Generalized Macdonald polynomials, spectral duality for conformal blocks and AGT correspondence in five dimensions
Received: February
Published for SISSA by Springer
Open Access 0 1 2 3
c The Authors. 0 1 2 3
0 7a Prospekt 60letiya Oktyabrya , Moscow , Russia
1 Institute for Nuclear Research of the Russian Academy of Sciences
2 25 Bolshaya Cheremushkinskaya street , Moscow , Russia
3 31 Kasirskoe chaussee , Moscow , Russia
We study five dimensional AGT correspondence by means of the qdeformed betaensemble technique. We provide a special basis of states in the qdeformed CFT Hilbert space consisting of generalized Macdonald polynomials, derive the loop equations for the betaensemble and obtain the factorization formulas for the corresponding matrix elements.
dimensions; Matrix Models; Duality in Gauge Field Theories; Conformal and W Symmetry

Generalized
Macdonald polynomials, spectral duality
for conformal blocks and
1 Introduction
2 Generalized Macdonald polynomials
3 qdeformed DotsenkoFateev integrals
4 Loop equations for qdeformed betaensemble
5 Spectral duality for conformal blocks
6 Comparison with topological strings
7 Conclusions
A Macdonald polynomials and Ruijsenaars Hamiltonians
B Five dimensional Nekrasov functions and AGT relations
C Ruijsenaars Hamiltonians and loop equations
D Useful identities
Introduction
conformal blocks of the Liouville theory. This connection has provided insight in two
dimensional CFT and gauge theory as well as other related areas [512]. In this paper we
version of the AGT duality has been checked for many cases in [13][15]. We will focus on
the simplest possible case of (qdeformed) Virasoro fourpoint conformal block.
We briefly introduce the approach that we will use to study the five dimensional AGT
correspondence. As in the four dimensional case the instanton part of the Nekrasov
parfact a finer structure consisting of several factorized terms, so that the whole sum can be
written as a sum over pairs of partitions (A, B):
zvect(A, B)
where zfund,vect are certain polynomials in the gauge theory parameters which we write
to the expansion of fourpoint conformal block in terms of a certain complete system of
zvect(A, B)
18], so that
zvect(A, B)
polynomials giving the desired expansion [19, 20]. However, for general t, q the basis turns
out to be a lot more elaborate: in particular the naive deformation of Schur polynomials to
Macdonald polynomials is not enough. In this paper we show that the right basis is in fact
given by the generalization of the Macdonald polynomials MAB depending on two partitions
(i.e. generalized Jack polynomials) were introduced in [21] and also studied in [22].
To compute the matrix elements in eq. (1.3) we use the qdeformed version of the
cuts in the DF integrands pulverize into a set of poles, so that the integrals can be taken
by residues. The sum over residues is captured by the Jackson qintegral, which is in fact
a sum of the form
dqzf (z) = (1 q) X qkaf qka =
k0
1 q
1 qaa (af (a)).
The matrix elements in this framework are given by the qdeformed Selberg averages
of the generalized Macdonald polynomials:
which provide the recurrence relations for the averages of any symmetric polynomials.
One of our main results is the remarkable factorized formula for the averages of
generalized Macdonald polynomials (4.12). It can be written schematically as
(zfund(A))(zfund(B))
[zvect(A, B)]1/2
and evidently leads to the AGT conjecture for fourpoint blocks (1.3). Though we were
not able to obtain a rigorous proof of this formula, we have checked it for several lower
polynomials. This is the only missing step in the proof of the five dimensional AGT
conjecture, however the conceptual picture is already apparent.
Let us clarify the relation between our study and the alternative approach to the AGT
duality proposed in [24, 25]. In these works the sum over residues in the DF integrals
without any basis decomposition was shown to be the sum over pairs of partitions:
qR+ t qR t , qR+ t, qR t ,
R+,R
miraculously into the Nekrasov partition function
However, this is not the Nekrasov function featuring in the AGT correspondence but a
spectral dual [2628] thereof. One can also notice that the expansion in eq. (1.7) is not in
We therefore have two different expansions of the qdeformed conformal block
connected by the spectral duality. The original expansion in terms of a special basis
corresponds to the AGT dual Nekrasov function and the spectral dual expansion in the
intermediate momentum corresponds to the manifest sum over poles in the DF integrals:
zvect(R+, R )
ZNek( = q2a, mf, q, t)
Generally the spectral dual Nekrasov functions describe gauge theories with
differNevertheless, the parameters of the theories are reshuffled by the duality, e.g. the dual
with the gauge theories. One can see that the AGT relation is a combination of the explicit
DF expansion [24, 25] and the spectral duality.
