Decomposing Nekrasov decomposition
V
Nekrasov decomposition
A. Morozov 0 1 3 4 5 6
Y. Zenkevich 0 1 2 3 5 6
0 31 Kashirskoe highway, Moscow , 115409 Russia
1 191 Bolshoy Karetniy , Moscow, 127051 Russia
2 Institute for Nuclear Research of Russian Academy of Sciences
3 25 Bolshaya Cheremushkinskaya , Moscow, 117218 Russia
4 Institute for Information Transmission Problems
5 National Research Nuclear University MEPhI
6 6a Prospekt 60letiya Oktyabrya , Moscow, 117312 Russia
AGT relations imply that the fourpoint conformal block admits a decomposition into a sum over pairs of Young diagrams of essentially rational Nekrasov functions  this is immediately seen when conformal block is represented in the form of a matrix model. However, the qdeformation of the same block has a deeper decomposition  into a sum over a quadruple of Young diagrams of a product of four topological vertices. We analyze the interplay between these two decompositions, their properties and their generalization to multipoint conformal blocks. In the latter case we explain how DotsenkoFateev allwithall (star) pair \interaction" is reduced to the quiver model nearestneighbor (chain) one. We give new identities for qSelberg averages of pairs of generalized Macdonald polynomials.
aITEP

Decomposing
HJEP02(16)98
1.1
1.2
1.3
2.1
2.2
3.1
3.2
3.3
4.1
4.2
4.3
4.4
4.5
1 Introduction
2
3.4 Identi cation with topological strings
4
Re nement
qSelberg measure
Generalized bifundamental kernel
Vertical slicing
Generalized Kostka functions
SeibergWitten theory and topological string pattern
Re nement and slicing invariance
Four point conformal block, no qdeformation
Multipoint case. Starchain duality
Resolution of the star/chain problem. From chain to star.
BifundaHorizontal slicing. DF representation and spectral dual Nekrasov function
5
Conclusions and discussion
A Fivedimensional Nekrasov functions and AGT relations
B Loop equations for matrix elements
C Open topological string amplitude on resolved conifold
D Useful identities E DotsenkoFateev integrals as contour and Jackson integrals
{ 1 {
Introduction
Conformal blocks [1{4] are among the most interesting and important quantities under
study in modern theoretical physics. Perturbatively they are de ned as series of matrix
elements in highest weight representations of Virasoro algebra, see [5, 6] for recent reviews.
Nonperturbatively they are examples of matrixmodel functions [7{11], associated with
peculiar conformal [12{14] (also known as DotsenkoFateev [15] or Penner [16{18]) matrix
models, and exhibit nontrivial and almost unexplored behavior in various regions of moduli
space [19]. Their modular transformations [20{32] are important for the study of knot
polynomials (Wilson loop averages in ChernSimons theory [33{35]), see [36{38] for a recent
outline. AGT relations [39{41] connect conformal blocks to LMNS quantization [42{45] of
the SeibergWitten theory [46{52] and express them in terms of Nekrasov functions [53{56].
Both the matrix model and Nekrasov function formalisms imply natural lifting of original
conformal blocks to (q; t)dependent quantities  looking from di erent perspectives this
can be either a  or a qdeformation, associated with 5d generalization of SeibergWitten
theory [57{59] and AGT relations [60, 61].1 It is at this level that the full duality pattern
gets clear and manifest.
Finally, as a quintessence of all this, conformal blocks are expressible through
topological vertices [67, 68]  and this will be the story we concentrate on in the present paper.
This relation involves not only the fullscale theory of Schur and Macdonald functions [73],
but also conceptually important notions of starchain duality and Selberg factorization.
The ideniti cation between qdeformed CFT blocks and topological vertices has been used
in [74] to prove the spectral duality [75{78] of the former. In the present paper we
generalize this identi cation to the higherpoint case. We also clarify the relation between
preferred direction in re ned topological strings and the basis of states in conformal eld
theory Hilbert space. Further generalizations to WN and elliptic cases would be given
in [79].
1.1
Conformal blocks and characters
Conformal blocks are best described by the version of DotsenkoFateev (DF) conformal
matrix model, introduced and investigated in [80{85]
where BU(1) is an explicit function representing the contribution of an extra free boson.
We nd it most convenient to use the number k of independent integration contours as
1For recent developments on the 6d generalizations of the AGT relations see [62{66].
{ 2 {
V 2( 1)
of bifundamentals in the gauge theory description below will be k
2. The parameters
of conformal block can be conveniently summarized in a diagram, such as one shown in
External dimensions
i = vi (vi +
1)
are parameterized by the \momenta" vi, while internal dimensions
0+v1+:::+va 1+ N1+:::+ Na 1
0+v1+:::+va 1+ N1+:::+ Na 1+
are expressed through the numbers Na of screening integrations, i.e. conformal block is
considered as analytical continuation of the integral in the number of integrations. It
is important for this description that the integral is of Selberg type [86] and analytical
continuation in Na is actually under control.
The next important fact [87] is that the interscreening coupling is reduced to a
gure 1.
e a=
square of
block to a bilinear combination of bicharacter Selberg averages [86] over x and y,
B4 =
X
A;B 
A[x] B[x]
A[y] B[y]
{z
ZAB
}
which are exactly calculable rational combinations of vparameters, and are basically
nothing but Nekrasov functions [53{56], labeled by arbitrary pairs A; B of Young diagrams.
This line of reasoning reduces AGT relation [39{41] between conformal block and
Nekrasov functions to HubbardStratanovich resummation of Selberg integrals [80{85].
There are important details, making the story a little more technically involved, especially
for
6= 1 (i.e. for the central charge c 6= 1) [74, 88{90], but in what follows we try
=exp
8
<
:
X 1 X
k k
a;b
xa
yb
=exp
(
X pk[x]pk[y 1] )
k
k
=XJ A[x]JA[y]
A
k
9
=
;
and
{ 3 {
N fund
U (N )
1 bifund
U (N )
1 bifund
U (N )
to separate concepts from technicalities, putting simpli ed general considerations before
exact, but overloaded, formulas.
After qdeformation (which in the SeibergWitten theory framework means going from
4d to 5d YangMills theories [57{59]), the integral remains basically the same, only the
integration is replaced by Jackson qintegration2 [91]:
is not bi linear, but rather quadri linear:
h A Bi =
X SARSRB
R
=)
B4
X
Y1;Y2;Y3;Y4
SAR1 SR1BSBR2 SR2A
(1.8)
 and this is the decomposition which is related to topological vertex [67{72] and geometric
engineering [92, 93]. The origin of two extra Young diagrams is simple: summation over
them substitutes integration over x and y variables in the de nition of averages in (1.6) 
this appears to be the right way to interpret the multiple Jackson integrals/sums in (1.7)
(see appendix E for technical details).
1.2
SeibergWitten theory and topological string pattern
To better understand the origin of the multicharacter decomposition let us investigate
the structure on the gauge theory side of the AGT duality. Conformal blocks correspond
to instanton partition functions of quiver gauge theories which are given by Nekrasov
formulas. The comblike (k + 2)point conformal blocks on a sphere correspond to linear
quiver theories, in which the gauge group is a product of (k
1) U(N ) factors and the
matter content is encoded in the quiver diagram as, e.g. in
gure 2.
Here a circle is a gauge group, a box denotes a collection of matter hypermultiplets,
a outgoing (resp. incoming) link connecting a circle with a box indicates that the
corresponding hypermultiplets transform as a fundamental (resp. antifundamental) under the
gauge group. The structure of the corresponding Nekrasov function is modelled after the
quiver diagram above:
ZNek=X
~
jY1j
1
jY~kjzfund(Y~1)
k
~
Ya
1
zvec(Y~1)
zbifund(Y~1;Y~2) zbifund(Y~k 1;Y~k)
1
zvec(Y~k) zfund(Y~k);
(1.9)
2Jackson qintegral is de ned as a sum R0a f (x)dqx = (1 q) Pk1 0 qkaf (qka).
{ 4 {
where the de nitions of the rational factors zfund;vect;::: are given in appendix A. The
structure of each term in the decomposition is linear, in particular for a (k + 2)point conformal
block there are k
1 vector multiplet contributions and k
2 bifundamental matter hy
permultiplets. Such quiver or chain decomposition of the conformal block is obtained
by inserting a special basis of states j ; Y~ i labelled by a pair of Young diagrams in the
intermediate channels of the block:
is then given by the Selberg average of a collection of orthogonal polynomials as in eq. (1.6).
For c = 1 the special basis which reproduces the corresponding factor in the Nekrasov
function (1.9) is given by Schur polynomials. We will compute the most general matrix element
using qSelberg averages and show that it is indeed given by the Nekrasov expression.
For 5d gauge theories compacti ed on a circle of radius R5 the structure of Nekrasov
function remains basically the same. The only change is that all the monomial factors in
the rational functions zfund;vec;::: are transformed into qanalogues roughly as x ! q
where q = e 2R5 . However, quite remarkably in this case Nekrasov partition function
1,
 or conformal block  turns out to have yet another interpretation. Gauge theory in
x
ve dimensions can be obtained by compacti cation of Mtheory on a toric CalabiYau
threefold. Partition function of the resulting theory is equal to the (re ned) topological
string partition function, which can be computed by the topological vertex technique as
follows.
