(p, q)webs of DIM representations, 5d \( \mathcal{N}=1 \) instanton partition functions and qqcharacters
HJE
(p; q)webs of DIM representations, 5d instanton partition functions and qqcharacters
J.E. Bourgine 0 1 2 4
M. Fukuda 0 1 3
K. Harada 0 1 3
Y. Matsuo 0 1 3
R.D. Zhu 0 1 3
Bunkyoku 0 1
Tokyo 0 1
Japan 0 1
0 Via Irnerio 46 , 40126 Bologna , Italy
1 85 Hoegiro , Dongdaemungu, Seoul , South Korea
2 Sezione INFN di Bologna, Dipartimento di Fisica e Astronomia, Universita di Bologna
3 Department of Physics, The University of Tokyo
4 Korea Institute for Advanced Study (KIAS), Quantum Universe Center , QUC
Instanton partition functions of N = 1 5d Super YangMills reduced on S1 can be engineered in type IIB string theory from the (p; q)branes web diagram. To this diagram is superimposed a web of representations of the DingIoharaMiki (DIM) algebra that acts on the partition function. In this correspondence, each segment is associated to a representation, and the (topological string) vertex is identi ed with the intertwiner operator constructed by Awata, Feigin and Shiraishi. We de ne a new intertwiner acting on the representation spaces of levels (1; n) (0; m) ! (1; n + m), thereby generalizing to higher rank m the original construction. It allows us to use a folded version of the usual (p; q)web diagram, bringing great simpli cations to actual computations. As a result, the characterization of Gaiotto states and vertical intertwiners, previously obtained by some of the authors, is uplifted to operator relations acting in the Fock space of horizontal representations. We further develop a method to build qqcharacters of linear quivers based on the horizontal action of DIM elements. While fundamental qqcharacters can be built using the coproduct, higher ones require the introduction of a (quantum) Weyl re ection acting on tensor products of DIM generators.
Quantum Groups; Supersymmetric Gauge Theory; Topological Strings; D

4
Quantum Weyl re ection and qqcharacters
Horizontal intertwiner and qqcharacter for the A1 quiver
4.1.3
Higher qqcharacters 4.2
Quantum Weyl re ection
De nition of the horizontal intertwiner Horizontal intertwiner as screening current and fundamental qq character
Horizontal intertwiner for the A2 quiver
Fundamental qqcharacter for the A2 quiver
Quantum Weyl re ection and the second qqcharacter
Generalization to the Ar quivers
Inclusion of fundamental/antifundamental matter elds
1 Introduction 2
DIM algebra and representations 3
Generalized AFS intertwiners
DIM algebra
Quantum torus and DIM
Vertical (0; m) representation
Horizontal (1; n) representations
Reminder on 5d N = 1 instanton partition functions
Discrete symmetries of DIM algebra
De nition of the generalized intertwiners
AFS lemmas
Gaiotto state
Vertical intertwiner
2.1
2.2
2.3
2.4
2.5
2.6
3.1
3.2
3.3
3.4
4.1
4.1.1
4.1.2
4.2.1
4.2.2
4.2.3
4.2.4
4.2.5
5
Perspectives
A Di erent expressions for the vertical representation B
Useful formulas for the horizontal representation
B.1 qbosons vertex operators
B.2 Commutation relations in horizontal representations
C
Derivation of the AFS lemmas
D Connection with quiver Walgebras E
Derivation of the qqcharacters for the A3 quiver
{ 1 {
Introduction
Duality has been one of the most fundamental issues in string/gauge theories, and it has
been studied from many di erent viewpoints and in various contexts. One of the standard
approaches to the problem is to assign a brane con guration to the gauge dynamics, and
interpret the duality in graphical ways. Such an approach has been taken in N = 2
superYangMills in 4 dimensions (and N = 1 in 5d). The corresponding graphical object, the
SeibergWitten curve, is expressed through various con gurations of D and NSbranes.
For example, supersymmetric gauge theories with N = 1 supercharges in
ve dimensions
can be engineered in type IIB string theory using the methods developed in [1]. Linear
quiver gauge theories with U(m) gauge groups are obtained from webs of (p; q)branes that
are bound states of p D5 branes and q NS5 branes [2, 3].
A useful algebraic tool to analyze brane con gurations is the topological vertex [4].
It has been introduced to reproduce the topological string amplitude on toric CalabiYau
manifolds. In fact, the toric diagram of CalabiYau threefold can be identi ed with the
(p; q)branes web diagrams of type IIB string theory [5]. From this identi cation, it is
possible to build the instanton partition function of the gauge theory using the machinery
of topological string theory. In order to recover the Nekrasov partition function [6] in a
general Omegabackground, it is necessary to re ne the de nition of the topological vertex
to include the graviphoton background [7]. The importance of this representation of gauge
theories in the context of the BPS/CFT correspondence was rst realized in [8, 9] where
the connection with the decomposition of conformal blocks was also investigated.
In [10, 11], Nekrasov partition functions have been studied from a di erent perspective,
namely through the representation of underlying quantum algebras: the spherical double
a ne Hecke algebra with central charges (SHc) [12] for the 4d gauge theory, and its quantum
deformation, the DingIoharaMiki (DIM) algebra [13{15] for the 5d gauge theory. The
representations of these algebras coincide with those of (quantum) WN algebra [12, 16{18]
while the basis of the representation coincides with the set of xed points which represent
the equivariant cohomology of the instanton moduli space. This observation was essential
in the proof of the AGT conjecture elaborated in [12]. In addition, the presence of these
algebras re ects the integrable nature of the BPS sector of the gauge theory. It led to
the construction of R and T matrices satisfying the standard RT T relation of quantum
integrable systems [19{22].
The main focus of our previous works [10, 11] was the derivation of the qqcharacters
from SHc/DIM algebras. These particular correlators of the gauge theory were rst
introduced in [23], and further studied in [24{28]. They have the essential property to be
polynomials, thus de ning a resolvent for the matrix model representing the localized gauge
partition function. These quantities generalize the qcharacters of quantum groups de ned
in [29, 30], and naturally associated to the T operators of integrable systems. In [10, 11],
we have shown that the representation theoretical properties of the Gaiotto state and the
intertwiner associated with bifundamental matter are directly translated into the regularity
property of qqcharacters.
{ 2 {
A di erent construction of qqcharacters has been presented by Kimura and Pestun
(KP) in [31] (see also [32]). While both constructions are based on quantum Walgebras,
the action of these algebras is seemingly di erent. In our approach, a copy of the DIM
algebra is associated to each node of the quiver diagram, so that a quantum Wm algebra is
attached to each gauge group U(m). On the other hand, the quiver Walgebras constructed
in [31] is based on a Lie algebra whose Dynkin diagram coincides with the gauge theory
quiver. In a sense, the two approaches are Sdual to eachother: in our approach the rank
is the number of Dbranes while it is the number of NS5branes in KP's work. On the
algebraic level, the Sduality is believed to be realized by Miki's automorphism [
14
] that
exchanges the labels (l1; l2) of DIM representations, here identi ed with the (p; q) charge
It was realized in [33, 34] that the two di erent pictures can be better understood
using the re ned topological vertex. Indeed, in [35], Awata, Feigin and Shiraishi (AFS)
have reconstructed this object using the generators of the DIM algebra where it plays the
role of an intertwiner between vertical (0; m) (associated to m Dbranes) and horizontal
(1; n) (associated to a NS5brane bound to n Dbranes) representations [35]. Hence, like
a string junction, it interpolates between the representations associated to di erent brane
charges. In this way, di erent representations of DIM algebra can be combined to form a
representation web [33, 34] that can be identi ed with the (p; q)web diagram engineering
the gauge theory.2 This presentation clari es the two approaches for the construction of
qqcharacters: KP employed DIM generators in horizontal representations [36] while we
used similar generators but in vertical representations [16, 33].
The purpose of this paper is to propose a uni ed method to build qqcharacters and
prove their regularity property. As suggested in [33], the method is based on insertions
of DIM operators in the horizontal representations. However, it also makes use of the
commutation of vertical actions which was instrumental in our previous derivation [11].
The link is made by a set of lemmas that intertwines horizontal and vertical actions on AFS
intertwiners. The AFS lemmas can be regarded as an uplift in the horizontal representation
space of the relation characterizing Gaiotto state and vertical intertwiners obtained in [11].
With this new method, we derive all the (higher) qqcharacter of linear quiver theories
with U(m) gauge groups, thus largely extending our previous results that were restricted
to fundamental qqcharacters. In order to achieve this general treatment of quivers, several
new insights were necessary. First, we generalized the AFS intertwiner to higher level m
of vertical representations, allowing us to treat gauge groups of arbitrary rank. Most of
the previous considerations, including results on integrability, were restricted to gauge
groups of rank one or two. Then, in order to build higher qqcharacters, a Weyl re ection
acting on (tensor products of) DIM generators has been introduced. Called quantum Weyl
1Due to a di erent choice of conventions for DIM representations, the usual representation of (p;
q)branes webs is rotated by 90 degrees, with the NS charge q in the horizontal direction and the Ramond
2In [33, 34] this structure was called a network matrix model. However, since we do not use the matrix
model presentation of partition functions, we prefer not to employ this terminology here.
{ 3 {
transformation, it keeps the qqcharacter invariant. In practice, it is used to build operators
commuting with a T operator, the vacuum expectation value (vev) of which reproduces
the instanton partition function. This commutation property is directly related to the
regularity property of the qqcharacter, thus providing another link with the manifestation
of integrability in supersymmetric gauge theories.
This paper is organized as follows. The second section provides the main properties
of DIM algebra and its vertical and horizontal representations. We put some emphasis
on the various duality properties, including the SL(2,Z) automorphisms. This section also
includes a brief reminder on N = 1 5d gauge theories. The third section starts from the
de nition of the AFS intertwiners and proposes a generalization obtained from horizontal
composition. The generalized intertwiners simplify the computation of amplitudes
associated to the braneweb. In sections 3.3 and 3.4, these intertwiners are used to reconstruct
the Gaiotto state and the vertical intertwiner built in [11]. In the fourth section, the
horizontal intertwiner is de ned by taking vertical contractions of generalized AFS vertices.
It is shown that it commutes with the coproduct of DIM generators. The quantum Weyl
transformation is de ned in the section 4.2 as an operation on the tensor product of
generators. It leads to a systematic method to construct qqcharacter which is the main result
of the paper. Finally, the details of computations can be found in the appendix, along with
several useful identities.
