Toric CalabiYau threefolds as quantum integrable systems. \( \mathrm{\mathcal{R}} \) matrix and \( \mathrm{\mathcal{R}}\mathcal{T}\mathcal{T} \) relations
Received: August
Rmatrix and
Hidetoshi Awata 1 2 6 7 8 9 10 11 12
Hiroaki Kanno 1 2 4 6 7 8 9 10 11 12
Andrei Mironov 0 1 2 3 5 7 8 9 10 11 12
Alexei Morozov 0 1 2 3 7 8 9 10 11 12
Andrey Morozov 0 1 2 3 7 8 9 10 11 12
g Yusuke Ohkubo 1 2 6 7 8 9 10 11 12
Yegor Zenkevich 0 1 2 7 8 9 10 11 12
h 1 2 7 8 9 10 11 12
Open Access 1 2 7 8 9 10 11 12
c The Authors. 1 2 7 8 9 10 11 12
Field Theories, Topological Strings
0 National Research Nuclear University MEPhI
1 Leninsky pr. , 53, Moscow, 119991 Russia
2 Nagoya , 4648602 Japan
3 Institute for Information Transmission Problems
4 KMI, Nagoya University
5 Theory Department, Lebedev Physics Institute
6 Graduate School of Mathematics, Nagoya University
7 60letiya Oktyabrya pr. , 7a, Moscow, 117312 Russia
8 Bratiev Kashirinyh , 129, Chelyabinsk, 454001 Russia
9 Kashirskoe sh. , 31, Moscow, 115409 Russia
10 Bol. Karetny , 19 (1), Moscow, 127994 Russia
11 Bol. Cheremushkinskaya , 25, Moscow, 117218 Russia
12 [39] B. Feigin, K. Hashizume , A. Hoshino, J. Shiraishi and S. Yanagida, A commutative algebra
Rmatrix is explicitly constructed for simplest representations of the DingCalculation is straightforward and signi cantly simpler than the one through the universal Rmatrix used for a similar calculation in the Yangian case by A. Smirnov but less general. We investigate the interplay between the Rmatrix structure and the structure of DIM algebra intertwiners, i.e. of re ned topological vertices and show that the Rmatrix is diagonalized by the action of the spectral duality belonging to the SL(2; Z) group of DIM algebra automorphisms. We also construct the T operators satisfying the RT T relations with the Rmatrix from re ned amplitudes on resolved conifold. We thus show that topological string theories on the toric CalabiYau threefolds can be naturally interpreted as lattice integrable models. Integrals of motion for these systems are related to qdeformation of the re ection matrices of the Liouville/Toda theories.
relations; Conformal and W Symmetry; Supersymmetric gauge theory; Topological

DIM algebra, generalized Macdonald polynomials and the Rmatrix
Re ned topological strings and RT T relations
Rmatrices: from
deformation to (q; t)deformation
A ne Yangian Rmatrix from generalized Jack polynomials
DIM Rmatrix from generalized Macdonald polynomials
1 Introduction 2 3 1.1
RT T relations in the toric diagram
Trivial diagrams on vertical legs
Arbitrary diagrams on vertical legs
4 Integrals of motion and compacti cation 5 Conclusion
A Explicit expressions for DIM Rmatrix
RMatrix at level 1
A.1.1 (q; t)deformed version
deformed version
RMatrix at level 2
A.2.1 (q; t)deformed version
deformed version
B.2 Higher Hamiltonians
Action of x+1 on generalized Macdonald polynomials
Introduction
Integrability plays an exceptional role in modern studies of quantum
eld theory and string
list of recent examples includes
SeibergWitten solution of N
complex integrable systems [3{6],
from integrable spin chains and models [7],
models with new solutions to YangBaxter equations [9],
topological string calculations and the study of Hurwitz functions [23{25].
kind of integrability appears.
emerging in the description of coproducts of group elements g^ 2 G
A(G) [28{31] for
quantum groups [32{36],
I) = R (g^
diagram corresponding to a DIM intertwiner [40, 41].
of the generalized Macdonald Hamiltonian with known eigenvalues.
The notation in this paper follows our paper [41].
+
n form the three central rows. n
into the zero mode of the raising generator.
