#### Refined toric branes, surface operators and factorization of generalized Macdonald polynomials

HJE
Re ned toric branes, surface operators and factorization of generalized Macdonald polynomials
National Research Nuclear University MEPhI 0 1 2 3 4 5 6 7 8
0 Moscow , 115409 , Russia
1 Moscow 117218 , Russia
2 Institute for Theoretical and Experimental Physics , ITEP
3 I-20126 Milano , Italy
4 INFN , sezione di Milano-Bicocca
5 Piazza della Scienza 3 , I-20126 Milano , Italy
6 Dipartimento di Fisica, Universita di Milano-Bicocca
7 Moscow 117312 , Russia
8 Physics Department, Moscow State University
We nd new universal factorization identities for generalized Macdonald polynomials on the topological locus. We prove the identities (which include all previously known forumlas of this kind) using factorization identities for matrix model averages, which are themselves consequences of Ding-Iohara-Miki constraints. Factorized expressions for generalized Macdonald polynomials are identi ed with re ned topological string amplitudes containing a toric brane on an intermediate preferred leg, surface operators in gauge theory and certain degenerate CFT vertex operators.
Topological Strings; Conformal and W Symmetry; M(atrix) Theories
1 Introduction
1.1
Re ned topological strings and branes
1.2 q-deformed CFT
2
Factorization of generalized Macdonald polynomials
3
4
1
2.1
2.2
2.3
2.4
2.5
3.1
3.2
Schur and Macdonald polynomial factorization. A reminder
Generalized Macdonald polynomials factorization on the general topological
locus
Factorization identities from matrix model averages
New formulas for skew generalized Macdonald polynomials
Gluing, traces and factorization of instanton sums
Toric brane on the intermediate leg and surface operators
Re ned topological amplitudes with branes
Degenerate elds and surface operators
Conclusions and further prospects
Introduction
The interplay between algebraic structures and geometry has been fundamental to the
development of mathematics in the recent decades. In particular, it has led to a
cornucopia of new results in mathematical physics. One of examples is the (re ned) topological
vertex function [1{9] which on the geometric side describes Gromov-Witten and
DonaldsonThomas invariants of toric Calabi-Yau (CY) three-folds, while from the algebraic point of
view it is the intertwiner of the Ding-Iohara-Miki (DIM) algebra1 [10, 11]. The second
famous example comes from the gauge theory: the equivariant cohomology of the
instanton moduli spaces (captured by Nakajima quiver varieties [12{14] and the corresponding
Nekrasov partition functions [15{22]) is acted on by a certain vertex operator algebra, which
turns out to be the WN -algebra of two dimensional conformal eld theory. This
correspondence between the geometric (moduli space) and algebraic (WN -algebra) objects is known
as the AGT relation [23{25] and has many known implications and generalizations [26{36].
These two examples are in fact directly related to each other and their relation can be
understood on both sides of the algebro-geometric correspondence. On the algebraic side the
equivariant cohomology (or more precisely K-theory [37]) of the instanton moduli spaces
is a tensor product of Fock representations of the DIM algebra [38], while the q-deformed
1Alternative names include quantum toroidal, elliptic Hall, spherical degenerate DAHA algebras, or
simply Uq;t(gbbl1).
{ 1 {
WN -algebra generators are built from the currents of the DIM algebra [39{41] and vertex
operators are combinations of topological vertices intertwining the action of the DIM
algebra and therefore of the WN -algebra [42]. On the geometric side the 5d gauge theory
is obtained by compactifying M-theory on the toric CY three-fold. The parameters of the
gauge theory correspond to Kahler moduli of the CY and the cohomology of the moduli
space of instantons is identi ed with the Hilbert space of M2 branes stretching between
toric xed points.
In this paper we explore a particular case of the algebro-geometric correspondence,
which is important for topological strings as well as for gauge theories. We consider re ned
toric branes wrapping Lagnangian submanifolds inside a toric CY three-fold [43]. As is
well-known, this setup corresponds to surface operators in gauge theory and to degenerate
elds of the WN -algebra [44{50, 53{55]. We will consider mostly the algebraic side of the
problem and relate the stack of re ned branes on the preferred leg of the toric diagram to a
particular intertwining operator of DIM algebra, which can be recast into a combination of
generalized Macdonald polynomials [56, 59]. The properties of the branes are related to the
remarkable factorization identities for generalized polynomials evaluated on a particular
submanifold in the brane moduli space called the topological locus.