2
2.1
DIM algebra and representations
DIM algebra
The DingIoharaMiki algebra E [
13, 14
] can be presented in terms of the Drinfeld currents
k are usually associated to points of a Z 2lattice representing the
elements of the algebra (see gure 1). We assign Z 2degree for generators as deg(xn ) =
( 1; n), deg( n ) = (0; n). The notations and conventions used here are mostly borrowed
from [21], up to minor di erences in the normalization of operators. The qcommutation
relations satis ed by the currents read
+(z)
(w) =
g(^w=z)
g(^ 1w=z)
(w) +(z)
+(z)x (w) = g(^ 1=2w=z) 1x (w) +(z);
(z)x (w) = g(^ 1=2z=w) 1x (w)
(z)
x (z)x (w) = g(z=w) 1x (w)x (z)
[x+(z); x (w)] =
(1
(1
q1)(1
q2)
q1q2)
(^ 1z=w) +(^1=2w)
(^z=w)
(^ 1=2w) ; (2.2)
with ^ a central element. This algebra has two independent parameters encoded in the
q with
= 1; 2; 3 under the relation q1q2q3 = 1. It is sometimes more convenient to use
instead the parameters q = q2 and t = q1 1, in particular in the context of representations
{ 4 {
group SL(2,Z) acts a 90 degrees rotation.
over Macdonald polynomials. We will also introduce the notation
These parameters appear through the functions
3
= q1=2 = t1=2q 1=2.
g(z) =
Y
of two operators, they are sometimes called scattering factors.
In this paper, representations of level (l1; l2) 2 Z
Z with a weight u will be denoted
(l1;l2) or sometimes simply (l1; l2)u. The levels are de ned through the representations of
u
the central element ^ and the zero modes 0 ,
(l1;l2)(^) = q3l1=2;
u
(l1;l2)( 0 )
u
(l1;l2)( 0+)
u
= q3l2 :
Here, we will focus on the socalled vertical representations (0; m) and horizontal
representations (1; n). The intertwiner de ned in the next section relates three di erent
representations, it will be portrayed as a threelegged vertex. Products and tensor products of such
operators can be described using diagrams resembling the (p; q)web diagrams of brane
con gurations in type IIB theory, they will be called representation webs. Note however,
that here diagrams are rotated by 90 , such that vertical lines (0; m) are associated to m
D5 branes and the horizontal line (1; 0) to a NS5 brane. Note also that we will take no care
of the precise slope of horizontal lines: vertical lines in the diagrams will always refer to a
vertical representation (0; m), while horizontal and inclined lines can represent any of the
horizontal representations (1; n). To avoid confusion, the representation space associated
to each line will be explicitly written on every gure.
{ 5 {
(2.3)
(2.4)
The DIM algebra is a Hopf algebra with the following coproduct:
tion the simplest representation with level (0; 0) which may be identi ed with the symmetry
of a quantum mechanical system. We consider the noncommutative torus generated by the
two operators U; V satisfying
V U = q1U V:
where q1 is not a root of unity. The enveloping algebra of U; V is generated by the elements
wrs = U rV s identi ed as the degree (r; s) generators de ning the algebra
[wr1s1 ; wr2s2 ] = (q1s1r2
q1s2r1 )wr1+s1;r2+s2 :
This algebra has a SL(2; Z) duality realized as the rede nition of the basis, U 0 =
In particular, the Stransformation is realized as S : (U; V ) ! (V; U 1).
AU aV b, V 0 = BU cV d (a; b; c; d 2 Z) which satis es V 0U 0 = q1U 0V 0 as long as ad
In this simple setup, the DIM algebra with (l1; l2) = (0; 0) is realized as
bc = 1.3
(0;0)(x+(z)) =
u
(0;0)(
u
(1 (V =z) 1)(1 (q1 1V t=z) 1)
(0;0)(x (z)) =
u
X xn z n =
n
= exp
X1 1
r=1
r
(1
1
1
1
q 1 (V =z)U 1
;
q2 r)(1
q3 r)V r
z r
{ 6 {
n2Z
with (z) = P
zn. In this representation, the expression (2.5) of the coproduct simpli es
as (u0;0(^) = 1. In the vector representation, the generators U; V are represented on a basis
labeled by a single integer,
(0;0)(U )ju; ii = ju; i + 1i;
u
(0;0)(V )ju; ii = uq1iju; ii ;
u
where u is the weight of the representation.
3A di erent but similar duality structure is realized by writing q1 =: e2 i 1 and de ning a SL(2,Z)
modular transformation for 1:
10 = ac 11++db . Then, there exists two generators U~ ; V~ they satisfy the
quantum torus algebra with q10 = e2 i 10 and that commute with the original generators [U nV m; U~ rV~ s] =
This duality is known as the Morita equivalence, it plays a fundamental role in noncommutative
geometry [37, 38], and is also relevant in more recent works such as [39].
(2.5)
(2.6)
(2.7)
!
(2.8)
(2.9)
HJEP1(207)34
It is of some interest to compare the DIM algebra with the loop algebra g^ of a Lie
algebra g. The generators of g^ (without the central extension) are de ned in terms of the
generator ta of the Lie algebra as Jna = taU n where U is a formal variable. DIM algebra
is a natural generalization of this setting in which two formal variables are introduced. It
sometimes referred to as a twoloop symmetry. The algebra (2.7) depending on a single
deformation parameter q1 can be extended by two central charges, c1; c2 [
14
],
In this formulation, the SL(2,Z) symmetry is manifest, and the Stransformation is realized
c1. The introduction of a second quantization
representation of the two central charges, (ul1;l2)(c1) = q3l2 ; u
parameter q2 leads to the DIM algebra [
14
]. One may identify the levels (l1; l2) with the
The vertical representation (0; 1) has been formulated in [15, 16], it is equivalent to the
rank m representation studied in [11] with m = 1 and up to a normalization. Here we
employ conventions similar to the ones used in [11], but with di erent states normalization
(the change of the convention is summarized in appendix A).
The (0; m) representations depend on a mvector of weights ~v = (v1;
vm) and act
on a space spanned by states in onetoone correspondence with mtuple Young diagrams
~ = ( (1);
; (m)),
(0;m)(x+(z))j~v; ~ ii =
~v
m 1z (m 1)
(0;m)(x (z))j~v; ~ ii =
~v
2m+1zm 1
~v
(z))j~v; ~ ii =
m
~ (z)
j~v; ~ ii:
X
(z= x) zR=esx zY~ (z) j~v; ~ + xii;
(z= x) zR=esx z 1Y~ (zq3 1)j~v; ~
xii;
(2.11)
where A(~ ) and R(~ ) denote respectively the set of boxes that can be added to or removed
from the set of the Young diagrams 1
;
; m. These expressions involve the functions
~ (z) and Y~ (z) that depend on a mtuple Young diagram. Their expression can be
factorized in terms of individual Young diagram contributions,
m
Y
l=1
Y~ (z) =
Y l (z);
(z) = Y (zq3 1)
;
Y (z)
~ (z) = Y
m
l=1
Y (z) = 1
(l) (z);
v
z
x2
Y S( x=z) = Q
Q
x2A( ) 1
x2R( ) 1
z 1 x
(zq3) 1 x
Here, each box x 2
position of the box in (l). The associated box coordinate reads x = vlq1i 1qj 1
~ is de ned by three integer labels (l; i; j) such that (i; j) indicates the
As in [11], it will be important to add a set of diagonal operators Y (z) such that
Y (z)j~v; ~ ii = Y~ (z)
j~v; ~ ii :
{ 7 {
(2.12)
; (2.13)
(2.14)
and ^(v0;m)(^) = 1. Strictly speaking, this is not a representation of the DIM algebra
because some of the qcommutation relations are no longer satis ed. Instead, it should be
seen as a representation on the dual states hh~v; ~ j that are orthogonal to the basis j~v; ~ ii,4
hh~v; ~ j~v; ~ 0ii = ~ ;~ 0 a~ 1;
such that we have the property
hh~v; ~ j ^~v
(0;m)(e) j~v; ~ 0ii = hh~v; ~ j
(0;m)(e)j~v; ~ 0ii
~v
for any element e of the DIM algebra. The norm of the states involves the coe cients
a~ which will play an important role in the construction of instanton partition functions.
They are de ned in terms of the vector multiplet contribution to the instanton partition
function Zvect.(~v; ~ ) as follows,
The notation [
] refers to an expansion in powers of z 1 of the quantity inside the
brackets. They will be used to de ne the qqcharacter.
The action on the bra states will be referred to as the dual vertical representation. In
this representation, the roles of x+ and x are exchanged:
hh~v; ~ j ^~v
(0;m)(x+(z)) =
hh~v; ~ j ^~v
(0;m)(x (z)) =
1
X
(z= x) zR=esx zY~ (z) hh~v; ~ + xj;
hh~v; ~ j ^~v
(0;m)(
The vector contribution Zvect.(~v; ~ ) will be de ned in the section 2.5 below, it is expressed
in terms of the Nekrasov factor (2.33) as a result of localization. As shown in [11, 40], the
Nekrasov factor obeys a set of discrete Ward identities. Consequently, the coe cients a~
also obey similar identities. They can be written in terms of the function Y~ (z) as
a~ +x =
a~ x =
a~
a~
(1
1
q1q2
q1)(1
1
q1q2
q2)
(1
q1)(1
q2)
m
x
m Res
z= x Y~ (z)Y~ (zq3 1) ;
1
z= x
m m 2 Res Y~ (z)Y~ (zq3 1):
x
4Due to the change of states normalization performed in appendix A, and since the original states were
orthonormal, the coe cient a~ is expected to be proportional to N (~ ) 2. The additional factors are chosen
to simplify the formulation of the AFS lemmas below.
m
l=1
{ 8 {
Horizontal representations [36] of level (1; n) act as a vertex operator algebra in the Fock
space of qbosonic modes with the commutation relations5
The representations involve the positive/negative modes of the vertex operator
and can be de ned in terms of the following operators:
(z) = V (z)V+(z);
(z) = V ( z) 1V+(z= ) 1;
' (z) = V (
1=2z)V (
3=2z) 1:
Explicitly, we have
Useful commutation relations involving these operators are presented in appendix B. The
horizontal representation (1; n)u reads
(1;n)(x+(z)) = u n
u
z n (z);
(1;n)( +(z)) =
u
n'+(z);
(1;n)(x (z)) = u 1
u
nzn (z);
(1;n)(
u
and (u1;n)(^) = . Similarly, it is possible to de ne the representation ( 1; n)u using the
same vertex algebra,
( 1;n)(x+(z)) = u 1 nzn (z 1);
u
( 1;n)( +(z)) =
u
n' (z 1);
( 1;n)(x (z)) = u
u
n
z n (z 1);
( 1;n)(
u
5Here we use parameters q; t instead of q to follow the convention of [35]: q = q2; t = q1 1. The oscillator
modes can be represented on symmetric polynomials as follows:
Macdonald(a k) = pk;
Macdonald(ak) = k
where pk denotes the powersum symmetric polynomials.