DIM algebra, generalized Macdonald polynomials and the Rmatrix
xn+ and xn with n 2 Z together with the \Cartan" generators
n, n 2 Z>0 and two central
can be, therefore, drawn as an integral point on the plane. The generators, xn+,
and their commutators form a lattice, which is sketched in
gure 1. The exact de nition
There is a nice representation of the DIM algebra on the Fock space Fu
izontal direction. There is also the vertical Fock representation Fu
(0;1), isomorphic to the
Young diagram Y , while x
n delete one box, and
n act diagonally. The central charges
vertical ones and vice versa. The action of S is illustrated in gure 1.
Let us construct a natural basis in the tensor product of horizontal modules.
which are the eigenfunctions
of the Hamiltonian
with eigenvalues
H1 MfAB =
ABMfAB
H1 =
DIM(x+(z))
AB = u1
In the simplest example, i.e. for the tensor product of two Fock modules Fu1
the spectral parameters uu12 .
The Hamiltonian H^1 is the zero mode of the raising generator, x
0+ in the
horizonof the rst Cartan generator
the Cartan generators
n+ acts diagonally on the ordinary Macdonald polynomials. The
1 . As we mentioned above, in the vertical representation
given by tensor products of the Macdonald polynomials M A(q;t)(a(1n) )ju1i
ju2i, which diagonalize x
0+ = S( 1+), can be thought of
Hn = u1
2) are automatically diagonal, see appendix B.
IIB Sduality exchanging NS5 and D5 branes, hence, our notation.
number of time sets and Young diagrams is correspondingly increased.
(x+) = x
op(x+) = 1
by an Rmatrix:
op = R
Hence, their eigenfunctions are also related:2
RAB
conservation law
jAj + jBj = jCj + jDj
which makes R blockdiagonal with nitedimensional blocks.
The coproducts
op di er only by permutation of the two representations on
obtained by a simple change of variables, exchanging u1 $ u2, A $ B and pn $ pn:
= MfBA
polynomials as functions of the variable uu12 , which we call Q, whereas in [86] we denoted uu21 as Q.
as was suggested in [88, 89], and this will be the approach we adopt here.
Re ned topological strings and RT T relations
to the nonselfdual Nekrasov
deformation. In order to reduce it to the ordinary
topothreefold into C
3 patches and
nd the universal amplitudes, trivalent re ned vertices on
topology S1
D2 sitting on the legs of the toric diagram. These boundary conditions are
Kahler parameters of the edges.
involving descendants to those of the primary
elds. In fact, one can show that this
following from the regularity of qqcharacters.
Also, in the NekrasovShatashvili limit
systems related to the gauge theory [13{16].
TARBP (Q; u; z) =
CalabiYau manifold [102{104].
R(12)R(13)R(23) = R(23)R(13)R(12)
RT (1)T (2) = T (2)T (1)R
trW T (1) trW T (2) = trW T (2) trW T (1)
V (vertical leg) and the auxiliary space W (horizontal leg), b) the Rmatrix acting on W
of motion.
rotate back using S
. We thus obtain the relation of the form
to the spectral dual frame using S, then make the permutation of the spaces and
R = S
encounter this relation when performing concrete computations of the Rmatrices.
horizontal directions manifest.
S. Shakirov [108].
ing on the direction of compacti cation, it becomes either elliptic or a
ne. Eventually,
Rmatrices: from
deformation to (q; t)deformation
the DIM Rmatrix.
de nes a trivial Rmatrix, which is proportional to the identity matrix:3
RCABD(u)jt=q / A B :
C D
ne Yangian Rmatrix from generalized Jack polynomials
of two strands 12.
to the Cartan subalgebra of the a ne Yangian:
( ) =
2 n=1
2 n=1
makes dual polynomials di erent. Notice that this term vanishes for
= 1, i.e. in the
corresponding to the eigenfunctions JABfp; pg are
2 n;m=1
2 n;m=1
(AB) =
[2];[1;1]. Thus, one
still needs higher Hamiltonians Hn with n
2 to uniquely specify the polynomials, which
J[1];; = (1
J;;[1] = (u
J[1];; = (1 + u
J;;[1] = (u
u)p1 + (1
polynomials, one has the following de nition of the opposite ones:
NAB(u1
NBA(u2
u2j jp; p) = JBA (u2
the Rmatrix is indeed given by
u2j jp; p) =
RCABD(u1
u2) JCD (u1
or, using the Jack scalar product,
RCABD(u1
u2) =
u2j jp; p) jJCD (u1
The Jack scalar product is de ned as
hf (pn)jg(pn)i = f
g(pn)jpn=0 :
CD6=AB
Notice the conjugate polynomial J oApB in the bra vector in eq. (2.7).
