Explicit examples of DIM constraints for network matrix models
Received: May
Explicit examples of DIM constraints for network matrix models
Hidetoshi Awata 1 2 6 7 8 9 10 11
Hiroaki Kanno 1 2 4 6 7 8 9 10 11
Takuya Matsumoto 1 2 6 7 8 9 10 11
Andrei Mironov 0 1 2 3 5 7 8 9 10 11
Alexei Morozov 0 1 2 3 7 8 9 10 11
Andrey Morozov 0 1 2 3 7 8 9 10 11
g Yusuke Ohkubo 1 2 6 7 8 9 10 11
Yegor Zenkevich 1 2 7 8 9 10 11
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
DotsenkoFateev and ChernSimons matrix models, which describe Nekrasov functions for SYM theories in di erent dimensions, are all incorporated into network matrix models with the hidden DingIoharaMiki (DIM) symmetry. This lifting is especially simple for what we call balanced networks. Then, the Ward identities (known under the names of Virasoro/Wconstraints or loop equations or regularity condition for qqcharacters) are also promoted to the DIM level, where they all become corollaries of a single identity.
Conformal and W Symmetry; Supersymmetric gauge theory; Topological
1 Introduction 2
Basic example: theme with variations
The main theme
Variation I: matrix elements in the free eld theory
Variation II: generating functions
Variation III: DF model
Variation IV: multi eld case
Variation V: ChernSimons (CS) model
Variation VI: correlators with vertex operators
Variation VII: Nekrasov functions
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.1
3.2
3.3
3.4
3.5
3.6
3.7
Variation VIII: network model level. Network as a Feynman diagram
Variation IX: balanced network model
2.10 Variation X: qdeformation 2.11 Variation XI: q; t; : : :deformations 2.12 Variation XII: deformation to nonunit Virasoro central charge
3
DIM calculus for balanced network model
DIM algebra
Bosonization in the case of special slopes
Relation to topological vertex
Building screening charges and vertex operators
Network partition function
Examples of conformal blocks Compacti ed network and the a ne screening operator
4
5
6
The action of Virasoro and DIM(gl1)
Vertical action of DIM
Conclusion
A Properties of the DIM algebras and their limits
A.1 Constructing DIM(gl1) from W1+1 algebra
A.2 Elliptic DIM(gl1) algebra
A.3 Rank > 1: DIM(gln) = quantum toroidal algebra of type gln
A.4 A ne Yangian of gl1 [139]
{ 1 {
u sitting on the horizontal leg of the topological
vertex (denoted by the dashed line) is the same as its action on the product of two representations
 the \vertical" jv and \diagonal"
=
uv. b) Appropriate contraction of two intertwiners is also
an intertwiner. This gives the vertex operator of the corresponding conformal eld theory with
deformed Virasoro symmetry, corresponding to a single vertical brane in gure 2.
1
Introduction
Nekrasov functions, describing instanton corrections in supersymmetric YangMills
theories [1]{[11], and AGT related conformal blocks [12{16] possess rich symmetries that can
be separated into large and in nitesimal. The former describe dualities between di
erent models, while the latter de ne equations on the partition functions in each particular
case. They are also known as \Virasoro constraints" [17, 18] for associated conformal or
DotsenkoFateev (DF) matrix models [19{30], which are further promoted to network
matrix models [31{34], looking like convolutions of re ned topological vertices [35{37] and
possessing direct topological string interpretation.
As conjectured in a number of papers throughout recent years [38]{[49] and
recently summarized in [50], in full generality the symmetry underlying the AGT
correspondence [51{53], is the DingIoharaMiki algebra (DIM) [54]{[69], in particular, the
innitesimal Ward identities are controlled by DIM from which the (deformed) Virasoro and
WK emerge as subalgebras in particular representations. In other words, the full symmetry
of the SeibergWitten theory seems to be the Pagoda triplea ne elliptic DIM algebra (not
yet fully studied and even de ned), and particular models (brane patterns or CalabiYau
toric varieties labeled by integrable systems a la [3, 4]) are associated with its particular
representations. The ordinary DF matrix models arise when one speci es \vertical" and
\horizontal" directions, then convolutions of topological vertices can be split into vertex
operators and screening charges, and the DIM algebra constraints can be attributed in the
usual way [70{79] to commutativity of screening charges with the action of the algebra in
the given representation. Dualities are associated with the change of the vertical/horizontal
splitting, or, more general, with the choice of the section, where the algebra acts [80{82].
All this is illustrated in pictures 1 and 2, which we borrowed from [50], and our purpose
in this paper is to provide very explicit examples of how these pictures are converted into
formulas. A great deal of these formulas already appeared in the literature. Putting them
together, we hope to illustrate their general origin and better formulate the remaining open
problems.
{ 2 {
Qm;1
QF;1
Qm;2
mf;2
mf;1
mf;2+mf;1
2
D4
D4
QF;2
QB;1
NS5
NS5
QB;2
mbif
a(2)
a(
1
)
1
NS5
Qm;3
QF;3
Qm;4
mf;2+mf;1
2
mf;2
mf;1
2 2 SU (2)
SU (2) 2 2
c)
i are exponentiated complexi ed gauge couplings, a(a)
are Coulomb moduli and ma are the hypermultiplet masses. b) The toric diagram of the CalabiYau
threefold, corresponding to the 5d gauge theory with the same matter content. Edges represent
twocycles with complexi ed Kahler parameters Qi, which play the same role as the distances between
the branes in a). c) The quiver encoding the matter content of the gauge theory. SU(2) gauge
groups live on each node and bifundamental matter on each edge. The squares represent pairs of
(anti)fundamental matter hypermultiplets.
The main scheme could be formulated as follows:
To build a functor
!
rankr Lie algebra G
quantized doublecenter doubleloop DIM(G);
(1.1)
perhaps, q123:::dependent and elliptic
To obtain a nonlinear Sugawara construction of stress tensor and other symmetry
generators from a comultiplication
DIM .