{ 9 {
(2.21)
(2.22)
(2.23)
(2.24)
(2.25)
(2.26)
(2.20)
By de nition, the vacuum state j;i(1;n)u is annihilated by the positive modes ak, and
'+(z)j;i(1;n)u = j;i(1;n)u . The dual vacuum state (1;n)uh;j is annihilated by negative modes,
and (1;n)uh;j' (z) = (1;n)uh;j. The normal ordering, denoted :
:, corresponds to write
all the positive modes on the right, and all the negative modes on the left. Correlators of
operators Oi(zi) acting in the Fock space are de ned as the vacuum expectation values
hO1(z1)
ON (zN )i = (1;n)uh;jO1(z1)
ON (zN )j;i(1;n)u ;
(2.27)
with the radial ordering jz1j > jz2j >
> jzN j.
2.5
Reminder on 5d N = 1 instanton partition functions
The quiver Super YangMills gauge theories with N = 1 in 5d reduced on S1 are
characterized by a simply laced Dynkin diagram
. Each node i 2
is associated to a vector
multiplet with gauge group U(mi), and an exponentiated gauge coupling qi. To each edge
< ij >2
corresponds a bifundamental matter multiplet of mass ij that transforms
under the gauge group U(mi)
U(mj ). In addition, a ChernSimons term of level i coupled
to the gauge group U(mi) can be introduced at each node i. Thus, each node bears two
integer labels (mi; i) with mi > 0 that will later be related to the levels (l1; l2) of DIM
representations. Extra matter elds in the fundamental/antifundamental representation of
will be denoted i(;fj) with j = 1
fi, and i;j
(af) with j = 1
f~i respectively.
the gauge group U(mi) can also be attached to each node, and the corresponding masses
The expression of the instanton contribution to the (Ktheoretic) partition function
re ects the particle content of the theory. It is written as a sum over mituple Young
diagrams ~ i, and each term is factorized into vector, ChernSimons, (anti)fundamental
and bifundamental contributions:
Zinst.[ ] = X
Y qj~ ijZvect.(~vi; ~ i)ZCS( i; ~ i)Zfund.(~ i(f); ~ i)Za.f.(~ i(af); ~ i)
i
f~ ig i2
Y
<ij>2
Zbfd.(~vi~ i; ~vj ; ~ j j ij );
This operation will be very useful in order to express the qqcharacters of the gauge theory.
where the (exponentiated) Coulomb branch vevs ~vi are related to the vacuum expectation
value of the scalar
eld in the gauge multiplets. From this expression, it is possible to
de ne a normalized trace of functions depending on the realization of the set of (tuple)
Young diagrams f~ ig as follows,
DF [f~ ig]
E
gauge
=
1
Zinst.[ ]
XF [f~ ig] Yqj~ ijZvect.(~vi; ~ i)ZCS( i; ~ i)Zfund.(~ i(f); ~ i)
i
f~ ig
Za.f.(~ i(af); ~ i)
i2
Y
<ij>2
Zbfd.(~vi~ i; ~vj ; ~ j j ij ):
(2.28)
(2.29)
HJEP1(207)34
The bifundamental contribution with U(m)
U(m0) gauge group can be decomposed
as a product of Nekrasov factors,6
Zbfd.(~v; ~ ; ~v0; ~ 0j ) = Y
Y N (vl; (l); vl00 ; (l0)0):
Various expressions of the Nekrasov factors have been written, the one given here has been
obtained by solving the discrete Ward identities derived in [11, 40],
N (v1; 1; v2; 2) =
Y
The vector multiplet contribution is expressed in terms of the Nekrasov factors as follows:
Y
Finally, the ChernSimons and fundamental/antifundamental contributions are expressed
in terms of a simple product over all boxes in the Young diagrams,
ZCS( ; ~ ) =
Y ( x) ;
x2~
Zfund.(~ i(f); ~ i) =
Za.f.(~ i(af); ~ i) =
1
Y
fi
Y
The instanton partition functions de ned in (2.28) are invariant under the rescaling
~vi !
i~vi, qi !
i
i qi, ~ (f)
i
!
i~ i(f), ~ (af)
i
!
i~ i(af) and
ij ! ( i= j ) ij .
This
invariance can be used to set the bifundamental masses to a speci c value which simpli es
the algebraic formulation developed here. Thus, from now on, all bifundamental masses
will be set to ij =
1
.
These N
= 1 supersymmetric gauge theories can be engineered in type IIB string
theory [1]. Linear quiver gauge theories with U(m) gauge groups are obtained from webs
of (p; q)branes that are bound states of p D5 branes and q NS5 branes [2, 3]. The branes
occupy the dimensions 01234 corresponding to the spacetime of the 5d gauge theory, plus
an extra onedimensional object (line) in the 56planes. In order to preserve
supersymmetry, the lines have the slope
x6= x5 = p=q, so that the worldvolume of D5branes with
charge (1; 0) occupy the dimensions 012346, i.e. they are vertical segments in the 56plane.
On the other hand, NS5 branes of charge (0; 1) are extended in the 012345 directions, and
correspond to an horizontal line in the 56plane. A representation of DIM algebra has been
6The Nekrasov factors enjoy the property
N (v2; 2; v1q3 1; 1) = ( v1) j 2j( q3v2)j 1j Y
x N (v1; 1; v2; 2):
(2.30)
associated to each brane of the (p; q)web diagram [33, 34]. Representations of level (l1; l2)
correspond to (l2; l1)branes so that horizontal (1; 0) representations are associated to NS5
branes and vertical (0; 1)representations to D5 branes. The topological vertex play the
role of creation/annihilation operators for the (p; q)branes, they will be identi ed with the
generalized AFS intertwiners in the next section.
In [
14
], Miki has introduced an automorphism of the DIM algebra that he denoted . Since
it can be identi ed with the action of Sduality on the (p; q)branes, it will be denoted by S
here. This automorphism leaves the DIM algebra invariant, but map degree (r; l) generator
into degree (l; r) and representations of di erent levels:
( 1; n)u, so that (u1;n)( 2 e) =
in [
14
]), the square of the automorphism takes a rather simple form:
and ^ $ ^ 1, or in terms of generating series,
+(z) $
n+ $
n, xn $ x n
(1=z) and x (z) $ x (1=z).
2 transforms horizontal representations (1; n)u into the representations
By examination of the commutation relations, it is possible to de ne another
transformation T acting on the Drinfeld currents as
T
Vertical representations are invariant under the action of T , and horizontal representations
of level n are mapped to horizontal representations of level n + 1. The operations S and
T obey the properties S
4 = 1 and (ST )3 = 1, so that they generate a group of SL(2; Z)
transformations. To some extent, this group can be identi ed with the modular group of
type IIB string theory. In particular, the Miki automorphism S would correspond to the
Sduality that rotates the (p; q)web diagrams by 90 , exchanging NS5 and D5 branes.
The second duality symmetry in DIM algebra is permutation of three parameters
q1; q2; q3, which is manifest at the level of algebra. This S3 symmetry is sometimes
referred to as a \triality" [18] in connection with higher spin gravity [41]. We note that
the representations of DIM are constructed with the reference to q1; q2. In this sense, the
exchange between q1 and q2 is manifest. In 2D CFT language, such permutation is realized
where
parametrizes the central charge c = (n
1)(1
Q2n(n + 1)) with
(l1; l2) = (l1; l1 + l2):
or, in terms of modes,
T
xk (z) = ^ 1
x
k 1
;
k = ^ 1
k
:
Again, the DIM algebra is invariant, but representations of di erent levels are mapped to
eachother,
Q = p
p
1. In terms of the vertical representation basis, it is realized by taking the
transpose of each Young diagram,
$
0. The other transformations, such as q1 $ q3, are
less obvious. When the parameters are suitably chosen, they are identi ed with the
levelrank duality [41{43] where the correspondence between basis is also more involved [18].
While the SL(2,Z) transformation may be regarded as a Mtheoretical target space duality
since it interchanges Dbrane and NSbrane, the S3 duality may be interpreted as a
worldsheet symmetry since it acts on the Hilbert space of equivalent 2D conformal eld theories.
From the viewpoint of super YangMills, q1; q2 represent the graviphoton background in
the Euclidean planes (01) and (23). In this sense, we sometime denotes the symmetry
q1 $ q2 as (01)(23). On the other hand q1 $ q3 does not have an immediate woldvolume
5 is obtained by replacing the parameters q by their
inverse, e ectively exchanging S(z) with S(q3z), and g(z) with g(z) 1 = g(z 1).7 The
transformation of DIM generators resemble the action of S2, except that x+ and x
are
not exchanged:
(z) $
(1=z), x (z) $ x (1=z), and the central parameter ^ remains
invariant. Thus, representations of level (l1; l2) are mapped to representations of level
( l1; l2) and vertical representations are left invariant.The 5 re ection of vertical (0; m)
representations follows from the transformation of the functions
m
l=1
Y~ (z) ! z m Y( vl) 1
Y~ (z 1);
~ (z) ! q3 ~ (z 1);
m
(2.39)
(1;n)(e) =
u
! q 1
(2.40)
(2.41)
provided that the weights transform as vl ! 1=vl so that x ! 1= x for x 2 ~ . On the other
hand, (1; n)u representations are sent to ( 1; n)u representations so that 5
( 1;n)( 5 e) where the transformation
u
5 sends the background parameters q
together with the modes ak ! tk jkjak and the weights u !
and (z) are exchanged but ' (z) remain invariant.