we expand JAB in the basis of monomial symmetric functions:
CACBD(uj )mC (pn)mD(pn)
NAB(uj ) = gAB(uj )
gAB(x) =
i) (2.11)
The normalization factors satisfy the identity NAB(uj )NBA( uj )jjJAjj2jjJBjj
2 =
particular, one has
jjJABjj2 = zAveBc (uj ):
can be associated to the
xed points in the moduli space of SU(2) instantons (or to the
then describe stable envelopes of the corresponding
xed points.
of generalized Jack polynomials:
( ) = B
identity block arising from the generalized Jack polynomials at the zeroth level:
J?;?(uj jp; p) = J ?;?(uj jp; p) = 1 :
The resulting 3
3 matrix
3 block of the standard rational Rmatrix
1 A
which is the only 3
3 rational solution to the YangBaxter equation. This discrepancy is
products of the Jack polynomials J
the generalized Kostka matrices:
A( )(pn)J B( )(pn). The basis is changed with the help of
KACBD(uj ) = hJ AB(uj )j(jJ C( )i
K CABD(uj ) = (hJ A( )j
At the rst level, we have
K(uj ) =
K (uj ) =
The Rmatrix in the factorized basis of the ordinary Jack polynomials is given by
Rord Jack = K
K =
( ) = B
K = B
whole expression becomes
Rord Jack =
where NAB
CD = NNABBA((uu12 uu21jj )) AC BD =
K is lower triangular.
(1+ ) 1 . In eq. (2.22) the term in the curly
an interpretation in the cohomology of the instanton moduli space [88, 89].
Parts of
product over [ 2 ; ]:
automorphism S
. Also, in [119] it was shown that the generalized Kostka matrices are in
decomposition (2.22) is a re ection of the decomposition (1.12), where
is accompanied
by multiplication with the diagonal matrix N .
(z)x+(z) (z). A pair
since c = 1 + 6(p
but the re ection matrix of the Liouville theory.
be described as eigenfunctions of a certain Hamiltonian H1
( ) sitting inside the a ne
Yan2 n;m=1
+ nmp(nk+)m
@p(nk)@p(mk)
2 n=1
( ) =
H(k1;k2) =
H((k)) =
di erence is that there are now r
1 Rmatrices, which permute the factors in the tensor
strands becomes a copy of the twostrand Rmatrix:
Ri(j ) =
where i;j permutes the ith and jth strands.
Ri(;j) = S
H1 = u1
[u1 1(z)+u2 2(z)]
+ u2exp@
the previous subsection.
Two strands.
As we described in the Introduction, the generalized Macdonald
polynoof Fock representations:
H1MfAB =
ABMfAB
AB = u1
Mf[];[1] = (1 t)(1 Q)p1+(1 t) 1
Mf[];[1] = (1 q) 1
4We again remind the reader of the change of convention Q ! Q 1 as compared to [86].
is the DIM coproduct and u denotes the horizontal Fock representation [41, 63,
However, all these Hamiltonians are contained in the expansion of H1.
already entrenched.
The generalized Macdonald polynomials at the rst level are given by4
Mf[1];[] = (1 t) 1
Mf[1];[] = (1 q)(1 Q)p1 (1 q) 1
X 1 qn zn(q=t)n=2 @
C (q; t) =
GAB(ujq; t) =
NAB(u1=u2jq; t)
NBA(u2=u1jq; t)
For the polynomials MAB (without tilde), the de nition of opposite polynomial is
polynomials is
It is given by
R = B
q(u1 u2)u2(qu1 tu2)
u2(qu1 tu2)
u1(qu2 tu1)
q(u1 u2)u2
versions of generalized Kostka matrices:
K =
Rord Mac = K
The resulting 2
2 block is the same as the block appearing in the standard trigonometric
Rmatrix:
Rord Mac = B u2(qu1 tu2)
u2(qu1 tu2) u2(qu1 tu2)
for the qdeformed Virasoro algebra.