One rst draws the toric diagram of the CY threefold and assigns to each internal edge
the complexi ed Kahler parameter Q of the corresponding twocycle. One also assigns a
Young diagram to each internal edge, and an empty diagram to each external edge. There
are in general only trivalent vertices in the diagram, and to each of them one assigns a
certain function CY1Y2Y3 (q)  the topological vertex [67, 68]  depending in a cyclically
symmetric way on three Young diagrams Ya residing on the adjacent edges and also on the
parameter q = e 1R5 :
CABC (q) =
A
where by qC+
P(i;j)2A 2(j
B
C
(1.10)
HJEP02(16)98
(1.11)
(A) =
we mean the in nite set of variables f
qC1 21 ; qC2 23 ; : : :g and
i). The partition function is computed by summing up over all the Young
diagrams with weights given by the product of all topological vertices and the
\propagators" of the form ( Q)jY jfY (q)n where n is the framing factor depending on the relative
orientation of the edges adjacent to the given edge.
= q 2
(A)
C (q ) X
AT=D qC+
B=D qCT+
;
D
{ 5 {
t
QF;4
t
q Qe4
t q
q Q4
QB;3
t
QF;3
t q
QB;3Q3=Qe3 q Qe3
t
t q
q Q3
QB;2
t
QF;2
t q
QB;2Q2=Qe2 q Qe2
t
t q
q Q2
QB;1
t
QF;1
t q
QB;1Q1=Qe1 q Qe1
t
q Q1
t q
is drawn using the recipe of geometric engineering. It is the crossing of N horizontal and
k vertical lines, which intersect as shown e.g. in
gure 3.
There is a natural decomposition of the toric diagram depicted on gure 3 which leads
to the same quiver structure as in
gure 2 and the Nekrasov expression (1.9). One should
perform the sums over all Young diagrams except those residing on the horizontal edges
marked with QB;i, which are related to positions of the vertex operators in the conformal
block and the gauge theory couplings
i
. In this way one obtains a sum over a chain
of pairs of Young diagrams of certain rational factors, which turn out to coincide with
zvect;fund;::: for t = q (we introduce t = q ). The resulting expression has exactly the form
of Nekrasov function (1.9). Moreover, each term in the Nekrasov decomposition can now be
decomposed into an in nite sum of simpler building blocks ZARB, related to the fourpoint
topological string amplitude on resolved conifold. In the language of CFT this leads to the
decomposition
h e 1; Y~1jV 2
for the matrix element in the l.h.s.
Another natural decomposition of the toric diagram  cutting along the vertical edges
marked with QF;i (related to Coulomb moduli of the gauge theory and intermediate
dimensions in the conformal block)  corresponds to the spectral dual Nekrasov function.
The gauge theory origin of this dual description is that in 5d instantons are BPS particles
as are the gauge bosons. Spectral duality exchanges these two sets of BPS objects and
therefore leads to a nontrivial identi cation between two gauge theories. We will show
{ 6 {
that the spectral dual decomposition of the toric diagram has a natural interpretation in
terms of DF integrals of qCFT  it is the sum featuring in the discrete Jackson integrals,
each vertical leg corresponding to a separate integration contour in (1.1). Therefore, the
spectral dual decomposition over horizontal lines of the diagram corresponds to the DF
integrals themselves, while the original Nekrasov decomposition is the sum over a complete
set of intermediate basis states in the CFT:
Bk+2 BU(1)
X
R1;:::;Rk
ZR1;:::;Rk ZR1;:::;Rk :
(1.13)
HJEP02(16)98
Our goal in this paper is to explain the relation between eq. (1.7), eq. (1.9), and
gure 3. We will learn that the identi cation between conformal block and Nekrasov function
requires a nontrivial rewriting of the Vandermonde determinant (which is the product of
allwithall form) into the sum of Nekrasov form (which is of nearestneighbour form).
We rst clarify the relation of the toric diagram and the DF integral schematically in the
simplest case of the fourpoint conformal block (k = 2). Extension to arbitrary k involves
an a priori nontrivial starchain identity, which is in fact the key to understanding DF
description of conformal blocks and relies upon the basic properties of representation
theory. Another crucial property is Selberg factorization  a mysterious conspiracy between
the integrands and integration measure in DF theory, between what is averaged and how
it is done. This property guarantees that the averages of certain polynomials over the
qSelberg measure factorize into products of linear factors depending on the parameters
of the integral. The last mystery is that the elementary building block in the
quadrilinear decomposition of conformal blocks, i.e. the topological vertex, is closely related to the
modular kernel and therefore to certain knot polynomials.
1.3
Re nement and slicing invariance
The calculation we have just described yields the Nekrasov function of the 5d gauge theory
with the particular choice of
deformation parameters, i.e. 1 =
2, or equivalently
t = q, which corresponds to c = 1 in CFT. To obtain the partition function in a general
background, one has to use re ned topological vertex3 [69, 70]:
A
t
B
q
C
CABC (t; q) =
= q
jjBjj2+jjCjj2 t jjBTjj2+jjCTjj2 M C(q;t) t
2 2
X
D
q jDj+jA2j jBj
t
AT=D q C
t
B=D t CT q
;
(1.14)
3There is a slight historical mismatch of notations between the re ned and unre ned vertices. Reducing
the re ned vertex (1.14) back to the unre ned case to compare with eq. (1.11) one needs to transpose all
the diagrams and add some simple factors CABC(q; q) = ( 1)jAj+jBj+jCjq
(A)+ (2B)+ (C) CATBTCT (q).
{ 7 {
legs in the diagram is marked with a double stroke and the other two bear t and q labels
on them. This is to indicate the right order of the indices and arguments of the re ned
vertex, which depends on two deformation parameters and is not cyclically symmetric as
was the case for t = q.
The calculations generally get more technically involved, though the strategy remains
the same. The only essentially new feature in this case is the naive loss of rotation symmetry
of the diagram: the vertical and horizontal lines are no longer equivalent. However, it turns
out that the symmetry in fact survives even for general t and q, though the individual
vertices and propagators are not symmetric. This statement came to be known as the slicing
HJEP02(16)98
invariance hypothesis. For toric geometries, which we consider, slicing invariance is also
equivalent to spectral duality [75{78] of the corresponding Nekrasov partition functions,
since the two sides of the duality are related to the 2 rotation of the whole toric diagram
including the choice of preferred direction. We look at di erent choices of \slicing" of the
toric diagram and relate them to di erent choices of the basis in conformal eld theory.
One slicing direction corresponds to the \naive" basis of Schur polynomials
A, the other
 to the basis of generalized Macdonald polynomials MAB. The rst set of polynomials
does not have factorized qSelberg averages and does not reproduce the Nekrasov factors,
while our calculations indicate that the second one does. Schematically
*
We investigate the connection between the two sets of polynomials and introduce
generalized Kostka functions KACBD transforming one basis into the other:
MAB =
X KACBD C D:
CD
(1.15)
(1.16)
These functions are e ectively performing the 2 rotation of preferred direction. In more
algebraic terms they are related to the abelianization map [98] acting on the basis in
Ktheory of instanton moduli space.
The paper is partitioned into a set of sections with increasing level of detail and
complexity.
After reviewing the basic steps of the construction at the simpli ed level
in section 2 we ll in the details and provide full edged formulas for the unre ned case in
section 3. We then treat the re ned case in section 4. We provide a summary and point
out future directions in section 5.
2
Basic steps
In this section we introduce our approach to DotsenkoFateev integral expansion without
qdeformation. We consider rst the most simple example of fourpoint conformal block
{ 8 {
and show how decompose the integrand in terms of Schur polynomials. Next we consider
the multipoint block and observe that a nontrivial starchain duality is required in this
case. We demonstrate this duality explicitly using skew Schur functions.
Four point conformal block, no qdeformation
In the case of fourpoint conformal block there are two contours of integration: C1 stretching
from 0 to 1 and C stretching from 0 to
in eq. (1.1) are divided into two groups: xi and yi and the interscreening pairings are
1
. Therefore, the variables in the integration
decomposed into a product
HJEP02(16)98
The vertex operator contributions also decompose into a product of two factors:
(z)2
!
(x)2
(y)2
N1+N2
Y
i=1
(1
N1+N2
Y
i=1
(1
zi)v1 !
zi)v2 !