To clarify the interplay between two \orthogonal" (\horizontal" and \vertical")
comultipilcations.
To apply the functor (1.1) to centralextended loop algebras G, starting from G =
(\gl(
1
), to obtain triplea ne Pagoda DIM algebras. One of the immediate problems
{ 3 {
is that the known construction of DIM(G) for nona ne glN algebras [54, 83{85]
already involves the a
something more sophisticated.
ne Dynkin diagrams, thus, for an a ne gdlN one can need
An additional light on the problem can be shed by comparative analysis of DIM(gl2),
DIM(gl3), DIM(so5), DIM(g2) and DIM(gcl1), rst four of them being explicitly
constructed, and by studying their various limits including the one to the a ne Yangian
and further to the standard conformal algebras (coset constructions of conformal eld
theories, [86]).
Actually, the rst three issues are actively studied by various authors (and there has
been already achieved a serious progress), and we do not achieve too much in the two last
challenging directions in the present paper, which can be considered as an introduction to
the problem. What we actually do, is search for a q; tdeformed network analogue of the
CFT Ward identity [12]
HJEP07(216)3
*
a
Y V^ a (za) T^+(z) Q^r
+
where < : : : > denotes the matrix element < vacj : : : jvac > between two vacua of operators
in the xed chronological order and in the chiral sector [87, 88]. Here V (z) is a primary
eld (vertex operator) in the free
eld c = 1 CFT, T (z) is its stressenergy tensor and
Q is the corresponding screening charge [70{72], which is the integral Q = Hx S(x) of the
screening current S(x).
The order of operators in (1.2) means that in the conformal correlator
**
Y V a (za)T+(z)Qr
a
++
(where << : : : >> denotes the chiral part of the CFT correlator) all jzaj > jzj and jzj > jxij,
where xi's lies on the integration contours of the screening currents.
The Ward identity (1.2) can be manifestly written as
z
2
a b
za)(z
r
i=1
zb)
S(xi)
+ X
++
a;i (z
= Pol(z)
a
za)(z
xi)
+
r
X
and the notation Pol(z) means a power series, i.e. any positive powers of z are allowed.
The underlined terms just contribute to Pol(z) (since jzaj > jzj) and can be omitted giving
(1.2)
(1.3)
(1.4)
nally
z
2
X
eld theory correlators, it is dictated by operator expansions and is especially simple
because a free
eld formalism is available for conformal theories. The rst one is actually
about matrix elements, and the di erence is that it depends on the ordering of operators,
while correlators do not. Another way to say this is that the projected stress tensor T+(z)
does not have a simple operator product expansion (OPE) with other operators, the
projection is a nonlocal operation and actually depends on the position: if T+(z) was placed
to the left of vertex operators V (za), the matrix element would no longer vanish. At the
same time, in this case the underlined terms in (1.4) also contribute (since jzj > jzaj), and
they exactly cancel nonzero matrix element leading to the same Ward identity (1.5).
These are trivial remarks for the oldfashioned eld theory, where the Ward identities
were discovered and treated as sophisticated recurrence relations between Feynman
diagrams, but in modern CFT we got used to the formalism based on the operator product
expansion and moving the integration contours, which provides a shortcut for the derivations.
Unfortunately, in the network models, only the operator approach is currently available,
and this is the reason why we need to develop the formalism from this starting point.
Still, some elements of the free eld formalism are already worked out in particular
representations of DIM, and for a special class of balanced network models, drawn as a
set of horizontal lines with vertical segments in between, see gure 3, a), one has a direct
counterpart of (1.2). In (extremely) condensed notation it looks like
*
Y
a
a [za]
a [za] T^+(z; uj )
Y
b
X
!+
and involves operators like
Y
I
I [zI ] Y
J
J [zJ ]
!
0
I;J
where
[ ; z]n
sign(n)
q i 1=2t1=2 iz
n
are the Miwa variables associated with the Young diagram
, and the DrinfeldSokolov
operator (generalized stress energy tensor = Miura transformation from
i(z))
T^ (z; uj ) = z1=2 log!
z 1=2 log!
=
K k
K
X
k=1
ui i(z!2(i 1)) :
k
Y uia : ia (z!2(a 1)) :
X
i
K
: Y
i=1
X
i1<:::<iM a=1
{ 5 {
de ning numerous ows, is a linear combination of all W
are also made from the annihilation and creation operators ^ n, ! = pq=t and T^ (z; uj )
K. Here
i(z)
(m) with m
depends on an additional parameter
rameters of DIM representation ui.
generating di erent W
(m)(z) and on spectral
pa(1.6)
(1.7)
(1.8)
(1.9)
(
1
)
1
z(2)
2
u1
u2
u3
(
1
)
3
z(2)
4
z(
1
)
2
c)
z(
1
)
4
(
1
)
4
(2)
4
y4
u1
to the left, through external vertical legs, which do not commute with T^ (z). Moreover,
now we can also consider deformations of the section which do not preserve verticality, like
the dotted one in
gure 3, c), and everything can still be calculated. This should provide
a qualitatively new insight into spectral dualities [89{94] associated with global rotations
of the network graph.
Nonbalanced networks, where the rightmost and leftmost branes in gure 4 are tilted
and the number of operators
di ers from that of
, can be considered as certain limits of
the balanced ones, but these limits are nontrivial and singular when, say, q; t
the point of view of representation theory these limits should have independent description,
making use of more complicated intertwiners. A full edged free eld description for them
comparable to the one in [95{97] for ordinary a ne case still needs to be worked out.
! 1. From
{ 6 {
u1
u2
u3
Restriction to the balanced networks is a great technical simpli cation, but it requires
a somewhat lengthy comments on what this means and whether this really restricts the set
of handy physical models.
