the short multiplet that appear
in the OPE of (C^0(0;0))n+1 that contributes to the index as (q2T )n+1
1 q
multiplet C^n( n2 ; n2 ) contributes the same amount so that it might be absent.
is absent. The short
(A1; D5) theory. It can be obtained from a sphere with a I2;3 puncture and a regular
puncture. We get
I(A1;D5)(a) =
X
I2;3
(a)
(4.21)
(4.22)
(4.23)
= 1+qT 3 +q2 T ( 3 + 1)+T 2 5 +q3 T ( 3 + 1)+T 2( 5 +2 3)+T 3 7
+ q4 T ( 3 + 1) + T 2(2 5 + 3 3 + 2 1) + T 3( 7 + 2 5) + T 4 9 + O(q5) ;
where n denotes the character for the ndimensional representation of SU(
2
). When we
take the HallLittlewood limit q ! 0 with t xed, we get
I(A1;D5)(a) =
2n+1(a)tn =
X
n 0
1
t
2
(1
ta2)(1
ta 2)(1
t)
;
which is the same as the HL index of the (A1; A3) theory given in (3.6) with n = 2 and
w = a2. This is nothing but the Hilbert series of C2=Z2.
The rst term of the index (4.21) comes from the conserved current of the SU(
2
) avor
symmetry. We nd that the index has the form
I(A1;D5) = PE
qT
1
q adj(a) + : : : ;
where the omitted term vanishes in the Schur limit.
5
N
= 1 class S theories
For every theories in N = 1 class S, the superconformal indices can be written in terms of
the correlation functions of a (generalized) topological eld theory on the UV curve [38].
See also [20, 63{65]. In this section, we generalize our discussions to the N = 1 case.
For the chiral multiplets with (J+; J ) = (0; 2) in the adjoint representation of G, we get
i
i
#
#
(5.1)
(5.2)
(5.3)
(5.4)
(5.5)
(5.6)
(5.7)
Ichi(p; q; z) = PE
" J
p 2 q 2
J+
(z)
(1
p
1 J2 q1 J2+
p)(1
q)
(z) #
;
where
is the character of the representation
of the gauge group. In the mixed Schur
limit, the hypermultiplets charged with (J+; J ) = (1; 0) gives the index purely a function
and the hypermultiplets with (J+; J ) = (0; 1) gives
The N = 1 index for the class S is de ned as
I(p; q; ; x) = Tr(
1
)F pj1+j2+ R20 qj2 j1+ R20 F Y xiFi ;
where R0 is the UV Rcharge and F is the global U(
1
) charge conserved for a generic class
S theory. One of the simpli cation limit of the above index is to take
= pq=p, called
the mixed Schur limit [38]. It is given as
where we used J
F ). In this limit, the chiral multiplet contribution of the index
where R is the character of the representation R of G.
bundle degrees (p; q) is given by
The index of the theory in class S corresponding to the UV curve Cg;n with normal
I(ai; p; q; ) =
X(C+)p(C )
i (ai) ;
n
q Y
i=1
where
labels the representations of
and
(z) is the wave function associated to the
puncture of color
. The wave function in the mixed Schur limit becomes
=
1
q
+(z) = PE
IYM = (p; p)(q; q)
=
X
m2Z 0
from the TQFT.
(5.9)
(5.10)
(z) = Ic(h0i;2)(z) +(z) ;
+(z) = Ic(h2i;0)(z)
(z) :
This explains why we attach
avor adjoint chiral multiplets to the oppositely colored
punctures.
conformal, N
Equipped with the wave functions for the irregular punctures, we can easily compute
the index in the mixed Schur limit. Note that even when the N = 2 counterpart is
non= 1 version can actually
ow to a SCFT in certain cases. For example,
SU(3) theory with Nf = 5 is nonconformal for N = 2, but it is in the conformal window
for N = 1 theory. We can indeed compute the indices for these cases using the result of
section 2. In section 5.2, we consider simplest examples.
