Holomorphic field realization of SH c and quantum geometry of quiver gauge theories
HJE
eld realization of SHc and quantum
JeanEmile Bourgine 0 1 2 5 6
Yutaka Matsuo 0 1 2 3 6
Hong Zhang 0 1 2 4 6
Symmetry, String Duality
0 Sogang University
1 Hongo 731 , Bunkyoku, Tokyo , Japan
2 Via Irnerio 46 , 40126 Bologna , Italy
3 Department of Physics, The University of Tokyo
4 Department of Physics and Center for Quantum Spacetime , CQUeST
5 INFN Bologna, Universita di Bologna
6 35 Baekbeomro , Mapogu, Seoul 04107 , Korea
In the context of 4D/2D dualities, SHc algebra, introduced by Schi mann and Vasserot, provides a systematic method to analyse the instanton partition functions of N = 2 supersymmetric gauge theories. In this paper, we rewrite the SHc algebra in terms of three holomorphic elds D0(z), D 1(z) with which the algebra and its representations are simpli ed. The instanton partition functions for arbitrary N = 2 super YangMills theories with An and A(n1) type quiver diagrams are compactly expressed as a product which are written in terms of SHc and the orthogonal basis introduced by Alba, Fateev, Litvinov and Tarnopolskiy. These building blocks are characterized by new conditions which generalize the known ones on the Gaiotto state and the CarlssonOkounkov vertex. Consistency conditions of the inner product give algebraic relations for the chiral ring generating functions de ned by Nekrasov, Pestun and Shatashvili. In particular we show the polynomiality of the qqcharacters which have been introduced as a deformation of the Yangian characters. These relations de ne a second quantization of the SeibergWitten geometry, and, accordingly, reduce to a Baxter TQequation in the NekrasovShatashvili limit of the Omegabackground.
of four building blocks; Gaiotto state; dilatation

Holomorphic
1 Introduction 2
Reformulation of SHc algebra
SHc algebra in terms of holomorphic elds
Rank N representations
Adjoint action of the vertex operators
Ward identities of SHc and qqcharacter
Nekrasov instanton partition function
Action of SHc operators on instanton partition functions
A1 quiver
qqcharacters of higher representations for the A1 quiver
Generalization to the AQtype quiver
Quantum SeibergWitten geometry
Summary and concluding remarks
3 Instanton partition function and SHc algebra
C Some calculations of commutation relations in the rank N representation 29
D Detailed analysis of the qqcharacters
D.1
Matter case
D.2 Second qqcharacter of the A1 quiver
D.2.1
Double action of SHc operators D (z)
D.2.2 Derivation of the second qqcharacter
1
Introduction
SHc is an algebra introduced by Shi mann and Vasserot in [1] (see also [2]) to describe
the equivariant cohomology of the instanton moduli space of N = 2 gauge theories in four
dimensions. It has been de ned as a spherical (symmetric) version of the degenerate double
a ne Hecke algebra (DAHA) which has been developed by Cherednick for many years [3].1
While DAHA encodes the algebraic (recursive) properties of Macdonald polynomials [4]
1In fact, SHc is a short notation introduced in [1] for central extension of the Spherical degenerate double
a ne Hecke algebra. It may be better referred to as Schi mannVasserot algebra, but we will use SHc in
the text.
{ 1 {
with two deformation parameters, degenerate DAHA is obtained by taking a limit q; t ! 1
such that one parameter
remains, with t = q
. In this limit, Macdonald polynomials
degenerate into Jack polynomials.
This algebra precisely describes the algebraic structure behind Nekrasov instanton
partition functions [5] with the Omega background R21
R22 with the identi cation
=
1= 2
and has been used to prove the 4D/2D correspondence which generalizes Alday, Gaiotto
and Tachikawa's proposal [6] (AGT conjecture) for various types of quiver gauge theories
 namely pure super YangMills theory [1], the gauge theories with fundamental [7] and
bifundamental hypermultiplets [8] (see also the recent preprint [9]). For pure super
YangMills, the theory is characterized by a Gaiotto state [10], a coherent state a liated to the
has also revealed itself particularly useful in the study of vortex dynamics [12].
In such developments, the important role devoted to SHc as a \universal" symmetry
came principally from the fact that it contains all WN algebras for arbitrary N , together
with an additional U(1) factor. The parameter
is identi ed with a deformation parameter
p
the combination Q = p
that de nes the central charge c = (N
1. In this sense, SHc should be regarded as a one parameter
1)(1+ Q2N (N +1)) of WN representations through
deformation of the W1+1 algebra. The latter is known to have realizations in terms of N
free fermions acting on a space of N tuple Young diagrams. These representations, that
we call here rank N representations, are identical to those de ned by Fateev and Lukyanov
in [13]. The correspondence between the two algebras has also been con rmed in more
general cases where the Hilbert space contains singular vectors. The most typical example
is the minimal models of WN . There, it has been demonstrated explicitly in [14] that
SHc reproduces the proper descriptions of the Hilbert space constrained by the socalled
NBurge conditions [15, 16]. This universality is essential when we have to treat a system
that contains gauge groups of di erent rank, as it is the case for quiver theories.
On the other hand, the action of SHc on instanton partition functions of quiver N = 2
gauge theories is very di erent from the representation of WN algebras. It is better
understood after the introduction of an orthonormal basis constructed by Alba, Fateev, Litvinov
and Tarnopolsky (AFLT basis) to prove the 4D/2D duality [
17, 18
]. AFLT basis should be
regarded as a generalization of Jack polynomials [19{22], it is the proper basis to describe
the action of degenerate DAHA. Instead of the description in terms of chiral primary elds
with di erent spins, it is de ned more abstractly through generators Dm;n with two indices
m; n. The rst index m 2 Z is identi ed with the index of the Virasoro generators Lm
while the second one n 2 Z 0 corresponds to the spin n + 1 of the generator. Since it is
a nonlinear symmetry with a reasonably complicated structure, we have to be careful how
to organize the generators. One of the authors [23] has recently found that holomorphic
expansion in terms of the second index, D 1(z); D0(z), gives a compact description of SHc
{ 2 {
through the study of the NekrasovShatashvili [24] limit of AGT conjecture. This turns
out to be very useful and is a main tool of this paper.
The focus of the paper is to provide an SHc description of Nekrasov partition functions
for general AQ and A(Q1)type quiver gauge theories. To do so, the action of SHc operators
on Gaiotto states, together with the adjoint action on the intertwiner operator describing
bifundamental elds, is worked out. These actions are conveniently expressed in terms
of the vertex operators Y(z) associated with the current D0(z). They extend the work
on the covariance of the partition function presented in [8] by giving us the possibility
to consider quiver theories with gauge groups of di erent ranks. As a consequence of
our results, several useful identities can be established among correlators of the gauge
theories. In particular, we were able to recover the expression of the qqcharacters recently
introduced by Nekrasov, Pestun and Shatashvili (NPS) [25]. For the simplest A1 case with
fundamental multiplets (4.10),
(z) =
i denotes an average weighted by the instanton partition function,
which is de ned in (4.4). The operator Y(z) is interpreted as an operator version of the
chiral ring generating function.
These characters, presented as further deformation of
the characters of Yangian algebras [26], encode in a compact form a recursion relation
among the instanton partition functions [27]. Here we show that SHc provides a proper
symmetry behind the qqcharacter formulae, as was already predicted by NPS, and that
the polynomiality property naturally follows from our description.
The qqcharacters de ne a double deformation of the SeibergWitten geometry in a
form of second quantization. In the above example, the SeibergWitten curve is expressed
as (5.1),
y + q
m(z)
y
N
= Y(z
`=1
a`):
SeibergWitten theory is wellknown to provide an e ective description of the infrared
sector of N = 2 gauge theories on R4 [28, 29]. The e ective Lagrangian is written in terms
of an holomorphic function, the prepotential, obtained from the knowledge of an algebraic
curve and a di erential form. This formulation is identi ed with the construction of nite
gap solutions for classical integrable hierarchies, the algebraic curve corresponding to the
spectral curve of the system [30].2 When the gauge theory is considered in the
NekrasovShatashvili (NS) limit 2 ! 0 of the Omegabackground, the associated integrable systems
are quantized, with the remaining parameter 1 playing the role of the Planck constant [24].
The algebraic curve becomes the Baxter TQequation of the quantum system [31, 32] (see
also [27, 33] for the extension to quivers), it is equivalent to a Schrodinger equation under
a quantum change of variables [34], in a form of ODE/IM correspondence [35].3 In this
2These
nite gap solutions can also be obtained from Hitchin systems.
3In this classical version of AGT correspondence, the Shrodinger equation is obtained as the
semiclasframework, the two complex variables of the algebraic curve become noncommutative,
thus de ning a rst quantization of the SeibergWitten curve [38{40]. In the full
Omegabackground, the qqcharacter is an operator acting in a Hilbert space of quantum states.
In the NS limit, the expectation value of this operator in the Gaiotto state (which plays the
role of a coherent state) becomes the Tpolynomial of the TQequation, while its de ning
relation in terms of vertex operators reproduces Baxter's relation. In this sense, the
qqcharacter formula presents a second quantization of the integrable system in which the
TQrelation emerges in the classical 2 ! 0 limit.
We organize the paper as follows. In section 2, we introduce the holomorphic eld
description of SHc algebra and the rank N representation. We also provide useful
expressions for the adjoint actions of the vertex operators. In section 3, after a brief review of the
general construction of the instanton partition functions, we introduce the building blocks
(Gaiotto states, avor vertex operator, intertwiner) with which the partition functions are
written as a product. We show that the Gaiotto state satis es stronger constraints which
are compactly expressed in terms of SHc
elds. A generalized intertwiner which connects
di erent rank gauge groups is also de ned. It satis es similar conditions as the Gaiotto
states and indeed it reduces to the Gaiotto state when the gauge group of one side is trivial.
The avor vertex operator is used to include the fundamental hypermultiplets in the gauge
theories. These results are used in section 4 to build an in nite number of constraints
among the correlation functions of the vertex operator Y. The new characterizations of
the Gaiotto states and the intertwiner play an essential role to give a closed and compact
expression for these constaints  written in the form of qqcharacters. Finally in
section 5 we present their interpretation as quantum SeibergWitten geometry along the line
of [23, 27, 41]. The concluding section proposes some perspectives for future research, and
several technical details are gathered in the appendix.
2
2.1
Reformulation of SHc algebra
SHc algebra in terms of holomorphic
elds
Z
The SHc algebra is de ned on a set of operators Dm;n with the double grading (m; n) 2
Z 0 [1]. The rst index is called the degree and the second index the order. The
algebraic relations involving D 1;n and D0;n are written as
[D0;n; D 1;m] =
D 1;n+m 1
; n
1 ;
[D 1;n; D1;m] = En+m
n; m
0 ;
[D0;n; D0;m] = 0 ; n; m
0 :
(2.1)
(2.2)
(2.3)
where Ek denotes a linear combination of powers of the generators D0;n that will be given
shortly. Additional relations can be found in [2], but they will not be used here. The
algebra is spanned by the operators of degree 0 and
1 upon the recursive use of the
following commutation relations,
1
m
D (m+1);0 =
[D 1;1; D m;0];
D m;n =
[D0;n+1; D m;0];
(2.4)
{ 4 {
1
n=0
1
n=0
for n
product of the WN algebra and a U(1) current. The Heisenberg generators (U(1) currents)
are related to Dm;0, the Virasoro generators to Dm;0; Dm;1, and operators Dm;n with n > 1
to the currents of spin n + 1 [1, 7].