Generalized Macdonald polynomials [56, 59] play the central role in the AGT
correspondence. They arise naturally in the study of the DIM algebra representations on tensor
producs of Fock modules. In [59, 60] matrix elements between generalized Macdonald
polynomials were computed using matrix model techniques. They turned out to reduce
to integral factorization identities, which provide a very explicit answer for the q-Selberg
averages in terms of Nekrasov functions.
In this note we would like to use these integral identities to prove new topological locus
factorization identity recently found in [61]. For special values of parameters the integrals
disappear and one is left with Macdonald polynomials evaluated at the topological locus.
The integral identity implies that those are still given by the factorized formulas. This
technique allows us to
nd several new identities for generalized Macdonald polynomials
on a more general topological locus. In this way we prove and generalize the results of [61].
We then connect the factorization of the polynomials to the re ned topological string
picture. To this end we interpret matrix model averages as topological string amplitudes
on toric CY threefolds with Lagrangian branes appropriately placed on the legs of the toric
diagram (for exact correspondence and explanation see [60]). Factorization of averages
relies on the particular properties of the branes residing on the preferred direction of the
diagram. The topological locus corresponds to a certain degenerate limit of the CY, which
models addition of a stack toric branes on one of the legs in the preferred direction.
Factorization of generalized Macdonald polynomials in this picture allows us to understand the
amplitudes with toric branes placed on intermediate preferred legs of the toric diagram.
One can also interpret factorization formulas for generalized Macdonald polynomials
in terms of CFT vertex operators in Dotsenko-Fateev (DF) representation. In this case the
radical simpli cation of the formulas occurs due to the particular choice of the dimensions
for which the vertex operators do not require any screening currents. In view of the AGT
relations this corresponds to a particularly simple surface operator in the corresponding
gauge theory.
{ 2 {
In the remaining part of the introduction we discuss the main points of this network
of correspondences in more detail. In section 2 we write down and prove the factorization
identities, in section 3 we connect this results with topological strings and gauge theories.
We present our conclusions in section 4.
1.1
Re ned topological strings and branes
Re ned topological string theory [8, 9] is a deformation of the topological string theory
living on a toric CY three-fold, which gives additional information on the spin content of
the D-brane BPS spectrum of type IIA string theory. Re ned amplitudes are computed
using re ned topological vertex, quite similarly to the ordinary topological vertex
computations [1{7]. A Young diagram Yi is assigned to each leg i. The vertices, always trivalent,
correspond to certain explicit combinations CYiYjYk (q; t) of symmetric functions depending
on three Young diagrams Yi;j;k on the adjacent legs i, j, k:
There is one crucial point in the computation of re ned amplitudes. Unre ned topological
vertex CYiYjYk (q; q) is cyclically symmetric in the three Young diagrams Yi, Yj , Yk, while
the re ned vertex CYiYjYk (q; t) for general q 6= t is not. The recipe above, therefore, includes
the choice of ordering of Young diagrams in each vertex. This choice is indicated by the
double ticks and the labels q and t on the corresponding legs in eq. (1.1). In what follows
we will usually omit the indices q and t. It turns out that the choices for the neighbouring
vertices should be coordinated, so that the only freedom remaining is the global choice of
the preferred direction (horizontal in eq. (1.1)) on the toric diagram. We omit here the
concrete expression for CYiYjYk (q; t), which can be easily found in the literature, not to
overcomplicate our presentation.
To get the nal answer for the amplitude one takes the sum over all the Young diagrams
Yi on the intermediate legs, each taken with weight2 ( Qi)jYij, where Qi denotes the
exponentiated complexi ed Kahler parameter of the two-cycle associated to the leg i. Let
us give the simplest example of two vertices glued together to form the resolved conifold
geometry:
Yi
i
t
j
Yj
q Yk
k
=
CYiYjYk (q; t)
(1.1)
2In general there are also framing factors which we will not need here.
=
X( Q1)jY1jCY1Y2Y3 (q; t)CY1TY4TY5T (t; q)
(1.2)
Y1
{ 3 {
=
X( Q)jY1jCY1?W (q; t)CY1??(t; q)
Y1
= Zopen ?
W Q1
?
?
?
?
(1.3)
(1.4)
HJEP09(217)
With the external lines one can associate either empty or non-empty diagrams which
do not take part in the sums. The former choice gives the closed string amplitude (partition
function), while the latter one gives the open string amplitude with stacks of toric branes
on the external legs determining the external diagrams, or \boundary conditions" for the
theory:
W
=
Q1
Q1
W
Zclosed(Q1) = Zopen ?