5.2
SU(
2
) SYM with Nf = 0; 1; 2; 3
In this section, we verify that the mixed Schur index is indeed reproduced by the generalized
TQFT we discussed. Pure YM. Let us compute the index naively by using the UV matter content. Here we only have a vector multiplet. We get [66]
I
dz
2 iz
(z)(qz 2; q)(pz 2; q) =
1
2
X
m;n2Z
p 21 m(m+1)q 12 n(n+1)
I
dz
2 iz
(pq) 21 m(m+1) =
X
I2; 1;+ I2; 1; ;
z2(m n)
(5.11)
The wave functions for the irregular punctures with color
choosing the N = 2 wave function with di erent arguments
Ik;n (q). In this limit, the structure constant can be also simply written as
=
Ik;n;+ =
are given by simply
Ik;n (p) and
Ik;n;
=
C+ = C (q)
C
= C (p) ;
where C (q) is the structure constant for N = 2 theory.
transforming under the adjoint of the avor symmetry:
Note that one can ip the color of the wave function by attaching a chiral multiplet
HJEP02(16)45
where we used the Jacobi triple product identity. We indeed get the mixed Schur index
SQCD with Nf = 1. This theory has a dynamically generated runaway superpotential,
therefore we cannot de ne proper superconformal index. Nevertheless, we compute the
index at the UV
xed point with incorrect Rcharges for the chiral multiplets. Namely,
we pick R = 12 for the chiral multiplets. This value is the correct Rcharge for the
massdeformed N = 2 SCFTs such as SU(N ) theory with 2N
avors.
I
2 iz
a
2
2
= 1 +
+ t4
t
y
+ t2
2a2
y2
1
a2 + y2
2
5
a2y2 + y4
+ t3
1
y2
+ O(t5) ;
Now, we apply the integral formula for the index to compute
INf =1(p; q; a) = (p; p)(q; q)
(z)
INf =1+1(p; q; a; b) = (p; p)(q; q)
which agrees with the TQFT expression
dz
2 iz
(z) 1
(q 2 z a ; q)(p 2 z b ; p)
dz
2 iz
(z) 1
(q 2 z a ; q)(q 2 z b ; q)
ITQFT(p; q; x; y) = X C
I2; 1;
+(x) +(y) ;
where p = ty; q = t=y. This result agrees with the TQFT on a sphere with one irregular
puncture I2: 1 and one regular puncture with (p; q) = (0; 0), which is given by
HJEP02(16)45
ITQFT(p; q; a) = X
I2; 1;+
(a) =
X
(
1
)mp 12 m(m+1) Rm(a) : (5.13)
1
(qa 2;0; q) m2Z 0
Note that we have only kept the diagonal subgroup of the full avor symmetry
U(
1
)L
U(
1
)R.
SQCD with Nf = 2. This theory without superpotential con nes with a deformed
moduli space, but we perform the computation with the same philosophy as before. Let
us take the (wrong) Rcharge 1=2 to compute the index. We have two di erent ways
to construct the theory, as we have discussed in the case of N
= 2 counterpart. Here
depending on the choice of the colors on the punctures we actually get di erent indices
because these choices determine the superpotential that are allowed [67].
Let us rst consider the case with two irregular punctures I2;0 of each color. This
con guration realizes the theory with a quartic superpotential between two quarks. We
write the index as
INf =2+0(p; q; a; b) = (p; p)(q; q)
It agrees with the TQFT expression
upon identifying a = xy; b = x=y.
ITQFT(p; q; a; b) = X
I2;0;+(a) I2;0; (b) :
Now, let us consider a 3punctured sphere realization of Nf = 2 theory. We have two
regular + punctures, and one irregular I2; 1 with
color. We pick the normal bundle
degrees to be (1; 0). This realizes the Nf = 2 theory without quartic superpotential, which
gives the index to be
1
1
;
:
I
I
(5.14)
(5.15)
(5.16)
(5.17)
SQCD with Nf = 3.