ZNek( = q2a, mf, q, t)
QF
= P
A,B QBA+B (zfund(A))2(zfund(B))2 = ZNek
zvect(A,B)
R+,R
QF corresponds to the vev of the scalar, Q1,2 represent the masses of the hypermultiplets and
Generally the spectral dual conformal block B
different conformal algebra (qWM instead of qVirasoro) compared to the original one B.
Again, the case we consider here is extremely simple: the number of points and the algebra
remain the same for the fourpoint conformal block of qVirasoro. The dimensions of the
We will mention a tentative application of this duality for conformal blocks in section 7.
Why do these new duality features become visible only in the five dimensional version
of the AGT correspondence and not in the original one? It turns out the in five dimensions
gauge theory partition function has fine structure, which can be effectively analysed from
the gauge theory.1 The geometry of this manifold is encoded in its toric diagram. For our
case of SU(2) gauge theory with four fundamental hypermultiplets the diagram is shown
in figure 1. The edges of the diagram correspond to the twocycles in the threefold while
Kahler parameters Qi of these cycles correspond to the gauge theory parameters.
The topological string partition function can be computed using the topological vertex
There are essentially two ways to carry out the sum over partitions in figure 1. One
can cut the diagram vertically, compute the sum over R
explicitly and leave the sum over
A, B in the final answer. This sum over pairs of partitions is nothing but the sum in the
Nekrasov function ZNek. Moreover, each half of the diagram corresponds to the matrix
element in the conformal block expansion (1.2):
However there is one more way to perform the summation: one can cut the diagram
horizontally and do the sums over A and B first. In this way one gets the spectral dual
corresponds to the sum in the DF integral
representation (1.7):
Q2
B
= qR+ t qR t , qR+ t, qR t 1/2
. (1.12)
The spectral duality in this picture is natural and corresponds to taking the mirror image
In section 2 we introduce the generalized Macdonald Hamiltonian, compute its
eigenpolynomials. In section 4 we derive the loop equations for the corresponding qdeformed
section 5 and compare our results to the topological string partition function in section 6.
We present our conclusions in section 7.
Generalized Macdonald polynomials
Generalized Macdonald polynomials are symmetric polynomials in two sets of variables
xi and xi labelled by pairs of partitions Y1, Y2 and depending on an extra parameter
DingIohara coproduct acting on the of the original Macdonald Hamiltonian [15]:2
H1gen =
t 1
ald eigenvalues
H1genMY(q1,tY)2 (Qp, p) = X2 Qa1 X(qYia 1)ti MY(q1,tY)2 (Qp, p).
a=1
i1
Eigenvalues of H1gen are nondegenerate, so the eigenfunctions are uniquely determined. In
practice one finds Macdonald polynomials by solving linear equations for the coefficients
of the polynomial eigenfunctions. One writes a general symmetric polynomial as a linear
Y 0 cY Y 0 pY 0 . Acting with H1gen involves
shifts of pn:
X(qn 1)zn
n1
This gives a rational function so it is straightforward to obtain the residues.
The properties of generalized Macdonald functions are analogous to those of generalized
Jack polynomials, though with minor variations.
1. Orthogonality, conjugation and normalization. There is a convenient way to
normalize generalized Macdonald polynomials:
through Macdonald polynomials themselves as follows
MY(q1,tY)2 (Qp, p) =
mY 1 (p)mY 2 (p)+
with respect to Macdonald scalar product
hMA~ , MB~ i = A~B~ CA01 CA02
hf (pk), g(pk)i = f
1 q
n
1 tn pk
pk=0
2We rescale pn by (q/t)n/2 compared to [15].
CA =
(i,j)A
1 qAij+1tAjTi ,
CA0 =
(i,j)A
1 qAij tAjTi+1 .
Sometimes it is more convenient to use a different normalization
1 qYiaj t(Y a)jTi+1
MfY~ (p) = GY 2Y 1 Q1
MfY~ (p) = GY 1Y 2 (Q)
a=1,2 (i,j)Y a
a=1,2 (i,j)Y a
where GAB is given in appendix B. This normalization is tailored so that the norms
of the polynomials are given by the vector part of the Nekrasov function:
Notice also that in this normalization generalized Macdonald polynomials depend
polynomially on the parameters Q, q and t.