? Q1
The dashed lines here denote toric branes. The number of branes in the stack sets the
maximal possible number of rows in the Young diagram W . The
nal answer for the
closed string amplitude does not depend on the choice of the preferred direction, though
open string amplitudes do.3
In the unre ned case there is also a natural way to put a stack of toric branes on
the internal leg (and indeed on any Lagrangian submanifold of the CY). However, in the
re ned case the study of branes on the external lines has been so far very limited (see [51],
though). In the present paper we will address this problem and propose a recipe to put a
stack of branes on the intermediate preferred leg. To do this we will employ the duality
between open and closed string amplitudes.
Open-closed duality in topological strings allows one to model stacks of toric branes
by closed string amplitudes [52{55]. The open string amplitudes should be packed in
the Ooguri-Vafa generating function, and the closed strings propagate in the modi ed
background containing additional vertical line in the toric diagram. Let us draw the dual
pictures in the simplest case of one toric brane. The diagram corresponding to the brane
can have at most one column, i.e. it is of the form4 W = [l]. We then have
X zl
{ 4 {
In the l.h.s. of eq. (1.5) z plays the role of the holonomy of the (abelian) gauge eld living
on the toric brane, while on the r.h.s. it is identi ed with the Kahler parameter of the
3In the algebraic approach of [38] the choice of the preferred direction is associated with the choice
of the slope of the coproduct
used in the de nition of the DIM algebra. The most relevant choices
used e.g. in [62, 63] where the \horizontal" coproduct
and the \vertical" (or perpendicular, or Drinfeld)
coproduct
?.
4Compared to the notation of [55] we use the transposed diagram W . Another way to obtain our
conventions from that of [55] is to exchange the equivariant parameters, q $ t 1. Using the terminology
of [43] this amounts to the exchange of q-branes and t-branes.
two-cycle obtained by adding an extra vertical line to the geometry. In general, for N toric
branes the Kahler parameter in the r.h.s. will change to q 1=2tN+1=2.
There are several points requiring clari cation in this approach which are absent for
ordinary topological string, i.e. for t = q, and appear only in the re ned case:
1. The additional vertical line necessarily intersects all the parallel legs coming out of
the diagram if there happen to be any (see gure 4 b) for an example). One expects
that the amplitude should be insensitive to these intersections since they have nothing
to do with the toric brane insertion. However, in the re ned case there is no way to
make a \trivial crossing" of two lines: no choice of the Kahler parameter gives the
desired result. One concludes that for several parallel legs the toric brane attached
to one of them also interacts with all the others.
2. Although there is no way to make a \trivial crossing" of lines one can make a crossing,
which models the trivial one in some situations. For example, this crossing can be
used to set the diagram on one side to vanish if the diagram on the other vanishes
(see gure 5 b), c)). However, it works only in one direction: either the left diagram
vanishes whenever the right one is empty or vice versa.
3. Because of these features of the re ned theory it is unclear how to put a toric brane
on the intermediate preferred leg.
The explanation of these puzzles will be the main focus of the present work. We will
show that the amplitudes in the presence of the toric brane on the intermediate leg can be
identi ed with generalized Macdonald polynomials evaluated on the topological locus.
1.2
q-deformed CFT
It was shown in [57{60], that certain re ned topological string amplitudes on toric CY
three-folds correspond to conformal blocks of the q-deformed Virasoro or WN -algebras.5
The horizontal legs of the toric diagram represent the Hilbert space of the CFT, on which
the conformal algebra acts, and the intersections with vertical legs give vertex operators
or intertwiners of the algebra (see
gure 1). Naturally, the sums over Young diagrams
living on the horizontal lines represent the sums over the complete basis of states in the
CFT Hilbert space. There is a natural choice for such a basis | the basis of generalized
Macdonald polynomials, which leads to explicit factorized matrix elements for the
vertex operators given by Nekrasov formulas. The sums over diagrams on the vertical lines
corresponds to the integrals over the positions of the screening currents appearing in the
Dotsenko-Fateev representation of the conformal blocks. Therefore, vertical lines
correspond not simply to vertex operators, but more concretely to the screened vertex operator
insertions [41, 60]. The topological loci, i.e. the submanifold of the moduli space on which
generalized Macdonald polynomials factorize into products of monomials, represent the
special set of parameters, for which the number of screenings is zero.
5More concretely, to get a conformal block one should consider only balanced toric diagrams, see [40, 41]
for details.
{ 5 {
W1
W2
Q1
Q2
QF
Y2
= hMY1Y2(QF Q2 1)j
V Q1
Q2
jMW1W2(QF Q1)i
HJEP09(217)
of CFT. Double lines in the r.h.s. denote the CFT Hilbert space on which Virq;t
Heis algebra acts.