N1
Y(1
i=1
N2
Y(1
i=1
N1 N2
Y Y
yi we can write the cross terms which we denote by
cross(x; y) = Y Y
1
N1 N2
Employing the Cauchy completeness identity (1.4) we get the expansion of the cross
contributions in terms of Jack polynomials:
1
(2 pnq n + v1q n + pnv2)A
jY1j+jY2jJY1 (pn)JY2
pn
JY1
q n
JY2 (q n);
(2.1)
v1
v2
0
X
n 1 n
n
=
X
Y1;Y2
where pn = PiN=11 xin, qn = PiN=21 yin.
polynomials
After this decomposition the DF integral becomes the double Selberg average of Jack
jY1j+jY2j JY1 (pn)JY2
pn
v1
v2
JY1
q n
JY2 (q n)
u1;v1;N1;
{ 9 {
u2;v2;N2;
(2.2)
where the averages are taken with respect to the measure
QiN=a1 (xiua (1
xi)va ). For general
the averages (2.1) do not give the Nekrasov expansion
(xjua; va; Na; ) =
2 (x)
of the conformal block (a more re ned basis of generalized Jack functions JAB depending
on a pair of diagrams is required [88{90]). However, for the special case
= 1 when
Jack polynomials turn into Schur functions the structure of Nekrasov sum is indeed
reproduced [80{85].
Thus, from the fourpoint case without qdeformation we learn that decomposing the
interscreening pairings in the DF integral in terms of characters and then taking the
Selberg averages produces Nekrasov representation of the conformal block. We now move to
the multipoint case where the starchain duality is required to obtain Nekrasov
decompoIf one approaches the multipoint case in a naive way one arrives at what seems to be
a paradox. The DF representation contains a product of all pairings between screening
operators, i.e. an expression of the form
(2.3)
(2.4)
(2.5)
(2.6)
Y
a<b
1
xa !2
i
x
b
j
:
Y zbifund(Ya; Ya+1) :
a
A=W =
def X cBW B
A
B C =
X cBC A
A
:
B
A
where cABC are the LittlewoodRichardson coe cients, describing multiplication of
repreHowever, the gauge theory corresponding to the multipoint comblike conformal block is a
linear quiver of the form depicted in gure 2, and its Nekrasov partition function contains
only the nearest neighbour pairings:
Thus the multilinear decomposition of the DF integral should also have the
nearestneighbor structure. In the fourpoint case there are only two term in the product, so
that allwithall (star) type interaction is the same as nearestneighbour (chain) one. But
how can one decompose the multipoint product (2.3) into a sum of nearestneighbour
products, how can star become equivalent to a chain?
2.2.2
Skew characters
The resolution of the paradox is technically based on the properties of skew characters,
Moreover, this is straightforwardly generalized to
A pn(x(1))+:::+pn(x(m))
A=W [y]= W pn(x(1))+:::+pn(x(m))
m
1
X CVAW CVBW A
V;W
B (pn)
B;W
A
)
=X
A
=exp
k k pk(z) pk(x)+pk(y)
A[z] A pn(x)+pn(y)
At x = y we can apply (2.6) to the l.h.s. to get a doubling rule
e.g. [1] (2pn) = 2 [1] (pn), [2] (2pn) = 3 [2] (pn) + [11] (pn), [11] (2pn) =
[2] (pn) + 3 [11] (pn),
[3] (2pn) = 4 [3] (pn)+2 [21] (pn), [21] (2pn) = 2 [3] (pn)+6 [21] (pn)+2 [111] (pn), [111] (2pn) =
2 [21] (pn) + 4 [111] (pn); : : : which can be further promoted to tripling, quadrupling and
higher multiplication formulas.
2.2.3
Resolution of the star/chain problem. From chain to star. Bifundamental
kernel
We claim that the chain of skew characters indeed reproduces the starlike structure of the
DF integrand. The basic building block of the chain decomposition is the bifundamental
kernel
X
C
NAB[y] =
A=C [y] B=C [y];
where
Y [x] =
Y [ pn(x 1)]. Two such kernels, averaged over the Selberg measure like
hNAB[y]NCD[y]i, correspond to a single bifundamental eld in Nekrasov partition function
of the gauge theory depending on two pairs of diagrams (A; B) and (C; D). Observe that4
N?A[x] =
A[x], NA?[x] =
A[x].
Qk) 1 Qi;j 1(1
Qkxixj 1).
4There is another curious identity, which would be useful for toric blocks: PA QjAjNAA[x] = Q
k 1(1
We start with the case of vepoint conformal block. Using the identities from the
previous section, we can rewrite the chain answer into the DotsenkoFateev (star) form:
X N?Y1 [x]NY1Y2 [y]NY2?[z]
Y1;Y2
Y1 [x] Y1=W [y] Y2=W [y] Y2 [z]=Y
i;j
xi
yj Y2;W
X
Inverting this short derivation, we see that it is an iteration of the twostep procedure,
which starts from m = k
1 with Fk 1[Y ] =
Y [z] and ends at m = 2 with x = y0.
In obvious notation:
XFmfYmg Ym [y0;y1;:::;ym 1]=
Ym
X
Ym;Wm 1
FmfYmg Ym=Wm 1
character is combined with the next product
X
Zm 1
Zm 1
HJEP02(16)98
!
(2.15)
(2.16)
(2.17)
Since, whatever are the sets u and w,
X
Z
and we are ready for the next iteration.
At k = 3 this can be pictorially represented as
FmfYmg
Ym=Wm 1
Y2=W1
6
r r
y
x
r
=
z
Y1=W1
:
Dots here stand for characters, which have labels according to the points at which they are
evaluated, and arrows point from
to
. The arrows are labeled by the Young diagram,
over which the sum is taken. The two encircled characters are evaluated at the same point,
and can be transformed into one using eq. (2.6), thus fusing the two dots. The resulting
trivalent vertex represents the LittlewoodRichardson coe cients, which depend on three
Young diagrams. Note that only dots at the same place which are both either starting or
endpoints of the arrows can be merged in this way.
Likewise at k = 4:
z = y3
r
r
r
r
y1
6
r
y2
r
rkx
r
r
W1
Ar r
y
=
=
z
z
Y3=W2
Y3=W2
W1
r
KA
A
Y2=W1A
Y2=W2
r
KA
A
Y2=W1 A
Y2=W2
6
r r
y2
6
r r
y2
rnx
r
A
A
Ar r?
y1
x
r r Cr rHHHHjr
C
r
r
r
C
C
C
C
C
Cr r r
y3
r
HJEP02(16)98
rr y1
r
z
r
Cr r
y3
r
r
OC
C
C
C
r
OC
C
C
Cr r
y2
rkrx
CrC rHHHHjr
C
C
C
C
C
C
C
C
rrmy1
r
rn
r
x
r
W3 r Y3
W2 r Y2
The main secret behind this derivation is that the structure constants in (2.6) are
always the same  do not depend on the number of \Miwa variables" yi in [y0; : : : ; ym]
 which allows to merge entire collections of points and parallel arrows in the examples
above. This conspiracy between characters and the structure constants adds to associativity
of multiplication and together they provide the starchain equivalence.
2.3
Factorization of Selberg averages
The \chain" decomposition of DF integrals (2.15) is also tied with the structure of the
Selberg averages. More concretely, the averages of the bifundamental kernels (2.13) are
given by the factorized formulas:
hNAC [x]NBD[ pn(x) v]iu;v;N; =1=( 1)jBj+jDj zb4idfund([A;B];[C;D];u=2+v=2+N;u=2; v=2)
;
CA4dCB4dCC4dCD4dG4AdB(u+v+2N )G4CdD(u)
(2.20)
where
CA4d =
(ArmA(i; j) + LegA(i; j) + 1) ;
G4AdB(x) =
x + Ai
j + BT
j
i + 1
x
Bi + j
1
AjT + i :
Y
zb4idfund A~; B~ ; ~a;~b; m
=
G4AdiBj (ai
b
j
m)
and ~a = (a; a), ~b = (b; b). This factorization means that expansion of the DF integrand
in terms of the bifundamental kernels NAB indeed reproduces the Nekrasov decomposition.
In the next section we will compute the qdeformed averages and show how to decompose
them even further to obtain topological vertices.
3
Complete formulas for t = q
In this section we esh out the basic formulas introduced in the previous section and
incorporate qdeformation into our framework. After qdeformation, we obtain the natural
identi cation of DF integral decompositions with topological vertices. We start with
especially symmetric example of the fourpoint conformal block of qVirasoro algebra and then
consider multipoint blocks. We calculate qdeformed Selberg averages of the skew
characters and identify the elements of the multilinear decomposition with topological string
amplitudes.
3.1
Four point conformal block (t = q)
The origin of the quadrilinear expansion of the four point conformal block is straightforward
to see: two diagrams come from the character decomposition and two more represent the
two integration contours in the DF integral (which becomes a sum in the qdeformed
case [74]). The corresponding toric diagram is depicted in gure 4 and the four diagrams
The DF representation is given by the sum over DF poles labelled by two partitions
are denoted by A, B, R+ and R .
R [94{96]5 (see appendix E for details):
B4 =
X
R
xR ;i = qR ;i i+N +1:
(xR+ ) cross(xR+ ; xR ) (xR+ );
(3.1)
(3.2)
5One should notice that since there are only N
variables x ;i the number of columns in the diagrams
R is not greater than N . This constraint is exactly parallel to a similar constraint in the topological
strings, where it appears for quantized values of the complexi ed Kahler moduli, corresponding to N ,
i.e. Q
= qN .