2nu 1. In this manner, (z)
The action of the 5 symmetry on instanton partition functions is closely related to
the re ection symmetry studied in [45], where it relates two dual TQ equations in the
NekrasovShatashvili limit. However, here the Coulomb branch vevs behave di erently,
since vl ! 1=vl. Vector and ChernSimons contributions transform as
Zvect.(~v; ~ ) ! q3 mj~ jZvect.(~v; ~ );
ZCS( ; ~ ) ! ZCS(
; ~ );
and the A1 pure U(m) partition function is invariant provided that the sign of the
ChernSimons level is ipped, and the extra q3factor is absorbed in the transformation q ! q3mq.
On the other hand, the bifundamental contribution transforms into itself, but with the two
nodes exchanged:
Zbfd.(~v1; ~ 1; ~v2; ~ 2j ) ! Zbfd.(~v2; ~ 2; ~v1; ~ 1jq3 1 0)
7Obviously, the two re ections (01)(23) and 5 commute. The composition
(01)(23) 5 acts on the DIM
parameters as the exchange q1 $ q2 1, or t $ q. This is a wellknown symmetry in the context of Macdonald
polynomials, see for instance [44].
where 0 is the re ection of
( 0 =
when
=
1). As a result, the 5 symmetry
for the instanton partition function of linear Ar quiver consists in re ecting the order of
the nodes 123
r ! r(r
1)
1. Hence, this S2symmetry can be interpreted as the
re ection symmetry of the (p; q)web diagram with respect to the horizontal (x5) axis. In
fact, the symmetry
5 also acts as a re ection along the horizontal axis in the graphical
representation of the DIM modes xn and
n (see gure 1).
3
De nition of the generalized intertwiners
The AFS intertwiner operator has been introduced in [35], it generalizes the free fermion
presentation [46] of the topological string vertex to the re ned case. It is built over bosonic
elds that coincide with those introduced in the horizontal representation of DIM algebra,
thus providing a direct link with the representation theory. The original intertwiner acts
in the tensor product of the representation spaces (0; 1)v and (1; n)u, and takes values
in the space (1; n + 1) uv. The vertical space (0; 1)v is spanned by states in one to one
correspondence with Young diagrams . Hence, the intertwiner is a vector in this space
with index ,
(n)[u; v] =
(n)[u; v]
;
(n)[u; v] : (1; n)u ! (1; n + 1) uv:
Both horizontal spaces (1; n)u and (1; n + 1) uv can be identi ed with the same Fock space
of qbosonic modes, and the elements
(n) are expressed in terms of the modes as follows:
The vacuum component is de ned in terms of a new vertex operator
(n)[u; v] = tn( ; u; v) :
;(v) Y
x2
( x) : :
;(v) = V~ (v)V~+(v);
V~ (z) = exp
X1 1
k=1
k 1
z k
q k
!
a k ;
which is related to the previous operator V (z) de ned in (2.22) as
V (z) = V~ (q1z)V~ (q2z)V~ (z) 1V~ (q3 1z) 1:
In fact, the vacuum operator can be obtained as a (normalordered) product of ( x)
factors associated to an in nite Young diagram since
1
Y
i;j=1
V~ (v) =
V (vq1i 1qj 1) 1
2
)
;(v) =: Y
1
i;j=1
(vq1i 1qj 1) 1 : :
2
(3.5)
Thus, this operator is associated to the perturbative part of the partition function,
extending the arguments developed in [47] for the degenerate limit relevant to 4d gauge
theories. Indeed, the prefactors obtained from the normal ordering of two vacuum
intertwiner and involving the function G(z) (de ned in appendix B) should be interpreted as
(3.1)
(3.2)
(3.3)
(3.4)
perturbative (one loop) contributions to the gauge theory partition function. However, to
keep our arguments simple, we will simply neglect these factors and no longer refer to this
interpretation here.
The normalization factor tn( ; u; v) is the vev of the operator
D (n)[u; v]E. It is chosen in order to recover the exact form of the AFS relations presented
below. Its explicit expression depends on the form of the vertical representation, which is
(n), i.e. the correlator
slightly di erent than the original one employed by AFS,
tn( ; u; v) = ( uv)j j Y( = x)n+1:
x2
(3.6)
The reason for this di erent choice of normalization is that Awata, Feigin and Shiraishi
were using the action on Macdonald polynomials to investigate the connection with the
re ned topological vertex [48]. On the other hand, here we have chosen to keep a certain
symmetry in the way the boxes of Young diagrams enter the formulas. It also makes the
connection with our previous results on qqcharacters more explicit [11].
The AFS intertwiner can be generalized to vertical representations of higher level,
(n;m)[u; ~v] : (0; m)~v
(1; n)u ! (1; n + m)u0 ;
with
where the vector in the vertical space (0; m)~v has components labeled by the mtuple ~
that reads
(n;m)[u; ~v] = tn;m(~ ; u; ~v) : Y
~
( x) :; tn;m(~ ; u; ~v) = (u0)j~ j Y( = x)n+1:
x2~
(3.8)
This operator can be constructed as a product of vertical level one intertwiners coupled in
the horizontal channel, as represented in the
gure 3. The contraction in the horizontal
channel simply corresponds to a product of operators in the qboson Fock space. However,
it is only possible if the weights of the intermediate representation spaces coincide. The
resulting product can be normal ordered, and commutations produce a bifundamental
contribution,
(n+1)[u2; v2] (n1)[u1; v1] =
2
G(v1= 2v2)
N (v1; 1; v2; 2)
: (n1)[u1; v1] (n+1)[u2; v2] :;
2
(3.9)
with the requirement u2 =
u1v1. The function G(z) is de ned in appendix, formula (B.4).
It only depends on the ratio v1=v2 and thus can be easily discarded. The (vertical) level
m intertwiner is obtained by repeating this operation m times,
m
=
m
Y
Qlm=1 tn+l 1( l; ul; vl) (n;m)[u; ~v];
~
(3.10)
with for each intermediate horizontal space the weight
The extra factors in (3.10) and in (3.15) below will be absorbed in the replacement of
products of a (l) by a~ in the de nition of the gauge theory operators (see the next section).
The AFS dual intertwiner can be generalized in the same way. It is de ned as the
operator8
HJEP1(207)34
(n;m) [u; ~v] : (1; n + m)u0 ! (1; n)u
(0; m)~v;
with
with vertical components
(n;m) [u; ~v] = tn;m(~ ; u; ~v) : Y
~
m
l=1
x2~
( x) :;
where:
;
(v) = V~ ( v) 1V~+(
1v) 1; tn;m(~ ; u; ~v) = ( u) j~ j Y( x= )n:
Again, it can be constructed from the original dual intertwiners of vertical level one as a
product in the horizontal channel using the relation
2
(n ) [u2; v2] (n +1) [u1; v1]
1
= G(v1=v2)( v2)j 1j( q3v1) j 2j x2 1 x
1
: (n +1) [u1; v1] (n ) [u2; v2] :
2
1 Q
x2 2 x
N (v2; 2; v1; 1)
Repeating the operation m times reproduces the dual intertwiner of level m,
m
Qlm=1 tn+m l( l; ul; vl) (n;m) [u; ~v];
~
with the intermediate horizontal weights ul such that ul =
vl+1ul+1 and um = u:
8Note that we have exchanged the role of u and v with respect to the original de nition so that ~v is
always the weight in the vertical space.
Q
m
(3.11)
(3.13)
(3.14)
(3.15)
(3.16)
Representation web. In order to form a particular web of representations relevant to
a given gauge theory, the AFS interwiners can be coupled in two di erent ways: either
along the horizontal (1; n) or the vertical (0; m) channels. These two contraction channels
will be discussed below. To anticipate, an horizontal contraction corresponds to the
operator product of the qHeisenberg algebra describing the horizontal representation. On
the other hand, the vertical contraction will be associated a scalar product that generates
the trace of a tensor product. These contractions are represented by joining the vertex
drawn in the gure 2 along horizontal/vertical legs.9 Taking only AFS intertwiners of rank
m = 1, we recover the (p; q)web diagram giving the brane con guration engineering the
gauge theory. From the NS5 brane perspective, the operators
and
are interpreted as
creation/annihilation operators of D5 branes since they increase/decrease the horizontal
level by one respectively. Branes charge conservation takes the form of representation
levels conservation, with horizontal edges oriented from left to right, and vertical edges from
bottom to top.
The introduction of intertwiners with higher rank allows us to simplify the diagram,
e ectively folding it m times in the horizontal direction. In this way, it describes the
creation/annihilation of m coinciding D5 branes in one go. The resulting web diagram
corresponds to the (p; q)web diagram of the gauge theory where all gauge groups have
been replaced by U(1) groups. Although the information contained in this diagram become
less visible, calculations are much more e cient in this setting.
Massdeformed intertwiners. For simplicity, the parameters of the gauge theory have
been rescaled in order to set all the bifundamental masses to
1. It is however
possible to keep arbitrary masses upon the introduction of massdeformed intertwiners. This
9We remind the reader that we call \horizontal" every segment that is not vertical.
can be achieved by using the twisted operators
(z) = V (z
;
; (v) = V~ (z
1) 1V~+(zq3 1
the interwiner products of the form
1
) 1 to construct the dual intertwiner (n;m) [u; ~v]. Then,
1 2 reproduces the bifundamental contribution
cou
;~
1) 1V+(zq3 1
1) 1 and
pled to the nodes 1 and 2 with an arbitrary bifundamental mass . Since such a
deformation brings only little new insight to our discussion, in the following we will keep all the
bifundamental masses set to
1 to lighten the notations.
In [35], a series of commutation relations were derived, involving the intertwiners ,
and
the DIM generators in the appropriate representations. These relations can be extended
HJEP1(207)34
to the generalized intertwiners de ned here. They are expressed formally as
u0
(1;n+m)(e) (n;m)[u; ~v] =
(n;m) [u; ~v] (u10;n+m)(e) =
for any element e 2 E of the algebra. A proof is brie y sketched in appendix C. To be a
little more explicit, the lemmas can be expressed as an action in the qboson Fock space
of the horizontal representation of DIM generators on the vertical components
(3.17)
~ of the
(3.18)
(3.19)
intertwiner operators:
( 1=2z) ~(n;m)[u; ~v]
=
z
m
m
m 1
~ (z)
~ (z)
m
X
~
(
~
~ +x
x (
~
1z) =
= zm 1
2m+1 X
(z= x) zR=esx z 1Y~ (zq3 1) ~(n;mx)[u; ~v] +
(
1=2z):
In these relations, the representation in the horizontal space of the DIM operators has
been omitted: for instance x (z) reads (u1;n)(x (z)) on the right of the intertwiner
~ ,
and
u0
(1;n+m)(x (z)) on the left.