On the l.h.s. it corresponds to the two horizontal lines. The circle in the r.h.s. represents the vertex
operator corresponding to the intersection with the vertical line in the toric diagram in the l.h.s.
The matrix element on the r.h.s. is computed in the basis of generalized Macdonald polynomials
MY1Y2 , which corresponds to the choice of horizontal preferred direction (lines marked by by double
ticks) on the l.h.s. Notice the relation between the Kahler parameters Q1;2, QF of the CY on the
l.h.s. and the parameters of the vertex operator VQ1=Q2 and the states on the r.h.s. .
2
2.1
Factorization of generalized Macdonald polynomials
Schur and Macdonald polynomial factorization. A reminder
Let us rst recall the familiar factorization formulas for Schur and Macdonald
polynomials. Schur polynomials sY (xi) are symmetric polynomials in the variables xi, i = 1 : : : N
labelled by Young diagrams Y . They can be understood as characters of nite-dimensional
irreducible representations RY of slN algebra corresponding to the Young diagrams Y :
sY (x) = trRY diag(x1; : : : ; xN )
FWoer upsauratlilcyulwarritveaalullessyomfmtehtericvaproialybnleosmliyailnsgasonfunthcteiotnospoolfotghiceaplolwoceurssupmn s=pn 11= AqPnn iN=S1chxuinr.
polynomials are given by very simple factorized formulas:
sY
1
1
An
tn
=
Y
(i;j)2Y
ti 1
1
1
Atj i
tYi j+YjT i+1
These expressions can be related to \quantum dimensions", or generating functions of the
values of the Casimir operators on the corresponding representations.
Macdonald polynomials MY(q;t)(pn) provide a natural generalization of Schur
polynomials, depending on two parameters q and t. Macdonald polynomials do not have immediate
group theory interpretation, but nevertheless have many properties similar to Schur
polynomials, to which they reduce for t = q. In particular, they still factorize on the topological
locus pn = 11 Atnn :
MY(q;t) 1
1
An
tn
=
Y
{ 6 {
Notice that the parameters of the topological locus for Macdonald polynomials are tied
with the deformation parameters, so that for given t the locus is one-dimensional. We will
see similar e ect in the following sections, where generalized Macdonald polynomials will
depend on an additional parameter which will enter the de nition of the topological locus.
Generalized Macdonald polynomials factorization on the general
topological locus
In this section we give general factorization formulas for generalized Macdonald
polynomials. Concretely, we have found a generalization of the factorization formula for
generalized Macdonald polynomials conjectured in [61] to a wider topological locus. The
identity reads:6
0
An
tn
;
= ( 1)jY2jt
t n
q
1
q
tn
The original formula which appeared in [61] is given by
1
qxqAi j tBjT i+1
1
qxq Bi+j 1t AjT+i
Y
(i;j)2B
(1
qArmY (i;j)tLegY (i;j)+1);
jY j =
l(Y )
X Yi;
i=1
jjY jj2 =
l(Y )
X Yi2
i=1
= ( 1)jY1jq jjY1jj2 jY1j Q jY2jqjjY2jj2 jY2jt
2
jjY2Tjj22 jY2j
G(?qY;t1)(Q 1)G(?qY;t2)(1)
CY01 (q; t)CY02 (q; t)G(Yq2;Yt)1 (Q 1)
It is obtained from eq. (2.4) in the limit A ! 1 (one should divide both sides by AjY1j+jY2j
to get a nite answer).
For completeness let us also give the factorization formula where the second argument
of the generalized Macdonald polynomial is nontrivial:
0
1
1
t B
q
tn
6Similar identity for A = Q has already appeared in [59] (see eq. (24) there). There was a minor typo
in [59] eq. (24): Kronecker symbol Y2? was missing in the r.h.s.
{ 7 {
(2.4)
(2.5)
(2.6)
(2.7)
(2.8)
where the integration measure is
and the Jackson q-integral is de ned as
hf (xi)iu;v;N;q;t =
def R01 dqN x (u; v; N; q; tjxi)f (xi)
R01 dqN x (u; v; N; q; tjxi)
(u; v; N; q; tjxi) = Y Y
xxji ; q
i6=j k 0 t xxji ; q
N 0
Y
i=1
u Y
(xi; q)1
k 0 (qvxi; q)1
1
A
Notice the asymmetry between the two arguments of the generalized Macdonald
polynomials: while the rst factorization formula (2.4) has nontrivial dependence on both Young
diagrams, the second one (2.8) actually reduces to the formula (2.3) for the ordinary
Macdonald polynomials. This peculiar feature can be traced back to the nontrivial choice of
coproduct in the DIM algebra [64, 65].