1
0
0
AT qn(N +1=2)q n
X
i=N++1
N +v
X
i=N +1
9
=
;
1
1
A
The contribution of the qSelberg measure (xju; v; N; q; q) =
can be evaluated explicitly and is written as follows:
(x) QiN=1 xiu Qvk=10(1
qkxi)
=qjRj(2N+u+v 1) (RN)(q ) (RN+v)(q )=qjRj(u 1) (RN)(q ) (RN+v)(q );
(xRju;v;N;q;q)
(x?ju;v;N;q;q)
cross(x;y)=YY
1
N+ N
i=1j=1
8
=exp<X
:n 1
=X
A;B
X
X
R+;R A;B
Using the identities
xi
yj
2 N v+ 1
Y
Y
j=1 n=0
n
n
pn+
1 q nv+
1 q n
q n 1
N+ v 1
Y Y
yj i=1 l=0
( q n)+pn
q n
1
qlxi
1 qnv
1 qn
qN+ N jAj+jBj A@q (N++1=2)npn+
qn(1=2 i)
A B q (N++1=2)npn
and
(AN)(x) denotes Schur polynomial in N variables xi. The cross contribution reads
one gets
where
(3.3)
is the open topological string amplitude for the resolved conifold. Pictorially SAB(Q) is
given by one corner of the toric diagram from
gure 4. It is also equal to the ChernSimons
(or WZW) Smatrix.
identity
A( pn) = ( 1)jAj AT (pn). Collecting all the contributions one obtains
where pn = PiN=+1 xin, qn = PN
j=1 yjn and we have employed the Cauchy identity and the
B4= (x?ju+;v+;N+) (x?ju ;v ;N )
qN+ N jAj+jBjqjR+j(u++v++2N+ 1)+jR j(u +v +2N
1) (RN++)(q ) (RN+++v+)(q )
(N )(q ) (RN +v )(q ) (BN+)(qR++ ) (ANT )(q R
R
) (AN++v+)(qR++ ) (BNT +v )(q R
(RN)(q ) = ( 1)jRjq NjRj (RTN)(q );
(AN)(q R
) = ( 1)jAj (ATN)(qRT+ )
SR+B(q N+ )SART (qN )SRT B(qN +v )SAR+ (q N+ v+ );
B4= (x?ju+;v+;N+) (x?ju ;v ;N ) X
X
qN+ N jAj+jBjqjR+j(u++v++2N+ 1)+jR j(u 1)
R+;R A;B
Q2
QF
R
R+T
QF
A
B
QB
Q2+
gauge theory with four fundamentals.
Using the AGT relations (A.5) one immediately obtains the identi cation between the
Kahler parameters of the toric CalabiYau, CFT and the gauge theory parameters:
QB =
qN+ N
=
QF = q2N++u++v+ 1 = qu
Q1+ = q N+ v+ = qa+m1+ ;
Q
1 = q
N
= qa+m2 ;
q 2a+m2+ m2 ;
1 = q 2a 1;
Q2+ = q N+ = qa m2+ ;
Q
2 = q
N +v
= qa m1 :
Let us state once more the result for the fourpoint qdeformed conformal block for
t = q.
This block can be simultaneously decomposed in two ways: DF integral and
the decomposition in terms of the complete basis of states. Using the simplest choice of
basis states (Schur functions) one gets a symmetric quadri linear decomposition in terms
of characters. Moreover, this decomposition is naturally identi ed with the corresponding
topological string amplitude, computed using the topological vertex technique. We now
move on to describe the multipoint case.
3.2
Multipoint (t = q)
In section 2.2.3 we understood the starchain transformation for ordinary DF integrals.
The qdeformed case goes along the same lines. Also, as in the fourpoint case above, we
obtain a natural interpretation of the objects featuring in the decomposition from the point
of view of the topological strings.
Using the starchain relation we rewrite the Vandermonde measure in the multipoint
DF integral as a sum over chains of skew Schur functions. We consider the corresponding
expansion of the kpoint DF integral in terms of k qdeformed Selberg averages of skew
Schur functions (cf. eq. (2.15)):
Bk+2BU(1) =
X
k
Y
fY~ag a=1
*
jaY~aj NYa 1;1Ya;1 [x]NYa 1;2Ya;2
pn
1
1
q nva +
q n
ua;va;Na;q;q
(3.8)
?
?
Z
Ya;1
Ya;2
Ya 1;2
Ya 1;1 Q1; Q2; QF ; q; q
!
=
Ya;1
Ya 1;2
Ya;2
Q1
HJEP02(16)98
equal to qdeformed Selberg average of two bifundamental kernels as in eq. (3.8).
where h: : :iua;va;Na;q;q denotes the qSelberg average with the corresponding parameters and
Y~0 = Y~k = (?; ?). Each average depends on two pairs of Young diagrams Y~a 1, Y~a and
corresponds to the element of the toric diagram depicted on gure 5. Notice that a certain
U(1) factor appears in the left hand site. It can be nicely eliminated in the fourpoint case,
though not for higher multipoints.
In the next section we show that each average in eq. (3.8) indeed reproduces the
bifundamental part of the Nekrasov function. The whole sum thus becomes the Nekrasov
function for linear quiver gauge theory, of the form depicted in gure 2.
3.3
Factorization of averages
Let us check that the DF averages of the bifundamental kernel NAB[x] indeed factorizes.
To do it we use the loop equations, which are given in appendix B. We obtain a formula
for the average of four Schur polynomials:
NAC[x]NBD
pn(x) 1 q nv
1 q n
;
(3.9)
(3.10)
(3.11)
(3.12)
and the de nition of G(AqB;t) and zb(qif;ut)nd are collected in appendix A. This indeed proves that
the average of four Schur polynomials gives the bifundamental Nekrasov contribution.
Identi cation with topological strings
We would like to further decompose the qSelberg average in eq. (3.9) to observe the
structure of the corresponding topological string amplitude from
gure 5. Notice that each
average contains a product of two bifundamental kernels NAB[x]. This corresponds to the
product of two fourpoint functions each having the form:
?
RT
Z C
A Q2;q;q =
?
A
C
Q2
RT
=X( Q2)jW jCATW T?(q)CCW RT (q)
W
=Z ?
?
?
? Q2;q;q q C 2 A
( Q2)jCj( 1)jRj (RN)(q )N A(NC) q R
;
(3.13)
HJEP02(16)98
where Q2 = qN . Recall the expression for the qdeformed Selberg measure (3.3), which
consists of two Schur polynomials. The product of two fourpoint amplitudes (3.13)
therefore gives exactly the product of qSelberg measure with two bifundamental kernels as in
the average (3.8). More explicitly, gluing two amplitudes (3.13) one obtains the amplitude
from
gure 5, which is given by:
R
?
?
R
R
?
?
?
?
?
?
?
?
?
?
?
?
Z DC
BA Q1;Q2;QF ;q;t
RT
=X( QF )jRjZ C R?T A Q2;q;q Z D ? B Q1;q;q
=Z ?
? Q1;q;q Z ?
? Q2;q;q q C A+2 D B
( Q1)jCj( Q2)jDj
X( QF )jRj (RN)(q ) (RTN v)(q )N A(NC)hq R iN D(TNBTv)hq RT i
=Z ?
? Q1;q;q Z ?
? Q2;q;q q C A+2 D B
( Q2)jCjQj1Dj( 1)jBjq ( 21 +N)(jAj jCj+jBj jDj)
X(QF q N v)jRj (RN)(q ) (RN+v)(q )N A(NC)hq R
21 N iN B(ND+v)h pn q R
12 N i
=Z ?
? Q1;q;q Z ?
? Q2;q;q q C A+2 D B
( 1)jBj+jCjq(2N+ 12 )jCjq( 21 v)jDjq ( 21 +N)(jAj+jBj)
Su;v;N;q;q N A(NC)[x 1]N B(ND)
where Q1 = q N v, Q2 = qN , QF = qu+v+N 1 and Su;v;N;q;q is the qSelberg integral
without character insertions. Notice how in the last equality we have used the fact that
the qSelberg averages can be calculated as sums over partitions R with the measure given
where
given by6
jjM E(q;t)jj2 =
CE (q; t)
CE0 (q; t)
=
Y
1
(i;j)2E 1
qEi j+1tEjT i
qEi j tEjT i+1
is the norm of Macdonald polynomials and generalized skew Macdonald polynomial is
M A(qB;t=)EF (Qjpn; pn) = M E(q;t)
n
1
1
q
MF(q;t)
n
1
1
q
M A(qB;t)(Qjpn; p):
One can immediately notice that for t = q
HJEP02(16)98
Ne A(qB;q;)CD(u; v; N jx) = NAC [x]NBD
pn(x)
1
1
qnv
qn
;
exactly reproducing the unre ned case (3.9). Generalized Macdonald polynomials are
obtained from the kernel by forgetting about one of the two pairs of Young diagrams:
Ne A(qB;t;)??(u;v;N jx)=MA(Bq;t) qu+1t 1
Ne ?(q?;t;)CD(u;v;N jx)=M C(qD;t) qu+v+1t2N 1 pn; pn
1 q nv
1 t n
:
t n
q
p n
1 tn
1 (t=q)n qnv;p n+
(t=q)n qnv
1 tn
One can also get the product of two ordinary Macdonald polynomials by setting the two
\crosswise" diagrams to be empty:
Ne ?(qB;t;)C?(u; v; N jx) = M B(q;t)
p n +
The most remarkable property of the generalized bifundamental kernel is that its
qSelberg average is factorized into a product of simple monomials (B.10). More concretely,
it is given by the bifundamental contribution to the Nekrasov function (hence the name of
the kernel). Schematically
DNe A(qB;t;)CD(u; v; N jx)
E
Averages of this kind can be obtained by using the loop equations for qSelberg integral (or
(q; t)matrix model). The full form of the average (4.7) and technical details are summarized
in appendix B. Thus, we prove that generalized bifundamental kernel is indeed the relevant
object to be averaged to get the chainlike decomposition of the DF integral. Let us now
try to nd similar objects in re ned topological strings.