(1;n)(x (z)) on the left and
u
written for the dual intertwiner,
~
(n;m) [u; ~v] +( 1=2z)
~
1=2z)
u0
m
~
=
1 X
For the dual intertwiner, x (z) is understood to be
(1;n+m)(x (z)) on the right. Symmetric relations can be
m
~ (z)
~ (z)
1z)
+( 1=2z) ~(n;m) [u; ~v] = 0
(
x+(
1=2z) ~(n;m) [u; ~v] = 0;
(z= x) zR=esx z 1Y~ (zq3 1)
(
1=2z) ~(n;mx) [u; ~v];
m
z
~v
(
m+1
X
~ +x
z
z
the insertion of an operator e 2 E.
~
(n;m) [u; ~v]x (z)
m
~ (z)x (z) ~(n;m) [u; ~v] =
In fact, it is possible to rewrite the r.h.s. of the relations involving x in a slightly more
condensed way,
~ (z) ~(n;m)[u; ~v]x+(z) = ~(v0;m)(x+(z))
x (
~
1z)
m
x+(
~v
1=2z)h ^(0;m)(x+(z))
~
(n;m) [u; ~v] ;
i
~ (z)x (z) ~(n;m) [u; ~v] = ^~v
(0;m)(x (z))
~
where the dot denotes the action in the vertical space. The AFS lemmas are represented
on the gure 4, in a very simpli ed manner. The insertion of the symbol z denotes the
action of DIM generators e 2 E , and two insertions the action of the coproduct.
3.3
Gaiotto state
The qdeformed Gaitto state is a Whittaker state for the qVirasoro (or qW)
algebra [49{53], it is a deformed version of the original Gaiotto state de ned for the
Virasoro algebra [54{57]. The intertwiners
and
de ned in the previous section can be
interpreted as an uplift of the Gaiotto state to the horizontal representation space.
Indeed, the Gaiotto state can be recovered by taking the vacuum expectation value in the
horizontal spaces,
~
~
(n ;m) [u ; ~v]j;i(1;n +m)u 0
hhG; ~vj = (1;n+m)u0h;j
(n;m)[u; ~v]j;i(1;n)u =
=
X a~
~
D (n ;m) [u ; ~v]E j~v; ~ ii;
~
X a~
~
D (n;m)[u; ~v]E hh~v; ~ j;
~
where we have used the de nitions
(n ;m) [u ; ~v] = hh~v; ~ j
~
(n ;m) [u ; ~v];
~
(n;m)[u; ~v] =
(n;m)[u; ~v]j~v; ~ ii
and introduced the closure relation in the vertical space
1 =
X a~ j~v; ~ iihh~v; ~ j:
~
(3.21)
(3.22)
(3.23)
(0; m2)~v2
(1; n2)u2
Horizontal intertwiner for the A2 quiver
The A2 quiver gauge theory, with gauge group U(m2)
U(m1) is described by an operator
TU(m2) U(m1) involving two vertical contractions (for the vector multiplet contributions),
and a single horizontal contraction (the bifundamental contribution):
TU(m2) U(m1) =Xa~1a~2 ~
with the constraints n2 = n1 and u2 = u1 in order to match the representation levels and
weights in the horizontal channel. The relevant con guration of DIM representations can
be seen on gure 9. Actually, this operator can be obtained from the combination of the
operators TU(m1) and TU(m2) associated to the two gauge groups, using a new product
to represent the horizontal contraction, TU(m2) U(m1) = TU(m1)
TU(m2). The new product
corresponds to the concatenation of two chains of tensor products of respective length r
and s, in order to form a chain of length r + s
1:
(a1
ar) (b1
bs) = a1
ar 1
arb1
b2
bs:
(4.20)
As before, the instanton partition function of the gauge theory corresponds to the vev
in the horizontal spaces, up to a factor of Gfunctions depending only on the Coulomb
branch vevs,
m1 m2
Zinst.[A2] = Y Y
G(vl(02)= vl(1))
TU(m2) U(m1)
=
X qj~1jqj~2jZvect.(~v1; ~ 1)ZCS( 1; ~ 1)Zvect.(~v2; ~ 2)ZCS( 2; ~ 2)Zbfd.(~v2; ~ 2; ~v1; ~ 1j
1 2
~1;~2
(4.19)
(4.21)
1);
with the identi cation q =
m u =u and
= n
n for the gauge coupling and
the ChernSimons level associated to the two gauge groups
= 1; 2.
Fundamental qqcharacter for the A2 quiver
A di erent qqcharacter is associated to each node of the quiver diagram, in correspondence
with antisymmetric representations of the Lie algebra. In the case of the A2 quiver, the
two relevant representations are the fundamental
and the fully antisymmetric
ones.
We focus rst on the construction of the fundamental qqcharacter
+(z), leaving the
antisymmetric representation for the next section. Within our conventions, the fundamental
representation is associated to the action of x+(z) on the rst node.
The commutation property (4.6) of the vertical channel is valid at each node in the form
TU(m2) U(m1) = 1
TU(m2) U(m1) = 1
1
~v1
(0;m1)(e)
1
~v2
(0;m2)(e)
The operator leading to the fundamental qqcharacter can be obtained using the double
coproduct
= (
1)
= (1
) :
The action of this double coproduct on the DIM generators reads
(z)) =
(1 ^ 1=2
^ 1=2 z)
1
(^
1
1 z)
^1=2
^ z)
^1=2
1 z)
+(1
^ 1=2 z)
1
1
+(1
1 z)
1
^
1 z)
^1=2 z)
1
^1=2 z)
Combining the general commutation properties of the vertical and horizontal contractions,
together with the AFS lemmas (3.17), it is possible to show that the horizontal action of
the generators commutes with the T operator:
(1;n1)
u
1
(1;n1+m1)
u0
1
= TU(m2) U(m1)
(1;n2+m2)
u0
2
(1;n1+m1)
u0
1
(e) TU(m2) U(m1)
(1;n2+m2)
u0
2
(1;n2)
u2
(e):
The proof is a tedious but straightforward calculation, using the same method as in the
single node case.
Introducing the operator X +(z) =
(x+(z)) that commutes with TU(m2) U(m1), the
fundamental qqcharacter can be written (omitting the horizontal representations):
= u =(q3m u0 ). Again, this quantity is a polynomial in z because X
+ commutes
with T . The power of z in the prefactor is xed by consideration of the asymptotic behavior.
(4.22)
(4.23)
(4.24)
(4.25)
(4.26)
1 in the
:
gauge
(4.27)
(4.28)
; (4.29)
Evaluating the correlators in the qbosonic Fock spaces, we recover the expression given
in [11] for the fundamental A2 qqcharacter with a bifundamental mass
=
vertical channel:12
*
Quantum Weyl re ection and the second qqcharacter
There are two ways to obtain the second qqcharacter. The simplest one is to consider the
insertion of the operator X (z) =
(x (z
1)) that also commutes with the T operator.
De ning
gives after evaluation of each horizontal actions,
(z) =
*
which is indeed the second qqcharacter of the A2 quiver found in [11]. In fact, because of
the re ection symmetry obeyed by the A2 quiver, we have
why we have obtained the correct qqcharacter. This property seems to be a consequence
of the 5 symmetry of the representation web combined with the S2 rotation of the DIM
(z) =
+(z) which explains
algebra.
There is a more natural way to derive the second qqcharacter, which is to start from
insertions of x+(z) in two of the three horizontal lines at the end of the diagram, with
spectral parameters
netuned to obtain the commutation with the T operator. Indeed,
the rst term of the qqcharacter, that is proportional to Y~ 2 (z
the insertion of the operator
2), can be obtained from
x+(z)
1
(4.30)
term [31].
second node to
on the left of the representation web. The other terms entering the expression of the
qqcharacters are known to be obtained by acting with the Weyl re ection on the rst
The Weyl re ection for the A2 quiver diagram sends the coweight w2 attached to the
w2 !2 w1
1
w2 !
w1
()
Y~ 2
!
2 Y~ 1
Y~ 2
1
!
1
Y~ 1
;
(4.31)
where i denote the roots of the Lie algebra with the Dynkin diagram A2. To each
intermediate expression has been associated a term of the qqcharacter. A similar transformation
12To simplify formulas, the labels corresponding to the two nodes have been exchanged here with respect
to the conventions employed in [11].
re ection on x+(z)
x+(z)
1
1
1
!
2
!
can be de ned on the DIM generator x+(z) involved in tensorial expressions like (4.30). It
will be called the quantum Weyl re ection. The quantum Weyl re ection with respect to
the root i consists in replacing the insertion of x+ in the ith tensor space with an insertion
of
in the ith space and x+ in the (i + 1)th space, together with the appropriate shifts
of the spectral parameters:13
x+(z) i 1
i
!
(^(1i=)2z) i x+(^(i)z)
(4.33)
where ^(i) = (1 )i 1
^( 1)r+1 i. The transformation is forbidden if two x+ were to
collide in the same space. In a sense, the generators x+ obey a fermionic statistics in
the tensor spaces. It is further assumed that the operators
are ordered on the left of
operators x+, although this fact does not modify the derivation of the qqcharacters.
Before applying the quantum Weyl re ection to obtain the operator relevant to the
second node of the A2 quiver, let us review how this transformation works in the known
cases. The A1 quiver is described by two horizontal spaces, it has a single weight w !
w
+(z) has only two terms. The application of the quantum Weyl
1 leads to the coproduct
(x+(z)).
Turning to the A2 quiver, the quantum Weyl re ections of
reproduces the three terms in the expression of the operator X +(z) constructed from the
application of the squared coproduct
. Thus, the quantum Weyl re ection de nes a
generalization of the coproduct that implements the action of an operator into three copies
of the initial space.
over the three terms in order to de ne
Now, we apply the Weyl re ection to the operator (4.30) of the A2 quiver, and sum
x+(z)
+
+
1 z)
1
1 z)
1 z):
1
(1
(1
^1=2
1 z)
x+(1
1 z)
^1=2
1 z)
(^1=2
(^1=2
1
1
1 z)
1 z)
1 z)
1
1 z)
x+(^
^
1 z)
(4.34)
After a tedious but straightforward computation, it can be shown that the operator
(x+(z)) commutes with T . The commutation of the operator de ned in (4.14)
with TU(m) (seen here as a subdiagram) is essential for the various cancellations to occur.