1 qYia+j t(Y a)jT+i1
MfY (p) , (2.8)
2. The inversion relation.
1 q
= (1)Y~  CY 1 CY 2
MY~(tT1,q1)
3. Cauchy completeness identity. Generalized Macdonald polynomials form a complete
basis in the space of symmetric functions in two sets of variables:
= exp
zvect(Y~ , Q)
k1 a=1
1 qk
4. Specialization identities. For q 1, t = q
become generalized Jack polynomials M~(q,t)(q2apk) J ~()(apk). For t = q the
Y Y
Hamiltonian H1gen turns into a sum of screening factors. Therefore, the eigenfunctions
consistent with the results of [19, 20], where Schur polynomials were found to be the
right basis for the case of t = q.
One has the following reduction to ordinary Macdonald polynomials: MY(q1,tY)2 (Q0, pk)
donald polynomials on the Weyl vector can also be found and are nicely expressed
through the ordinary Macdonald polynomials:
Q pk =
1 Qn
1 tn , pk = 1 tn
1 qtnn !
Q pk = 0, pk = 1 tn
1 Qn
= MY(q1,t)
= MY(q2,t)
1 Qn
1 tn
pk =
pk = 1 tn
1 Qn
M[2],[] =
M[1,1],[] =
M[1],[1] =
M[],[2] =
M[],[1,1] =
= C
i=1 j=1 k=0
noninteger values.
The first few generalized polynomials are
M[1],[] = p1,
M[],[1] = p1 q (Q 1)
p1(q t)
1q
v+1
2 2
(q + 1)(t 1)p21 +
2(qt 1)
(q 1)(t + 1)p2
2(qt 1)
p21(q t) (qtQ + qQ tQ 2t + Q)
2q (qQ 1) (Q t)
(q 1)(t + 1)Qp2(q t)
2q (qQ 1) (Q t)
(q + 1)(t 1)p21(q t) q2 + qtQ qt tQ
2q2 (Q 1) (qt 1) (q Q)
(q 1)(t + 1)p2(q t) q2 qtQ + qt tQ
2q2 (Q 1) (qt 1) (q Q)
(q + 1)(t 1)p1p1(q t)
(qt 1) (q Q)
p21(q t) q t2Q + tQ t
2q2 (Q 1) (tQ 1)
p2(q t) q t2Q tQ + t1
2q2 (Q 1) (tQ 1)
p1p1(q t)
q (tQ 1)
2 2
N v+1
yj j=1 n=0
q yj i=1 l=0
(q + 1)(t 1)p21 +
2(qt 1)
(q 1)(t + 1)p2
2(qt 1)
v1
v1
N+ v1
B =
dqN z Y
i6=j k=0
N+Y+N
v+1
1qkzi
qdeformed DotsenkoFateev integrals
In the DotsenkoFateev approach conformal blocks are expressed in terms of multiple
contour integrals of the degenerate field insertion.
What is specific to the qdeformed
case is that these integrals can be taken by residues and reduce to multiple Jackson
qintegrals (1.4). We use the qintegral formalism which turns out to be more convenient
throughout this paper. The fourpoint conformal block is given by the following integral3
Y (1 qkxi) (q,t)(y) Y yi
Y (1 qkyi)
The completeness (2.11) of the generalized Macdonald polynomials can be employed
in the last line of eq. (3.1) and gives the following expansion
i=1 j=1 k=0
= exp
n1 n 1 qn
N v+1
yj j=1 n=0
N+ v1
q yj i=1 l=0
pn qn 1 tn
1 qnv+
qn + pn qn
1 tn
1 qnv =
tn 1 qnv+
= X A+B CACB MA(Bq,t) q2a qn pn 1 tn , pn
A,B CA0CB0
M A(qB,t) q2a qn, qn
t n 1 qnv
1 tn
new parameter a has appeared in the last lines of eq. (3.2). This is an extra parameter of
of the Selberg average according to eqs. (B.3). To check the AGT correspondence (1.3)
one should therefore check that
1 qnv+
hf (x)i =
R dqNx (q,t)(x) QiN=1 xiu Qvk=01(qkxi 1)f (x)
R dqNx (q,t)(x) QiN=1 xiu Qv
k=01(qkxi 1)
and the parameters of the Selberg sum are identified with the gauge theory parameters
with the help of eqs. (B.3). In the next section we develop the loop equations for the
qdeformed betaensemble (3.4) in order to check eq. (3.3).