6One can ask why we \subtract" ordinary Macdonald polynomials from the generalized ones to obtain
the skew polynomials. In fact eq. (B.10) is independent of the concrete choice of \subtracted" polynomials
as long as they constitute a complete system and the Cauchy identity holds. We will return to this issue in
section 4.4.
; (4.4)
(4.5)
(4.6)
(4.7)
4.3
We start from the re nement of the basic building block, i.e. the fourpoint amplitude (3.13)
and set preferred direction to be vertical. The general re ned amplitude depending on four
Young diagrams is given by eq. (C.1). For our purposes we need the following
specializations:
?
R
Z
t
pn(t )
pn
Q2
r q
t
t
CT=E
pn(q R
t ) + pn
Q2
r q
t
t
r t
Q 1
q 2 t
pn(qRt )
q
n
tn pn
;
(4.8)
and
R
?
Z
t q
AT
1
1
t
n
qn
R
t q
BT
CT
Q2
R
?
t
?
X
E
AT=E
DT
Q1
q
?
?
?
t
?
= Z
? Q1; q; t q jjRjj2 t jjRT2jj2 ( Q1)jDjM (t;q)
2
RT
pn(q )
pn
Q1
r t
q
q
X
F
BT=F
DT=F
q
n
tn pn(q R
1
1
t
n
qn
r q
r t
t
q
t
t
t )
pn
Q 1
1
pn(qRt )
pn
Q1
;
(4.9)
To make contact with DF integrals we make the identi cation q qt Q1 = t1 N q v 1,
q qt Q2 = tN .
and (4.9) each give precisely one \half" of the qSelberg measure (4.1).
We immediately notice that two Macdonald polynomials in eqs. (4.8)
Skew Schur functions in eqs. (4.8), (4.9) can be rewritten through the discrete qSelberg
\integration" variables xR;i = qRi+1tN i in the following way:
Z CT
AT tN+ 12 q 21 ;q;t
=Z ?
? tN+ 12 q 21 ;q;t ( 1)jAj q 21 t N jAj
qt2N 12 jCj
q jjR2jj2 t jjRT2jj2 M R(q;t) pn(t ) pn tN t
X
E
t jEj
A=E
1 t
1 qn
Z DT
BT t 21 N q v 12 ;q;t
=Z ?
? t 21 N q v 12 ;q;t ( 1)jBj q 1t 12 N jBj
q v+ 32 t 1 jDj
q jjRjj2 t jjRTjj2 M (t;q) pn(q ) pn t
2 2
RT
1 N q 1 v
q
X
F
t jF j
q
B=F
n
1 t
1 qn
pn(xR;i)+ t
1 t n
t n
q
pn(xR;i)
1
t
1 t n
C=E (p n(xR;i));
(4.10)
HJEP02(16)98
D=E
t n
q
1 qnv
1 tn
:
(4.11)
?
R
?
?
R
?
?
?
=X( QF )jRjZ CT
AT tN+ 12 q 21 ;q;t Z DT
BT t 21 N q v 12 ;q;t
R
?
Su;v;N;q;tZ ?
? tN+ 12 q 21 ;q;t Z ?
? t 21 N q v 12 ;q;t
(x?ju;v;N;q;t)
?
?
( 1)jAj+jBj q 21 t N jAj
qt2N 12 jCj
q 1t 12 N jBj
q v+ 32 t 1 jDj
Let us glue two fourpoint functions (4.8) and (4.9) to obtain the qSelberg average:
Z
DT
t
R
t q
q Q2
CT
QF
t
q Q1
t q
BT
?
?
?
R
where Su;v;N;q;t is the qSelberg integral without insertions. We have used the identity (4.1)
HJEP02(16)98
to obtain the qSelberg measure evaluated at discrete points labeled by R from the
product of two Macdonald functions, and made the identi cation QF = qu+v+ 12 tN 2 . The sum
3
over representations R corresponds to the qSelberg average as shown in appendix E. Notice
also that the two pairs of Schur functions under the average are skew, just as in the de
nition of the bifundamental kernel (2.13), and should be summed over the \intermediate"
representations E and F .
At this point one observes that the expression under the average is not the generalized
xi ! q
i
bifundamental kernel (4.3), which would give the bifundamental contribution as an average.
Instead it is simply a product of two pairs of Schur polynomials. However, closer look
n
reveals that the arguments of the Schur polynomials exactly match (up to the factors 11 qtn )
the arguments of the generalized bifundamental kernel after the elementary transformation
1 vx 1 as can be seen e.g. from eq. (B.11). If we could expand the generalized
Macdonald polynomials in terms of Schur polynomials with the same arguments, this would
produce a transformation between the expression under the average in eq. (4.12) and the
average of generalized bifundamental kernel (4.3). This average (but not the original one in
eq. (4.12)) in turn is given by the bifundamental Nekrasov function, as shown in eq. (4.7)
and in appendix B.
Going further, we notice that the bifundamental Nekrasov contribution is given by the
re ned topological amplitude with horizontal preferred direction, whereas the preferred
direction is vertical in the diagrams (4.8), (4.9) and thus in the average (4.12). This leads
us to the relation between horizontal and vertical slicing of the re ned amplitude. To get
this relation we will need to introduce generalized Kostka functions, which are the expansion
coe cients of the generalized Macdonald polynomials in terms of Schur polynomials.
4.4
Generalized Kostka functions
The basis of generalized Macdonald polynomials can be reexpanded in terms of Schur
M A(qB;t)(Qjpn; sn) =
X KACBD(Q; q; t) C (pn) D(sn);
C;D
MA(Bq;t)(Qjpn; sn) = jjM A(q;t) 2
jj jjM B(q;t)jj2 X KACBD(Q; q; t) C
1
1
q
n
tn pn
D
1
1
(4.13)
q
n
tn sn ;
(4.14)
*
q
A=E
n
1 qn
C=E(p n(xR;i))
B=F
D=E
pn(xR;i)
1 t n
t
1 t
1 qn pn(xR;i)+ t
1 t n
t n
1 qnv +
q
1 tn
;
(4.12)
C;D
analogy with the ordinary Kostka polynomials,7 de ned as
where the coe cients KCADB(Q; q; t) can naturally be called generalized Kostka functions by
As an explicit example we give here generalized Kostka functions for the rst level:
M A(q;t)(pn) =
X KAB(q; t) B(pn):
B
KACBD(Q; q; t) jAj+jBj=jCj+jDj=1 =
KACBD(Q; q; t) jAj+jBj=jCj+jDj=1 =
1
0 !
11 Qqt 1
;
1
0
1 qt !
(1 Q)
:
1
(4.15)
(4.16)
(4.17)
q
B
1
1
t
n
qn rn
:
(4.18)
1 qn
We would like to use generalized Kostka functions to transform the two pairs of Schur
polynomials inside the average in eq. (4.12) into two generalized Macdonald polynomials
and thus obtain the bifundamental kernel. The Schur functions in eq. (4.12) are skew and
are summed over the \intermediate" representations E and F , so we use the following
identity, which is the consequence of the Cauchy completeness theorem, to transform the
sum over Schur polynomials into that over Macdonald ones:
X
E
A=E (pn) B=E
X
E
0
X 1 1
n 1
n 1
M E(q;t)
1
1
t
qn rn
q
1
1
q
A(pn) B
1
1
A(pn)
M E(q;t)
t
n
qn rn
n
1
1
Now the sum over \intermediate" representations in the last line includes the combination
of Macdonald functions depending on the conjugates of the power sums pyn = n
which exactly reproduces the sum in the de nition of generalized bifundamental
kernel (4.3). The Schur polynomials
A, B remaining in eq. (4.18) are no longer skew and
can be transformed into generalized Macdonald polynomials using the generalized Kostka
functions, as in eqs. (4.13), (4.14).
7Our de nition of generalized Kostka functions can be modi ed slightly to turn them into polynomials.
This is achieved by using a di erent normalization of generalized Macdonald polynomials, called MfA(qB;t) in
eqs. (19), (20) in [74].
Eventually, we obtain the connection between re ned topological string amplitude with
vertical slicing and bifundamental Nekrasov function:
KWCD1W2 (qQF Q1) 1; q; t Z
CT
DT
BATT Q1; Q2; QF ; q; t KY1AYB2 (qQF Q2) 1; q; t
?
?