DIM is a quantization of double loop (double a ne) algebras, and the existing free
eld formalism, which we are going to expose and exploit in the present paper, explicitly
breaks the symmetry between the two loops. Bosonized/fermionized are only the Chevalley
generators, in the case of DIM there are many, still they depend on one of the two loop
parameters, while the other loop is associated with their multiple commutators and is
described very di erently: in terms of Young diagrams parameterizing states in the Fock
space. This breaks the symmetry of the DIM algebra: the SL(2; Z)automorphisms acting
on the square lattice of the generators and introduces asymmetry between horizontal and
vertical directions in the planar graphs which are used to de ne the network models,
and makes the spectral dualities interchanging these two directions highly nontrivial. In
particular, allowed networks look like in nite \horizontal" lines, connected by vertical
segments, see gure 4, a), and not vice versa. We call these lines horizontal, though they
can have varying slopes, however, they have a nontrivial projection on the horizontal axis,
i.e. are strictly nonvertical. In the original brane theory interpretation these horizontal
lines depict the Dbranes, while vertical are the N S branes, from this point of view our
description applies only to the conformal models (Nf = 2Nc) with de nite Nc = M = #
of horizontal lines. Quiver models
SU(Ni) with di erent Ni can seem excluded, but in
fact they appear after application of the spectral duality: a 90 rotation of the graph, see
Fig, 4, b). After this rotation, the in nite horizontal lines get associated with the in nite
N S branes, while the vertical segments with Dbranes between them. This pattern looks
more relevant from the gauge theory point of view, but we emphasize that our free elds live
on the in nite horizontal lines, the threevalent vertices (the DIM algebra intertwiners
and
, also known as topological vertices) act as operators in the Fock spaces horizontally,
while the third vertical edge carries a Youngdiagram label, not converted into operator
language. In result these vertices can look like ? or >, but not like ` or a.
All these restrictions can be lifted by switching from Fock to MacMahon modules,
which are representations of DIM spanned by 3d partitions, but such a description is only
combinatorial so far, no generalization to the full edged doubleloop free eld formalism
is available yet. This is what makes tedious the consideration of dotted sections in
gure 3, c). We brie y touch this issue at the very end of this text, but detailed presentation
is postponed to the future work. Our main purpose here is to describe the powerful free
eld formalism for the balanced network as a straightforward generalization of that for
the ordinary conformal theories, and explain how the DIM algebra becomes the
symmetry of generic Nekrasov functions generalizing the Virasoro/W symmetry of the ordinary
conformal blocks and DotsenkoFateev matrix models.
In the next section 2, we explain how the elementary theory of a harmonic oscillator
can be straightforwardly developed and lifted to description of generic networks, i.e. of
generic Nekrasov functions. In section 3, in the simplest examples we demonstrate the
actual formalism in full detail. It is important that most complications come from
sophisticated notation, which are largely no more than a change of variables (normalization
{ 7 {
M
L
M + 1
M + 2
M + 3
L 1
L 2
L 3
a)
Bending of the \horizontal" lines due to tension from the vertical segments is re ected in their
slopes marked above them. b) Spectral duality acts by rotating the diagram a). After rotation
one can identify the conventional HananyWitten (or brane web/geometric engineering) setup with
NS5 and D5branes ().
of creation and annihilation operators). The really big change comes in section 4, when
one looks at the symmetry : it is indeed essentially deformed. But this deformation
actually simpli es things, reducing all the symmetries to the action of the DIM generators,
while the Sugawara construction of Virasoro and Woperators and of their sophisticated
qdeformations is no more than the simple comultiplication rule. At last, at section 5 we
brie y discuss the spectral duality action on symmetry generators. Finally, the appendix
contains further details about various DIM algebras and their representations. At present
stage of development, di erent parameters are treated as providing di erent algebras, but
further studies can promote them to parameters of di erent representations of a single
uni ed algebra (like the triplea ne elliptic Pagoda DIM algebra anticipated in [50]).
Notation. Throughout the text we use the notation
(1.10)
2
Basic example: theme with variations
We assume some familiarity with [50] and do not repeat the general logic, leading to Ward
identities like (1.2) in DF and network matrix models.
2.0
The main theme
Screening charge Q^, acting on the Fock space F
= n
Pols( n)
o
e T0 , is
Q^ =
I
S^(x)dx = resx=0 S^(x);
!
r q
t
{ 8 {
it gives
where the calculation involves
1
2
X
m1;m2
m1 m2
I I
and
i.e. the power of Q^ acts as a character of rectangular Young diagram. This is the old result
by [98{104]. The rectangular diagrams arise from the Cauchy formula
r
Y exp
i=1
X
n>0
nxin !
n
= exp
X 1
n>0
n n
r
X x
n
i
i=1
!
=
X
f g
[~x]
with a sum over all Young diagrams
(actually, with no more than r lines) after the
Vandermonde projection
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
(2.8)
(2.9)
HJEP07(216)3
S^(x) = : ep2 ^(x) : = exp
nxn !
n

{z
Pn xn nf g
}
n>0
where nf g are the characters of symmetric representations [n] of sl algebras (the Schur
polynomials in this particular case). Applied to a highestweight state (i.e. the one
annihilated by all negative modes a^ n =
Residue is nonvanishing, because x2@0 converts jm + 1 > into x m 1. Similarly
which is a direct generalization of (2.5).
Since the screening charge commutes
with the Virasoro generators
^
Ln =
X(k + n) k
k
+ X k(n
k)
(2.10)
Yr I
i=1
dxi
xm+r
i
(~x)2
[~x]
;[mr]
[L^n; Q^] = 0
{ 9 {
one has
In application to (2.6), this gives while the action of gives just the size of the Young diagram:
L^nQ^r m + r
= Q^rL^n m + r
= 0 for n > 0
n > 0
L^0 =
X k k
k
[mr]
In the Miwa parametrization n = Pi Xin, this turns into the statement about the Calogero
eigenfunctions. Also Qr m + r
E are singular vectors in Verma modules and (2.12) can
be considered as the simplest version of BPZ equations for correlators with degenerate
elds, [12].