It is useful to assemble the generators in the form of holomorphic elds [23],
D 1(z) =
X z n 1D 1;n;
D0(z) =
X z n 1D0;n+1;
E(z) = 1 + +
1
X z n 1En;
n=0
(2.5)
(2.6)
(2.7)
(2.8)
(2.9)
(2.10)
holomorphic elds,
[D0(z); D 1(w)] =
to introduce
where + = 1+ 2. We use here the Omegabackground equivariant deformation parameters
1; 2 [5, 42] instead of the SHc deformation parameter
=
1= 2 in order to simplify the
comparison with gauge theories.4 It is noted that these Laurent series are vanishing at
z = 1, in which they are di erent from usual holomorphic elds in CFT.
We rewrite the de ning properties of the generators D 1;n and D0;n in terms of the
D 1(w)
This de nition of the eld
(z) resembles the mode expansion of an holomorphic free
bosonic eld in CFT and the series D0(z) can be interpreted as the associated current. As
we noted, however, the usual U(1) current in CFT is expanded as a sum over the degree
as J ( ) = P
n2Z (coe :)D n;0
In addition, elds at di erent points are commuting, [ (z); (w)] = 0, as a consequence
of (2.3). In this sense, the interpretation of the complex variable z is di erent from the
holomorphic coordinate of a Riemann surface but it should rather be seen as the spectral
parameter of an integrable model. We will come back to this description later.
The following dressed combination of vertex operators will play a central role in our
reformulation of the SHc algebra and the correspondence with gauge theories,
n 1 while (2.7) is expanded with respect to the order.
Y(z) = ec(z)e (z 1)e (z 2)e
(z)e
(z +):
The function c(z) encodes the dependence in the in nite number of the central charges cn
(n
0) of the algebra. It expands as
The generating series E(z) can now be expressed using the newly de ned vertex operator,
c(z) = c0 log(z)
1
X
n=1
cn :
nzn
E(z) = Y(z + +)Y(z) 1:
4The correspondence between the convention of [8] and this paper is summarized in appendix A.
{ 5 {
Among possible representations of SHc, the best studied one is the rank N representation
where the Hilbert space is spanned by a basis labeled by N tuple Young diagrams Y~ =
(Y1;
; YN ). The representation is characterized by N complex numbers a` (` = 1;
; N )
that de ne the central charges cn through the relation
N
`=1
ec(z) = Y(z
a`):
To emphasize the dependence of the representation in the parameters a` through the central
charges, they will be included in the notation of the vector basis j~a; Y~ i. These vectors form
an orthonormal basis of the representation space,
be identi ed with the generalized Jack polynomials introduced in [21] and studied in [22].
The vacuum state is obtained by taking the N tuple of empty Young diagrams denoted ~;,
it satis es
D0;nj~a;~;i = D 1;nj~a;~;i = 0; or D0(z)j~a;~;i = D 1(z)j~a;~;i = 0:
(2.13)
The Hilbert space spanned by j~a; Y~ i will be denoted V~a. When several Hilbert spaces are
considered, an extra label ~a will be inserted on the notation of the operators D~ra(z) to
specify in which space V~a they act. The rank N representations of SHc are equivalent to
the representations of WN
U(1) [1, 14].
The action of SHc generators of degrees 1 on the state j~a; Y~ i involves the N tuple
Young diagram with a box added/removed.
As such, they can be seen as an analog
of creation/annihilation operators while the total number of boxes in Y~ represents the
number of particles (later identi ed with the instanton charge). Following [23], N tuple
Young diagrams Y~ with a box x added/removed will be denoted Y~
x (respectively).
We further introduce the sets A(Y~ ) and R(Y~ ) containing all the boxes that can be added
to/removed from the Young diagrams composing Y~ . In gure 1, we illustrate the locations
of the boxes in the sets A(Y ) and R(Y ) with the example of a single Young diagram Y .
The boxes x 2 Y~ are characterized by a triplet of indices (`; i; j) where ` = 1
N and
(i; j) 2 Y` gives the position of the box in the `th Young diagram. To each box x is
associated a complex number x depending on the central charges using the map
x = (`; i; j) 2 Y
~
!
x = a` + (i 1) 1 + (j
1) 2 2 C:
With these de nitions, the action of the spanning subalgebra takes the simple form [8, 23]
D+1(z)j~a; Y~ i =
D0(z)j~a; Y~ i = X
X
x2A(Y~ ) z
x2Y~
z
1
{ 6 {
which are equivalent to their component form (n
x ;
i
We note that the second relation in (2.16) implies that the moments of x2Y~ are the
eigenvalues of the commuting charges D0;n. In the (generalized) CalogeroSutherland system,
D0;n plays the role of in nite commuting charges and
x is interpreted as the momentum
E(z), e (z). However it is reversed for the operators D 1(z),
of each particle. The interpretation of z as the spectral parameter is natural in this sense.
The left action of SHc generators on bra h~a; Y~ j is identical for the diagonal operators D0(z),
h~a; Y~ jD+1(z) =
X
2 R(Y )
X
The series E(z) is also diagonal on the states j~a; Y~ i, with eigenvalues given by the function
(z)2 at z = x with x 2 A(Y~ ) or R(Y~ ):
x(Y~ ) in the action (2.15) of D (z) correspond to the residues of this
Eventually, the action of the vertex operator is expressed in terms of a product over the
boxes of Y~ ,
e (z)j~a; Y~ i = QY~ (z)j~a; Y~ i;
with
QY~ (z) =
Y (z
x2Y~
x):
The speci c combination of vertex operators entering in the de nition (2.8) of Y(z) leads
to a remarkable simpli cation of its eigenvalues
Y(z)j~a; Y~ i = Y(z
N
`=1
a`) Y (z
x2Y~
(z
x
1)(z
x)(z
x
x
+)2) j~a; Y~ i = Q
Q
x2A(Y~ )(z
We note that there is a cancellation of factors between the numerators and the denominators
in the middle term, and the resulting expression in the r.h.s. bears contributions only from
the edges of the Young diagrams. Taking the ratio (2.10) de ning the operator E(z), we
recover the expression (2.18) for the function
(z)2:
In appendix C, we provide an explicit computation of the commutation relations of D0(z),
D 1(z) in the rank N representation.
Finally we would like to mention the existence of an automorphism of representation.
Under the shift of ~a, ai ! ~a0 = ~a + ~e where ~e = (1; 1;
; 1), the representation (2.16)
implies that
D~+a+1 ~e(z)j~a + ~e; Y~ i =
D~a+1 ~e(z)j~a + ~e; Y~ i =
X
spectral ow in the context of W1+1algebra [43].
2.3
Adjoint action of the vertex operators
The coe cients appearing here may be identi ed with the representation of D~a 1(z
implies that there is an automorphism of the algebra by shifting the variable z: Dr
~a+ ~e(z)
) for r = 0; 1. This shift symmetry of the representations is referred as the
In order to prepare for the computations necessary in the next sections, we would like to
evaluate the commutation relations between the vertex operators e
(z) and the elements
spanning the SHc algebra. The generators of degree zero form a commutative subalgebra,
as a consequence the eld
(z) commute with the series D0(z).
The evaluation of the adjoint action on D 1 is slightly more involved. We introduce a
vertex operator depending on two nite sets of points zi and wj with i 2 I, j 2 J ,
U (fzig; fwj g) := exp
(zi)
We claim the following identities:
U (fzig; fwj g) 1D1(u)U (fzig; fwj g) = Pu=1;zi2I
U (fzig; fwj g) 1D 1(u)U (fzig; fwj g) = Pu=1;wj2J Q
X
i2I
{ 8 {
X
j2J
!
(wj ) :
" Q
Q
" Q
with the projector Pu=1;zi2I acting on functions of the variable u as
Here Pz+ picks up the positive powers of a Laurent series in z, namely for a function
powers of f (z). The second term in (2.27) also plays the role to remove singularities at
u = zi. One may use a contour integration to write these projections in a compact form,
where the contour C is de ned by jwj = R with R < Mini(jzij). The formula (2.26)
formally resembles an OPE in CFT, up to the existence of the projection operator which
is necessary here since there is no singularity except for u = 0 on the left hand side.
Before proving (2.26), it may be instructive to give some speci c examples which will
be used later. The rst one is when one of the sets I; J is null:
(2.28)
(2.29)
;
(2.30)
(2.31)
(2.32)
e
PiM=1 (zi)D (z0)e PiM=1 (zi) =
X D (zi) Y
e PiM=1 (zi)D (w)e
PiM=1 (zi) = Pw D (w) Y(zi
M
i=0
"
j(6=i) zj
M
i=1
1
z
i
;
#
w) :
The other one is the adjoint action of Y(z) which is de ned as a product of vertex operators
with shifted arguments:
1
Y(z)
D 1(w)Y(z) = S(w
z)D 1(w) +
1
Y(z + +)
Y(z + +)D1(w)
= S(z
w)D1(w)
where S(z) denotes a scattering factor S(z) de ned as
of the SHc algebra. We use the property
Proof of the formula (2.26).
To end up this section, we would like to give a short
derivation of the identity (2.26). Rather than working with the commutator (2.6) directly,
it is easier to evaluate the action on the states j~a; Y~ i which form a faithful representation
S(z) =
(z + 1)(z + 2) :
z(z + +)
QY~ x(z)
QY~ (z)
= (z
x) 1;
{ 9 {
which is a direct consequence of (2.20), and the action (2.15) of D (z) on the states j~a; Y~ i.
It follows that
U (fzig; fwj g) 1D1(u)U (fzig; fwj g)j~a; Y~ i =
X
The product in the r.h.s. can be rewritten as a sum over single poles in x, with an extra
polynomial term, using the algebraic identity,
(u
Q
Here an are the coe cients appearing in Laurent expansion of the l.h.s. in , they depend
on the parameter zi, wj and u:
P+
(u
Q
can be used to reform D1(z), while the polynomial part in
x gives the transformations D1;n, and (2.33) becomes
Q
Since the states j~a; Y~ i generate a faithful representation, the equality of vectors can be
lifted at the level of operators
U (z; w) 1D1(u)U (z; w) =
Q
Q
The expression for U (z; w) 1D 1(u)U (z; w) is similarly obtained and is written as (2.37)
with the substitution of the variables zi $ wj . The right hand side of (2.37) can be
simpli ed by analyzing the rst term: the second term cancels the poles (and the constant
part) at u = 1 of the rst term, while the third term cancels the simple poles at u = zi.