!
X
A;B;C;D
2 Q17!t 21 N q v 12 3
Su;v;N;q;t Z
Q3 t q QB;2
t q
R2
Q2 t q QB;1
Q1 t q
t q
R1
? t 21 N q v 12 ; q; t
?
?
?
(x(?)ju; v; N; q; t)
( 1)jAj+jBj qv 23 t N jAj qvt2N 12 jCj qv 2t 12 N jBj q 12 t 1 jDj
D
NeAB;CD(u; v; N jq1 vxR1)E
(q;t)
zbifund [A; B]; [C; D]; u v 12 2 N+ ; u 21+ ; v2
CA0(q; t)CB0(q; t)CC0 (q; t)CD0(q; t)G(AqB;t)(q u v 1t1 2N )G(DqC;t)(qu+1t 1)
Note that generalized Kostka functions in this formula depend on the \distance" (in the
sense of Kahler parameters) between the pairs of horizontal external legs of the toric
diagram. Let us also point out that our Kostka functions are qdeformation of the coe cients
of the abelianization map acting on the instanton moduli space.
4.5
Horizontal slicing. DF representation and spectral dual Nekrasov function
Let us glue three pieces (4.8) together horizontally to obtain the DF integrand for vepoint
conformal block and its AGT dual  U(2)2 quiver gauge theory. The resulting amplitude
is equal to \half" of the total DF measure DF(x(1); x(2); x(3)) evaluated at discrete points
xa;i = qRa;i+1tNa i. We get:
Z ?
? ? ?
? Q1;Q2;Q3;QB;1;QB;2;q;t
(4.19)
Z ?
Y
1 2 N1 N2
YY
m=0 i=1j=1
4
4
Y
2 2 N1
Y
m=0 i=1
1
? tNa+ 12 q 21 ;q;t q
jjR2ajj2 t jjRaTjj2 M R(qa;t) pn(t ) pn tNat
2
1
1qm xR2;j
N2 N3
Y Y
xR1;i j=1k=1
1q m
1 2q m
Y
xR2;j i=1k=1
1 2qm xR3;k
xR1;i
N1 N3
YY
2q m
xR2;j
1
1
3
5
:
3
5
(4.20)
Three Macdonald polynomials in the second line can be thought of as a \half" of the three
qSelberg measures, corresponding to three integration contours in the DF representation.
Moreover, the measure (4.20) can be evaluated explicitly and also gives the \half" of
the spectral dual Nekrasov function with gauge group U(3), cf. (A.1) (the other half of the
factors comes from the lower half of the diagram):
HJEP02(16)98
Z ?
? ? ?
? Q1;Q2;Q3;QB;1;QB;2;q;t
q jjR1jj22+jjR3jj2
tjjR1Tjj2+ jjR2Tjj2 Z?(Q1)Z?(Q2)Z?(Q3)Z?(QB;2)Z?(QB;2Q2Q3)Z?(QB;1)
2
CR01(q;t)CR02(q;t)CR03(q;t)Z?
q qt QB;2Q2 Z?
q qt QB;2Q3 Z?
q qt Q1QB;2
(Q2B;1QB;2Q2)jR1j( QB;2)jR2jZ?(Q1Q2QB;1)Z?(QB;1QB;2Q2)Z?(Q1Q2Q3QB;1QB;2)
Z?
q qt Q2QB;1 Z?
q qt Q1Q2QB;1QB;2 Z?
q qt QB;1QB;2Q2Q3
G(Rq1;t?)
G(Rq2;t?)
r q
r q
t Q1 G(Rq1;t?)
t Q2 G(Rq2;t?)
r q
r q
t QB1;1 G(Rq1;t?)
t QB1;2 G(Rq2;t?)
r q
t
r q
(QB;1QB;2Q2) 1
t Q1Q2QB;1
G(Rq3;t?) q qt Q3 G(Rq3;t?) q qt QB;2Q2Q3 G(Rq3;t?) q qt Q1Q2Q3QB;1QB;2
G(Rq3;tR)2(QB;1Q3)G(Rq2;tR)1(Q2QB;1)G(Rq3;tR)1(QB;1QB;2Q2Q3)
(4.21)
?
where Z?(Q) = Z
? Q; q; t . This is a manifestation of the spectral duality for
Nekrasov functions [75{78]: while vertical slicing of the toric diagram gives Nekrasov
function for U(2)3 quiver gauge theory, the horizontal slicing yields its spectral dual  gauge
theory with a single U(3) gauge group. In the language of conformal blocks [74] this means
that both the Jackson integral and the sum over complete basis of generalized Macdonal
polynomials have the form of Nekrasov decompositions, which are spectral dual to each
other. For re ned topological strings only one Nekrasov decomposition can be obtained for
a given choice of preferred direction  the cut should dissect the preferred edges. If one cuts
along a di erent direction the amplitudes do not reproduce the Nekrasov functions, as can
Nekrasov/generalized Macdonald decomposition (1.10)
Vertical slicing, horizontal preferred direction (4.21)
Conformal block Bk, gure 1
qSelberg measure (4.1)
DF integral (1.1)
Decomposition in Schur polynomials
Rotation of preferred direction by 2
Topological string
Closed string amplitude on toric CY Ztop, gure 3
Two fourpoint conifold amplitudes (3.13), (3.14)
Horizontal slicing, vertical preferred direction (3.16), (4.20)
Vertical slicing, vertical preferred direction (4.12)
Generalized Kostka function (4.16), (4.17)
be seen in eq. (4.12). However, there is still a way to see the dual decomposition: preferred
direction can be changed with the help of generalized Kostka functions (4.16), (4.17).
HJEP02(16)98
5
Conclusions and discussion
We have investigated the connection between qdeformed conformal blocks and topological
strings. This connection arises in the following way. Due to the AGT relation conformal
blocks are equal to Nekrasov partition functions, which can be obtained by the geometric
engineering technique, as compacti cations of type IIA strings (or, more generally,
Mtheory) on toric CY threefold. String partition function on the threefold is equal to partition
function of topological strings.
We obtain an explicit dictionary between the objects in CFT and elements of the
corresponding toric diagram, summarized in table 1. For the case of t = q we introduce
the bifundamental kernel (2.13), compute its qSelberg averages (2.20) and show that they
reproduce Nekrasov partition function. We also study spectral duality of conformal blocks
and generalize the statements of [74] to multipoint blocks. Most importantly, we study the
evertroublesome case of t 6= q, where we introduce generalized bifundamental kernel (4.3).
We compute the average of the generalized kernel  it satis es the most general of all the
so far encountered \factorization of averages" type identities (B.10)  and is again given
by Nekrasov function. We interpret the change of preferred direction of re ned topological
strings as a change of basis between generalized Macdonald and Schur polynomials, which
is performed by generalized Kostka functions (4.13), (4.14).
Of course the expansion we have considered is not limited to the case of U(2) gauge
theories and Virasoro conformal blocks. U(N )/WN story goes along the same lines. In
this setting qSelberg integrals are replaced by the AN qSelberg integrals, their measure
being given by the product of several basic building blocks (C.1). Generalized
Macdonald polynomials, bifundamental kernels and Kostka functions can also be found for the
U(N ) case.
It would be extremely interesting to understand the relation of the
character/topological string decomposition of conformal blocks from the point of view of SeibergWitten
integrable systems. One relation is provided by the quantum spectral curve for DF
integrals [97], which in NekrasovShatashvili limit reproduces the quantum spectral curve
(Baxter TQ equation) of the relevant SeibergWitten system, the XXZ spin chain. Of
course, a general method to obtain quantum spectral curves from the toric data is
desirable. Let us also mention that in this way one can study the mirror symmetry between
the Bmodel CY, encoded in the spectral curve and the Aside toric CY described by the
topological vertex formalism.
In the four dimensional limit generalized Kostka polynomials coincide with the
coe cients of the abelianization map acting on the
xed points in the cohomology of the
instanton moduli space. Explicit combinatorial expressions for these coe cients were
obtained in [98]. It would be interesting to understand these formulas from the point of view
of re ned topological strings.
The product of generalized Kostka matrices turns out to be an interesting algebraic
object. We can reason in the following way. Let us rst expand generalized Macdonald
polynomials in terms of products of Schur polynomials using the Kostka matrix. Then we
exchange the two Schur polynomials and apply the reverse Kostka transformation. Thus we
obtain another set of generalized Macdonald polynomials. However the two sets are clearly
related. Recall that generalized Macdonald polynomials are eigenfunctions of the operator
H1gen =
(H1), which is given by the DingIohara coproduct, acting on trigonometric
Ruijsenaars Hamiltonian H1. The second set of generalized Macdonald polynomials is
obtained by acting on the same Ruijsenaars Hamiltonian with the opposite coproduct
As in any quasitriangular Hopf algebra, there is an Rmatrix performing the transformation
from one coproduct to the other. The two sets of generalized Macdonald polynomials are
therefore also related by the same Rmatrix. This is the Ktheoretic version of the instanton
Rmatrix [99, 100] with spectral parameter being the parameter of generalized Macdonald
polynomials. The implications of this observation and the relation between toric CY and
integrable systems will be studied elsewhere.