This theory can be realized by a sphere with two regular punctures
of + color and one irregular puncture I2;0 with
color, and normal bundle degrees (p; q) =
(1; 0). It splits 3
avors into 2 + 1 with a quartic superpotential interaction. The index
INf =2+1(p; q; a; b; c) = (p; p)(q; q)
I
2 iz
(z)
1
(q 2 z a ; q)(q 2 z b ; q)(p 2 z c ; p)
1
1
: (5.18)
We nd that it agrees with the TQFT expression
ITQFT(p; q; x; y) =
X C
+(x) +(y) I2;0; (c) ;
(5.19)
upon identifying a = xy; b = x=y.
Acknowledgments
The author would like to thank Abhijit Gadde, Ken Intriligator, Yuji Tachikawa and
Wenbin Yan for useful discussions and correspondence. The author is grateful for the hospitality
of the Simons Center for Geometry and Physics during the 2015 Simons Workshop in
Mathematics and Physics and also Korea Institute for Advanced Study. This work is supported
by the US Department of Energy under UCSD's contract desc0009919.
Open Access.
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
Approximation, arXiv:0907.3987 [INSPIRE].
[3] J. Kinney, J.M. Maldacena, S. Minwalla and S. Raju, An Index for 4 dimensional super
conformal theories, Commun. Math. Phys. 275 (2007) 209 [hepth/0510251] [INSPIRE].
[4] C. Romelsberger, Counting chiral primaries in N = 1, D = 4 superconformal eld theories,
Nucl. Phys. B 747 (2006) 329 [hepth/0510060] [INSPIRE].
[5] A. Gadde, E. Pomoni, L. Rastelli and S.S. Razamat, Sduality and 2d Topological QFT,
JHEP 03 (2010) 032 [arXiv:0910.2225] [INSPIRE].
[6] A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, The 4d Superconformal Index from
qdeformed 2d YangMills, Phys. Rev. Lett. 106 (2011) 241602 [arXiv:1104.3850] [INSPIRE].
[7] A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, Gauge Theories and Macdonald
Polynomials, Commun. Math. Phys. 319 (2013) 147 [arXiv:1110.3740] [INSPIRE].
[8] D. Gaiotto, L. Rastelli and S.S. Razamat, Bootstrapping the superconformal index with
surface defects, JHEP 01 (2013) 022 [arXiv:1207.3577] [INSPIRE].
[9] L. Rastelli and S.S. Razamat, The superconformal index of theories of class S,
arXiv:1412.7131 [INSPIRE].
(2012) 450 [arXiv:1206.5966] [INSPIRE].
869 (2013) 493 [arXiv:1210.2855] [INSPIRE].
(2015) 456 [arXiv:1505.06565] [INSPIRE].
86 (2012) 065015 [arXiv:1109.0283] [INSPIRE].
[13] T. Kawano and N. Matsumiya, 5D SYM on 3D Deformed Spheres, Nucl. Phys. B 898
HJEP02(16)45
[14] F. Benini, T. Nishioka and M. Yamazaki, 4d Index to 3d Index and 2d TQFT, Phys. Rev. D
[15] L.F. Alday, M. Bullimore and M. Fluder, On Sduality of the Superconformal Index on Lens
Spaces and 2d TQFT, JHEP 05 (2013) 122 [arXiv:1301.7486] [INSPIRE].
[16] S.S. Razamat and M. Yamazaki, Sduality and the N = 2 Lens Space Index, JHEP 10 (2013)
048 [arXiv:1306.1543] [INSPIRE].
[17] N. Mekareeya, J. Song and Y. Tachikawa, 2d TQFT structure of the superconformal indices
with outerautomorphism twists, JHEP 03 (2013) 171 [arXiv:1212.0545] [INSPIRE].
[18] M. Lemos, W. Peelaers and L. Rastelli, The superconformal index of class S theories of type
D, JHEP 05 (2014) 120 [arXiv:1212.1271] [INSPIRE].
(2015) 173 [arXiv:1309.2299] [INSPIRE].
[19] O. Chacaltana, J. Distler and A. Trimm, Tinkertoys for the Twisted DSeries, JHEP 04
[20] P. Agarwal and J. Song, New N = 1 Dualities from M5branes and Outerautomorphism
Twists, JHEP 03 (2014) 133 [arXiv:1311.2945] [INSPIRE].