The existence of such terms is natural since the left hand side of (2.37) is not singular at
these points. The procedure of removing the unwanted poles is performed by the projector
at in nity, the following property should be employed,
de ned in (2.27), and (2.37) produces (2.26). It is noted that to analyze the pole
(2.33)
(2.34)
(2.35)
(2.36)
(2.37)
P+ r( )
u
=
P+r( )
Pu+ ur(u)
(2.38)
for any meromorphic function r(z). It implies in particular
Instanton partition function and SHc algebra
Nekrasov instanton partition function
Q
Y~1; Y~Q i=1
Class S gauge theories with N = 2 supersymmetry are obtained by compacti cation of the
six dimensional N = (2; 0) theory on a Riemann surface. They are classi ed by a quiver
diagram where each node i is in correspondence with the simple group component SU(Ni)
of the total gauge group G =
iSU(Ni). Thus, to each node corresponds a gauge multiplet
containing a vector, two fermions and a scalar
eld in the adjoint representation. The
arrows i ! j of the quiver represents bifundamental matter elds, i.e. a chiral multiplet
containing a fermion and a scalar eld, with mass mij , and transforming in the fundamental
representation of SU(Ni)
SU(Nj ). In addition, a number N~i of fundamental (or
antifundamental) matter elds can be attached to each node i. They consist in chiral multiplets
of masses m(f) with f = 1
i
N~i, encoded in the N~ivector m~i (see gure 2).
The instanton partition functions of class S theories have been evaluated using
localization in the Omegabackground [5]. The theory is considered on the Coulomb branch
where the adjoint scalar elds take nonzero vacuum expectation values. These complex
parameters will be denoted a(`i) with ` = 1
Ni, they form the Nivector ~ai attached to the
node i. Localization provides a sum over nested integrals that can be computed by residues.
The residues are in onetoone correspondence with the boxes of the Nituple Young
diagrams for each node i of the quiver. The resulting formula is a sum over realizations of
these diagrams weighted by the multiplets contributions [42, 44{47]:
Zinst. =
X
~
Y qjYijZvect.(~ai; Y~i)Zfund.(~ai; Y~i; m~i)
i
Zbfd.(~ai; Y~i; ~aj ; Y~j jmij ); (3.1)
Y
i!j2EQ
where Q is the number of nodes in the quiver, EQ its set of links, and jY~ j denotes the total
number of boxes in the Ntuple Young diagram Y~ . The instanton counting parameter qi
corresponds to the exponentiated gauge coupling at the node i, suitably renormalized in
asymptotically free theories .
It is known that the contribution from each representation can be systematically
derived from that for the bifundamental representation. Taking a bifundamental eld of mass
m12 coupled to the two gauge groups SU(N1) and SU(N2), the contribution reads:
Zbfd.(~a; Y~ ;~b; W~ jm12) =
Y gY`;W`0 (a`
b`0
m12)
Y
N1 N2
`=1 `0=1
Y
(i;j)2
Y
(i;j)2
j + 1)) :
(3.2)
(3.3)
fund. m~1
~
N1
SU(N1)
vevs a~1
fund. m~2
~
N2
SU(N2)
vevs a~2
bf. m12
SU(NQ)
vevs a~Q
Here i is the height of ith column and 0i is the length of ith row of Young diagram
(see
gure 3).
The other building blocks can be written from (3.2) as follows.
Fundamental hypermultiplets transforming under the gauge group SU(N ) and the
avor group SU(N~ ), with masses m1;
mass m12 = 0, for the rst node N1 = N~ , ~a1 = m~ := (m(1);
and for the second node N2 = N , ~a2 = ~a and Y~2 = Y~ arbitrary:5
; mN~ : we take a vanishing bifundamental
; m(N~ )) and Y~1 = ~;,
Zfund:( m~; ~a; Y~ ) = Zbfd.( m~; ~;; ~a; Y~ j0):
Y~2 = ~;,
Antifundamental hypermultiplet: in a symmetric way, we take m12 = 0, for the rst
node N1 = N , ~a1 = ~a and Y~1 = Y~ and for the second one N2 = N~ , ~a2 =
m~ and
Zaf:( m~; ~a; Y~ ) = Zbfd.(~a; Y~ ; m~; ~;j0) = Zfund:(
+
m~; ~a; Y~ ):
Adjoint hypermultiplet: we take N1 = N2 = N , Y~1 = Y~2 = Y~ and ~a1 = ~a2 = ~a,
Vector multiplet: inverse of the adjoint hypermultiplet with zero mass,
Zadj:(~a; Y~ jm) := Zbfd.(~a; Y~ ; ~a; Y~ jm):
Zvect.(~a; Y~ ) := Zbfd.(~a; Y~ ; ~a; Y~ j0) 1:
(3.4)
(3.5)
(3.6)
(3.7)
HJEP04(216)7
We note that the fundamental matter can be seen as bifundamental matter where one
of the two gauge groups is taken in the weak coupling limit, e ectively becoming a avor
group. In this limit, the corresponding exponentiated gauge coupling q is sent to zero,
~
and due to the presence of the factor qjY j, only empty Young diagrams contribute in the
5We have chosen to shift the de nition of the fundamental masses by + in order to simplify formulas:
as they are equivalent to fundamental contributions with shifted masses.
summations. As a result, the contribution of a fundamental matter multiplet is derived
from the bifundamental contribution by attaching to each set of N~i fundamental avors an
N~ituple of empty Young diagrams.
The fact that the various contributions to the partition function of the di erent
multiplets are derived from the bifundamental contribution implies important consequences for
their SHc realization that will be presented in the next section.
3.2
Action of SHc operators on instanton partition functions
In this paper we focus on the linear quiver AQ and its a ne version A(Q1), they are
characterized by the set of arrows EQ = fi ! i + 1; i = 1
Q
1g and EQ = fi ! i + 1; i = 1
Qg
respectively, with the identi cation of indices modulo Q. One of the goal of this paper is
to formulate the action of SHc operators on the (a ne) linear quiver instanton partition
function. For this purpose, we need to rewrite the partition function in terms of elements
of the representation theory of SHc: Gaiotto states, intertwiner and the vertex operator
( gure 4).
Gaiotto state. To each node i of the quiver diagram is associated a vector space of
representation Va~i spanned by the vectors j~ai; Y~ii where Y~i takes values in all the possible
realization of Nituple Young diagrams. The set of complex parameters ~ai is xed in each
Va~i and de ne the central charges of the representation of SHc. The Gaiotto state has been
introduced in [10] as a speci c Whittaker vector of the Virasoro algebra with respect to the
maximal nilpotent subalgebra fLn; n > 0g. This algebra is spanned by the two elements
L1 and L2, and the Gaiotto state is de ned up to a normalization by the conditions,6
L1jGi =
jGi;
L2jGi = 0;
(3.8)
where
is a constant. This de nition has been generalized to the case with
fundamental avors [48], and to higher rank [49, 50], and eventually implemented in the space of
6As a consequence of the Virasoro commutation relations, the second condition implies LnjGi = 0
for n > 2.
X q
This state is known to provide the instanton partition function of pure N = 2 SYM (A1
where the operator D = D0;1 counts the number of boxes in Y~ , Dj~a; Y~ i = jY~ jj~a; Y~ i. It is
identi ed with L0 in Virasoro algebra up to the zero mode and qD may be regarded as the
propagator in string theory. In the following, we refer to the operator of the form qD as
the dilatation operator. It satis es,
qDD 1(z) = D 1(z)qD 1
:
In terms of SHc operators written in the form of holomorphic elds, the Gaiotto state
has a new characterization. This is one of the main results of the paper:
hG; ~aj
jG; ~ai
U ( m~)
representation of SHc (which contains a Virasoro subalgebra) [1],
These formulae are a consequence of a more general result, presented in (3.22) and (3.23)
below, and proven in appendix B.
These new expressions contain more information than the previously known relations
given in (3.16). They reveal themselves powerful enough to derive several useful relations,
presented in [27], among instanton partition function for arbitrary (Atype) quiver
diagrams. The asymptotic of the operators Y(z) at in nity is deduced from (2.21): Y(z)
zN
since jA(Y~ )j
jR(Y~ )j = N for any N tuple Y~ . Expanding the rst relation at in nite
spectral parameter z allows to recover the characterization of the Gaiotto states in [7, 8],
D 1;njG; ~ai = 0;
D 1;N 1jG; ~ai = p
jG; ~ai;
D 1;N jG; ~ai = p
1
2, and the last property has been obtained using the formula (A.3)
in [23]. These identities suggest to see the Gaiotto state as a (partial) coherent state in
the physical sense of eigenstate of the annihilation operators D 1;n.
Flavor vertex operator.
Due to the presence of the empty Young diagram in the
de nition (3.4), the fundamental matter contribution can be written in a simpler form,
0 N~
X
f=1
1
where QY~ (z) denotes the eigenvalue of the vertex operator de ned in (2.20). This
expression implies that the vertex operator can be used to insert fundamental multiplets in the
quiver gauge theories.
U ( m~)j~a; Y~ i = Zfund.( m~; ~a; Y~ )j~a; Y~ i: (3.18)
This operator generates the modi ed Gaiotto states in the presence of fundamental
multiplets, as studied in [7]. Since it plays the role to add the contribution of fundamental
hypermultiplets with
avor group SU(N~ ), it will sometimes be referred to as the avor
vertex operator. The instanton partition function for this theory can be written
Zinst = hG; ~ajqDU ( m~)jG; ~ai =
X qjY jZvect.(~a; Y~ )Zfund.( m~; ~a; Y~ ):
~
(3.19)
It is noted that the vertex operator U ( m~) commutes with the dilatation operator qD.
Intertwiner.
Up to now, only partition functions of N = 2 theories with a single gauge
group have been reproduced. To address the case of bifundamental matter coupled with
multiple gauge groups, the construction of a new operator V12(~a1; ~a2jm12) : V~a2 ! V~a1
is required. This operator intertwines two SHc representations speci ed by ~a1; ~a2, with a
di erent rank N1 for V~a1 and N2 for V~a2 ,
V12(~a1; ~a2jm12) =
Zbfd.(~a1; Y~1; ~a2; Y~2jm12) j~a1; Y~1ih~a2; Y~2j;
(3.20)
Zfund.( m~; ~a; Y~ ) =
m(f)) =
Y( 1)jY~ jQY~ (m(f));
(3.17)
~
N
f=1
where a renormalized version of the bifundamental contribution has been used,
Zbfd.(~a1; Y~1; ~a2; Y~2jm12) =
Zvect.(~a1; Y~1)Zvect.(~a2; Y~2)Zbfd.(~a1; Y~1; ~a2; Y~2jm12):
(3.21)
Several algebraic properties of the intertwiner operator were studied from the viewpoint of
SHc in [8], in relation with a recursion formula satis ed by Zbfd..7
The intertwiner satis es a set of identities which resemble the conditions (3.12){(3.15)
for the Gaiotto states:
D~a11(z)V12(~a1; ~a2jm12)
V12(~a1; ~a2jm12)D~a21(z
m12)
D~+a11(z)V12(~a1; ~a2jm12)
V12(~a1; ~a2jm12)D~+a21(z + +
Here the notation Y(i) (i = 1; 2) represents the action of the vertex operor Y in the space
under the action of SHc. The proof of the formulae is summarized in appendix B.
V~ai . These formulas characterize the transformation of the bifundamental contribution
In the 4D/2D correspondence, the intertwiner is described as a vertex operator of the
form, V = V COV Toda where V CO is the CarlssonOkounkov vertex [11] for the U(1) factor
and V Toda is the vertex operator of Toda eld theory associated with the WN algebra.
This construction, however, has some limitations.