Equation (2.12) provides a simple example of the Ward identity for the state Q^rjm +
r >, which can be promoted to identity for the matrix element in conformal eld theory,
i.e. in the abstract Fock module and corresponding Sugawara energymomentum tensor
(which we denote by Gothic letters),
[mr] =< m
insertion of the intertwining operator, see below. We are now ready to formulate the main
rjC^T (z)Q^rjm + r > by additional
theme of the present paper:
A trivial symmetry property (2.9) gives rise to a nontrivial equation for the matrix
element (2.12), provided one can calculate (2.6).
In what follows we extend this simple example to matrix elements of an arbitrary
network of intertwining operators, what allows to reveal in a rather explicit form the
hidden DIM symmetry of the SeibergWitten/Nekrasov theory.
We continue in this section with variations on the main theme, developing it at
conceptual level. Next sections will describe technical details of the story.
2.1
Variation I: matrix elements in the free eld theory
mode operators a^ n = n=p2, n > 0, i.e. contains Q
Actually, in theory of free eld (z), the bra vacuum state is annihilated by all the negative
n>0 (a n) in holomorphic
represen< m
rj Qr jm + r >: this matrix element would not depend on
to introduce a special intertwining operator
tation. Thus, one can not simply convert (2.6) into a statement that [mr]f g is equal to
at all. The way out is
which converts the bra vacuum into the coherent state
C^fpg = exp
X pna^n !
n>0
n
hmj
! hmj C^fpg
(2.11)
(2.12)
(2.13)
(2.14)
(2.15)
Symmetry (2.9) actually holds for all n 2 Z.
We will also need a \current"
T^(z) =
X
n2Z
^
Ln
zn+2
L^ n =
k
+
[L^n; L^m] = (n
m)L^n+m +
with
n
J^ n = p ;
2
J^n = p
2n
with the property
This allows us to rewrite (2.6) as
hmj C^fpg a^ n = pn hmj C^fpg
Among many complications as compared with (2.6), there is p2, which re ects the fact
that the character is extracted here from the screening charge in a single eld (\current")
section 3.2 of [72] and section 2.6 below) which involves two scalar elds, and p
realization. A more adequate kind of formulas arise within the fermionic realization (see
2 is a result
of basis rotation to their symmetric combination.
Variation II: generating functions
We can make from particular Virasoro generators L^n a single operator (stress tensor)
(2.17)
(2.18)
(2.19)
(2.20)
(2.21)
(2.22)
(2.23)
(2.24)
Positive and zero modes with n 0 are given by (2.10) and (2.13) respectively, negative
X
n2Z
J^0 = p
2
^
Jn
zn+1
[J^n; J^m] = n n+m;0
[L^n; J^m] =
mJn+m
T^(z) = : 1 J^(z)2 :
2
The two operators are related by the Sugawara relation
where normal ordering puts all pderivatives to the right of all p's (in each term of the
formal series).
The generating functions satisfy the commutation relations
[J^(z); J^(w)] = 0(w=z)
(x) =
X x
n
In terms of generating functions, the Ward identity (2.12), i.e. the corollary of
symmetry (2.9) becomes
or, in other words, a regularity constraint
z2 T^(z)
z2 T^(z)
m
r
p
2
zJ^(z)
m
p
2
r
zJ^(z)
This will be the typical form of Ward identities (regularity condition for qqcharacters) for
network Nekrasov functions Z generalizing the simple character [mr].
2.3
Variation III: DF model
Expressions (2.6) and (2.1) together imply the integral representation of the matrix element
[mr]f g = D
m
r C^f n= 2g Q^r m + r
E
=
Gf jxg = exp
which is the archetypical example of DF or conformal matrix model [70{72, 102{104].
Ward identity (2.27), which is a trivial corollary of commutativity (2.9) looks now like
a notsoobvious set of integral identities:
z2 T^(z)
=
*
X
k;i z
m
r
p
2
kxik+1
xi
zJ^(z)
D
1
E
+
r
X
DFm;r
xixj
xi)(z
Actually there are two standard ways to derive the l.h.s.:
Y(xi
i<j
xj )2 =
D
1
E
DFm;r
n nf g
xj )
(m
r) X
i
xi
(z
xi)
+
+
DFm;r
FFm
(2.25)
(2.26)
(2.27)
(2.28)
(2.29)
(2.30)
(
1
) by using bosonization, which is the simplest version of free eld (FF) formalism, i.e.
the Wick rule for decomposition of correlators into pair ones,
T^(z) 1
D
E
DFm;r
=
*
C^( n=p2)
T (z)
Yr I
ep2'(xi)dxi
where the index m refers to a special way of handling the zero mode of ' and '(z)
refers to the scalar eld acting in the abstract Fock module, and
(2) by a change of integration variables xi = xin+1 in the multiple integral (2.28), [17,
18, 73{79]: in this case we get the identities in a slightly di erent form:
* r
X 2 i
i<j
xn+1
xi
xn+1
j
xj
+ X
i;k
r
i
kxik+n + X(n + 1
m
r)xin
= 0;
n > 0
+
DFm;r
(2.31)
In this paper we actually need an outdated and tedious third way:
(
3
) the operator formalism based on an explicit calculation of commutators arising when
the stress tensor is carried from the left to the right through the screening operators:
this is what we are now doing, starting from section 2.0 and this is what in the
simplest case brought us to the Ward identity in the form (2.29).
Both the OPEbased and changeofintegrationvariables/totalderivative approaches
should also work in the network model context, but they still need to be developed.
2.4
Variation IV: multi eld case
The network matrix models can be considered as associated with networks of branes
(branewebs [105{110]), which being projected onto the 4 5 plane look like segments with di erent
slopes. From the point of view of YangMills theories, interpretation of the di erent slopes
is di erent. Surprisingly or not, it is also di erent at the present level of understanding of
the DIM symmetry. Throughout the section, we distinguish only between the horizontal
and vertical segments, while intermediate slopes appear in this section only in sections 2.8
and 2.9. Our next variations introduce and describe the associated notions.