From the form of eqs. (2.4) and (2.8) one could have suspected that there is a general
two-parametric factorization formula involving both A and B. However, it turns out that
this is not the case, as we explain below.
Factorization identities from matrix model averages
Before presenting more generalizations of the factorization formulas for generalized
polynomials let us give here a short and simple proof for the factorization identities obtained so
far. To this end we will employ the integral factorization identities discovered in [59, 60].
These identities give explicit factorized answers for q-Selberg averages of generalized
Macdonald polynomials. The Selberg average of a symmetric function f (xi) is given by
the following matrix integral:
HJEP09(217)
Z a
0
dqxg(x) = (1
q)a X qng(qna)
One example of the factorized identity for the average considered in [60] is
M A(qB;t)
q u 1t p n +
(t=q)n
qnv
1
tn
; p n
t n 1
q
1
(t=q)n
tn
= ( 1)jAjq 2jBj+ujAjtjBj jAjtP(i;j)2B i+2 P(i;j)2A iq
P(i;j)2A j
u;v;N;q;t
=
GA? t N q u GA? t
N 1qv+1 GB? t N 1
q GB? t
N 2qu+v+2
CA0(q; t)CB0(q; t)G(BqA;t) (qu+1t 1)
:
(2.12)
Notice that the parameters of the measure also enter the arguments of the generalized
polynomials under the average sign. Let us make a peculiar specialization of eq. (2.12) and
take N = 0. What does it mean to have zero number of integrations? There is of course
no general answer, but for Selberg averages the de nition we consider seems very natural
and can be obtained from analytic continuation in N . The generalized polynomial under
{ 8 {
1
1
the average is written in terms of power sums pn = PiN=1 xin of integration variables. For
N = 0 power sums contain zero terms and therefore should vanish. It is also evident that
for N = 0 there are no integrations neither in the numerator, nor in the denominator in the
de nition of the average and the integration measure is absent. Thus the l.h.s. of eq. (2.12)
reduces to the generalized Macdonald polynomial evaluated at the point pn = 0, while the
r.h.s. gives the correct factorized answer, coinciding with eq. (2.4).
Notice that the topological locus parametrized by u and v in eq. (2.12) and by Q and A
in eq. (2.4) is two-dimensional. This will always be the case in our considerations since the
original integral depends on three parameters, u, v, and N and we have to put N to zero.
HJEP09(217)
More identities can be obtained by using the symmetry of the Selberg measure
(u; v; N; q; tjxi) under the change of parameters:
Of course, if the function f itself depends on the parameters u, v or N one has to
revariables (2.13) in eq. (2.12) and setting Ne = 0 we get the identity (2.8).
place them with ue, v or Ne respectively to get the same average. Making the change of
e
Summarizing, the factorization identities for generalized Macdonald
polynomials (2.4), (2.8) follow from the integral identity (2.12) in the limit N = 0. In the next
section we will give more factorization identities involving skew generalized Macdonald
polynomials. They are proven using a similar argument.
hf (x)iu;v;N;q;t = hf (x)iue;ve;Ne;q;t :
Since
the same,
(u; v; N; q; tjxi) =
(ue; ve; Ne ; q; tjxi), the average of any function f (x) remains
2.4
New formulas for skew generalized Macdonald polynomials
We can also take the specialization N = 0 in more general integral factorization formulas
from [60] (see eqs. (93), (94) there). The identities we obtain in this way involve two skew
generalized Macdonald polynomials. Skew generalized Macdonald polynomials are de ned
similarly to the usual skew Macdonald polynomials:
MY(q1;Yt)2=Z1Z2 (Qjpn; pn) = M Z(q1;t)
n
1
1
q
M Z(q2;t)
n
1
1
q
MY(q1;Yt)2 (Qjpn; pn)
{ 9 {
(2.13)
(2.14)
(2.15)
where MZ
details let us write down the nal results:
(q;t) are ordinary Macdonald polynomials. Without giving too much technical
1
1
t