One issue is the technical di culty
to de ne the transformation properties of V Toda for higher spin generators. We have to
face a nonlinear expression in terms of W generators or Toda elds which is usually not
manageable. A more serious issue is the impossibility to de ne an intertwiner between WN
and WM Toda systems with N 6= M since there is no obvious correspondence between the
generators in WN algebras with a di erent N (see, for example [51, 52], for attempts to
explore such a setup). At the level of SHc, the correspondence between the generators for
representations of a di erent rank becomes obvious and the transformation properties (3.22)
and (3.23) are compact and tractable. Furthermore, it was con rmed in [7, 8] that these
conditions contain the modi ed Ward identities for the U(1) current and the Virasoro
operator for V = V COV Toda when N1 = N2. In this sense, our characterizations of the
intertwiner is a natural generalization of the conventional vertex operators in Toda eld
theories to study the 4D/2D correspondence.
Gaiotto state from intertwiner.
As brie y recalled in the previous subsection, the
study of the instanton partition functions for miscellaneous eld content can be reduced to
the analysis of the bifundamental hypermultiplet. This fact has an important consequence
for the SHc realization that we explain here. We rst consider the special case N1 = N ,
N2 = 0 and m12 = 0 for the intertwiner. Here the rank 0 representation means a trivial
representation which consists of one state  the vacuum j; i (empty slots means that we
have no Fock space). Since Zbfd.(~a; Y~ ; ; j0) = Zvect.(; ) = 1, we nd after omitting the trivial
bra vacuum,
V12(~a; j0) = jG; ~ai:
(3.24)
7In [8] it was assumed that the ranks of the two representations are the same. However, the computation
performed there can be straightforwardly generalized to the case N1 6= N2.
SU(N )
vevs ~a
Taking the opposite case of a rank zero representation for the rst node produces the
bra Gaiotto state in a similar way. In this sense, the recursion properties of the Gaiotto
states (3.12){(3.15) are straightforward consequences of (3.22) and (3.23). We note that
we can take D(2)(z) = 0 for the trivial representation and the action operator Y on V2 is
replaced by 1.
Inclusion of (anti)fundamental hypermultiplet is also straightforward. From (3.4)
and (3.5), after
xing N1 = N , N2 = N~ , m12 = 0 and ~a2 = m~ + +, we obtain that the
action of the intertwiner on the vacuum produces the Gaiotto state with a avor vertex
operator inserted:
V12(~a; m~ + +j0)j m~ + +; ~;i = U ( m~)jG; ~ai :
Full partition function.
We now have all the elements to write down the instanton
partition function of any linear quiver as a product of operators,
Zinst =
( hG; ~a1jq1DU ( m~1)V12(~a1; ~a2jm12)q2DU ( m~2)V23(~a2; ~a3jm23)
TrV~a1 q1DU ( m~1)V12(~a1; ~a2jm12)q2DU ( m~2)V23(~a2; ~a3jm23)
jG; ~aQi
VQ1(~aQ; ~a1jmQ1) for A(Q1):
for AQ ;
~
Y
~
Y
(3.25)
(3.26)
(3.27)
To each arrow i ! j of the quiver is associated the intertwiner Vij (~ai; ~aj jmij ), and to each
node i an operator qiDU ( m~i). For the linear quivers, the resulting operator is sandwiched
between the Gaiotto states attached to the rst and the last node. On the contrary, a trace
is directly obtained for the a ne quiver from the intertwiners, it is de ned as
TrV~a
=
Xh~a; Y~ j
As an example, the partition function of N
bifundamental elds of mass m reads
= 2 theory represented in
gure 5 with
Zinst = TrV~a qDV11(~a; ~ajm) =
X qjY jZvect.(~a; Y~ )Zbfd.(~a; Y~ ; ~a; Y~ jm)
~
=
X qjY jZvect.(~a; Y~ )Zadj(~a; Y~ jm);
~
(3.28)
thanks to the orthonormality property of the states j~a; Y~ i.
Alternative expressions.
Although it will not be used in this paper, we would like
to provide, as a side remark, a new expression for the bifundamental contribution (3.2)
involving the vertex operator Y(z). This expression is a consequence of the property (B.1)
expressing the variation of Zbfd.(~a; Y~ ;~b; W~ jm) under the addition of a box in the Young
diagrams Y~ . It turns out that the right hand side of (B.1) is independent of the actual
content of boxes in the Young diagrams Y~ . As a result, this formula can be used to build
recursively Zbfd.(~a; Y~ ;~b; W~ jm) from Zbfd.(~a; ~;;~b; W~ jm), adding boxes one by one. Since,
Zbfd.(~a; ~;;~b; W~ jm) can be further identi ed with a fundamental contribution of mass ~a
m,
it is shown that
Zbfd.(~a; Y~ ;~b; W~ jm) = h~b; W~ jU (~a
m)j~b; W~ i
~b; W~ jY( x
m + +)j~b; W~ E
= h~a; Y~ jU (~b + m
+)j~a; Y~ i
Y D~a; Y~ jY( x + m)j~a; Y~ E
Y D
x2Y~
x2 W~
where the second equality has been obtained by exploiting the symmetry under the
exchange of (~a; Y~ ) $ (~b; W~ ) and m $
+
formulae for the vector contribution and the Gaiotto states can also be deduced,
m. As a special case of this expression, new
*
X
~
Y
Zvect.(~a; Y~ ) =
~a; Y~ U (~a
+) 1 Y
Y( x) 1 ~a; Y~
+
;
Ward identities of SHc and qqcharacter
In the previous section, we have seen that the instanton partition function for any AQ
type quiver gauge theories can be written by combining Gaiotto states, dilatation
operators,
avor vertex operators and intertwiners as in (3.26).
The behavior of these
states/operators under the action of SHc generators has been characterized through the
set of relations (3.12){(3.15), (3.11), (2.26) and (3.22){(3.23). As a side result, one obtains
a series of consistency conditions by inserting D 1(z) in the correlator and evaluating the
inner product in two di erent ways,
(hG; ~a1jO1D 1(z)) O2jG; ~aQi = hG; ~a1jO1 (D 1(z)O2jG; ~aQi) ;
(4.1)
where Oi denotes a combination of avor vertex operator, dilatation operators and
intertwiners. These conditions may be regarded as the Ward identities for the correlation
functions of SHc. Since it is written as a generating function with parameter z, it gives
an in nite number of constraints. In the following, we evaluate the explicit form of these
identities. We observe that their structure takes the form of a double quantum deformation
of the character formulae for AQ, the socalled qqcharacter proposed by Nekrasov, Pestun
and Shatashvili [25, 41]. In the next section, we will discuss another interpretation of these
formulae as a quantum deformation of the SeibergWitten curve.
(3.29)
(3.30)
HJEP04(216)7
We start from the expression (3.10) of the instanton partition function for pure SYM
with SU(N ) gauge group, and consider the insertion of the operator of D 1(z),
hG; ~ajD 1(z)qDjG; ~ai; evaluated in two di erent ways as in (4.1). After the use of the
identities (3.11), (3.12), (3.14), we arrive at,
hG; ~ajPz
qDjG; ~ai = 0 :
q
Y(z)
The insertion of Y has the e ect of adding extra factors to the partition function. For
HJEP04(216)7
example, from (2.21),
hG; ~ajY(z + +)qDjG; ~ai =
X qjY j
~
Zvect.(~a; Y~ ):
We will use the following notation for the expectation value of the Gaiotto state:
1
Zinst Y~
h
i =
X qjY jZvect.(~a; Y~ )h~a; Y~ j
~
(4.2)
(4.3)
(4.4)
(4.5)
(4.6)
(4.7)
(4.8)
This de nes an average of operators acting on states j~a; Y~ i that is normalized to h1i = 1.
The relation (4.2) is rewritten in the form:
P
z
This condition is the generating function of an in nite number of constraints on the
instanton partition function. At the same time, this formula implies that
(z) := hY(z + +)i +
= Pz+(Y(z + +))
q
Y(z)
has no negative powers of z in the Laurent expansion at z = 1. We note that Y(z) behaves
as Y(z)
zN as z ! 1. It implies that (z) thus de ned is a polynomial in z of degree
N . The expression
y + 1=y is reminiscent of the character of sl(2) for the fundamental
representation. The formula (4.6) is deformed by two parameters 1;2 and was referred as
a fundamental qqcharacter in [25, 27, 41] for the quantum deformed Yangian Y (sl(2)).
The inclusion of fundamental hypermultiplets with N~
avor is a straightforward
generalization. The only necessary modi cation is to insert a avor vertex operator U ( m~) in
front of jG; ~ai. The commutator with D 1(z) is obtained from (2.29):
D 1(z)U ( m~) = U ( m~)Pz [D 1(z)m(z)] ;
m(z) =
~
N
Y(z
f=1
m(f)) :
Inserting this relation between two Gaiotto states (with an operator qD), and then
evaluating the action of D 1(z) through (3.12) and (3.14) leads to
P
z
where the average acquired an extra factor Zfund.( m~; ~a; Y~ ),
1
i =
X qjY jZfund.( m~; ~a; Y~ )Zvect.(~a; Y~ )h~a; Y~ j
~
After including the fundamental hypermultiplets, the qqcharacter is modi ed to
(z) =
As a consequence of (4.8), Pz (z) = 0 and the qqcharacter is again a polynomial of degree
N in z. A more detailed discussion of the qqcharacter in the presence of fundamental
hypermultiplets is presented in appendix D.