The
rst one is horizontal branes. These are associated with di erent free elds.
Generalization of the DF model to K eld case provides WK constraints for models with
K horizontal branes. An additional procedure can be applied to separate a
\centerofmass" eld: this explains why in the previous subsection 2.3 the number of elds was one
rather than two.
The multi eld conformal model [72] is de ned as
*
m~
~r
Y ^
a=1
Caf n(a)=p2g Q^raa m~ + ~r
= h1iDFm~;~r
+
(2.32)
where the screening charges now carry additional indices labeled by K
1 simple roots
~ a of slK . They are actually associated with segments of the vertical branes ending on
two adjacent horizontal branes, gure 3, a), in accordance with the decomposition ~ a =
~ea+1 ~ea. In other words, a better labeling of Q is by pairs of indices ab, each corresponding
to a particular horizontal (in fact, any nonvertical, see section 2.5) brane.1 Now the matrix
model partition function depends on K sets of times, one of which is associated with the
\center of mass" and actually decouples in the DF model (2.28), thus it was actually
suppressed in that formula. However, this is not always true: the decoupling will not take
1To avoid possible confusion, note that in [50] an \orthogonal" labeling rule was used, treating horizontal
edges of the network as segments between the vertical ones.
place already in the ChernSimons deformation of (2.28) in section 2.5, and all the M sets
of times will be relevant in generic DIM considerations. This phenomenon is familiar in the
CFT approach to Nekrasov functions, where relevant is the Heis + V irasoro symmetry
and its generalizations rather than the V irasoro alone. This is also re ected in appearance
of \100 in the popular notation W1+1.
Algebraically, the multi eld generalization is controlled by the comultiplication
DIM ,
which builds all the symmetry generators from a single element of DIM:
current algebra
Virasoro
#
#
#
#
W3
: : :
WK
#
: : :
Z
(rJ )2 + : : :
K
a=1
X Ja = 0
This comultiplication adds new scalar elds, and nonlinearity of the usual 4d Sugawara
formulas is mostly due to elimination of the centerofmass eld; what makes this
possible is the exponential form of symmetry generators beyond 4d. Somewhat symbolically,
the Sugawara formulas for the stress tensor (at the second level of DIM) arise from the
expansion of characters (in fact, qcharacters)
underlined in the rst two lines are terms appearing due to the centerofmass reduction
Cab is the Cartan matrix for slK , which the K = 1 limit describes a di erence Laplace
2
operator r . Other Woperators made from higher powers of J arise in the same way at
K = 2 :
K = 3 :
: : :
K :
: : :
Tsl2 =
Tsl3 =
1
2
1
3
eJ + e J
= 1 +
2
1 J 2 + : : :
eJ1 + eJ2 J1 + e J2
= 1 + (J12
J1J2 + J22) + : : :
1
3
1
K!
X
a;b=1
TslK = 1 +
CabJaJb + : : :
K = 1 :
Tsl1 = 1 + const
(2.33)
(2.34)
(2.35)
higher levels of DIM, i.e. after several applications of the comultiplication
DIM , e.g. at
K = 3 the second generator of the W3algebra is
so that the standard W3generator is a di erence
dxiGf jxige (log xi)2 Y(xi
xj )2
i<j
The parameter
controls the brane slope, it vanishes for the horizontal branes, while for the
vertical ones it becomes in nite and the story gets a separate twist, see section 2.6 below.
From the point of view of DIM symmetry of the network model, the Virasoro/Ward
constraints should look similar with and without these logarithmic terms, in the sense that
they should be always dictated by the Wick theorem hidden in the algebraic structures
of DIM. There is, however, a crucial di erence: in this case, the U(
1
)mode should not
decouple for nontrivial slopes, and two sets of times survive (see section 2.6). This is
re ected in the fact that one needs to consider Gf jxg depending on n>0 and n<0 in (2.38),
Gf jxg = exp
1
X
n2Z
nxn !
n
in order to construct the Ward identities. Then, a counterpart of (2.31) for (2.38) looks
somewhat di erent [122, 123, 125{127]:
i
i
*
(n
r + 1) Xr xin + Xr xin (log (xijq))0 + 2 X xin+1
i<j
xi
xn+1
j
xj
+ X
k;i
kxin+k
+
= 0 (2.40)
where q = exp( 21 ) and (xjq) = P1
Variation VI: correlators with vertex operators
The vertical branes are associated with insertions of vertex operators into the DF and CS
models. A particular instance of the vertex operator is the screening current. As already
mentioned in section 2.4, screening charges are segments of vertical branes between the
two neighbour horizontal ones, and they can be considered as contractions of two
vertex operators attached to these two branes. However, the relevant operators are special,
Variation V: ChernSimons (CS) model
The brane slopes show up in a specially designed 4d limit as additional squarelogarithmic
terms (log xi)2 in the action of the DF matrix model (2.28), giving rise to what is often
called the CS matrix model [111{124]:
namely, they are e
with
=
1: a kind of \fermion vertices" (in fact, intertwining
operators)
= e
. Accordingly, the screening charges should be associated with bilinears
a+(x) b (x), \nonlocal" in the vertical direction:
Q^ab =
I
a+(x) b (x)dx
(2.41)
exponentials rather than @ like currents, as well as the emergency of peculiar p
This nonlocality explains, among other things, why the screening currents are \naturally"
2 in (2.1)
coming from the 45 rotation of the basis 1
; 2 into 1
instead of x and the screening charge is a convolution of these indices (see s.3.2
of [72] for details). Interchanging of + and
labels changes the screening charge to
the dual one (in algebraic terms, this corresponds to using instead of a positive root the
corresponding negative one): as usual in conformal matrix models [70{72], the use of dual
charges is unnecessary. In fact, one can connect every screening charge with a simple root:
one can associate with each end of leg a a basis vector ~ea, then, the screening charge Qa;a+1
corresponds to a simple root ~ a = ~ea+1
~ea.