Acknowledgments
We thank Andrey Smirnov for clarifying the concept of instanton Rmatrix to us. We
would also like to thank the anonymous referee for many insightful comments, which led
to serious improvement in the presentation of the results. Our work is partly supported
by grants NSh1500.2014.2, 153120484Molaved and 153120832Molaved, by RFBR
grants 130200478, by joint grants 155250034YaF, 155152031NSCa, by
140192691Inda and by the Brazilian National Counsel of Scienti c and Technological Development.
Y.Z. is supported by the \Dynasty" foundation stipend.
A
Fivedimensional Nekrasov functions and AGT relations
The Nekrasov partition function for the U(N ) theory with Nf = 2N fundamental
hypermultiplets is given by
~
A
ZN5de;kU(N)=X jA~j zfund(A~; m~+;~a)zfund(A~; m~ ;~a)
zvect(A~;~a)
~
A
=X jA~j QiN=1QfN=1fA+i (qmf++ai )fAi (qmf +ai )
zvect(A~;~a)
(A.1)
where fA (qx) = Q(i;j)2A 1
q xt (i 1)q (j 1) , zvect(A~; ~a) = QiN;j=1 G(Aqi;At)j (qai aj ) and
G(AqB;t)(qx) =
Y
1 + ;
v+ =
n+ =
m2+ ;
qxqj 1t1 i = fA (q x);
qxq1 jti 1 = fA+(qx);
u =
v =
1 +
m1
2a ;
m2 ;
n = a + m2 ;
(A.2)
(A.3)
(A.4)
(A.5)
b2,
(A.6)
a2 = a. Masses ma, vevs ai, radius R5 of the fth dimension and 1;2 all
have dimensions of mass. In this paper we set the overall mass scale so that 1 =
2 = 1 and q = e R5. The t parameter in Macdonald polynomials is related to q by t = q
More generally, one can consider quiver gauge theories with gauge groups U(N )k and
bifundamental matter hypermultiplets as shown in gure 2. The corresponding Nekrasov
~
Ya
ZN5de;kU(N)k =X j1Y~1j
jY~kj Y YfY+1;i qmf++a1;i
k
N N
f=1i=1
zbifund Y~k 1;Y~k;~ak 1;~ak;mbifund;k 1 zvec(Y~k;~ak) f=1i=1
where the bifundamental contribution is given by zbifund(Y~ ; W~ ; ~a;~b; m) = QiN=1 QjN=1 G(Yqi;Wt)j
qai bj m .
B
Loop equations for matrix elements
We would like to compute the qSelberg average of a function f (pn) which is polynomial
both in pn and p n. To do this we use an improved version of the loop equations obtained
in [74].
Concretely, we use the fact that the qintegral of a total qderivative vanishes:
1
1
zbifund Y~1;Y~2;~a1;~a2;mbifund;1
N N
Y YfYk;i qmf +ak;i
We will write a instead of ~a = (a; a) for N = 2. The AGT relations for N = 2 are:
Z 1
0
{ 32 {
1)g(x) = 0;
(B.1)
if g(1) = g(0) = 0. We write down the following judiciously chosen total derivative:
Z
N
dqN x X 1
i=1 xi
2
1)xi 4 z
xi q Y xi txj Y
xi j6=i xi
N
xj k=1
v 1
a=0
x
u Y(qaxk
k
1)
q;t(x)f (x)5 = 0:
The di erence with [74] is that we assume f (pn) to be a Laurent polynomial, i.e. the
function of pn for n both positive and negative. Writing down the action of the qderivative
in eq. (B.2) we get the following identity:
*
1 1
q z
+t2N 1qu+1 qv 1 1
z
*
X 1 t n qnz npnA
n 1
Res =04
f (pn+(qn 1) n)exp@
0
n 1
0
X 1 tn n
n
n
X 1 t n npnA5
n 1
13
0
X 1 t
n
n 1
n
n
13
p nA5
z npnA
1
1
tN f (pn)
z
f (pn)
=0:
p n, e.g.:
p 1 +
v
1 t
More generally
MA p n+
MB(pn)
tn=qn qnv
1 tn
The expression in the average in eq. (B.3) looks complicated and not too suitable for explicit
calculations. However, expanding in powers of z and taking f (pn) to run over products
of pn (with n both positive and negative) one gets the recurrence relations determining
the averages of any symmetric function. Let us note that the expansion in positive and
negative powers of z lead to the same recurrence relations as it should. In addition to
the usual factorized formulas for the generalized Macdonald polynomials these equations
give the averages of the products of two Macdonald polynomials, one in pn the other in
p1
t N+1(1 tN )(1
qutN 1)(1
q1+utN )(1
q1+vtN 1
q2+u+vt2N 2)
Y tiq 1(1 qArmA(i;j)tLegA(i;j)+1) 1 Y
q1+j+utN 1(qArmB(i;j)tLegB(i;j)+1 1) 1
G(Aq?;t)(q u)G(Bq?;t)(qu+v+2t2N 2)
;
(B.2)
One can also transform the averages of positive power sums pn to negative ones p n and
vice versa:
which leads to
hf (xi)iu;v;N;q;t = hf (q1 vxi 1)i u v 2+2 2 N;v;N;q;t
(B.6)
MA pn +
q nv
1 t n
q(jAj jBj)(1 v)
MA p n +
We remind the result from [74]:
M A(qB;t) qu+v+1t2N 1 pn; pn
MB(p n)
tn=qn
qnv
1 tn
MB(pn)
1
q nv
1 t n
= ( 1)jAjq (v+1)jAj (u+2v+3)jBj+P(i;j)2A j+2 P(i;j)2B jtjCj (2N+3)jBj P(i;j)2B i
We also give an alternative average of generalized Macdonald polynomial (notice the
difference in shifts of the power sums pn)
M A(qB;t) q u 1t p n +
1 tn
; p n
= ( 1)jAjq 2jBj+ujAjtjBj jAjtP(i;j)2B i+2 P(i;j)2A iq
t n 1 (t=q)n
1 tn
P(i;j)2A j
u;v;N;q;t
f
A tN qu f
A t1 N q v 1 f
B tN+1q 1 f
B t2 N q u v 2
CA0(q; t)CB0(q; t)G(BqA;t) (qu+1t 1)
:
(B.9)
which gives all the averages above as special cases:8
Finally, we were able to nd the most general factorized formula for the average of two
generalized Macdonald polynomials (or generalized bifundamental kernel NeA(qB;t;)CD(u; v; N jx)),
:
(B.7)
u v 2+2 2 N;v;N;q;t
hNeA(qB;t;)CD(u; v; N jx)iu;v;N;q;t
*
X
E;F
t jEj+jF j
MA(Bq;=tE)F
M C(qD;t=)EF
qu+1t 1
qu+v+1t2N 1 pn; pn
jjM E(q;t) 2
jj jjMF(q;t)jj2
1
t n
q
8We have checked this formula up to the third level.
= ( 1)jBj+jCjq 2jAj (v+1)jCj+ujBj (u+2v+3)jDjtjAj jBj+jCj (2N+3)jDj
p n
1 tn
1 (t=q)n qnv; p n +
1 t n
q nv +
u;v;N;q;t
P
R
Z
A
B Q; q; t =
Q t q
t R
where CA0(q; t) = Q(i;j)2A(1
qAi jtAjT i+1).
hNeA(qB;t;)CD(u; v; N jx)i u v 2+2 2 N;v;N;q;t
= hNeA(qB;t;)CD(u; v; N jq1 vx 1)iu;v;N;q;t
= q(v 1)(jAj+jBj jCj jDj)
Re ned open string amplitude depends on four Young diagrams A, B, R and P and the Kahler
parameter Q of the conifold.
HJEP02(16)98
CA0(q; t)CB0(q; t)CC0 (q; t)CD0(q; t)G(AqB;t)(qu+1t 1)G(DqC;t)(q u v 1t1 2N )
; (B.10)
Again, one can use the symmetry (B.6) to write eq. (B.10) in an alternative form:
*
X
E;F
t jEj+jF j
jjM E(q;t) 2
jj jjMF(q;t)jj2
MA(Bq;=tE)F
M C(qD;t=)EF
q u v 1t1 2N
q u 1t p n; p n
t n
pn
1 (q=t)n
1 t n ; pn +
q nv
1 t n
t n 1
1 tn
qnv +
u;v;N;q;t
:
(B.11)
C
Open topological string amplitude on resolved conifold
In this appendix we write down the basic building block of the toric diagrams related to
5d quiver gauge theories. It is given by an open re ned topological string amplitude in the
resolved conifold background depicted in gure 7.
Using the IKV re ned topological vertex one gets the following answer for this
amplitude:
Z A B Q;q;t
C
? Q;q;t q jjRjj2 2 jjPjj2 t jjPTjj2 2 jjRTjj2 q jAj 2jBj M R(q;t)(t )MP(tT;q)(q )G(RqP;t) r q
X( Q)jCj AT=CT pn(t q R) pn
B=C pn(q t P T ) pn
r qt Qt q P
r qt Qq t RT
? Q; q; t = Qi;j 1 1
Qqi 21 tj 21 is the closed re ned string amplitude
One can perform a
op transformation on the conifold geometry.