Loop equations for qdeformed betaensemble
Let us write down the loop equations for the DF integral. They provide the recurrence
the average of any symmetric function.
One first observes that the qintegral of a total qderivative is zero
tn 1 qnv
AGT
zvect(A, a)
dqz z (1 qzz )g(z) = 0
i=1 xi
xi q Y xi txj Y
v1
where f (x) denotes a symmetric polynomial in xi corresponding to the insertion of extra
vertex operators. Eq. (4.2) in this case is valid as a power series in negative powers4 of z.
The qderivative of the qVandermonde is given by
qxixi q,t(x) = q
N1
Y (qxi xj )(txi xj )
j6=i (xi xj ) xi qt xj
Using this expression one can rewrite eq. (4.2) as follows:
* N tN1qu+1(qvxi 1) qxii f (x)
X
z qxi
Y txi xj
j6=i xi xj
(xi q)f (x) Y xi txj
q(z xi)
= 0, (4.4)
where h. . .i denotes the qSelberg average. Let us cast this equation into a more convenient
form. We first observe that the sum in eq. (4.4) can be written as a contour integral
powers of xi one expands eq. (4.2) in positive powers of z and obtains the recurrence
4Positive powers give rise to total derivatives with nonvanishing boundary values.
( q)f (pn) YN txj #+
= 0, (4.5)
X 1 tn
qnznpn
= 0.
where the contour Cx encircles all the points xi. Deforming the contour we pick up the
powers of z and will not affect the recurrence relations for pn, so this term can be safely
dropped. Other residues give
q 1
f (pn) exp
X 1 t
znpn
q
q z
+ tN1qu+1 qv1 z
f (pn + (1 qn)zn) exp
n1
f (pn) exp
(1tn)pn+
n1
X 1 znqn(1 tn)pn
t k 1 qkv !+
1 tk
q(Y 1)ijt(Y 1)jTi+1
q1j+u+vt2i2(tNi+1qj1
1)(tN+2iqjuv2 1)
(qv1ti)(tNiqu+j
1)(tNi+2qjv1 1) (B.3)
q(Y 2)ijt(Y 2)jTi+1
1
Let us demonstrate how the recurrence relations work for the simplest example.
Conage of p1:
hp1i =
q tN 1 tN1qu+1
1
(t 1) (t2N2qu+v+2 1)
= 0.
GY 2Y 1 (t1qu+1)
1
k=1 (i,j)Y k
als [19, 20]:5
hMY (pn)i =
hMY (pn)i =
the factorized expressions
MY(q1,tY)2 qu1t qk, qk
(i,j)Y 1
(i,j)Y 2
Qf2=1 Q2k=1 fYk (mf + ak)
GY 2Y 1(q2a)
(i,j)Y
qti1 1 tNi+1qj1 1 qu+jtNi
The following simple relation for the negative power sums can be immediately deduced
hpY iu,v,N = qY (1v)hpY iuv2+22N,v,N .
One can also check that the averages of generalized Macdonald polynomials are given by
and analogously
1qkv+
= (1)Y 2qP(i,j)Y 1 i+P(i,j)Y 2(2i+2aj+1)
q(Y 2)ijt(Y 2)jTi+1
1
q(Y 1)ijt(Y 1)jTi+1 1
qu+v+jt2N++2i(tN+iqu++j
1)(tN+i+1qj1 1) (B.3)
GY 1Y 2 (t2N++1qu+v+1)
Combining eqs. (4.12), (4.13) we obtain the desired result (3.3), though with additional
MA(1qA,t2) q2a qn pn
1 qnv+
1 qnv
(i,j)Y 1
(i,j)Y 2
k=1 (i,j)Y k
This relation gives the proof of the five dimensional AGT conjecture for fourpoint
conlas (4.12), (4.13) for the qSelberg averages. In appendix C we sketch some ideas which
might lead to the proof of these identities. We have performed computerized checks of
eqs. (4.12) and (4.13) on the first three levels.
Spectral duality for conformal blocks
and use combinatorial identities to cast them into the form of Nekrasov function.