In operator formalism the correlator of vertex operators is just a matrix element of an
ordinary product of linear operators. A generic vertex operator is constructed from the
primary eld V (x) and is labeled by the Young diagram :
^
V
= L^
V (x)
V^ (x + z) = ezL^ 1 V^ (x)e zL^1
with L^
= Qi L^
i . The conjugation with L^ 1 moves it to an arbitrary point z:
However, in CFT the positions of operators does not matter: they can be considered as
located at points in the complex zplane, or, more generally, on a Riemann surface (in the
latter case same traces need to be taken in operator formalism).
Still, location of the stresstensor insertion does matter: in the Riemann surface picture,
it is associated with a choice of a contour encircling the vertex operator insertions, and
correlator depends on the homology class of this contour. Changing the class is equivalent to
commutation of T (z) with the vertex operator, which is read o the commutation relations
[Ln; V (x)] = xn+1V 0 (x) +
2(n + 1)xnV (x)
(2.44)
and those of the Virasoro algebra. This is what we did in the derivation of (1.5) placing
the stresstensor to the left, and to the right of vertex operators.
Centralchargepreserving comultiplication
is provided by the MooreSeiberg comultiplication
MS.
The action of Virasoro algebra
MS , which is given by the ordinary
Leibnitz rule on the negative modes T , but the positive modes act di erently:
1
X zn+1 k
k=0
n + 1
k
!
MS (Ln)R1
R2 =
Lk 1R1
R2 + R1
LnR2
(2.45)
(2.42)
(2.43)
This comultiplication can be read o the conformal Ward identities, [128] and celebrates
two important properties:
It is parameterized by an arbitrary parameter z,
it does not change the central charge, in contrast with the comultiplication in the
DIM algebra that we use below.
We de ne the Nekrasov function as partition function of the DF/CS network matrix model
depending on parameters ~ i, zi and Na, associated respectively with external legs (assumed
vertical), horizontal and vertical edges of the graph : schematically,
Z
=
Y V^~ i (zi) exp
i=1
K 1
X Q^a;a+1
a=1
!+
DFNa
(2.46)
and this partition function describes the A1quiver with obvious modi cations for more
sophisticated quivers, [50] (changing the number of vertex operators and adding more
screening charges that di er by the choice of the integration contours). The right numbers
of screening charges are automatically selected from the series expansion of the exponential
by zero mode conditions.
On the gauge theory side, this data describes the theory with the gauge group SU(K)
and 2K fundamental matter hypermultiplets (i.e. zero
function).
Here the numbers
Na are the Coulomb moduli, the hypermultiplet masses are parameterized by the vertex
operator parameters ~ i and the positions of vertices (rather their doubleratio) control the
instanton expansion in the gauge theory. Note that this theory is characterized by zero
function, all other cases are obtained by evident degeneration. The case of adjoint matter
hypermultiplets is described by the elliptic DIM algebras2 [140{143] and is out of scope of
the present paper. The other quiver theories, say Ak are described, on the physical side,
by a product of k gauge groups: Qik SU(ni) with i = 2ni
ni 1
ni+1 bifundamental
hypermultiplets for each i transforming under the gauge groups SU(ni) and SU(ni+1).
There are also 0 and k fundamental hypermultiplets that are transformed under SU(n1)
or SU(nk) (we put n0 = nk+1 = 0). These theories have also zero
functions, other
cases can be obtained by a degeneration of hypermultiplet masses. Note that the Nekrasov
network partition functions typically contain additional singlet elds, which corresponds to
U(K) instead of SU(K) group. The contribution of this singlet factorizes out and reduces
just to a simple multiplier in the Nekrasov function.
While exponentiation of bosonized screenings Q = H e can look somewhat arti cial,
the same procedure is very natural in the fermionic version Q = H
+
: this adds
bilinear terms to the free fermion action, i.e. leaves it quadratic. This is the reason for
integrability, and in bosonized version this is re ected in integrable properties of Toda like
systems with exponential actions.
Exponentiation of fermionic screenings makes a new interesting twist after the
qdeformation in section 2.10, see eq. (2.50) below.
2By DIM algebras in this paper we mean both DIM and its limits like a ne Yangian [49, 129{139].
n = v1
v2 = 0
Q
a)
K
v2 = 0. The \horizontal" line with spectral parameters u and v is shown
in blue. The length of the intermediate edge is determined by the ratio of the spectral parameters
on the adjacent edges, Q = uv
. b) An example of a 3d Young diagram which contributes to the
vertex C[1];[2];[2;1]. The vertex C[1];[2];[2;1] is given by the weighted sum over all 3d Young diagrams
with three
xed asymptotics shown in blue.
2.8
Variation VIII: network model level. Network as a Feynman diagram
Network model is de ned for a planar 3valent graph
with edges parameterized by slopes
and lengths. Slopes are given by pairs of numbers (X1; X2), see
gure 5, and lengths by
parameters Q. The 2component vectors X~ are conserved at the vertices of : X~ v 0 + X~ v 00 +
X~ v 000 = 0 at each vertex v; this is a stability condition for the braneweb. The graph
with
this structure describes a la [3, 4] the tropical spectral curve of the underlying integrable
system, but for our purposes it can be considered just as a Feynman diagram with cubic
vertices and momenta QX~ on the edges, associated with some e ective ChernSimonstype
eld theory. Expressions Z for this Feynman diagram (Nekrasov partition function or
genagators
eralized conformal block) is build by convolution of vertices CIJK (X~ 0; X~ 00; X~ 000jq) and
prop
IJ (Q), where indices I; J; K are Young diagrams, and CIJK are, in turn, \(re ned)
topological vertices" [35{37] given [144{148] by sums over 3d (plane) partitions with three
boundary conditions described by three ordinary Young diagrams I; J; K, see gure 5, b).
In the generic network matrix model, the exponentials of screening charges no longer
turn into exponential of \fermions": it produces an elementary 3valent vertex (=re ned
topological vertex) providing the true DIM intertwiner. Automatic is now not only
adjustment of the number of screenings, but also matching between their
+ and
constituents.