In the case N~ < N , the ratio m(z)=Y(z) has no polynomial part and the character
(z) equals the average of the operator Pz+Y(z + +). As a result, explicit expressions for
the qqcharacter can be obtained by expansion of Y(z + +) at in nity using the
properties (2.8), (2.21):
(4.9)
(4.10)
(4.11)
(4.12)
(4.13)
+
:
(4.14)
N
`=1
N
`=1
N
`=1
Y(z + +) = Y(z + +
a`) 1
d
1 2 dz
D0(z) + higher terms in
= Y(z + +
a`) + 1 2zN 2 D + O(zN 3):
In the average (4.4) the operator D with eigenvalue jY~ j can be replaced by a logarithmic
qderivative,
(z) = Y(z + +
a`) + 1 2zN 2q@q log Zinst + O(zN 3) :
Specializing to N = 1 and to N = 2 with a1 =
a2 = a, we deduce the following expressions
U(1) :
SU(2) :
(z) = z + +
(z) = (z + +)
2
a;
4.2
qqcharacters of higher representations for the A1 quiver
In a series of recent lectures, Nekrasov proposed a generalization of the qqcharacter for
higher representations of Y (sl(2)) [25]. Higher qqcharacters involve a set of complex
parameters 1
;
; r 2 C, and they are de ned as
r(zj 1;
r) =
X
ItJ=f1; ;rg
qjJj Y S( i
i2I
j2J
j )
*
Y
i2I
Y(z + + + i
) Y m(z + j )
j2J Y(z + j )
Here S(z) is the scattering factor (2.31). It is claimed in [25] that the expectation value
of these operators is again a polynomial in z. This proposal has been veri ed using our
formalism in appendix D for the second character of pure SU(N ) SYM in the restricted
The polynomiality is further obtained in the cases N = 1 and N = 2 by employing the
explicit expression (4.11) of the operator Y(z + +). In both cases, it was found that
2(z1; z2) = Pz+1 Pz+2 hY(z1 + +)Y(z2 + +)i +
Generalization to the AQtype quiver
For simplicity here we will only treat explicitly the case of the A2 quiver without
fundamental matter elds. For any operator O we introduce the index
space V~a in which the operator acts, and we associate the expectation value
= 1; 2 labeling the
O
D ( )(z)E =
1
~ ~
X qjY1jqjY2jZvect.(~a1; Y~1)Zvect.(~a2; Y~2)Zbfd.(~a1; Y~1; ~a2; Y~2jm12)
1 2
h~a ; Y~ jO (z)j~a ; Y~ i:
To derive the qqcharacter relations, we consider the commutation relation (3.22) between
the SHc generating series D (z) and the intertwiner operator. We consider the operator
insertion of the following type,
hG; ~a1jD~a11(z)q1DV12(~a1; ~a2jm12)q2DjG; ~a2i;
hG; ~a1jq1DV12(~a1; ~a2jm12)q2DD~+a21(z)jG; ~a2i;
and then using the action of the SHc modes on Gaiotto states, it is possible to derive the
following identity, obtained respectively from the former and latter expressions:
cases N = 1 and N = 2. The second character can be rewritten using the shifted spectral
variables z1 = z + 1, z2 = z + 2
,
;
z12
+
Y(z1)
with z12 = z1
z2. This condition is equivalent to (4.14). From the insertion of two
operators D1(z1)D 1(z2) within two Gaiotto states, it is possible to show that
Pz1 Pz2 2(z1; z2) = 0:
P
z
P
z
Y
Y
(2)(z) =
(1)(z + +) + q1 Y
(2)(z + +) + q2 Y
(2)(z + +
m12)
Y(1)(z)
(1)(z + m12)
Y(2)(z)
+ q1q2 Y(2)(z
1
1
+ q1q2 Y(1)(z + m12
*
*
Y
Y
(1)(z + +) + q1 Y
(2)(z + +) + q2 Y
(2)(z + +
m12)
Y(1)(z)
(1)(z + m12)
Y(2)(z)
+ q1q2 Y(2)(z
+ q1q2 Y(1)(z + m12
m12)
+)
1
1
+
+
= 0;
= 0:
m12)
+)
+
+
;
:
These identities imply that the two following qqcharacters are polynomials in z:
(4.15)
(4.16)
Generalization of these formulae to the AQ quiver with fundamental multiplets is
straightforward. For example, the rst one is generalized to
;
j =
j 1
X mk;k+1;
k=1
(4.22)
with 0 = 1 = 0, Y0 = YQ+1 := 1 and the mass polynomials mi(z) = QfN~=i1(z
associated to the fundamental multiplet of the node i, with avor group SU(N~i). We note
m(fi))
that for a linear quiver with N = N1
N2
NQ, the rank of the avor group need
to satisfy N~i
2Ni
Ni+1
Ni 1 with N0 = NQ+1 = 0. In this setup, the character
(1)(z) is a polynomial of degree at most N .
As already stated, in the weak coupling limit q2 ! 0 the second node of the quiver
diagram acts as a set of N2 fundamental avors of mass a(`2) coupled to the rst node.
This relation can also be observed at the level of characters. As can be seen from (4.18)
in this limit only the empty N2tuple Y~2 = ~; contribute to the sum, Zvect.(~a2; ;) = 1 and
~
Zbfd. ! Zfund.. We further notice that the operator Y
as such, can be identify with (2)(z), it reproduces a mass polynomial with masses a`
(2)(z + +) becomes polynomial and,
N2
Y(z
`=1
(2)(z) !
a(2) + +) =: m(z):
`
In this weak coupling limit, the rst equation in (4.21) becomes the equation (4.10) for the
massive qqcharacter, with an extra shift of the fundamental masses by m12.
5
Quantum SeibergWitten geometry
In the limit 1; 2 ! 0, the Omegabackground reduces to R4 and the infrared theory
is characterized by a complex algebraic curve. This curve, together with a di erential
form, determines the prepotential of the theory through the SeibergWitten relations. It
is also associated to the spectral curve of a classical integrable system in the Bethe/gauge
correspondence (see for instance [30] and references inside). For simplicity here, we focus
our discussion on the case of a single node with gauge group SU(N ) and a number N~ of
fundamental multiplets. In this case, the algebraic curve can be written in the form
(2)
+,
(4.23)
y + q
m(z)
y
N
= Y(z
`=1
a`):
(5.1)
This expression should be compared with the de nition (4.6) of the qqcharacter. It is then
appealing to interpret the qqcharacter as a double deformation of the SeibergWitten
geometry, where the expectation value of the operator Y(z) reduces to the complex parameter
y of the curve E(y; z) = 0, while the qqcharacter
(z) reproduces the gauge polynomial
in the r.h.s. of (5.1). This is indeed the case, as we will demonstrate shortly.
The discussion becomes even more illuminating if we introduce the intermediate
background R21 R2 obtained in the NekrasovShatashvili limit 2 ! 0 of the Omegabackground.
This 1deformation of the Euclidean background is known to be responsible for the
quantization of the classical integrable system associated to the N = 2 gauge theory [42]. In
this background, the SeibergWitten curve is replaced by a Baxter TQequation that has
been derived in [31, 32] (the derivation was later extended to quivers in [27, 33]),
T (z)Q(z) = Q(z + 1) + qm(z)Q(z
1);
Q(z) = Y(z
ur);
r
where T (z) and Q(z) denote respectively the Baxter T and Qpolynomials. The
TQequation can be recast in a form more similar to the original SeibergWitten curve (5.1)
by the introduction of the ratio Y (z) = Q(z)=Q(z
(5.2)
(5.4)
(5.5)
(5.6)
(5.3)
In this form, it readily reproduces (5.1) in the limit 1 ! 0. In order to show that the
qqcharacter de nes a sort of second quantization of the SeibergWitten geometry, we will
take the NS limit and reproduce the TQrelation (5.4), the operator Y(z) being reduced
to the rational function Y (z), and the qqcharacter to the Tpolynomial.
To perform the NS limit, we will follow the procedure described in [23] (see also [27])
and rst rederive the Bethe equations. In the NS limit, the sum over Young diagrams
entering the expression (3.19) of the partition function is dominated by a Young diagram
with in nitely many boxes.9 This critical Young diagram minimizes the summation
and its pro le is obtained by solving the discrete saddle point equations:
qjY~ +xjZvect.(~a; Y~
+ x)Zfund.( m~; ~a; Y~
+ x)
qjY~ jZvect.(~a; Y~ )Zfund.( m~; ~a; Y~ )
= 1;
8x 2 A(Y~ ):
Taking into account the variation of the vector and fundamental contributions, we nd
q
1 2
m( x) Q
Q
Following [23], we now consider only Young diagrams with in nitely high columns, and
such that a box can be added to (or removed from) each column. Up to 2corrections,
the images under x of a box x 2 A(Y~ ) and the box immediately below x0 2 R(Y~ ) are
8The TQequation can also be written in an operatorial form,
y^ + qm(z)y^ 1 Q(z) = T (z)Q(z); y^ = e 1@z
T (z) = Y (z + 1) + q
1):8
m(z)
Y (z)
:
where y^ is a shift operator. Here the noncommutativity of the variables y^ and z becomes manifest and the
previous relation de nes a quantum curve. This di erence equation is actually equivalent to a Schrodinger
equation under a quantum change of variables [34]. This correspondence goes under the name of bispectral
duality [38{40] and can be seen as a degenerate version of the AGT correspondence relating the gauge
theory in the NS background with the semiclassical Liouville/Toda theory.
9This argument is similar to the one employed by Nekrasov and Okounkov in [42] to perform the limit
1; 2 ! 0. The main di erence here is that the critical Young diagram doesn't have a continuous pro le
but is instead described by a stepfunction where the plateaux are given by the Bethe roots.
equal, they de ne the set of Bethe roots ur =
x 2 A(Y~ ), except for N extra boxes (one for each diagram) that lie on the top right of
the diagrams, and for which
x = ` with l = a` + n` 1 and n` the number of columns
for the Young diagram Y` .It is emphasized that these extra boxes are necessary to ful ll
the relation jA(Y~ )j = jR(Y~ )j + N between the cardinal of the two sets. The number of
columns n` in each diagram will play the role of a cuto sent to in nity at the end of the
computation. Under this identi cation, and taking into account the factor
1 2 from the
box y of coordinate y =
x
2 just below x, we nd in the limit 2 ! 0:
x for x 2 R(Y~ ).10 This is true for all boxes
1 = q
m(ur)
(ur) (ur + 1) s=1 ur
M
Y ur
s6=r
us
us + 1
1
(z) = Y(z
(5.7)
HJEP04(216)7
and the number of Bethe roots is M = P` n`. These equations resemble the Bethe
equations of an inhomogeneous sl(2) XXX spin chain with a twist parameter q.11 The
TQequation associated to this system of Bethe roots reads
T (z)Q(z) =
(z) (z + 1)Q(z + 1) + qm(z)Q(z
1);
(5.8)
eigenvalues:12
Introducing the Qpolynomial as in (5.2), it is indeed possible to show that the r.h.s. is
a polynomial of degree M + 2N , with M zeros at z = ur as a consequence of the Bethe
equations (5.7). The TQequation (5.2) is reproduced by further sending the number of
Bethe roots M to in nity, together with the cuto s ` after a proper rescaling of the T
and Q polynomials. More details on this limit will be provided in the work [53] to appear.
The NS limit of the expectation value (4.9) of operators is also dominated by the
single state j~a; Y~ i, and diagonal operators in the basis j~a; Y~ i can be identi ed with their
hOi
D~a; Y~ jOj~a; Y~ E
due to the simpli cation Zinst
Zvect.(~a; Y~ )Zfund.( m~; ~a; Y~ ). From the action (2.21) of
to /removed from Y~
by Bethe roots, we nd
the operator Y(z) on states j~a; Y~ i, replacing the coordinate x of boxes that can be added
D~a; Y~ jY(z)j~a; Y~ E
Q(z) (z)
Q(z
1)
~a; Y~ j Y(z) j~a; Y~
(5.9)
Q(z
Q(z) (z)
1) :
(5.10)
for an appropriate choice of masses mf .
point equations.
Denoting
NS(z) the limit of the qqcharacter (z), the identity (4.6) reproduces the
TQequation (5.8) with the Tpolynomial T (z) =
NS(z) (z) (the presence of the extra cuto
factor (z) will be explained in [53]).
Our observation can be easily generalized to apply to linear quiver gauge theories.
The NS limit for the A2 quiver has been performed in [23]. Using the same procedure, the
qqcharacter identity (4.21) reproduces the TQrelation for an inhomogeneous sl(3) XXX
spin chain characterized by two sets of Bethe roots (equation (7.15) of [27]).
10In a Young diagram
, the image x of x = (i; i) 2 R( ) is given explicitly by x = a + (i 1) 1 +
( i 1) 2, it is
nite in the limit 2 ! 0 since i tends to in nity such that 2 i remains
nite.
11In fact, in the superconformal case N~ = 2N , it exactly reproduces the inhomogeneous XXX spin chain
12This is true for wellbehaved operators for which the insertion does not modify signi cantly the saddle
In this paper, we developed a holomorphic eld representation of SHc algebra. It has
the merit to express the commutation relations and the
nite rank representations of the
operators in a compact form. Instanton partitions for AQ and A(Q1)type quiver gauge
theories can be expressed concisely in terms of Gaiotto state, intertwiner operator, and
a newly introduced
avor vertex operator that insert the contribution of fundamental
hypermultiplets. A new characterization of the Gaiotto state and intertwiner has been
established using the adjoint action of SHc holomorphic elds. It provided the in nite set
of constraint on the instanton partition function from the chiral ring generating function
proposed in [27]. These constraints are summarized in simple algebraic relations which
were referred as the qqcharacter.