The corresponding qqcharacter is de ned as
13The quantum Weyl transformation can be de ned in a similar manner on the generator x (z), i.e. in
such a way that it reproduces the coproduct in the case of the fundamental representation:
1 i x (z)
i
!
x (^(i+1)z) i +(^(1i=+21)z)
:
(4.35)
(4.36)
(4.32)
The evaluation of the vev in the horizontal spaces reproduces the expression of
(z) given
in (4.29), showing that indeed
(z).
The results obtained for the A2 quiver are easily generalized to linear quivers with an
arbitrary number of nodes r. The T operator is constructed using r
the horizontal channel, rendered by the product
de ned in (4.20),
1 contractions in
where the order of indices labeling gauge groups has been reversed for convenience.
(4.37)
HJEP1(207)34
Schematically, it reads
U(1) =
X
Y a~
~ 1; ~ r s=1
1
1
2
r
;
(4.38)
where we have omitted all the weights and level parameters. This expression implies
the constraints ns = ns+1 and us = us+1 in order to match the representation spaces
in horizontal channels. Then, up to a prefactor of Gfunctions, the vev reproduces the
instanton partition function (2.28) for the quiver
n
ns for the ChernSimons levels, and qs =
= Ar under the identi cation s =
s ms us=us for the gauge couplings.
As before, the fundamental qqcharacter, attached to the rst node, can be obtained
by multiple applications of the coproduct. The Ar fundamental coproduct
is de ned
recursively as
= ( ( 1)r 1
) ( ( 1)r 2
(4.39)
and acts on the DIM generators as follows:
r+1
X
s=1
r+1
s=1
(z)) =
(1( ^ 1=2)r z)
^(s) = (^ )s1( 1)r s, and the tensorial transpose de ned as (a1
The fundamental coproduct of DIM generators commutes with the T operator,
(e) TU(mr)
U(1) = TU(mr)
U(1)
(e);
where we have omitted to indicate the horizontal representations. As a result, the
fundamental qqcharacter de ned as
(4.41)
is a polynomial. Explicit evaluation of the correlators for each horizontal space provides
the formula
i=1
2) + X
s=1 s i=1
1
Y qiz i 2mi ms+Pjs=i+1 j zm1 ms
Y qiz i 2mi mr+Pjr=i+1 j zm1 mr
mr
!
!
1
Y~ r (z r 1)
+
gauge
:
ms Y~ s+1
(z s 2)
Y~ s (z s 1)
(4.43)
(4.44)
The qqcharacter associated to the sth node corresponds to the antisymmetric
representation denoted by the Young diagram (s) with s boxes, all in the rst column. The
corresponding operator X(+s)(z) is obtained by application of quantum Weyl re ections on
Note that since the operator is evaluated in horizontal representations, the position of the
central element ^ in the arguments of operators is somewhat arbitrary here. As an
illustration, the A3 quiver is treated in details in the appendix E. This construction can also be
applied to the generator x (z). Because of the re ection symmetry of the quiver diagram,
the corresponding qqcharacters are expected to obey the relation
(s)(z) =
(r+1 s)(z).
In fact, it is possible to de ne qqcharacters associated to arbitrary representations
of the Lie algebra. To a representation labeled by a Young diagram
is associated the
+
operator X +(~z) obtained by taking the product over the columns
X( i)(zi) de ned previously. This construction works if the
i of the operators
rst column of the Young
diagram contains at most r boxes. By construction, these operators commute with the T
operator of the gauge theory, and the vev
by a monomial of the variables zi.
X +(~z)T
is a polynomial up to multiplication
4.2.5
Inclusion of fundamental/antifundamental matter elds
Matter elds are introduced by semiin nite D5 branes that are vertical edges in the
representation web. These can be inserted either in the bottom or top part of the diagram,
leading to fundamental ( ) or antifundamental (
) matter respectively. It is wellknown
in gauge theory that such matter elds can be obtained by introducing extra gauge groups,
sending the corresponding gauge coupling q to zero. This constraints the Young diagrams
~ associated to this gauge group in the partition function expansion (2.28) to be empty,
hence generating the contributions
Zfund.(
1~ (f); ~ ) = Zbfd.(~v; ~ ; ~ (f); ~;j
1);
Za.f.( ~ (af); ~ ) = Zbfd.(~ (af); ~;; ~v; ~ j
1):
(4.45)
In this spirit, we can regard the massive A1 quiver as the limit of the A3 quiver as
q1; q3 ! 0. This procedure corresponds to send two NS5 branes at in nity. Taking the
massive A1 qqcharacter
+(z) obtained in [11],
formula from appendix E for the A3 qqcharacter
sending the gauge couplings q1; q3 ! 0 while q2 = q is held
+(z) associated to the second node, and
xed, we indeed recover the
since
zmY~ (z
Y~ (z)
gauge
;
z
1( l(f)) 1) = pfund.(z
2);
(af)z 1) = pa.f.(z);
l
;
(n;f~) [u; ~(af)]
~
and we have dropped the label 2 of the middle node.
In our formalism, the gauge coupling q is obtained as a ratio of horizontal weights u=u .
The limiting procedure q ! 0 corresponds to send either u to zero for some intertwiner
or u to in nity for the dual intertwiner
. In either case, the normalization coe cients,
tn;m or tn;m, vanishes except when the Young diagrams ~ are empty. The case of the
,
antifundamental matter is the easiest one to consider. Indeed, it is observed from the
AFS lemma that since R(~;) = ;, the vacuum intertwiner
the T operator is simpli ed as the extra
; can be decoupled,14
x+(z). As a result, an additional horizontal contraction with this operator, as represented
on the gure 10, does not spoil the commutation with the operator X +(z). In this case,
;
commutes with the action of
TU(a(mf) ) =
X a~
~
(n ;m) [u ; ~v]
~
(n;m)[u; ~v] ~(n;f~) [u; ~ (af)]:
~
;
(4.48)
14Said it otherwise, the action of x+(z) on ; being proportional to u ! 0, the extra horizontal channel
can be dropped.
(4.46)
(4.47)
HJEP1(207)34
On the other hand, in the case of fundamental matter, it does not seem possible to
fully decouple the extra horizontal channel, and we are forced to de ne the T operator
within three di erent Fock spaces,
X a~
~
~
;
(n;f) [u; ~ (f)]
;
(n;f)[u; ~ (f)] ~(n ;m) [u ; ~v]
~
~
(4.49)
in order to observe the commutation relation with the operator X
is related to the noncommutation of ~ with x+(z), it can be solved by considering the
+(z).15 This problem
commutation with the operator X (z) instead. However, the problem persists if both
fundamental and antifundamental matter are introduced, in which case the only solution
is to consider a third horizontal channel with a trivial vertical contraction as in (4.49).
The treatment of fundamental matter here is rather di erent from the usual brane
description. In particular, we do not observe a limitation on the number of elds in this
algebraic construction, which may be an e ect of the presence of ChernSimons terms. It
would be advisable to achieve a deeper understanding of the precise di erence between the
two constructions.
Since our understanding of fundamental matter is based on gauging the avor group,
the generalization of these results to all linear quivers would require to construct arbitrary
quiver theories, which is way beyond the scope of our paper. However, we hope to be able
to address this issue in a near future.
5
We have proposed an algebraic method to derive qqcharacters of linear quiver N = 1
gauge theories with U(m) gauge groups. It is based on the insertion of DIM generators in
a tensored horizontal representation, symmetrized in order to de ne an operator commuting
with the T operator of the gauge theory. This method provides an e cient way to derive
the explicit expression of the qqcharacters as correlators in the gauge theory.
There are several directions in which this work can be extended. The most natural one
is the treatment of DEtype quivers. In the case of Dtype quivers, the brane construction of
Kapustin [60] involving an orientifold brane seems relevant. Progress along this direction
will be reported elsewhere. A
ne quivers could also be considered. There, the extra
compact dimension seems to impose the consideration of a ring of tensor spaces in the
horizontal representations in which an in nite number of quantum Weyl transformations
can be applied. A much harder problem would consist in studying gauge theories with
DEtype gauge groups, i.e. Sp(m) or SO(m) groups. The recent construction of Hayashi
and Ohmori [61] could be helpful in this context.
In [62], a deformation of the re ned topological vertex has been introduced, that
corresponds to a further (q; t)deformation of the horizontal representation. It would be
interesting to further study the underlying algebraic structure.
15Taking the limit u1 !
1 in the A3 operator X +(z), the dominant terms are those with a x+(z)
generator inserted in the rst space. They are of order
u1 and reproduce the two terms in the massive
{ 36 {
HJEP1(207)34
We hope that the generalized intertwiners introduced here will also be useful in the
description of the underlying integrability, leading to a generalization of the Rmatrix
construction [21, 22].
Finally, the action of a similar quantum algebra has been observed in the context
of higher spins [63], and it would be interesting to investigate the role played by these
fundamental objects that are interwiners and qqcharacters.
Acknowledgments
J.E.B. would like to thank A. Sciarappa and Joonho Kim for discussions. In the early
stages of this project, he has been supported by an I.N.F.N. postdoctoral fellowship within
the grant GAST, and the UniToSanPaolo research grant Nr TOCall320120088
\Modern Applications of String Theory" (MAST), the ESF Network \Holographic methods for
strongly coupled systems" (HoloGrav) (09RNP092 (PESC)) and the MPNS{COST Action
MP1210. He also wishes to thank Tokyo University for their generous nancial support
during his stay. YM is partially supported by GrantsinAid for Scienti c Research
(Kakenhi #25400246) from MEXT, Japan. RZ is supported by JSPS fellowship and he is also
grateful for the hospitality during his stay in KIAS.