Let us examine the qSelberg integral featuring in the DF representation of the
qdeformed CFT (3.4)
kNN
zvect(A, a)
. (4.14)
of the conformal block (3.1) as a sum over pairs of partitions
B
R+,R
BR+R ,
B
Using the identities from appendix D one immediately obtains a compact expression
not affect the final answer):
BR+R =
B
(i,j)R+ qR+,ijtR+T,ji+1
1
qR+,ij+1tR+T,ji 1
(i,j)R
1 qj2tN++2i 1 qj+v1tN+i+1
qR,ijtRT,ji+1
qR,ij+1tRT,ji 1
1 qjv+1tiN+1
1
GR+R
Qk=
Qf2=1 fR+k ak + m,+ fRk ak + m,
f f
zvect(R+, R, ak)
where f , GAB and zvect are given in appendix B,
a+ = a = a
2a = 1 + (N+ N + 1),
m1,+ = a + v + 1 + (N 1),
m1, = a + 1 + N ,
m2, = a + v+ + 2 + N 2,
Since we have proven both horizontal and vertical equalities in eq. (1.9), the diagonal
line should also be true. We therefore obtain a (slightly indirect) proof of the spectral
duality for the SU(2) Nekrasov partition functions. Eventually, this implies that all the
dualities in the diagram (1.10) are valid.
Comparison with topological strings
Firstly we notice that the vertical half of the diagram from figure 1 indeed corresponds
of Schur functions):
q2i+m2+akj
f=1 fYk (mf + ak)
QF = q2a,
Q1 = qm1a,
Q2 = qm2a .
This equality tells us that the vertical half of the toric diagram can be identified with the
Of course, the computations are the same for the horizontal halves of the diagram,
which correspond to the DF integrands (see eq. (1.12)):
A R+
B R+
k=
f=1 fR+k
zvect R+, R, ak
Conclusions
the remarkable factorization formula for the averages of generalized Macdonald
polynomials. We have proven the spectral duality for SU(2) Nekrasov functions and fourpoint
vertex operators.
It would be interesting to better understand other features of the CFT in the language
of topological strings. In particular, the modular properties of conformal blocks should
be encoded in the topological string partition function. The spectral duality in this case
provides an alternative expansion, which can be used to find the nonperturbative results
(cf. [41, 42]).
Of course extensions of the above approach to the six dimensional gauge theories and to
integrable systems, most interestingly to the double elliptic ones [43, 44], should also be
Acknowledgments
The author would like to thank A. Mironov, Al. Morozov, And. Morozov, S. Shakirov and
hospitality of the International Institute of Physics, Natal, Brazil where part of this work
and by D. Zimins Dynasty foundation stipend.
Macdonald polynomials and Ruijsenaars Hamiltonians
In this appendix we very briefly review some essential properties of the (trigonometric)
Ruijsenaars Hamiltonians and list some useful expressions for them.
Macdonald polynomials are the eigenfunctions of the set of commuting Ruijsenaars
Hamiltonians Hk, which can be thought of as a maximal commutative subalgebra inside
the DingIohara algebra [15]:
HkMY = ek
1)tNi
MY
where ek is the elementary symmetric polynomial of N variables and
Hk =
= t 2
k(k1)
xia xj
txia xj qPa xia ia .
1i1<...<ikN a=1 j6=i1...ik
Note that for generic q and t the spectrum of H1 is nondegenerate and no higher Hk are
needed to solve for the eigenfunctions. However, we give some expressions for higher Hk
for the sake of completeness.
Let us derive a compact expression for Ruijsenaars Hamiltonians acting on symmetric
terms of power sums is given by
The shift operators commute and we get
Hk = t 2
k(k1)
1i1<...<ikN a=1 j6=i1...ik
xia xj a=1
txia xj Y ePn1(qn1)xina pn .
One can see that this sum can be expressed as a sum over residues:6
Hk =
k(k1)
where all k integrals are taken over the same contour encircling the points xi. The origin
of the tdeformed factorial lies in the useful symmetrization formula
Y tx(i) x(j) = [k]t! .
Using (A.6) once again for za variables we get
Hk =
k(k1)
Yk dza Y za zb Y Yk tza xi ePka=1 Pn1(qn1)zan pn .
N
We expand the rational factors containing xi in terms of power sums
Hk =
k(k1) +kN I
(t1)kk!