[21] O. Chacaltana, J. Distler and A. Trimm, Tinkertoys for the E6 theory, JHEP 09 (2015) 007
[22] O. Chacaltana, J. Distler and A. Trimm, Tinkertoys for the Twisted E6 Theory,
[arXiv:1403.4604] [INSPIRE].
arXiv:1501.00357 [INSPIRE].
arXiv:1509.00847 [INSPIRE].
[23] D. Xie, General ArgyresDouglas Theory, JHEP 01 (2013) 100 [arXiv:1204.2270] [INSPIRE].
[24] Y. Wang and D. Xie, Classi cation of ArgyresDouglas theories from M5 branes,
[25] P.C. Argyres and M.R. Douglas, New phenomena in SU(3) supersymmetric gauge theory,
Nucl. Phys. B 448 (1995) 93 [hepth/9505062] [INSPIRE].
[26] P.C. Argyres, M.R. Plesser, N. Seiberg and E. Witten, New N = 2 superconformal eld
theories in fourdimensions, Nucl. Phys. B 461 (1996) 71 [hepth/9511154] [INSPIRE].
[27] M. Buican and T. Nishinaka, On the superconformal index of ArgyresDouglas theories,
J. Phys. A 49 (2016) 015401 [arXiv:1505.05884] [INSPIRE].
[28] M. Buican, S. Giacomelli, T. Nishinaka and C. Papageorgakis, ArgyresDouglas Theories and
Sduality, JHEP 02 (2015) 185 [arXiv:1411.6026] [INSPIRE].
Symmetries, J. Phys. A 49 (2016) 045401 [arXiv:1505.06205] [INSPIRE].
JHEP 01 (2016) 040 [arXiv:1506.00265] [INSPIRE].
HJEP02(16)45
Inequality, arXiv:1509.05402 [INSPIRE].
(2014) 036 [arXiv:1212.1467] [INSPIRE].
[arXiv:1307.5877] [INSPIRE].
03 (2013) 006 [arXiv:1301.0210] [INSPIRE].
[36] F. Benini, Y. Tachikawa and B. Wecht, Sicilian gauge theories and N = 1 dualities, JHEP
[37] I. Bah, C. Beem, N. Bobev and B. Wecht, FourDimensional SCFTs from M5Branes, JHEP
[38] C. Beem and A. Gadde, The N = 1 superconformal index for class S xed points, JHEP 04
[39] D. Xie, M5 brane and four dimensional N = 1 theories I, JHEP 04 (2014) 154
[40] D. Xie and P. Zhao, Central charges and RG ow of stronglycoupled N = 2 theory, JHEP
[41] D. Gaiotto, Asymptotically free N = 2 theories and irregular conformal blocks, J. Phys.
Conf. Ser. 462 (2013) 012014 [arXiv:0908.0307] [INSPIRE].
[42] M. Taki, On AGT Conjecture for Pure Super YangMills and Walgebra, JHEP 05 (2011)
038 [arXiv:0912.4789] [INSPIRE].
[43] C.A. Keller, N. Mekareeya, J. Song and Y. Tachikawa, The ABCDEFG of Instantons and
Walgebras, JHEP 03 (2012) 045 [arXiv:1111.5624] [INSPIRE].
[44] L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville Correlation Functions from
Fourdimensional Gauge Theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219]
[45] N. Wyllard, A(N1) conformal Toda eld theory correlation functions from conformal N = 2
SU(N ) quiver gauge theories, JHEP 11 (2009) 002 [arXiv:0907.2189] [INSPIRE].
[46] G. Bonelli, K. Maruyoshi and A. Tanzini, Wild Quiver Gauge Theories, JHEP 02 (2012) 031
[arXiv:1112.1691] [INSPIRE].
[47] D. Gaiotto and J. Teschner, Irregular singularities in Liouville theory and ArgyresDouglas
type gauge theories, I, JHEP 12 (2012) 050 [arXiv:1203.1052] [INSPIRE].
superconformal eld theories, JHEP 03 (2013) 147 [arXiv:1301.0721] [INSPIRE].
superconformal eld theories, JHEP 10 (2012) 054 [arXiv:1206.4700] [INSPIRE].