Screening charges are substituted by vertical lines between pairs of horizontal brains,
H exp(~ ij ~), involving two free elds associated with the corresponding branes.
Slopes of the horizontal branes enter the matrix model description through (log xi)2
terms in the action, see (3.45) in s.3. The coe cient is made out of the skew product
(see gure 5 a))
Xv1 ^ X~ v2
~
(2.47)
where X~ v1 , X~ v2 are associated with the external horizontal lines, one incoming, the
other one outgoing. In the case with several horizontal lines, see e.g. (3.42), one has
to consider X~ v1 , X~ v2 for di erent horizontal lines, and the answer in this case does
not depend on the concrete choice of these lines.
We described in this subsection a generic network model. One can consider its
particular case: the model that gives rise to the quiver gauge theory (as described in the previous
subsection). In this case (for any quiver gauge theory), one can construct a Ktheoretic
version of the Nekrasov functions, Z , [149{152]. They coincides [36, 37, 153, 154] with
the re ned partition functions in the corresponding geometry, which can be constructed
via the re ned topological vertex.
Another possibility is to consider the quiver theories with zero
functions (so that
all other can be obtained via various limiting procedures from these) and all gauge groups
coinciding, ni = n 8i. These theories are associated with so called balanced networks
and can be immediately described within the representation theory of DIM algebras, and
the requirement of all gauge groups having the same rank is implied by a possibility of
immediate extension of DIM to the elliptic DIM: this latter describes the quiver gauge
theories with adjoint matter, where the condition ni = n is inevitable. We discuss the
issue of balanced networks in the next subsection.
2.9
Variation IX: balanced network model
As usual, the q; tdeformation leads to overloaded formulas, but in fact it drastically
simplies them by providing a very clear and transparent interpretations and unifying seemingly
di erent ingredients. Namely, everything gets controlled by the DIM symmetry: the edges
of graph carry DIM representations, the topological vertices C become their intertwiners,
and symmetries (stresstensor and its Wcounterparts) are just the generators of DIM
acting in tensor products of representations and thus de ned by powers of the comultiplication
DIM (which is di erent from
MS).
An exhaustive description of the network models depends on development of
representation theory for the double a ne algebra DIM, and it is not yet brought to the generality
level of [95{97] for ordinary a ne algebras. In particular, at the moment, it is not
immediate to describe within the DIM framework an arbitrary DF or CS matrix model. However,
among the DF matrix models there is a subclass that is directly lifted to rather peculiar
networks, which we call balanced which are controlled by an analogue of the level one
representations of KacMoody algebras and allow a drastically simpli ed bosonization and
even fermionization. As we already mentioned the balanced networks correspond to special
quiver gauge theories with zero functions.
the DIM(gl1) algebra. This is a very powerful method, but it is only at the rst stage of
development. There are several restrictions which should be consequently lifted at the next
stages. If one considers them as a consequent speci cation of representation types, the list
should be read in inverse order.
The construction that admits fermionization of intertwiners
and
at the level one
of DIM is much similar to that for the level k = 1 KacMoody algebras. Hence, one
could expect a straightforward generalization to arbitrary level a la [95{97] involving
analogues of the b; csystems. Note, however, that the requirement on the level does
not restrict the value of the Virasoro central charge regulated by : all matrix models
ensembles and, hence, the generic Liouville and WK conformal blocks are
already handled by the existing formalism. Also, at this level the di erence disappears
between the vertex operators (in particular, the screening charges) and the stress
tensors (including the Woperators): all these are described by exponentials of the
free elds, the di erences emerge only in the limit q; t
! 1.
The formalism is best developed for the intertwiners, which act as operators between
the two \horizontal" Fock modules F
is the \vertical" leg associated with F
(1;L) and F
(1;L 1), while the third representation
(0; 1). Such a nonsymmetricity is inevitable
since the resulting topological vertex of [35] is still asymmetric and remembers about
the distinguished vertical direction. Technically this restricts consideration to the
balanced networks, what makes many important models, including the quiver ones,
treatable only via additional application of the spectral duality.
A better treatment should involve in nitely many free elds, giving rise to MacMahon
type modules, what should also allow one to de ne skew intertwiners, where all the
three legs are nonvertical. An existing description of the MacMahon modules is pure
combinatorial, in terms of 3d Young diagrams (plane partitions). A naive free eld
formalism would involve elds depending on two coordinates instead of one, and this
requires a fargoing generalization of holomorphic elds used in the ordinary 2d CFT.
Such a formalism is now developing, also with the motivation coming from MHV
amplitudes, but its incorporation into the DIM representation theory is a matter of
future. Still, it seems important for a full understanding of the spectral dualities
and of generic networks, including the sophisticated ones from [165{169]. They can
be treated by the existing formalism, but it leaves the underlying symmetries well
hidden: they show up only in answers, but not at any of the intermediate stages.
A further challenge is further generalization from DIM(gl1) to DIM(gln) and the
triplePagoda algebras DIM(gcl1) and DIM(gcln). An intriguing problem (see appendix
A3) is that already DIM(gln) is built from the a
ne Dynkin diagram of gcln, thus,
the triplea ne generalization should involve more sophisticated Dynkin diagrams.
We hope that the present text can serve as a good introduction in the DIMbased
generalization of conformal theories, where the conformal blocks are the generic Nekrasov
functions and the Ward identities are the associated regularity conditions for qqcharacters.
We hope that it will help to attract more attention to emerging challenging problems, which
we have just enumerated. Technical means for this seem to be already at hand.
Acknowledgments
work on this project.