More strands.
Again the discussion here is exactly parallel to the previous section, only
is given by
H1 = u1
(x+(z))
+u2exp@
+u3exp@
:::+uN exp@
+ p(nN)(q=t)(1 N)=2 z n
zn(q=t)n=2 @
zn(q=t)n
(1 tn=qn)p(n1)+p(n2)(q=t) n=2 z n
(1 tn=qn) p(n1)+(q=t) 1=2p(2) +p(n3)(q=t) n z n
n
(1 tn=qn) p(n1)+(q=t) 1=2p(2)+:::+(q=t)(2 N)n=2p(M 1)
n n
zn(q=t)(1 N)=2 @
satis ed, as shown in appendix A.1.1.
resolved conifold.
RT T relations in the toric diagram
placed on each external line:
TARBP (Q;u;z)=
and the generalized Macdonald polynomials are de ned as its eigenfunctions:
H1MfA1:::AN =
A1:::AN MfA1:::AN
A1:::AN =
are the intertwiners of DIM algebra [40, 41], jsA; ui denote the basis of
polynomials in the vertical Fock space (hence, the sign j) and
P denotes the matrix
element of
for the Macdonald polynomial MP on the vertical leg. Such building blocks
in the algebraic form, we have:
simpli cation that we use in this subsection and lift in the next one.
the automorphism S
. This corresponds to the change of basis in the tensor product of
moved two Rmatrices to the r.h.s. of eq. (3.3)):
similar to the general case, which requires one additional observation.
relations in any integrable system and can be drawn as follows:
: (3.3)
nomials is the transposed matrix of B(12):
M;;;;[1] M;;[1];; M[1];;;
M;;;;[1] M;;[1];; M[1];;;
where ki
(12) are the proportionality constants between M A(1B2)C and (R12)(MABC ).
In the same way, from the formula
(x0+) = R23(
M;;;;[1] M;;[1];; M[1];;;
M;;;;[1] M;;[1];; M[1];;;
the representation matrix of u1u2u3 (R23) is
B(23) := B@
k(12) = 1;
k(23) =
k(12) =
k(23) =
1. Now we consider the basis change from MABC to power sum symmetric
B(ij) := tA(u1; u2; u3) tB(ij) tA 1(u1; u2; u3):
e
Then Be(ij) have the following form
must not appear, one gets the equations b(212) = b(12) = 0. Similarly, b(123) = b(223) = 0. By
3
solving these equations, one can see that
In this way, one obtains an explicit expression of the RMatrix at level 1
; (A.14)
k(12) =
k(23) = 1:
B(13) = B
Indeed, one can check that they satisfy the YangBaxter equation
B(12)B(13)B(23) = Be(23)B(13)B(12):
e e e e e
Of course, the same equations for B(ij) also follow. Incidentally,
t2u2q qt + t2u2u3q qt + qtu22 + qtu1u2
2
2qtu1u2
2qtu2u3 + t2u1u2
u3(q t)p qt (tu1p qt tu2p qt qu1+tu2)(tu2 qu3)
q2t(u1 u2)(u1 u3)(u2 u3)
2qtu1u3 + qtu2u3
2qtu2u3 + t2u2u3
u3)(qu2
x =
y =
matrix of ( u1
u2 )(R) is the 2
2 matrix block at the lower right corner of tB(12)
u2 )(R) = B
q(u1 u2)(qu1 tu2)
deformed version
Hence, the deformed version of Rmatrix R
( ) is immediately obtained from the results
of the last paragraph. For example, for the representation u1
u2 and in the basis of
generalized Jack polynomials,
( ) =
J;;[1] = p(12)
J[1];; = p(11):
R(12) = BB
R(23) = BB
(u1 u3)(u3 u2)
(u1 u2)(u1 u3)(u2 u3)
( 1)( +u1 u2 1) 2 2 +u21+u22 2u1u2+1 A
+u1 u2+1)
2 2 +u22+u23 2u2u3+1
+u2 u3+1)
k2 = lim k(12) =
k3 = lim k(12) =
and the generalized Jack polynomials are
(u1 u2)(u1 u3)(u2 u3) (u1 u2)(u1 u3)(
In the basis of power sum symmetric functions,
( ) = B
A.2.1 (q; t)deformed version
M;;;;[2] M;;;;[1;1] M;;[1];[1] M[1];;;[1] M;;[2];; M;;[1;1];; M[1];[1];; M[2];;;; M[1;1];;;;
= A
M;0;;;[2] M;0;;;[1;1] M;0;[1];[1] M[01];;;[1] M;0;[2];;
sentation matrix of R. First of all, we choose B(12) at level 2 to be of the form
0 1 0 0 0 0 0 0 0 0 1
B 0 1 0 0 0 0 0 0 0 C
B b31 b32
B b61 b62 b63 b64
B b71 b72 b73 b74
b91 b92 b93 b94
0 0 0 0 0 C
obtained in this way satis es the YangBaxter equation.