q
tn
n
;
1
1
1
B n
tn A
=
t
qB
q
BQ
Y~ ; W~
zbifund
(q;t)
G(Yq1;Yt)2 (Q)G(Wq;2tW)1
qjjW2jj2 jW2jt
1
Q 2 ; t BQ
q
1
2
; q B
t
q
t BQ
t n
1
2
1
2
jjY2jj2 jY2j tjjY2Tjj2 jY2j ( t)jW1j q jjW1jj2 jW1j
2 2
tn
n
t
q
1
1
(2.16)
(2.17)
(2.18)
(2.19)
1
where the conjugate generalized polynomial is de ned as
and the norm of Macdonald polynomial is given by an explicit expression
jjMY jj2 =
CY0 (q; t)
CY (q; t)
CY (q; t) =
(1
qArmY (i;j)+1tLegY (i;j))
The bifundamental Nekrasov function is given by
Y~ ; W~
zbifund(Q; P; M ) = G(q;t)
Y1W1
Q
M P
G(q;t)
Y1W2
G(q;t)
Y2W1
1
M QP
G(q;t)
Y2W2
1
M QP
There is one more identity similar to eq. (2.16):
X
Z1;Z2 jjMZ1 jj2jjMZ2 jj
q
tQ
Y~ ; W~
zbifund
t
jW2j
(q;t)
2 M
jjW2Tjj2 jW2j
2 jY1j jjY1Tjj2 jY1j
Qt2 jY2j
qjjW2jj2 jW2jt
1
Q 2 ; t AQ
q
1
2
; t A
q
G(Yq1;Yt)2 (Q)G(Wq;2tW)1 tAQ
q
A n
t n
1
; 0A
jjY2jj2 jY2j tjjY2Tjj2 jY2j ( A)jW1j q jjW1jj2 jW1j
2 2
CY01 (q; t)CY02 (q; t)CW01 (q; t)CW02 (q; t)
Identities (2.16), (2.20) are more general than eqs. (2.4), (2.8) and reduce to them in special
cases. For Y1;2 = ? eq. (2.16) reduces to eq. (2.8) and for W1;2 = ? it reduces to eq. (2.4).
In eq. (2.20) the situation is reversed, i.e. for Y1;2 = ? it reduces to eq. (2.4) and for
W1;2 = ? it reduces to eq. (2.8).
Gluing, traces and factorization of instanton sums
The new identity (2.16) allows one to glue several factorized expressions together and
then use Cauchy completeness to obtain a factorized answer for the full sum of factorized
terms. As a simplest example we can take the trace over Young diagrams Y~ = W~ in
the identity (2.16). In the language of gauge theory this corresponds to making a circular
quiver representing a U(
2
) adjoint theory, while for topological strings this gives the partial
compacti cation of the base of the toric bration. In each case, to get a meaningful result
we have to set spectral parameters of the generalized Macdonald polynomials equal to each
other. For eq. (2.16) this means taking B = qt . Thus, we set B = qt , Y1 = W1 and Y2 = W2
in eq. (2.16) and take the sum over Young diagrams Y1;2 with weight
jY1j+jY2j
jjMY1 jj2jjMY2 jj
2
. The
r.h.s. of eq. (2.16) then takes the form of Nekrasov instanton partitions function for a
particular value of the adjoint hypermultiplet mass:
X
jY1j+jY2j
X
Y1;Y2 jjMY1 jj2jjMY2 jj2 Z1;Z2 jjMZ1 jj2jjMZ2 jj
2 MY1(qY;2t=)Z1Z2 @Q
1
1
t n
q
tn ;
1
1
t n 1
q
tn
A
Now we notice that the l.h.s. of eq. (2.21) does not depend on the choice of basis in
the space of symmetric polynomials, since it is a trace over this space. This immediately
implies that the l.h.s. is in fact independent of Q. Choosing the basis of ordinary Macdonald
polynomials we nd that the sum factorizes into a product of two identical sums:
X
jY1j+jY2j
Y1;Y2 jjMY1 jj2jjMY2 jj2 Z1;Z2 jjMZ1 jj2jjMZ2 jj
so that the double sum in the r.h.s. turns into a single one:
jY j
X
Y jjMY jj2
X
q
Z jjMZ jj2 MY(q=;Zt) @
t jZj
0
1
1
t n 1
q
tn
AMY(q=;Zt) (0) =
X
Y
jY j
=
Y
MY(q1;Yt)2=Z1Z2 (Qj0; 0) = 6X
One can immediately notice that
jY j
Y jjMY jj2
X
q
Z jjMZ jj2 MY(q=;Zt) @
0
1
MY(q=;Zt)(0) = Y Z jjMY jj2;
MY(q=;Yt)(pn) = jjMY jj
2
t jZj
and
0
0
1
1
1
t
q
t n
q
tn ;
t n 1
q
tn
1
1
AMY(q=;Zt) (0)7 (2.22)
(2.21)
t n 1
A
q
tn
3
2
5
(2.23)
1
t
q
k
(2.24)
Eventually, we get the factorized answer for Nekrasov instanton partition function:
X
1
t
q
This is in fact nothing but the partition function of the corresponding 2d CFT, which
contains two bosonic eld, hence power two in the r.h.s. The example we have described
is of course a trivial one, since each term in the l.h.s. simpli es due to the identity
However, gluing two or more bifundamental contributions from eqs. (2.16), (2.20) together
one gets nontrivial factorization identities for linear quiver gauge theories. Also one can
take the trace to obtain circular quivers with several nodes.