We employ the following symmetry of the closed string amplitude:
Z
? Q; q; t = Z
? Q 1; q; t :
HJEP02(16)98
The opped open string amplitude is related to the original amplitude with Kahler
(C.1)
(C.2)
(C.3)
(C.4)
(D.1)
(D.2)
(D.3)
where Z
on the conifold.
parameter reversed:
Z opped A
B Q;q;t =
R
P
where
MY(t;q)(pn) = ( 1)jY jhY T (q; t)M (q;t)
Y T
hY T (q; t) =
CY0 (q; t) =
CY0 T (q; t)
CY0 (t; q)
Y
1
(i;j)2Y 1
Y
(i;j)2Y
1
qYi j tYjT i+1 :
1
1
q
n
tn pn ;
tYi j+1qYjT i
tYi j qYjT i+1
;
Q 1pq 1t 1 112 q jjRjj2 jjPjj2+j2jATjj2 jjBTjj2 t jjRTjj2 jjPTjj22+jjAjj2 jjBjj2
q jAj 2jBj
( Q)jRj+jP j+jAj+jBjZ A
B Q 1;q;t :
P
R
The multipliers in eq. (C.3) combine with the change of framing in the adjacent edges
induced by the
op. The answer for any closed string amplitude, which includes the
opped part is given simply by
Z opped(Q; Qadjacent; Qi) =
1
Q 1pq 1t 1 12 Z(Q 1; QQadjacent; Qi);
where in the right hand side the original Kahler parameter of the conifold is reversed and
the Kahler parameters of the twocycles adjacent to the opped conifold are shifted by Q.
D
Useful identities
One has the following identity for the power sum symmetric functions pn(x) = P
i 1 xin:
pn(qY t ) =
t n=2 1
1
n
tn pn(t Y T q )
where i = 12
i and Y is a Young diagram.
Macdonald polynomials satisfy the following \transposition" identities:
Combining eqs. (D.1) and (D.3) we get the identity, which will be useful in re ned
topological string computations:
M (t;q) pn(q t ) pn(Qt ) = ( 1)j jh T(q; t)M (qT;t) pn(t
T
q ) pn(Qq ) : (D.4)
The following identity involving Nekrasov functions is also useful:
YN1 YN2 1
Qqj Wi 12 ti YjT 21
1
Qqj 21 ti 21
G(YqW;t)(q 21 t 21 Q)
G(Yq?;t)(Qq 21 tN1 21 )G(?qW;t)(QqN2)
;
(D.5)
and in particular for N1;2 ! 1 (we assume jqj; jtj < 1):
One can exchange the diagrams in Nekrasov function using the identity
i;j 1
Qqj Wi 12 ti YjT 21
1
Qqj 21 ti 21
= G(q;t)
Y W
r q
Q :
= ( Q)jAj+jBjq jjBjj2 2 jjAjj2 t jjATjj22 jjBTjj G(AqB;t)
r q
Q 1 ;
G(BqA;t)
r q
Q
where jjRjj2 = Pi Ri2.
G(Rq?;t) r q
G(?qP;t) r q
and also
t
When one of the diagrams is zero, there is a nice expression in terms of Macdonald
Q =
Y (1 Qqj 21 t 12 i)=
Q =
Y (1 Qq 21 jti 21 )=
M R(q;t) 1 Qnq n2 t n2
t n2 t 2
n
M R(q;t)
t n21 t n2
MP(tT;q) 1 Qnt n2 qn n2
q n2 q 2
MP(tT;q)
q n2 q 2
1 n
M R(q;t) pn(t ) pn Qp q t
t
M R(q;t)(t )
MP(tT;q) pn(q ) pn Qq t q
q
MP(tT;q)(q )
(D.6)
(D.7)
;
(D.8)
;
(D.9)
(D.10)
M R(q;t) t
t
CR0(q; t)
DotsenkoFateev integrals as contour and Jackson integrals
In this appendix we show that the qdeformed DotsenkoFateev integrals can be understood
both as the contour integrals (as in [60, 61, 94{96]) or as Jackson integrals, i.e. discrete sums
(as in [74, 91]). More concretely, we show that in both description the DF representation
essentially reduces to the sum over (tuples of) Young diagrams.
Let us rst consider the contour integral description for the qdeformed (M + 2)point
Virasoro conformal block on a sphere:
BM+2 BU(1) =
I
C1;:::;CM
dN1x
dNM x (q;t)(x)Vu(x; 0)Vva(x; za);
(E.1)
where the contour Ca encircles the points xi = zaqmtn for n; m
Vv(x; z) =
Y YN 1
k 0 i=1
1
q
k v z
qk xzixi ;
Vu(x; 0) = Y xiu;
N
i=1
0, and t = q ,
(q;t)(x) = Y Y
N
i6=j k 0
1
1
and N = PM
of Young diagrams Ra, a = 1; : : : ; M , i.e. xa;i = xa;i(R~ ) = zaqRa;i tNa i, i = 1; : : : ; Na so
a=1 Na. As shown in [94{96], the poles of the integrand are labeled by M tuples
that the integral is reduced to the sum over residues. One can compute the ratio of the
residues corresponding to the given set of diagrams Ra and the set of empty diagrams
tqqkkxxxxjiji ;
(E.2)
HJEP02(16)98
= llnnqt , u and va are positive integers, so that the
i=1
! N
1
Yxi (1 N) Y Y xi qkxj :
i6=jk=0
=Resx=x(?~)
(q;t)(x)Vu(x;0)Vva (x;za)
X
R1;:::;RM
Resx=x(R~)
(q;t)(x)Vu(x;0)Vva (x;za)
Resx=x(?~) (q;t)(x)Vu(x;0)Vva (x;za)
The rst term is an inessential normalization constant N , which can be calculated
separately. One can evaluate the ration of the residues by simply evaluating the integrands at
the poles:
BM+2 BU(1) = N
X
R1;:::;RM
(q;t)(x(R~ ))Vu(x(R~ ); 0)Vva (x(R~ ); za) :
(q;t)(x(?~))Vu(x(?~); 0)Vva (x(?~); za)
Since there are only Na variables xa;i(R~ ), the sum in eq. (E.4) is actually over the partitions
Ra having at most Na columns. The ratio of the integrands turns out to be given by
the Nekrasov formula for the SU(M ) gauge theory with 2M fundamental hypermultiplets
depending on the M tuple of partitions R~ [94{96]. The vacuum moduli and masses of
the gauge theory are related to the momenta of the primary elds in the conformal block,
which can be expressed in terms of u, va and Na. In particular, since Na are integers,
the gauge theory is at a particular point in the vacuum moduli space, where ak + mi are
integers. At this point the contributions of the fundamental hypermultiplets vanish for all
partitions Ra having more than Na columns, which conforms with the DF result.
Now we consider the Jackson integral version of the DF representation:
(q;t)(x)Vu(x;0)Vva (x;za)
The Jackson integral is de ned as follows
We assume that the parameters integrand turns into a (Laurent) polynomial:
Vv(x;z)=( z)vq v(v2+1) YNxi v Y
v 1
i=1
1 qk+1 xi ;
z
BM+2 BU(1) =
Z zM
dNM x
q
(q;t)(x)Vu(x; 0)Vva (x; za);
Z z1 dqN1 x
0
Z a
0
dqxf (x) = (1
q) X aqnf (aqn):
n 0
In this case all the integrals are wellde ned geometric progressions, and evaluating them
amounts to evaluating the integrand at discreet points xa;i(na;i) = zaqna;i with na;i 2 N:
X
na;i 0
(q;t)(x(~n))Vu(x(~n); 0)Vva (x(~n); za);
One immediately notices that some terms in the sum vanish, e.g. if na;i = na;j , then
(q;t)(x) = 0. More generally, for
(q;t) not to vanish, the corresponding na;i and na;j
should di er at least by . This gives rise to the following form of na;i for nonvanishing
xi;a(Ra) = zaqRa;i+(Na i) ;
(E.9)
where Ra are Young diagrams (we have used the symmetry of the integrand and chosen a
particular ordering of xa;i). The sum in eq. (E.8) is reduced to the sum over M tuples of
Young diagrams. Of course one can rewrite this sum using the same trick as in eq. (E.3):
(q;t)(x(R~ ))Vu(x(R~ ); 0)Vva (x(R~ ); za)
(q;t)(x(?~))Vu(x(?~); 0)Vva (x(?~); za)
(q;t)(x(R~ ))Vu(x(R~ ); 0)Vva (x(R~ ); za) :
(q;t)(x(?~))Vu(x(?~); 0)Vva (x(?~); za)
(E.10)
X
R1;:::RM
X
R1;:::RM
We arrive at the same sum over Young diagrams Ra (each with no more than Na columns)
as in eq. (E.4), obtained from contour integrals.
Thus, the two approaches to the DF integrals give exactly the same decomposition,
leading to a sum over Young diagrams. In the main sections of the paper we relate this
sum to the sum in the topological vertex formalism.
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.
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