The qqcharacters describe a quantum version of the SeibergWitten geometry [41]
in the NS limit 2 ! 0. In this setup, the 2deformation introduces a form of second
quantization of the TQsystem in which the Tpolynomial is replaced by an operator acting
in the Hilbert space of the rank N representation. This should be compared with the
recent results obtained in [54, 55], where the subleading corrections to the NS limit have
been derived. These corrections are compatible (up to a quantum correction) with the
second quantization of the NS action that is interpreted as the YangYang functional of
the underlying quantum integrable system. By comparing these two di erent approaches, a
uni ed interpretation for the 2deformation of quantum integrable systems should emerge.
There are some obvious generalizations of the current study for future work. In [27],
similar algebraic relations for the chiral generating functional were proposed for ADE type
quiver gauge theories. In order to describe the bifurcation in the quiver diagram, we need
to nd a SHc description of trivalent vertex which takes of the form:
~
jbii
=
X
~
W
Zvect.(~b; W~ ) 1=2j~b; W~ i
j~b; W~ i
j~b; W~ i :
Here we added extra labels ;
to specify the Hilbert spaces. With the help of such
operator, one may give the partition for the D4 quiver partition function, for example, as
3
i=1
i hG; ~aijqi
D0(i;)1 V i i (~ai;~bjmi)
qD0(1;1) j~bii 1 2 3
4
=
X
Y~1;Y~2;Y~3; W~
3
i=1
~ ~
q4jW jZvect.(~b; W~ ) Y qijYijZvect.(~ai; Y~i)Zbfd.(~ai; Y~i;~b; W~ jmi) :
(6.1)
(6.2)
In order to derive the qqcharacter for such extended cases, we need to
nd an analog
of (3.22), (3.23) for the trivalent vertex. At this moment, however, this seems not so
simple and we would like to leave it for future study. Another possible direction proposed
in [27] is the 5D version of the current analysis. It corresponds to the algebra studied by
many authors in [56{59] and has implications in 4D [60, 61]. Since the building blocks
are already known (for example, [62, 63]), it would not be so di cult to perform a similar
analysis in such setup. In addition, our formulation of SHc seems particularly suited to the
generalization to the sixdimensional background R21
2
R 2
R23 in which the instanton
partition function of N = 2 theories are expressed as a sum over plane partitions [64].
In a di erent perspective, it is important to clarify the relations with integrable models.
On one hand, a proposal of NPS in [27] suggests a connection between quantum geometry
and the representation of the Yangian associated to the quiver Dynkin diagram. On the
other hand, Maulik and Okounkov have proposed in [65] the expression of the Yangian
of gbl(1). In their formalism, the Dynkin diagram is regarded as the
nite lattice of a
spin system, where the spin degree of freedom is actually described by a free boson Fock
space. In [66], a short summary of [65] and a possible supersymmetric generalization were
presented.
While the two approaches are very di erent, the coproduct de ned by the
authors of [65] coincides with the one employed in [1]. Our analysis which relates the NPS
qqcharacter [27] to the SHc algebra [1] could provide an interesting link between the two
Yangians.
Note added.
After the rst version of this paper was submitted to arXiv, N. Nekrasov
published a paper [67] where he studied DysonSchwinger equations for the instanton
partition functions. He evaluated the e ect of adding pointlike instantons, and express this
e ect by an operator Y which is identical to ours. There seems to be a direct relation with
our analysis and we hope to provide a more detailed comparison in the near future.
Acknowledgments
J.E. Bourgine thanks I.N.F.N. for his postdoctoral fellowship within the grant GAST,
which has also partially supported this project, together with 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 MPNSCOST Action MP1210. YM would like to thank Satoshi
Nakamura and R.D. Zhu for their comments and discussions. He is partially supported
by GrantsinAid for Scienti c Research (Kakenhi #25400246) from MEXT, Japan. HZ
thanks Chaiho Rim for instructions and comments. He is supported by the National
Research Foundation of Korea(NRF) grant funded by the Korea government(MSIP)
(NRF2014R1A2A2A01004951).
A
Comments on the notations
In this paper, in order to ease the comparison with the gauge theory, we use the omega
background parameters 1; 2 instead of the CFT parameter
=
1= 2 in [7, 8]. Since
some results of these papers are used here, we summarize the correspondence between
the notations in this appendix. As we use two parameters instead of one, we need some
rescaling and shift of parameters to compare with the results there. Adding a tilde to the
notations in [7, 8], the comparison goes as follows:
D0;n+1 = ( 2)nD~ 0;n+1;
a` =
(x)jx2A(Y`) =
2a~` + +;
2(a~` +A~t(Y`));
D 1;n = ( 2)nD~ 1;n;
En = ( 2)nE~n ;
z = 2= ~;
(x)jx2R(Y`) =
2(a~` +B~t(Y`)):
cn = (
2)nc~n; (A.2)
(A.1)
(A.3)
We note that under the rescaling (A.1), the algebra (2.1){(2.3) remains the same.
HJEP04(216)7
Proof of the recursion formulae for Gaiotto states and intertwiner
Since the Gaiotto state can be derived from the intertwiner (3.25), it will be su cient to
prove (3.22), (3.23). We need a few formulae to characterize the behavior of the instanton
partition function building blocks (here bifundamental and the vector contributions) under
the variations of the number of boxes in the N tuple Y~ .
Zbfd.(~a; Y~ + x;~b; W~ jm)
Zbfd.(~a; Y~ ;~b; W~ jm)
Zbfd.(~a; Y~
x;~b; W~ jm)
Zbfd.(~a; Y~ ;~b; W~ jm)
Zbfd.(~a; Y~ ;~b; W~ + xjm)
Zbfd.(~a; Y~ ;~b; W~ jm)
Zbfd.(~a; Y~ ;~b; W~
xjm)
Zbfd.(~a; Y~ ;~b; W~ jm)
Zvect.(~a; Y~ + x)
Zvect.(~a; Y~ )
Zvect.(~a; Y~
Zvect.(~a; Y~ )
= Q
= Q
Q
=
=
q
X
Y~ ; W~
Q
m)
y
y + +
m)
m)
y + m
y + m
+)
1
1
1 2 Q
Q
Q
y6=x
y6=x
y)( x
y)( x
y)( x
y)( x
y
+)
+)
(B.1)
(B.2)
(B.3)
(B.4)
(B.5)
(B.6)
(B.7)
These formulae were used in [8] to prove the recursive properties of the quiver gauge
theories. Essentially the same computation shows up here. We evaluate the action of
D (z) on the intertwiner:
X
Y~ ; W~
V (~a;~bjm) =
Zbfd.(~a; Y~ ;~b; W~ jm)j~a; Y~ ih~b; W~ j ;
with Zbfd.(~a; Y~ ;~b; W~ jm) :=
Zvect.(~a; Y~ )Zvect.(~b; W~ )Zbfd.(~a; Y~ : ~b; W~ jm). We rst
evaluate the action of D 1(z) on the intertwiner from the left. It is easily deduced from its
action on states j~a; Y~ i,
D~a 1(z)V (~a;~bjm) =
Zbfd.(~a; Y~ ;~b; W~ jm)
xih~b; W~ j:
(B.8)
X
z
x
An alternative expression can be obtained after noticing that the inverse images of a state
j~a; Y~ i under the mapping D 1(z) are the states j~a; Y~ + xi for x 2 A(Y~ ) because the action
of the operator removes one box. Since the states j~a; Y~ i form a basis of the vector space
V~a, it is possible to write
D~a 1(z)V (~a;~bjm) =
X
X
Y~ ; W~ x2A(Y~ )
x(Y~ + x)
z
x
Zbfd.(~a; Y~ + x;~b; W~ jm)j~a; Y~ ih~b; W~ j:
(B.9)
It is convenient to rewrite this expression as follows, using the fact that
D~a 1(z)V (~a;~bjm) =X
X
x(Y~ ) (2.19):
x(Y~ ) Zbfd.(~a; Y~ +x;~b; W~ jm)
Zbfd.(~a; Y~ ;~b; W~ jm)
= p
x(Y~ ) Zbfd.(~a; Y~ +x;~b; W~ jm)
y6=x
+) Q
y)
Q
y
m+ +)
We evaluate the ratio for Zbfd. from (B.1) and (B.5) and put the explicit form of
The action of D 1 from the right can be evaluated similarly,
V (~a;~bjm)D~b 1(z0) =
X
X
Y~ ; W~ x2R( W~ ) z0
x( W~ ) Zbfd.(~a; Y~ ;~b; W~
xjm)
Zbfd.(~a; Y~ ;~b; W~ jm)
From (B.4) and (B.6), the factor in the middle takes the form:
x( W~ ) Zbfd.(~a; Y~ ;~b; W~
Zbfd.(~a; Y~ ;~b; W~ jm)
xjm)
= p
1 2
Q
Q
y2R(W~ ) ( x
y6=x
y + +) Q
Q
y)
(B.10)
:
m)
(B.11)
(B.12)
y +m
+)
y + m)
:
(B.13)
1
To add (B.10) and (B.12), we use the following identity to simplify the formula (we put
z0 = z
m)
X
1
Q
y2R(Y~ )( x
y6=x
Q
= P
z
X
x2R( W~ )
1
m
Q
Q
y2A( W~ )( x
Q
y2R(W~ ) ( x
+) Q
y)
y6=x
Q
y2A( W~ )( x
y)
y2R(Y~ )(z
Q
y2A(Y~ )(z
y)
+) Q
Q
y2A( W~ )(z
y2R( W~ )(z
m + +)
m)
y
Q
y + m
+)
y + m)
m + +) !
y
m)
:
(B.14)
This formula is obtained by comparing the residue of the poles on both sides. Finally, we
note that by using (2.21), we obtain
P
z
Q
y2R(Y~ )(z
Q
y2A(Y~ )(z
+) Q
y)
Q
y2A( W~ )(z
y2R( W~ )(z
y
m + +) !
y
m)
j~a; Y~ ih~b; W~ j
= P
z
Y
1
(1)(z) j~a; Y~ ih~b; W~ jY
m + +) :
(B.15)
After combining everything, we prove (3.22). The proof of (3.23) is completely parallel
and we omit it here.
13In this expression, and the analysis hereafter, the correct choice of sign is veri ed by comparing with
the direct action of SHc generators on states with a small number of boxes.
tation
For completeness, we present here some explicit computations for the commutators of
holomorphic generators.
[D0(z); D1(w)]. From (2.15),
D1(w)D0(z)j~a; Y~ i =
D0(z)D1(w)j~a; Y~ i =
X
x2Y~
X
X
X
y(Y~ )
x w
y(Y~ )
j~a; Y~ + yi;
j~a; Y~ + yi:
y2A(Y~ ) x2Y~ +y
The di erence is the inclusion of the box y in Y~ + y. So we obtain the rst relation in (2.6).
[D0(z); D1(w)] j~a; Y~ i = X
[D1(z); D 1(w)].