Part of the results of the paper were announced in the workshop \Progress in Quantum
Field Theory and String Theory II" (March 2731, 2017 Osaka City University). We would
like to thank the participants of the workshop, especially H. Awata, H. Itoyama, H. Kanno,
Y. Zenkevich with whom very useful discussion was made.
A
Di erent expressions for the vertical representation
In [11], a di erentlooking vertical representation has been employed. At the level (0; 1),
e(z)jv; i = z 1
f (z)jv; i =
(z)jv; i =
X
x2R( )
X
x2A( )
(z= x) x( )jv; + xi
(z= x) x( )jv;
x ;
i
(z)] jv; i;
1 = (1
q1)(1
q2)(1
q3);
(A.2)
with function
residues
x( )2 =
tions as follows:
16Here the generators have been multiplied by a constant factor without altering the commutation
relae(z) ! z 1p(1 q3)ve(z); f (z) ! zp(1 q3)vf (z);
(z) ! (1 q3)v
(z):
(A.1)
The de nition of the function
(z) has also been modi ed in order to re ect this change of normalization.
(z) de ned in (2.12), and the coe cients being the square root of the
Resz= x
(z). However, the normalization of the states can be
modi ed by an arbitrary factor: letting jv; ii = N ( )jv; i, we have in general
e(z)jv; ii = z 1
f (z)jv; ii =
X
N ( + x) jv; + xii;
(z= x) x( )
N ( )
N (
x) jv;
Note that the action of the Cartan is not modi ed since they are diagonal in this basis.
Choosing the normalization factor as17
N ( ) =
p
1
Zvect.(v; )
Y
q3)v1=2 !
1=2
1
x
;
N ( )
N ( + x)
x( )
Y (q3 1 x)
;
using the fact that
Res
z= x2A( )
Res
z= x2R( )
(z) = Y ( xq3 1)
(z) =
1
1
Res
z= x2A( ) Y (z)
Res
Y (zq3 1);
Y ( x) z= x2R( )
e(z)jv; ii =
f (z)jv; ii =
X
X
x2R( )
1
(z= x) zR=esx zY (z) jv; + xii;
(z= x)Y
x(q3 1 x)jv;
xii:
Y
x(zq3 1) = Y (zq3 1)=S(q3 x=z),
The second relation simpli es after a careful treatment of the limit z !
x in the expression
f (z)jv; ii =
1q3
X
(z= x) zR=esx z 1
Y (zq3 1)jv;
xii:
and the property
x( ) =
x), the new representation can be written
Zvect.(v; + x)
Zvect.(v; )
= (1 q3)2v
1 2x
1
Y ( xq3 1) zR=esx Y (z)
1
= (1 q3)2v
1 2x
x( )2
Y ( xq3 1)2 :
tion [11],
Finally, we notice that the coe cient in front of the commutator [e; f ] is di erent from the
one in (2.2) for [x+; x ]. In order to recover the same convention, we need to multiply
f (z) ! (1
1q3
q3)2 f (z);
(z) !
1
1
(z):
Under the identi cation of the renormalized f (z) with x (z), and e(z) with x+(z), we end
up with the vertical representation (2.11). In addition, an extra cosmetic factor of
has been added in front of x (z) and
(z) in order to simplify some expressions. It is
important to stress that our renormalized vertical representation here does not coincide
with the one used in AFS's paper in which the normalization of the intertwiners
and
17The recursive property is inherited from the discrete Ward identity obeyed by the vector
contribuxii;
;
(A.3)
(A.5)
(A.6)
(A.7)
(A.8)
(A.9)
1
(A.4)
B.1 qbosons vertex operators
The vertex operators , and ' satisfy the relations
(z) (w) = S(w=z) 1 : (z) (w) :;
(z) (w) = S( w=z) : (z) (w) :;
'+( 1=2z) (w) =
'+( 1=2z) (w) =
S(z=w)
S(w=z) : '+( 1=2z) (w) :;
S(z=w)
S(w=z) : '+( 1=2z) (w) :;
' ( 1=2z) (w) =: ' ( 1=2z) (w) :;
' ( 1=2z) (w) =: ' ( 1=2z) (w) :;
(z) (w) = S(z=w) 1 : (z) (w) :;
(w) (z) = S( z=w) : (z) (w) :;
(w)'+( 1=2z) =: '+( 1=2z) (w) :;
(w)'+( 1=2z) =: '+( 1=2z) (w) :;
(w)' ( 1=2z) =
(w)' ( 1=2z) =
SS((zw==wz)) : ' ( 1=2z) (w) :;
SS((zw==wz)) : ' ( 1=2z) (w) : :
Explicitly, the vacuum intertwiners read
;(v) = exp
;(v) = exp
!
qk
vka k exp
kvna k exp
q k
1
q k
!
v kak ;
kv kak ;
!
they obey the relations
1
1
1
1
1
(z) ;(w) =
;(w) (z) =
(z) ;(w) = (1
;(w) (z) = (1
1
w=z
1
: (z) ;(w) :;
: (z) ;(w) :;
w=z) : (z) ;(w) :;
z=( w)) : (z) ;(w) :
1
2w=z
w=z
'+( 1=2z) ;(w) =
: '+( 1=2z) ;(w) :;
;(w)'+( 1=2z) =: '+( 1=2z) ;(w) :;
' ( 1=2z) ;(w) =: ' ( 1=2z) ;(w) :;
;(w)' ( 1=2z) =
1
z=w
: ' ( 1=2z) ;(w) :;
;
(z) ;(w) = (1
(w) (z) = (1
(z) ;(w) =
w=z) : (z) ;(w) :;
z=( w)) : (z) ;(w) :;
: (z) ;(w) :;
1
2w=z
(B.1)
(B.2)
w=z
2w=z
1
z=w
(w) (z) =
: (z) ;
(w) :;
'+( 1=2z) ;
(w) =
: '+( 1=2z) ;
(w) :;
(w)'+( 1=2z) =: '+( 1=2z) ;
(w) :;
' (
1=2z) ;
(w) =: ' (
1=2z) ;
(w) :;
(w)' (
1=2z) =
: ' (
1=2z) ;
(w) : :
Note also the properties with
G(z) = exp
and the fact that
;(z) ;(w) = G(w= 2z) :
;(z) ;
(w) = G(w=( z)) 1 :
;(z) ;(w) :;
;(z) ;
(w) :;
(z) ;
(w) = G(w=z) :
(w) ;(z) = G(z=( w)) 1 :
;
(z) ;
(w) :;
;(z) ;
!
;
;
=
1
Y
i;j=1
k
t k)
1
zq1i 1qj 1 ;
2
'+( 1=2z) =: (z) ( z) :;
' (
1=2z) =: (z) (
1z) : :
B.2
Commutation relations in horizontal representations
The simplest relations are the commutations between the operators
and the
intertwiners, they can be derived easily by combining the properties given previously and the
formula (2.12):18
(n;m)[u; ~v] +
(n;m)[u; ~v]
1=2z) ~(n;m)[u; ~v] =
(n+m) ~ (z) : '+(
1=2z) ~(n;m)[u; ~v] :
( 1=2z) ~(n;m)[u; ~v] =
n+m : ' ( 1=2z) ~(n;m)[u; ~v] :
+( 1=2z) ~(n;m) [u; ~v] =
~ (z) 1 : '+( 1=2z) ~(n;m) [u; ~v] :
1=2z) ~(n;m) [u; ~v] =
n : ' (
1=2z) =
n : '+(
1=2z) ~(n;m) [u; ~v] :
1=2z) ~(n;m)[u; ~v] :
( 1=2z) =
~ (z) 1 : ' ( 1=2z) ~(n;m)[u; ~v] :
~
(n;m) [u; ~v] +( 1=2z) =
n m : '+( 1=2z) ~(n;m) [u; ~v] :
~
1=2z) =
n m
~ (z) : ' (
1=2z) ~(n;m) [u; ~v] :
In these expressions, the representation (1; n + m)u0 of the DIM generator is understood
(but omitted) on the left of the operator
~
(n;m), while the representation on the right is
(1; n)u. The two representations are exchanged for the dual operator: (1; n)u is on the left
18Operators are supposed to be radially ordered.
(B.3)
(B.4)
(B.5)
(B.6)
(n;m) while (1; n + m)u0 is on the right. Similar expressions can be derived for x :
~
zn+mY~ (z)
: (z) ~(n;m)[u; ~v] :
x+(z) ~(n;m) [u; ~v] = u n
z nY~ (z
1) : (z) ~(n;m) [u; ~v] :
x (z) ~(n;m) [u; ~v] =
n
u nY~ (zq3 1) : (z) ~(n;m) [u; ~v] :
(n;m)[u; ~v]x+(z) =
(n;m) [u; ~v]x+(z) = u n
(n;m) [u; ~v]x (z) =
zn+m
~ (z)
~ (z) 1 u0 n+m
zn+mY~ (z)
n
u n
Y (zq3 1) : (z) ~(n;m) [u; ~v] :
C
Derivation of the AFS lemmas
The proof of the relations involving
(z) is a matter of writing the commutation
relations (B.6). Hence the focus here is on the generators x (z). We rst examine the product
of x+(z) and
(n;m)[u; ~v], the proof is based on the following decomposition for the function
1
zY~ (z)
X
x2A(~ ) z
1
Res
x z= x zY~ (z)
:
1
As a consequence, we can write the right product of x+(z) on
in (B.7) as
x+(z) ~(n;m)[u; ~v]
= u0 n+mz n m+1
= u0 n+mz n m+1
+ u0 n+mz n m+1
X
x2A(~ ) z
X
x2A(~ ) z
X
x2A(~ )
1
1
Res
Res
x z= x zY~ (z)
x z= x zY~ (z)
1
1
: (z) ~(n;m)[u; ~v] :
: ( x) ~(n;m)[u; ~v] :
x
1
Res
z= x zY~ (z)
:
This expression is valid for jzj > j xj, however the second line of the last equality has no
x and can be analytically continued to jzj < j xj. This is not true for the rst
line, and the fraction should be expanded in positive powers of z. A similar expression can
(B.7)
HJEP1(207)34
(C.1)
(C.2)
be obtained for jzj < j xj by considering the left product of x+(z) on
~ (z) ~(n;m)[u; ~v]x+(z)
+ u0 n+mz n m+1
X
x2A(~ )
x
1
Res
z= x zY~ (z)
:
Taking the di erence of the two, the terms with no singularity cancel eachother. The
remaining expression is a di erence of expansions in powers of z and z 1 that forms a
delta function,
x+(z) ~(n;m)[u; ~v]
~ (z) ~(n;m)[u; ~v]x+(z)
= u0 n+mz n m
X
x2A(~ )
(z= x) zR=esx zY~ (z)
1
: ( x) ~(n;m)[u; ~v] :
Then, since
(n;m) is built as a product of operators ( x) for all x 2 , the vertex operator
~
: ( x) ~(n;m) : can be written as
~ +x
(n;m). Taking into account the prefactor
tn(~ ; u; v)
tn(~ + x; u; v)
n+1
u0 n+1 ;
we recover the AFS lemma in the form (3.18).