Examples of the generalized Macdonald polynomials.
(q t)u3(tu3q3 tu2q2+tu3q2 u3q2 t2u3q+tu2)
qt(qt 1)(u2 u3)(qu3 u2)
(q+1)(q t)2(t 1)u32(q2u3 tu2)
p qt t2(qt 1)(u2 u3)(u1 qu3)(u2 qu3)
(q t)2p qt u23(qu3 t2u2)
q2t(tu1 u3)(u2 u3)(tu2 u3)
q(q+1)(q t)(t 1)u2
p qt t(qt 1)(u1 qu2)
(q 1)(q+1)(q t)(t 1)(t+1)u3
(qt 1)2(qu3 u2)
(q t)u3( qu2t3+u3t2+qu2t+q2u3t qu3t qu3)
qt(qt 1)(u2 u3)(tu2 u3)
(q 1)(q t)(t+1)u3
(q t)u3(qu3 tu2)(qu3 t2u2)
q2t(tu1 u3)(tu2 u3)(u3 u2)
(q t)u3(qu3 tu2)
qt(qu1 u3)(u3 u2)
(q t)u2(tu2q3 tu1q2+tu2q2 u2q2 t2u2q+tu1)
qt(qt 1)(u1 u2)(qu2 u1)
(q 1)(q+1)(q t)(t 1)(t+1)u3(qu3 tu2)(q2u3 tu2)
qt(qt 1)2(u2 u3)(qu3 u1)(qu3 u2)
C
C
(q t)u3(qu3 tu2)(qu3 t2u2)( qu1t3+u3t2+qu1t+q2u3t qu3t qu3) C
q2t2(qt 1)(u1 u3)(tu1 u3)(tu2 u3)(u3 u2) CC
C
C
a39 C
C
C
(qq(1q)t(q1)t()u(2t+1u)3u)3(t(uqu33 ut1u)2) CCC
C
(q 1)((qq+t1)1()q2(qt)u(2t 1u)1()t+1)u2 CCC
C
(q t)u2( qu1t3+u2t2+qu1t+q2u2t qu2t qu2) CC
qt(qt 1)(u1 u2)(tu1 u2) C
C
(q 1)(q t)p qt t(t+1)u2 CC
q(qt 1)(u1 tu2) CC
C
0 CC
C
A
(q 1)(q+1)(q t)(t 1)(t+1)(q
Qt2)( qt3 qt2+qQt2+t2+q2t qQ)
q(q Q)(Q 1)t(t Q)(qt 1)2
q2(Q 1)2t2(t Q)(qt 1)(Qt 1)
Q(q t)(Qtq3 Qq2+Qtq2 tq2 Qt2q+t)
(qQ t)(q2Q t)(qt 1)
(q 1)q(q+1)(Q
t)(t 1)t(t+1)
(q 1)(q+1)Q(q t)2(t 1)(t+1)(Qq2+Qq Qtq+tq t2 t)
(Q t)(qQ t)(q2Q t)(qt 1)2
(Q 1)Q(q t)t
(qQ t)(qQ t2)
Q(q t)( qt3+Qt2+qt+q2Qt qQt qQ) C
(qQ t)(qt 1)(qQ t2)
Q(q t)2(Qq2+Qtq2 t2q Qtq+tq t2) CC
q(qQ 1)(qQ t)t(qQ t2)
q(q Q)(qQ 1)t(t Q)(Qt 1)
q(q Q)(qQ 1)t(t Q)(Qt 1)
q2(Q 1)(qQ 1)2(Q t)t
qu23t + qu1u2t
u1u2t + qu1u3t
u1u3t qu2u3t + u2u3t + qu2u3
1)(q t)2(t + 1)u2u3 u2u3q2
u23q tu1u2q + u1u2q tu1u3q + u1u3q + tu2u3q
u2u3q + tu2u3
The representation matrix of R in the basis of generalized Macdonald polynomials
v53 =
v44 =
v34 =
v35 =
v55 =
r11 =
r41 =
(q 1)(t+1)(q t)p q q3Q+q2 Q3(t 1)+Q2 t2
Qt+t+1 +q5Qt Q3t2
Q2 t2+t+2 +Q t
3t+5 +2t 1
q4Qt Q3t+Q2 3t2
7t+2 +Q
13t2+11t 5
1)(qt 1)(qQ
q4Qt(6t2
4t+1)+q3t2 Q3 t2
4t+6 +Q2
5t2+11t 13 +Q 2t2
7t+3 +t +qQt3 Qt2+(Q
Qt4+q2t2 Q3(t 2)t+Q2
5t2+3t 1 +Q 2t2+t+1