3
Toric brane on the intermediate leg and surface operators
In this section we will demonstrate that factorization identities we have obtained can be
thought of as the amplitudes of re ned topological strings in the presence of a stack of toric
branes. We will also comment on their relation with surface operators in gauge theory and
degenerate vertex operators in 2d CFT.
3.1
Re ned topological amplitudes with branes
Selberg averages such as eq. (2.12), which we have used in our proof of factorization in
section 2.3, can be identi ed with re ned topological string amplitudes on toric CY depicted
in
gure 2. Kahler parameters Q1;2 and QF of the CY are related to the matrix integral
parameters u, v, N as follows:
Q1 = t 21 N q 12 v;
Q2 = t 21 N q 21 ;
QF = qu+v+ 32 tN 32
(3.1)
instead of u, v and N to get the second possible identi cation between the parameters.
In eq. (3.1) one can also use ue, v and Ne obtained by the change of variables (2.13)
e
Factorization happens for a special value of the parameters corresponding to N = 0 |
the topological locus. On this locus one of the resolved conifold pieces in the toric diagram
of the CY degenerates, i.e. its Kahler parameters becomes q qt or q qt as shown in gure 3.
For unre ned amplitudes degenerate resolution factorizes into a product of two pieces
as shown in gure 4. Each piece is given by an explicit factorized formula, which coincides
with the factorized answer for the polynomial. However, in the unre ned case the answers
for the amplitudes from
gure 4 b) are not very interesting since generalized Macdonald
polynomials in this case reduce to products of Schur functions, and the factorization
identities turn into the well-known formulas for quantum dimensions (2.2). They reproduce
the known amplitudes in the presence of the stack of toric branes on the intermediate leg
in the unre ned theory.
W1
W2
Q1
Q2
QF
Y2
DP
EF MY1Y2=EF MW1W2=EF
E
u;v;N
Macdonald polynomials. The number of integrations N and the parameters of the integral u; v are
expressed through the Kahler parameters Q1;2 and QF according to eq. (3.1).
HJEP09(217)
W1
W2
q t
q
p q Q 1
t
p q
t
b)
Y1
Y2
=
p qt in the CY corresponds to setting
The lower resolved conifold piece with Kahler parameter p q B 1 can be transformed by geometric
transition into geometry containing a stack of M toric branes, where tM
= qt B 1
. b) Setting
t
N =
v in the Selberg average. This average is
related by the symmetry (2.13) to the integral with N = 0 number of integrations shown in a). For
Q1 = q t A and QF = p qt (AQ) 1 one arrives at the factorization formula (2.20). This amplitudes
q
corresponds to a stack of M 0 toric branes on the upper horizontal leg with M 0 given by tM0 = A.
For re ned amplitudes degenerate conifold geometry does not split into two parts.
Moreover, there are two di erent degenerations of the resolved conifold with Kahler
parameter either q qt or q qt as shown in gure 5 a). The di erence between these two situations
is evident from
gure 5 b) and c): when some of the legs are empty the amplitudes do
factorize and give the same result as in the unre ned case.
The amplitudes from gure 3 are still given by the factorized expressions, though they cannot be separated into two noninteracting parts as shown in gure 4 b).
We argue that this is the natural de nition of the stack of toric branes placed on the horizontal leg of the diagrams. For a single horizontal leg the corresponding geometry is shown in { 13 {
Y1
Y2
into a product of two separate non-interacting lines. Notice that there is no preferred direction in the
unre ned case. b) Though the values of the polynomials on the topological locus are factorized into
a product of monomials, they do not factorize into a product of independent terms corresponding
to two horizontal lines. This happens only in the unre ned limit t ! q.
6
=
A
B
W1
W2
a)
b)
c)
A
A
?
B
Q = 1
C
q t
q
2 one gets di erent decoupling conditions, b) and c).
the diagrams are empty, the amplitude reduces to the unresolved one. For two choices of Kahler
W1
W2
D
C
?
C
D
?