D1(z)D 1(w)j~a; Y~ i =
w
y(Y~ )
j~a; Y~ i:
X
X
=
X
X
x2R(Y~ ) y2A(Y~ x)
x2R(Y~ ) y2A(Y~ )
+
X
y)
j~a; Y~ i =
X
y2A(Y~ ) z w
1
y(Y~ )
z
y(Y~ ) !
x(Y~ ) y(Y~
x z
x(Y~ ) y(Y~
x z
x)
x)
j~a; Y~
x + yi
j~a; Y~
x + yi
x(Y~ ) x(Y~
x z
x)
j~a; Y~ i;
where we have assumed for Y~ a generic form such that A(Y~
explicit form of x(Y~ ), one may prove for y 2 A(Y~ ):
x) = A(Y~ ) [ fxg. From the
y(Y~
x)2 = y(Y~ )2 S( x
S( y
y) ;
x)
x)2 = x(Y~ )2 ;
where S(z) is the scattering factor which appeared in (2.31). After combining them,
D1(z)D 1(w)j~a; Y~ i becomes,
X
X
x2R(Y~ ) y2A(Y~ )
x(Y~ ) y(Y~ ) S( x
w
x z
y
S( y
x)
j~a; Y~ x+yi+ X
x2R(Y~ ) (w
x)(z
(C.1)
(C.2)
j~a; Y~ i
(C.3)
(C.4)
(C.5)
x)
D 1(w)D1(z)j~a; Y~ i is evaluated similarly, assuming that R(Y~ + x) = R(Y~ ),
x z
S( x
S( y
x)
x+yi+X
(w
x)(z
In [D1(z); D 1(w)], the rst term cancels and the second term gives E(z) Ew( w)) j~a; Y~ i after
+(z
the use of the relations (2.18), (2.19) and (2.22). Degenerate situations should be evaluated
case by case, but the general conclusion remains unchanged.
x)
(C.7)
D
D.1
Matter case
Detailed analysis of the qqcharacters
There are di erent possibilities for the insertion position of the SHc operator D 1(z), the
simplest option is to insert it on the left of the mass operator as in hG; ~ajqDD 1(z)U ( m~)jG; ~ai,
and then use the braiding relation (4.7) to move it to the right. The two formulas (3.12)
and (3.14) for the action of D 1(z) provides the identity (4.8). However, this is not the
unique choice for the insertion of the D 1(z), a second possibility is to insert it on the right
of the mass operator,
deduced from (2.29):
hG; ~ajqDU ( m~)D 1(z)jG; ~ai:
A new braiding relation is needed in order to move the SHc operator to the left, which is
U ( m~)D 1(z) = B
0
N~ N~
X D 1(m(f)) Y
f=1 z
m(f0) AC U ( m~):
The action of D 1(z) on the Gaiotto state is then computed from (3.12) and (3.13), leading
to the identity
1
m(z) Pz hY(z + +)i
~
N
X
f=1 z
1
~
N
Y
f06=f
1
D
m(f0) Y(m(f) + +)
(m(f))E = q
(D.1)
(D.2)
1
Y(z)
;
(D.3)
where the operator (z) = Pz+Y(z + +) has been introduced to simplify the expression.
As a result, the fundamental qqcharacter de ned in (4.6) obeys
(z) = h (z)i + m(z) X
~
N
f=1
1
~
N
Y
f0=1 m(f)
f06=f
1
D
m(f0) Y(m(f) + +)
(m(f))E : (D.4)
In the last term, the apparent poles are cancelled by the zeros of m(z) and the r.h.s. is a
polynomial. The compatibility with the identity (4.8) implies the following equality,
m(z) X
~
N
f=1
z
1
~
N
Y
f0=1 m(f)
f06=f
1
D
m(f0) Y(m(f) + +)
(m(f))E = Pz+q
m(z)
Y(z)
: (D.5)
D.2.1
Double action of SHc operators D (z)
As a preliminary step it is necessary to derive the action of two operators D 1(z1) and
D1(z2) with di erent arguments on a Gaiotto state jG; ~ai. The method is the same as in
the case of a single operator, and upon using the property
h~a; Y~ + xjY(z + +)j~a; Y~ + xi = S(z
x)h~a; Y~ jY(z + +)j~a; Y~ i;
(D.6)
with S(z) the scattering factor de ned in (2.31), it is possible to write down
HJEP04(216)7
D 1(z1)D1(z2)jG; ~ai
1
X q
1 2 Y~
Zvect.(~a; Y~ ) X
x2A(Y~ )z1
1
x
Q
y2R(Y~ )( xy
y2A(Y~ ) xy
where the sign ambiguity has been xed by comparing the coe cient of the vacuum state
with the direct action of the SHc generators. Inserting the pole decomposition of
S(z2
z1
x) =
x
1 2
+z12 z2
1 2
x
+(z12
+) z2
1
x + +
;
with the shortcut notation z21 = z2
z1, it is possible to perform the summation over
1
Xq
1 2 ~
Y
Zvect.(~a; Y~ ) Pz2S(z21) Y(z2 + +) +
Y(z1)
1 2 Y(z2 + +)
+z12
Y(z2)
1 2
+(z12
+) j~a; Y~ i;
where it has been used that the last two terms in the brackets have no polynomial part at
in nity, and consequently in the last term the two factors Y(z2 + +) have cancelled each
other. This result can be written in the compact form
D 1(z1)D1(z2)jG; ~ai =
11 2 Pz2S(z21) Y(z2 + +) +
Y(z1)
1
+z12
Y(z2 + +)
Y(z2)
1
+(z12
The action of the commuted operators D1(z2)D 1(z1) is derived from the same method,
D1(z2)D 1(z1)jG; ~ai =
1
Y(z1)
1 2
S(z21) Pz2Y(z2 + +)
Pz1Y(z1 + +) +
+z21
Pz1Y(z1)
+(z21 + +) jG; ~ai:
This expression is simpli ed employing the following identity,
Pz2 [S(z21)Y(z2 + +)] = S(z21)Pz2Y(z2 + +)+
+1z221 Pz+1Y(z1 + +)
1 2
+(z21 + +) Pz+1Y(z1);
Q
1
x)Y(z2 + +)j~a; Y~ i;
(D.7)
(D.8)
(D.9)
+) jG; ~ai:
(D.10)
(D.11)
(D.12)
obtained by decomposition of S(z21) as a sum over single poles and of Y(z2 + +) into
positive and negative powers. As a result, we nd
Pz2 S(z21) Y(z2 + +) +
+z12
Y(z1 + +)
Y(z1)
1
+(z12
+) jG; ~ai:
(D.13)
Taking the di erence between (D.10) and (D.13), the commutation relation (2.6) between
D 1(z1) and D1(z2) is recovered, with the action of E(z ) on Gaiotto states given in (2.10)
by the ratio of Y operators with shifted arguments.
D.2.2
Derivation of the second qqcharacter
The expression of the second qqcharacter follows from the consideration of the symmetrized
action (D.10) of two SHc operators inside two Gaiotto states,
hG~ ; ~ajqD [D 1(z1) D1(z2) + D 1(z2) D1(z1)] jG; ~ai:
This quantity can be computed either using the right action (D.10) of two operators on
the Gaiotto state, or from the right (3.12), (3.13) and left (3.14) actions of a single SHc
operator. These two possible ways of calculation furnish the following identity:
2q 1
1 2
Pz1 Pz2 hY(z1 + +)Y(z2 + +)i
1
1
1 2
Y(z2 + +)
Y(z2)
Pz1 S(z12)
Y(z1 + +)
Y(z1)
Y(z1 + +)
Y(z2)
Pz2 S(z21)
1
1 2
2
z12
2
2
+
:
The second line involves the commutator of D 1(z1) with D1(z2) evaluated in the Gaiotto
states average. The same quantity can also be computed by direct right (3.12), (3.13) and
left (3.14) actions on Gaiotto states, leading to a second identity:
1
+z12
Y(z2 + +)
Y(z2)
Y(z1 + +)
Y(z1)
q 1
1 2
1 2
Pz1 Pz2 hY(z1 + +)Y(z2 + +)i
1
Y(z1)Y(z2)
:
Replacing the average of the commutator in the rst identity, and introducing the positive
part
(z) = Pz+Y(z + +) produces
0 = q 1 h(Y(z1 + +)
(z1))(Y(z2 + +)
(z2))i + Pz1 S(z12)
+ Pz2 S(z21)
Y(z2 + +)
Y(z1)
+ q
1
Y(z1)Y(z2)
2
z12
+
Y(z1 + +)
Y(z2)
This relation implies for the second qqcharacter (4.15),
Pz1 Pz2 2(z1; z2) =
2
z12
+
(D.14)
(D.15)
(D.16)
(D.18)
This is not enough to conclude on the polynomiality of the second qqcharacter, because
of the presence of the crossterms
Pz1Pz+2 2(z1; z2) =
(z2) Y(z1 + +)
(z1) + q
Y(z1)
(z1 + +)
(z21 + +)Y(z1)
:
On the other hand, we know the explicit expression of the second qqcharacter for small
N , and can use it to deduce the expression of 2(z1; z2). First, it is noted that after the
introduction of the orthogonal projector in (D.17), 2 can be rewritten as the average of
2(z1; z2) = h 2(z1; z2)i ;
(z1) (z2) + Pz+1
(z1) Y(z2 + +) + q
+ Pz+2
(z2) Y(z1 + +) + q
(D.19)
(D.20)
(D.21)
(D.22)
(D.23)
(D.24)
(D.25)
z12
2 1 2 :
2
+
q
Y(z2)
+ 2
z12
S(z12)
Y(z2)
+
2 1 2
z12
2
+
+ 1 2 Y(z2)
q
2
+
1
Zinst
At rst sight, it is not clear whether this quantity is a polynomial and we had to check it
case by case using the explicit expression of the polynomial operator (z) given in (4.13)
for N = 1; 2.
Case N = 1: in this case (z) is a scalar and can be taken out of the vacuum expectation
values, i.e. h (z)
i =
(z) h
i
Pz+1S(z12) (z1) = (z1) and as a result
. Since it is a polynomial of degree one, it satis es
h 2(z1; z2)i =
(z1) (z2) + (z1) (z2) + (z2) (z1) + q 2
where (z) is the fundamental qqcharacter given in (4.6). In this simple case, (z) = (z)
which provides the nal result
h 2(z1; z2)i = (z1) (z2) + q 2
Case N = 2: in this case, (z) is a polynomial of degree two that satis es Pz+1S(z12) (z1)
= (z1) + 1 2. It follows that
(z1) (z2) + (z1) Y(z2 + +) +
+ (z2) Y(z1 + +) +
Y(z1)
+ 1 2 Y(z1)
+ q 2
z12
The explicit expression of (z) deduced from (4.11) allows to show that
(z1) Y(z2 + +) +
Y(z2)
+ 1 2 Y(z2)
1
Zinst
Dz1 (Zinst (z2)) =
Dz1Dz2Zinst:
with the shifted derivative
Using this expression we arrive at
Dz = (z + +)
2
(z) = h (z)i =
2(z1; z2) = h 2(z1; z2)i =
1
Dz1 Dz2 + 2q z122
1 2
2 Zinst; h (z1) (z2)i =
replacing z1 = z + 1 and z2 = z + 2, this quantity is obviously a polynomial in z.
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.
HJEP04(216)7
DzZinst :
(D.26)
(D.27)
[1] O. Schi mann and E. Vasserot,Cherednik algebras, Walgebras and the equivariant
cohomology of the moduli space of instantons on A2, Publ. MathParis 118 (2013) 213
arXiv:1209.0429.