q(q+1)Q(q t)(t 1)
Qq3+Q2tq3
2Q2t2q2+Qt2q2+Q2tq2+2Qtq2
2tq2+Q3t2q 3Q2t2q+t2q 2Q2tq+2Qtq+tq+Q2t3+Q2t2
Qq2 + Qtq2 + Qt2q + Qq + 2Qtq
2t2 + Qt
t) Q2q3 + Qq3 + Q2tq3
Qtq3 + Q2q2
Qq2 + Q2tq2
2Qt2q + 2t2q
2Qtq + 2t3
1)(qt 1)(qQ
2(qQ t)(q2Q t)(qQ t2)
(Q 1)Q(q t)p qt t(2Qq2 t2q+tq t2 t)
2(qQ t)(q2Q t)(qQ t2)
2(qQ t)(q2Q t)(qQ t2)
(Q 1)(q t)p qt t(Qq2+Qtq2+Qq Qtq 2t2)
2(qQ t)(q2Q t)(qQ t2)
1)Q(q t)(t 1)t2
1)Q(q t)(t 1)t
1)t(qt 1)(Q
1)t(qt 1)(Q
1)Q(q t)t2(t+1)
(q 1)q(Q
1)Q(q t)t(t+1)
Q(q t) Qq3
Q2q2+2Q2t2q2
2Qtq2+2tq2
Q2t3q+Qt3q+t3q
Q3t2q+3Q2t2q t2q+2Q2tq
q5Q2t+q4Q Q2 2t3+2t2
5t2+t +t2(t+1) +q3t Q4t2+Q3
7t2+t+2 +Q2t t3+13t2+11t+3
Qt t3+3t2+4t+2 +t4
Q3 2t3+4t2+3t+1 +Q2 3t3+11t2+13t+1 +Qt 2t3+t2
qQt4 Q2(t+1)+Q t
t2+2t+2
The representation matrix of R in the basis of power sum symmetric functions,
1)Q(q t)(t 1)t
q2(q+1)(Q
(q 1)q(Q
1)Q(q t)t(t+1)
q2(q+1)(Q 1)Q(q t)(t 1)
q(Q 1)(q t)(Qq2+Qtq2+Qq Qtq 2t2)
q(q+1)(Q 1)Q(q t)(t 1)t q(Q 1)Q(q t)(2Qq2 t2q+tq t2 t) C
Q2tq2 + Qtq2 2Qt2q + 2t2q + 2Qtq 2t3
Q2tq2 Q2t3q + 3Qt3q t3q + Qt2q Qt4
Qt3q + t3q
Qt2q + t2q 2Qtq + Qt3 + t3 Qt2 + t2
t)(q2Q t)(qQ
Qt2q + t2q 2Qtq + Qt3 + t3 + Qt2 t2
t)(qQ t2)
The representation matrix of R
( ) in the basis of generalized Jack polynomials
( ) =
s33 =
s44 =
(a 1)2a( +1)
2+ +2a+2)
(a 1)a2(a+1)( +1)
(a 1)a(a
( +1)( a+
+2)( +1)2
1) (a3+ a2 a2+ 2a 6 a+4a+ 3 3 2+2 )
+2)( +1)(a+ )
( +1)2( a+
2a3 + 3 2a2
a + 14 a
4a 3 + 7 3 + 16a 2 +
1)(a + 1)(a
a(a + 1)(a
4 + 2a3 + 5a2 + 8a + 4
a(a+1)(a+ )
a(a+1)2(a
1)(a3+ a2 a2+ 2a 6 a+4a+ 3 3 2+2 )
)( +1)(a+ )
1)(a3+ a2 a2+4 2a 6 a+a 2 2+3
(a 2 +1)(a
( +1)( a+
1)( a+2
1)(a3+ a2 a2+4 2a 6 a+a 2 2+3
(a 1)(a 2 +1)(a
+1)(a+ )
(a 3 +3)(
(a+1)(a 2 +1)(a
a(a 2 +1)(a
+1)( +1)
4 5 8a 4 8 4+5a2 3+20a 3+ 3 2a3 2 5a2 2 16a 2+7 2+a4
The representation matrix of R( ) in the basis of power sum symmetric functions
where a = u1
2a(a 2 +2)(
(a 2 +1)(a +1)(a
a3 2 a2+2a2+ 2a 3 a+a 2 3+7 2 7 +2
(a 2 +1)(a +1)(a +2)
( 1)(2a2 4 a+4a+2 2 5 +2)
( a+ 2)( a+ 1)( a+2 1)
(a 2 +1)(a