B = qt
A = qt
A = B = 1
A
Q1 = Q2 = qt
the two loci.
strip geometry which can also be thought of as Coulomb moduli space of the gauge theory. The
moduli space is parametrized by Q, A and B, which are transformed into Q1, Q2 and QF using
the formulas Q1 = q qt A, Q2 = p q B 1, QF = p qt (QA) 1. There are two distinct topological
t
loci, fA = 1g and fB = 1g (shown in red and blue respectively), which intersect on the line
A = B = 1. There are also two special lines, A = qt and B = qt , each on its own locus, where and
1. In the unre ned limit t ! q the special lines coalesce with the intersection of
eq. (1.5). However, if we have several horizontal legs, the vertical line will intersect several
of them. Figure 3 describes precisely this situation:
gure 3 a) models the stack of M
branes on the lower horizontal line, and
gure 3 b) represents the stack of M 0 branes on
the upper horizontal line. The intersection of the vertical line with the second horizontal
leg is degenerate and in the unre ned limit gives the trivial crossing from
gure 4. In the
re ned case the crossing trivializes when the corresponding diagram on the left or right of
the crossing is empty as depicted in gure 5 b), c).
Let us recapitulate our main point. A stack of re ned toric branes sitting on a
preferred leg of the diagram interacts with all other parallel legs. The resulting amplitude
is given by the factorized value of generalized Macdonald polynomials evaluated on the
topological locus.
3.2
Degenerate elds and surface operators
Topological loci can be given a natural gauge theory interpretation. The Kahler moduli
space of the CY is identi ed with the Coulomb moduli space of the 5d gauge theory. The
topological locus corresponds to the root of the Higgs branch inside the Coulomb branch. In
other words, degeneration of the resolved conifold pieces of the toric diagram allows one to
deform the geometry instead. This deformation corresponds to going on the Higgs branch.
The identi cation can also be seen directly by identifying the parameters of the
corresponding gauge theory with Kahler parameters of the CY. The geometry in gure 2
corresponds to a single bifundamental eld of mass m charged under two SU(
2
) gauge groups,
HJEP09(217)
SU(
2
)L and SU(
2
)R. The parameters of the theory are the Coulomb moduli QL = qaL ,
QR = qaR and the exponentiated mass Qm = qm. They are given by the following formulas:
QL = (Q1QF ) 2 ;
1
QR = QF Q2 1 21 ;
Qm =
r t
q
Q1Q2 1 21
(3.2)
One can immediately see that on the topological locus where we have either QL = QRQm
or QR = QLQm. This indeed corresponds to the origin of Higgs branches.
It is well-known that gauge theory at this point is equivalent to a theory on a defect
associated with surface operator. Factorization formulas allow us to identify partition
function of this theory with the values of the generalized Macdonald polynomials on the
One nal interpretation of the factorization formulas is given by the thedegenerate
vertex operators in q-deformed 2d CFT. According to the AGT relations, gauge theory we have
just described corresponds to vertex operator in the Liouville theory. The Selberg integrals
used in section 2.3 are interpreted as integrals in the DF screening charges. Naturally, if
N = 0, the screening charges are absent and we return to pure bosonic vertex operator
V Q1 . Matrix elements of this operator in the generalized Macdonald basis are given by
generalized Macdonald polynomials evaluated on the topolgical locus. Schematically this
can be written as follows:
hMY1Y2 (QR)jVQm jMW1W2 (QRQm)i =
MY1Y2=Z1Z2 (QR)MW1W2=Z1Z2 (QRQm)jtop locus
X
Z1Z2
As usual degenerate eld obeys a di erence equation. Thus, generalized Macdonald
polynomials taken on the topological locus should also obey this equation. We hope to clarify
(3.3)
this point in the future.
4
Conclusions and further prospects
In this paper we have presented new factorization identities for generalized Macdonald
polynomials. We proved the identities using the technique of matrix models and related
them to re ned topological string amplitudes in the presence of a stack of toric branes. We
have also identi ed the corresponding gauge theories and CFT vertex operators.
It would be interesting to understand better the meaning of the factorization identities
directly in the DIM algebra. Also we would like to investigate the di erence equations
satis ed by the polynomials on the topological locus. Nekrasov-Shatashvili limit of our
construction might help to understand better the surface operators corresponding to toric
branes on the intermediate legs of the toric diagram.
Acknowledgments
The author thanks Y. Kononov and N. Sopenko for discussions. The work of the author
was supported in part by INFN, by the ERC Starting Grant 637844-HBQFTNCER and
by RFBR grants 17-01-00585, 15-31-20484-mol-a-ved, 15-51-52031 NSC, 15-51-50034 YaF,
16-51-53034 GFEN, 16-51-45029 Ind.
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 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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