HJEP02(16)45
branches of 3d N = 4 gauge theories, JHEP 01 (2014) 005 [arXiv:1309.2657] [INSPIRE].
99 [arXiv:1403.6107] [INSPIRE].
4d N = 2 (An; Am) SCFTs, Nucl. Phys. B 894 (2015) 439 [arXiv:1403.6523] [INSPIRE].
network, JHEP 12 (2013) 092 [arXiv:1309.3050] [INSPIRE].
arXiv:1509.00033 [INSPIRE].
[arXiv:1510.03866] [INSPIRE].
056 [arXiv:1303.0836] [INSPIRE].
m = 1,
[1] D. Gaiotto , N = 2 dualities , JHEP 08 ( 2012 ) 034 [arXiv: 0904 .2715] [INSPIRE].
[2] D. Gaiotto , G.W. Moore and A. Neitzke , Wallcrossing, Hitchin Systems and the WKB [10] M. Aganagic , H. Ooguri , N. Saulina and C. Vafa , Black holes, qdeformed 2d YangMills and nonperturbative topological strings , Nucl. Phys. B 715 ( 2005 ) 304 [ hep th/0411280] [11] T. Kawano and N. Matsumiya , 5D SYM on 3D Sphere and 2D YM, Phys. Lett. B 716 [12] Y. Fukuda , T. Kawano and N. Matsumiya , 5D SYM and 2D q Deformed YM , Nucl. Phys. B [29] M. Buican and T. Nishinaka , ArgyresDouglas Theories , S1 Reductions and Topological [30] M. Buican and T. Nishinaka , ArgyresDouglas Theories , the Macdonald Index and an RG [31] C. Beem , M. Lemos , P. Liendo , W. Peelaers , L. Rastelli and B.C. van Rees , In nite Chiral Symmetry in Four Dimensions, Commun. Math. Phys. 336 ( 2015 ) 1359 [arXiv: 1312 .5344] [32] C. Beem , W. Peelaers , L. Rastelli and B.C. van Rees , Chiral algebras of class S, JHEP 05 [33] M. Lemos and W. Peelaers , Chiral Algebras for Trinion Theories , JHEP 02 ( 2015 ) 113 [34] C. Cordova and S.H. Shao , Schur Indices, BPS Particles and ArgyresDouglas Theories , [35] S. Cecotti , A. Neitzke and C. Vafa , RTwisting and 4d/2d Correspondences, [48] H. Kanno , K. Maruyoshi , S. Shiba and M. Taki , W3 irregular states and isolated N = 2 [49] G. Andrews , A. Schilling and S. Warnaar , An A2 Bailey lemma and RogersRamanujantype identities , J. Am. Math. Soc . 12 ( 1999 ) 677 .
[50] P.C. Argyres , K. Maruyoshi and Y. Tachikawa , Quantum Higgs branches of isolated N = 2 [51] K.A. Intriligator and N. Seiberg , Mirror symmetry in threedimensional gauge theories , Phys.
Lett . B 387 ( 1996 ) 513 [ hep th/9607207] [INSPIRE].
[52] P. Boalch , Irregular connections and KacMoody root systems, arXiv: 0806 . 1050 .
[54] S.S. Razamat and B. Willett , Down the rabbit hole with theories of class S , JHEP 10 ( 2014 ) [56] D. Gaiotto and E. Witten , Sduality of Boundary Conditions In N = 4 Super YangMills Theory , Adv. Theor. Math. Phys. 13 ( 2009 ) 721 [arXiv: 0807 .3720] [INSPIRE].
[57] K. Maruyoshi , C.Y. Park and W. Yan , BPS spectrum of ArgyresDouglas theory via spectral [63] A. Gadde , K. Maruyoshi , Y. Tachikawa and W. Yan , New N = 1 Dualities, JHEP 06 ( 2013 ) [64] P. Agarwal , I. Bah, K. Maruyoshi and J. Song , Quiver tails and N = 1 SCFTs from