1) C C C C
Macdonald polynomials explicitly realize the spectral dual basis to j~u; ~ i in [142].
Action of x 1 on generalized Macdonald polynomials
We use the notation
X(1)(z) =
X Xn(1)z n = (uN1;):::;uN (x+(z)):
mials M~ are de ned to be eigenfunctions of X0(1) with the eigenvalues
e(1) =
1)t i= :
, in terms of the product of the monomial symmetric
m (N) . Their integral forms Mf~ are de ned by
G (j); (i) (uj =uijq; t) Y
M~ is renormalized as M~ = m~ +
functions m~ = m (1)
Mf~ = M~
k=1 (i;j)2 (k)
DIM algebra, that the action of (uN1;):::;uN
where T is the transposed of Young diagram
and we use the Nekrasov factor (2.34). It is
S(a) on the integral forms Mf~ are the same as
where (i; j) 2
R(~ ) respectively. We also use the notation
(`) are the coordinates of the box of the Young diagram
for the AFLT basis in this original sense.
j~j=j~ j 1
y6=x
y6=x
x y 1(q=t))
y x 1(q=t))
j~j=j~ j+1
c~~ ;~ Mf~ ;
i.e. X(11)M~ = P~ c~ ;~
( )
?
c~( +;~)(q; tju1; : : : ; uN ) =
and for the triple (`; i; j), we put
(`;i;j) = ( 1)N+`p `+21 t(N `)iq(` N+1)j QkNN=1``u`1+k ;
(+)
generators f1 and e1 in [142] respectively, i.e., those of x1 and x
are the spectral duals of x1+ and x 1
1+ in our notation, which
M~ , we can further conjecture that
5 for N = 1, for j~ j
3 for N = 2; 3 and for j~ j
with respect to X(11) has been also checked for the same sizes of ~ .
Higher Hamiltonians
For each integer k
Hk = [X(11); [X0(1);
According to [27], Hk are spectral dual to
k+ and consequently mutually commuting:
functions of all Hk, i.e. HkM~ = e(k)M~ . and Hk can be regarded as higher
Hamiltoni~
where for the partition
we de ne
e(k) =? (1
1z i=1
B+(z) =
i=1 (1
5 for N = 1, for j~ j
3 for N = 2, for j~ j
2 for N = 3 and for j~ j
N = 4 in the k
Acknowledgments
with A. Smirnov.
Open Access.
This article is distributed under the terms of the Creative Commons
any medium, provided the original author(s) and source are credited.
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