[2] N. Arbesfeld and O. Schi mann, A presentation of the deformed W1+1 algebra,
[3] I. Cherednik, Double a ne Hecke algebras, Cambridge University Press (2005).
[4] I.G. Macdonald, Symmetric functions and Hall polynomials, Oxford University Press (1998).
[5] N.A. Nekrasov, SeibergWitten prepotential from instanton counting, Adv. Theor. Math.
Phys. 7 (2003) 831 [hepth/0206161] [INSPIRE].
[6] L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville Correlation Functions from
Fourdimensional Gauge Theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219]
[7] Y. Matsuo, C. Rim and H. Zhang, Construction of Gaiotto states with fundamental
multiplets through Degenerate DAHA, JHEP 09 (2014) 028 [arXiv:1405.3141] [INSPIRE].
[8] S. Kanno, Y. Matsuo and H. Zhang, Extended Conformal Symmetry and Recursion Formulae
for Nekrasov Partition Function, JHEP 08 (2013) 028 [arXiv:1306.1523] [INSPIRE].
[9] A. Negut, Exts and the AGT Relations, arXiv:1510.05482 [INSPIRE].
[10] D. Gaiotto, Asymptotically free N = 2 theories and irregular conformal blocks, J. Phys.
Conf. Ser. 462 (2013) 012014 [arXiv:0908.0307] [INSPIRE].
[11] E. Carlsson and A. Okounkov, Exts and vertex operators, arXiv:0801.2565.
[12] T. Fujimori, T. Kimura, M. Nitta and K. Ohashi, 2d partition function in
background and
vortex/instanton correspondence, JHEP 12 (2015) 110 [arXiv:1509.08630] [INSPIRE].
[13] V.A. Fateev and S.L. Lukyanov, The Models of TwoDimensional Conformal Quantum Field
Theory with Zn Symmetry, Int. J. Mod. Phys. A 3 (1988) 507 [INSPIRE].
[14] M. Fukuda, S. Nakamura, Y. Matsuo and R.D. Zhu, SHc realization of minimal model CFT:
triality, poset and Burge condition, JHEP 11 (2015) 168 [arXiv:1509.01000] [INSPIRE].
[15] M. Bershtein and O. Foda, AGT, Burge pairs and minimal models, JHEP 06 (2014) 177
[arXiv:1012.1312] [INSPIRE].
01 (2012) 051 [arXiv:1109.4042] [INSPIRE].
[16] K.B. Alkalaev and V.A. Belavin, Conformal blocks of WN minimal models and AGT
correspondence, JHEP 07 (2014) 024 [arXiv:1404.7094] [INSPIRE].
of the conformal blocks arising from AGT conjecture, Lett. Math. Phys. 98 (2011) 33
[18] V.A. Fateev and A.V. Litvinov, Integrable structure, Wsymmetry and AGT relation, JHEP
arXiv:1404.5304 [INSPIRE].
114 [arXiv:1407.8341] [INSPIRE].
[19] R.P. Stanley, Some combinatorial properties of Jack symmetric functions, Adv. Math. 77
[20] B. Estienne, V. Pasquier, R. Santachiara and D. Serban, Conformal blocks in Virasoro and
W theories: Duality and the CalogeroSutherland model, Nucl. Phys. B 860 (2012) 377
[21] A. Morozov and A. Smirnov, Towards the Proof of AGT Relations with the Help of the
Generalized Jack Polynomials, Lett. Math. Phys. 104 (2014) 585 [arXiv:1307.2576]
[22] A. Smirnov, Polynomials associated with xed points on the instanton moduli space,
[23] J.E. Bourgine, Spherical Hecke algebra in the NekrasovShatashvili limit, JHEP 01 (2015)
[24] N.A. Nekrasov and S.L. Shatashvili, Quantization of Integrable Systems and Four
Dimensional Gauge Theories, arXiv:0908.4052 [INSPIRE].
[25] N. Nekrasov, Crossed instantons and qqcharacter, at CMSA colloquium, 6 Mar 2015.
[26] H. Knight, Spectra of Tensor Products of Finite Dimensional Representations of Yangians, J.
Algebra 174 (1995) 187.
arXiv:1312.6689 [INSPIRE].
[27] N. Nekrasov, V. Pestun and S. Shatashvili, Quantum geometry and quiver gauge theories,
[28] N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N = 2
supersymmetric QCD, Nucl. Phys. B 431 (1994) 484 [hepth/9408099] [INSPIRE].
[29] N. Seiberg and E. Witten, Electricmagnetic duality, monopole condensation and
con nement in N = 2 supersymmetric YangMills theory, Nucl. Phys. B 426 (1994) 19
[Erratum ibid. B 430 (1994) 485] [hepth/9407087] [INSPIRE].
[30] A. Marshakov, SeibergWitten theory and integrable systems, World Scienti c (1999).
[31] R. Poghossian, Deforming SW curve, JHEP 04 (2011) 033 [arXiv:1006.4822] [INSPIRE].
[32] F. Fucito, J.F. Morales, D.R. Paci ci and R. Poghossian, Gauge theories on
backgrounds
from non commutative SeibergWitten curves, JHEP 05 (2011) 098 [arXiv:1103.4495]
[33] F. Fucito, J.F. Morales and D.R. Paci ci, Deformed SeibergWitten Curves for ADE Quivers,
JHEP 01 (2013) 091 [arXiv:1210.3580] [INSPIRE].
JHEP 05 (2013) 047 [arXiv:1212.4972] [INSPIRE].
R205 [hepth/0703066] [INSPIRE].
[34] J.E. Bourgine, LargeN techniques for Nekrasov partition functions and AGT conjecture,
[38] A. Mironov, A. Morozov and S. Shakirov, Matrix Model Conjecture for Exact BS Periods
and Nekrasov Functions, JHEP 02 (2010) 030 [arXiv:0911.5721] [INSPIRE].
[39] A. Mironov and A. Morozov, Nekrasov Functions from Exact BS Periods: The Case of
SU(N ), J. Phys. A 43 (2010) 195401 [arXiv:0911.2396] [INSPIRE].
[40] A. Mironov and A. Morozov, Nekrasov Functions and Exact BohrZommerfeld Integrals,
JHEP 04 (2010) 040 [arXiv:0910.5670] [INSPIRE].
[41] N. Nekrasov and V. Pestun, SeibergWitten geometry of four dimensional N = 2 quiver
[42] N. Nekrasov and A. Okounkov, SeibergWitten theory and random partitions, Prog. Math.
gauge theories, arXiv:1211.2240 [INSPIRE].
244 (2006) 525 [hepth/0306238] [INSPIRE].
Commun. Math. Phys. 170 (1995) 337.
534 (1998) 549 [hepth/9711108] [INSPIRE].
[43] E. Frenkel, V. Kac, A. Radul and W. Wang, W1+1 and W (gl(N )) with central charge N ,
[44] A. Losev, N. Nekrasov and S.L. Shatashvili, Issues in topological gauge theory, Nucl. Phys. B
[45] G.W. Moore, N. Nekrasov and S. Shatashvili, Integrating over Higgs branches, Commun.
Math. Phys. 209 (2000) 97 [hepth/9712241] [INSPIRE].
[46] R. Flume and R. Poghossian, An Algorithm for the microscopic evaluation of the coe cients
of the SeibergWitten prepotential, Int. J. Mod. Phys. A 18 (2003) 2541 [hepth/0208176]
[arXiv:1012.1352] [INSPIRE].
[47] S. Baek, P. Ko and W.Y. Song, SUSY breaking mediation mechanisms and (g
2) ,
B ! Xs , B ! Xs`+` and Bs !
, JHEP 03 (2003) 054 [hepph/0208112] [INSPIRE].
[48] A. Marshakov, A. Mironov and A. Morozov, On nonconformal limit of the AGT relations,
Phys. Lett. B 682 (2009) 125 [arXiv:0909.2052] [INSPIRE].
[49] H. Kanno and Y. Tachikawa, Instanton counting with a surface operator and the chainsaw
quiver, JHEP 06 (2011) 119 [arXiv:1105.0357] [INSPIRE].
[50] H. Kanno and M. Taki, Generalized Whittaker states for instanton counting with
fundamental hypermultiplets, JHEP 05 (2012) 052 [arXiv:1203.1427] [INSPIRE].
[51] S. Kanno, Y. Matsuo and S. Shiba, Analysis of correlation functions in Toda theory and
AGTW relation for SU(3) quiver, Phys. Rev. D 82 (2010) 066009 [arXiv:1007.0601]
[52] N. Drukker and F. Passerini, (de)Tails of Toda CFT, JHEP 04 (2011) 106
41 (1997) 181.
products of Fock modules and Wn characters, arXiv:1002.3113 [INSPIRE].
HJEP04(216)7
Root of Unity Limit and Instanton Partition Function on ALE Space, Nucl. Phys. B 877
(2013) 506 [arXiv:1308.2068] [INSPIRE].
from AGT correspondence and Macdonald polynomials at the roots of unity, JHEP 03 (2013)
019 [arXiv:1211.2788] [INSPIRE].
Algebra, JHEP 01 (2010) 125 [arXiv:0910.4431] [INSPIRE].
DingIohara algebra and AGT conjecture, arXiv:1106.4088 [INSPIRE].
theory, arXiv:1507.00685 [INSPIRE].
arXiv:1211.1287 [INSPIRE].
093A01 [arXiv:1504.04150] [INSPIRE].
[17] V.A. Alba , V.A. Fateev , A.V. Litvinov and G.M. Tarnopolskiy , On combinatorial expansion [35] P. Dorey , C. Dunning and R. Tateo , The ODE/IM Correspondence, J. Phys. A 40 ( 2007 ) [36] A. Mironov , A. Morozov , B. Runov , Y. Zenkevich and A. Zotov , Spectral Duality Between Heisenberg Chain and Gaudin Model , Lett. Math. Phys. 103 ( 2013 ) 299 [arXiv: 1206 .6349] [37] A. Mironov , A. Morozov , Y. Zenkevich and A. Zotov , Spectral Duality in Integrable Systems from AGT Conjecture , JETP Lett . 97 ( 2013 ) 45 [arXiv: 1204 .0913] [INSPIRE]. [53] J.E. Bourgine and D. Fioravanti , to appear. [54] J.E. Bourgine and D. Fioravanti , Mayer expansion of the Nekrasov prepotential: The subleading 2order , Nucl. Phys. B 906 ( 2016 ) 408 [arXiv: 1511 .02672] [INSPIRE]. [55] J.E. Bourgine and D. Fioravanti , Finite 2 corrections to the N = 2 SYM prepotential , Phys.
Lett . B 750 ( 2015 ) 139 [arXiv: 1506 .01340] [INSPIRE]. [56] J. Ding and K. Iohara , Generalization of Drinfeld quantum a ne algebras , Lett. Math. Phys. [57] K. Miki , A (q; ) analog of the W1+1 algebra , J. Math. Phys. 48 ( 2007 ) 123520 . [58] B. Feigin , E. Feigin, M. Jimbo , T. Miwa and E. Mukhin , Quantum continuous gl1: Tensor [59] B. Feigin , E. Feigin, M. Jimbo , T. Miwa and E. Mukhin , Quantum continuous gl(1): Tensor