Cx a=1 za a6=b za tzb
H1 = h[1] ,
H2 =
H3 =
2 h[1,1] 2 h[2] ,
6 h[1,1,1] 2 h[2,1] +
h[1k] =
h[k] =
h[2,1] =
tkN+ k(k21)
(t 1)k
C
(za zb)2
a=1 za a<b (tza zb)(za tzb)
ePka=1((za)(tza))ePka=1(+(qza)+(za)),
(tk 1)
(t 1)(t2 1)
C
C
dz e(z)(tkz)ePka=01(+(qtaz)+(taz))
2 dz1 dz2 (z1 z2)(tz1 z2)
z1 z2 (z1 tz2)(t2z1 z2)
7To be precise one should ensure the convergence of the expansions. We assume that za are radially
ordered, i.e. z1 < . . . < zk.
understood as the commutative subalgebra in the DingIohara vertex algebra.
n1 zn pn and (z) = P
n1
als degenerate into Shur polynomials which are independent of q. In this limit all the
Hamiltonians can be explicitly expressed through the first one. For example:
h[1k](q) = (h[1](q))k,
h[k](q) = h[1](qk),
h[2,1](q) = h[1](q2)h[1](q).
The first integral also simplifies and becomes
h[1](q) =
: e(z) qN+zz 1 e(z) :
q 1
that h[1](q) with different values of q commute:
[h[1](q), h[1](q0)] = 0 .
h[1](q)R = qN 12 CR(q) +
qN 1
q 1
where CR(e~) = P
tion of the Casimirs CR(n) = P
i1
Ri i + 12 n
(i + 12 )n .
i1 e~(Rii+ 12 ) e~(i+ 12 )
= P
n
n0 ~n! CR(n) is the generating
func
Five dimensional Nekrasov functions and AGT relations
multiplets is given by
ZN5dek = X A~ QiN=1 QfN=1 fA+i(mf+ + ai)fAi(mf + ai)
zvect(A~, ~a)
G(AqB,t)(x) =
(i,j)A
(i,j)B
1 qxqAijtBjTi+1
1 qxqAijtBjTi+1
(i,j)B
(i,j)A
1 qxqBi+j1tAjT+i =
1 qxqBi+j1tAjT+i .
fermionic construction of GL() Casimirs.
The AGT relations for N = 2 are:
u+ = m1+ m2+ 1 + ,
v+ = m1+ m2+ ,
v = m1 m2 ,
Ruijsenaars Hamiltonians and loop equations
Let us rewrite the loop equations (4.4) in terms of the Ruijsenaars Hamiltonian (A.2).
We first write down a useful identity:
i=1 z xi j6=i xi xj
Y txi xj =
(1 t)z 1
which is proven by expanding the right hand side as a sum over poles in z.
Using eq. (C.1), one can rewrite eq. (4.4) as follows
tN1qu+1
n1 qn 1
z qtN
qz(1 t)
z q
f (x) qz(1 t) j=1
pn f (x)+
z xj
where H1 is the first Ruijsenaars Hamiltonian.9 The expansion of eq. (C.2) in negative
of any symmetric polynomials in xi. In fact, only two of the infinite family of constrains are
needed. Indeed, the Virasoro generators Ln with positive n can be obtained by commuting
L1 and L2. The same holds for the qdeformed case. We may therefore consider only the
tN1qu+vH1f (x) +
* tN1qu+2
tN1qu+v+1
q 1
[H1, p1] f (x) +
1 tN
1 t
f (x) q1p1f (x) = 0, (C.3)
For f (x) a Macdonald polynomial the Hamiltonian H1 acts diagonally. However
multiplication with pn produces a sum over multiple Macdonald polynomials. To see this one
should use the Pieri idenity:
where the sum is over diagrams W such that the skew diagram W \Y is a horizontal strip
(i.e. there are no more then one box belonging to each column) of k boxes; CW \Y (resp.
RW \Y ) is the set of columns (resp. rows) intersecting W \Y and
completely in terms of the averages of Macdonald polynomials. We are sure that one can
obtain a proof of the identities (4.9), (4.10), (4.12), (4.13) along these lines, though we
were not able to find it.
Useful identities
bY (i, j) =
(i,j)A
Qqj1t1i
(i,j)B
Qqjti
YN vY1 1 Qqk+Riti
i=1 k=0
1 Qqkti
k=0 i=1 j=1
(i,j)R
1 Qqktji
where GRP is given by eq. (B.2).
Y1 YN+ YN 1 Qqk+RiPj tji
1 qRij+1tRjTi
1 qRijtRjTi+1 = GRR(1)
qjti1
(i,j)R
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