A similar argument can be employed to treat the action of x (z), with the poles
located at the points z =
1
x2R(~ ), and the operator : (z) ~(n;m): simpli ed using the
property (B.5) of the appendix B.19 However, in this case, a more elegant proof is also
and
(n) that can be found in [64] (formula (6.15)). By de nition, we have
possible. It is based on the formula for the commutation relation between the modes xk
k =
I dz k 1
0 2i
x (z)
so that
19The following property is useful here,
I
jzj>j xj
X
I
x2R(~ )
2i
dz k 1
x 1 2i
x (z) ~(n;m)[u; ~v]
dz zk+n+m 1
u0 n+m Y~ (z
I
z=0
jzj<j xj
2i
1) : (z) ~(n;m)[u; ~v] :
~
dz zk 1 (n;m)[u; ~v]x (z)
z=Res1 f (z) =
1 Rz=es f (z 1):
(C.3)
(C.5)
(C.7)
(C.6)
The second equality is the consequence of several cancellations between poles, such that
only the poles of Y~ (z
1) will contribute. The expression for the product of operators is
taken from (B.7). The contour integral can be reduced to the residue contributions of the
integrand, which simpli es thanks to the properties (B.5) and (C.6) to give
k+m 2
2m k+1 Res Y~ (zq3 1) ~(n;mx)[u; ~v] +
z= x
1=2 x): (C.9)
Summing over the index k with the spectral parameter at the power z k, we recover the
AFS lemma (3.18). This short computation gives some insight on the interpretation of the
AFS lemma: it is valid for each power of z in a formal expansion.
D
Connection with quiver Walgebras
In [31, 32], Kimura and Pestun have introduced quantum Walgebras based on the Dynkin
diagram
of simple Lie algebras of ADE type. These algebras are constructed upon a set
of qbosonic modes s(ki) with k 2 Z and i 2
that obey the commutation relations
[s(ki); s(i0k)0 ] =
1 1
q
k
k k;k0 cii0 ;
[k]
k > 0;
where c[iki0] denotes the kth Adams operation applied to the massdeformed Cartan matrix.
For instance, in the case of the A3 quiver with bifundamental masses ii0 =
1, this matrix
reads
0 1 + q3k
k 1 + q3k A
1
k C :
Since this algebra is also acting on Nekrasov partition functions, it should be related to
the DIM algebra considered in our paper. The aim of this appendix is to highlight this
connection. It is based on the decomposition of the tensor product of two (1; 0) DIM
representations into qHeisenberg qVirasoro algebras. This decomposition has been described
by Mironov, Morozov and Zenkevich in [33], and this appendix is just a reformulation of
their results in our notations.
In order to simplify the discussion, we will neglect the role of zero modes,
TU(1) = tr
will also restrict ourselves to U(1) gauge groups at each node of the quiver diagram. It
is an easy exercise to extend the argument to more general cases. We rst focus on the
A1 quiver for which the T operator is built as a vertical contraction of two intertwiners,
. Since two horizontal spaces are involved, we need two copies of the
qbosonic modes in order to represent the horizontal action of the intertwiner and its dual.
We denote these modes a(ki) with i = 1; 2. By de nition, modes with a di erent value of the
label i commute, while modes with the same label obey the commutation relation (2.21):
;,. . . We
[a(ki); a(i0k)0 ] = k
1
1
q
t
k k;k0 i;i0 ;
k > 0:
(D.1)
(D.3)
(z) ': S(z) 1S(q2z) :;
S(z) =: exp
!
X zks k : :
k2Z
Taking the product over the boxes x 2 , several cancellations occur, and the nal result
is expressed in terms of a product over each column i of height i
,
: Y
x2
( x)
( x) : ' : Y S(vq1i 1q2 i ) :
i
where we have neglected the boundary terms S(vq1i 1) that can be taken care of using zero
modes. In the r.h.s. , the product is taken over the elements of the set X de ned in [31],
and we can formally identify the state jZT i representing the partition function with the
action of TU(m) over the horizontal vacuum states:
jZT i '
X : Y S(vq1i 1q2 i ) : j;i ' TU(1) (j;i
j;i) :
i
The modes sk can be used to build the stressenergy tensor of the qVirasoro algebra.
The orthogonal combination kbk = a(1) + jkja(k2), which by de nition commutes with sk,
k
obeys the qbosonic commutation relation
[bk; b k0 ] =
1 1
k 1
t
k k;k0 (1 + q3k) k > 0:
(1; 0)
(1; 0) = qHeisenberg
qVirasoro:
It leads to identify the modes ksk =
jkja(k1) + a(k2). They indeed obey the
commutation relation (D.1) with the deformed A1 Cartan matrix c[k] = 1 + q3k. As a result, the
operator (D.4) can be expressed in terms of the screening operator de ned in [31],
(D.5)
(D.7)
(D.8)
(D.9)
(D.10)
(D.11)
The operator T involves a trace over Young diagram realizations of a product over the box
content of the diagram. Each factor contains the following operator evaluated at z =
for some x 2 ,
(z) = exp
X1 1
k=1
t k zk a(1k) k
exp
k=1
z k a(1) k
k
It is also interesting to rewrite the coproduct of x+(z) in terms of the modes bk; sk:
: (x+(z)) : ' : exp
X
1
k2Z 1 + q3j j
k
z k
bk : Y (z
1)+ : Y (z ) 1 :
where, following Kimura and Pestun, we have introduced the operator
Y (z) =: exp
X z k
k2Z
yk :;
yk =
t
1
1 + q3k sk:
In this expression, a di erent set of modes a(i) is attached to each tensor space, with
i = 1
r + 1. It leads to identify the modes as follows:
ks(ki) =
jkja(ki) + a(ki+1);
kbk =
jkjia(ki):
Under this identi cation, the modes s(ki) reproduce the commutation relation (D.1) with
the deformed Cartan matrix of the Ar Dynkin diagram. In addition, they all commute
with the modes bk. Thus, for a general linear quiver, we have the formal decomposition
(1; 0) (r+1) = qHeisenberg
Wr:
E
Derivation of the qqcharacters for the A3 quiver
The qqcharacters associated to the three nodes of the A3 quiver are labeled by the Young
diagrams ,
r 1 r
r
( x)
( x)A
X
(D.12)
( x)A :
(D.13)
(D.14)
Hence, up to a U(1) factor, we recover in
(x+(z)) the operator T of Kimura and Pestun,
identi ed with the fundamental current (stressenergy tensor) of qVirasoro [59].20
For a general linear quiver diagram Ar, the modes s(ki) i = 1
r are associated to
the nodes of the diagram. On the other hand, the T operator is written as an (r + 1)th
tensorial product
X
Y
(1); ; (r) x2 (1)
U(m1) = tr
12 r 1
( x)
1 2
1 0
Y
x2 (2)
x2 (r)
( x)
:1=2x:2
x:1
: 1=2 :3=2x:3
:1=2x:2
x:1
20Note that we have chosen to denote the Qoperator of Kimura and Pestun as T since the partition
function is obtained as the vev of this operator. On the other hand, their Toperator has been denoted X
to emphasize the fact that it comes from the generators x
of the DIM algebra, and that the TQrelation
only holds if we forget about the di erence of representations.
1 +
:3=2
1
: 1=2 :3=2
: 1=2x:1
: 3=2 :1=2x:2
: 3=2 :1=2x:2
: 3=2 :1=2 5=2
+
x:3
:1=2
:5=2
1
1 +
:1=2 :5=2
: 1=2x:1
: 1=2x:1
: 1=2x:1
x:+2 +
:3=2x:3
:1=2
:1=2
x:1
1 +
:1=2
:3=2
:3=2
: 1=2 :3=2
:1=2x:2
:3=2
x:2
(1; n3)u3
(n3;m3) [u3;~v3]
(1; n3 + m3)u03
(0; m3)~v3
(1; n1 + m1)u01
where we have introduced the shortcut notations x:+k = x+( kz), :k =
( kz). Note
that the argument of operators has been simpli ed taking advantage of the fact that they
act in the horizontal representations where ^ becomes . After a long and tedious
computation, it is possible to show that these expressions do commute with the operator
TU(m3) U(m2) U(m1) represented on gure 11. In practice, we have used a short program in
Python to perform the algebraic manipulations.
TU(m3) U(m2) U(m1)
2
u01u1 2n1+2n1+2m1 zn1+n1+m1+m2
3
u01u1u02 2n1+2n1+2m1+2n2+2m2 zn1+n1+n2+m1+m2+m3
we nd the following expressions after evaluation in the four independent Fock spaces,
*
1zm1Y~1(z
2) + q1z 1 Y~2(z
1
+ q1q2
Y~1(z)
1 2+2m1 m2z 1+ 2+m1 m2 Y~3(z)
Y~2(z )
1 2+2 3+2m1+2m2 2m3 z 1+ 2+ 3+m1 m3+
E
TU(m3) U(m2) U(m1)
(E.2)
gauge
2+m2 (z ) 2+ 3+m1+m2 m3 Y~ 1 (z
2+m2 (z ) 1+ 2+ 3+m2 m3
1 2m1 (z ) 1+2 2+ 3+m1 m3 +
Y~ 2 (z 2)
Y~ 2 (z)
Y~ 3 (z )
1
Y~ 2 (z)
Y~ 1 (z )Y~ 3 (z )
+ q1q22q3
1 2
3
2
3
3
2)+q2 1
1 z 1+ 2 Y~ 3 (z
Y~ 1 (z )
*
3zm3 Y~ 3 (z
2) + q3 2
m2 z 3+m2 Y~ 2 (z
gauge
1
Y~ 3 (z)
+ q2q3 1
2+2m1 z 2+ 3+m1 Y~ 1 (z)
Y~ 2 (z )
+ q1q2q3 2 1+ 2
z 1+ 2+ 3 +
Y~ 1 (z 2)
gauge
:
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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