A.M.'s and Y.Z. are grateful for remarkable hospitality at Nagoya University during the
Our work is supported in part by GrantinAid for Scienti c Research (# 24540210)
(H.A.), (# 15H05738) (H.K.), for JSPS Fellow (# 2610187) (Y.O.), JSPS GrantinAid
for Young Scientists (B) # 16K17567 (T.M.) and JSPS Bilateral Joint Projects
(JSPSRFBR collaboration) \Exploration of Quantum Geometry via Symmetry and Duality"
from MEXT, Japan. It is also partly supported by grants 153120832Molaved (A.Mor.),
153120484Molaved (Y.Z.), moladk 163260047 (And.Mor), by RFBR grants
160100291 (A.Mir.) and 160201021 (A.Mor. and Y.Z.), by joint grants 155150034YaF,
155152031NSCa, 165153034GFEN.
A
Properties of the DIM algebras and their limits
In this appendix, we describe the algebraic structures of DIM algebras and their
degenerations.
A.1
Constructing DIM(gl1) from W1+1 algebra
Let us discuss how one can construct DIM(gl1) starting from the algebra of di erence
Algebra W1+1. Consider the algebra W1+1 (as usual, 1 + 1 refers here to adding the
Heisenberg algebra to W1
) given by the generators Wnk = W (znDk), n 2 Z; k 2 Z 0,
where D = z@z. One can consider the central extension of this algebra:
[W (znDk); W (zmDl)] = W [znDk; zmDl] + c n+m;0
n;kl;
n;kl =
( Pjn=1( j)k(n
0
j)l; n > 0
n = 0
or, in the di erent basis of Wnk = W (znDk) with D
tD (see (A.22)),
[W (znDk); W (zmDl)] = (tmk
tnl)W zn+mDk+l
c n+m;0 tk+l
tmk
tnl
1
nct nk n+m;0 k+l;0, see (A.3).
c
t
k
1
Note that, if k + l 6= 0, the second term in the right hand side of (A.2) can be absorbed
into the rst term by rede ning the generators W (Dk) with k 6= 0: W (Dk) ! W (Dk)
, k 6= 0. However, at k + l = 0 this term can not be absorbed and is equal to
(A.1)
(A.2)
Algebra W1+1. The next step is to consider the algebra W1+1 = n
Zo, which is a double of the W1+1 and may have two central extensions:
[W (znDk); W (zmDl)] = (tmk
tnl)W zn+mDk+l + t nk(nc1 + kc2) m+n;0 k+l;0 (A.3)
Automorphisms. The algebra W1+1, (A.3) has the evident automorphisms ; ~ and
(Wnk) = t 21 n2Wnk+n ;
(c1) = c2 ;
~(c1) = c2 ;
(c1) = c1 + c2 ;
(c2) = c1 ;
~(c2) = c1 ;
(c2) = c2 :
x0n; xm =
x0n; x0m =
n
n
n qc1n
q
n q(n jnj)c1=2xn+m
q c1n
q 1
n+m;0
In particular, and form SL(2; Z) acting on two central charges c1 and c2 .
Heisenberg subalgebras.
By the commutation relations (A.3), it is easy to see that it
contains a Heisenberg subalgebra generated by fWn0; c1gn2Z satisfying
[Wn0; W m0] = nc1 n+m;0 :
From the viewpoint of the root lattice of W1+1
, this can be seen as the vertical embedding
of the Heisenberg algebra. By using the automorphisms
and
in the above, it is easy to
nd the horizontal and the embedding with arbitrary slope
2 Z as follows;
[W0n; W0m] = nc2 n+m;0 ;
[Wn n; Wmm] = nt n2(c1 + c2) n+m;0 :
Chevalley generators and Serre relations. The generators Wn ;0 = W (znD 1;0) form
a closed subalgebra:
Wn+; Wm
Wn0; Wm
= (tm
t n)W m0+n + (nc1 + c2)t n
n+m;0
= (1 t n)Wm+n
Wn0; W m0 = nc1 n+m;0
One can generate the whole algebra from this subalgebra provided the Serre relations are
added:
h
Wn ; [Wn+1; Wn 1] = 0
i
Quantization: from W1+1 to DIM(gl1).
This algebra can be deformed with the
deformation parameter q. Let us denote the deformed (properly rescaled) generators through
Wn0 ! xn, Wn ! xn . Then,
0
(A.4)
(A.5)
(A.6)
(A.7)
(A.8)
where
and
Introducing the series of generators,
(z) =
(1
X
X
q1 = t2; q2 = q 2t 2; q3 = q2
(q1q2q3 = 1) (A.10)
k z k
q c2 exp
1
X x n
0 z n
n=1
!
k q c1k=2z k;
x (z) =
X xn z n
n2Z
we immediately come to the DIM(gl1) algebra of section 3.1 upon identi cation q1 = q,
q2 = t 1.
in its terms as
Free eld realization.
At the values of central charges (c1; c2) = (1; 0), the constructed
DIM algebra has the deformed a ne U(
1
) subalgebra so that the generators are realized
x+(z) = exp
x (z) = exp
+(z) = exp
(z) = exp
X 1
n>0
X 1
n>0
X(qn
n>0
X 1
n>0
t n
t n
n
1)(1
t n
n
(1
znpn
!
! nznpn
exp
!
X(qn
n>0
exp
! 2n)!n=2 znpn
!
!
!
X(qn
n>0
1)! n
!
(A.9)
(A.11)
(A.12)
(t 1
N
1) X
Y t 1z
i
i=1 j(6=i)
z
i
zj
zj zinq Di = (qt) n 4 n t N x n
n;0
(A.15)
with n
0. Similarly, at the values of central charges (c1; c2) = (2; 0) this DIM algebra
contains a qdeformed subalgebra (Virasoro
U[(
1
)) (and is realized by two free elds), at
(c1; c2) = (3; 0) it contains a qdeformed subalgebra (W (
3
)
U[(
1
)) (and is realized by three
free elds), etc.
After the Miwa transform of variables pn = PiN zin, these expressions reduce to the
Macdonald operators
(t 1
i=1 j(6=i) zi zj
N
1) X
Y t 1zi zj zinq Di
2 iz
X 1 t n znpn
n
exp
X 1 t n
n>0
n
z np n
exp
X(q n
n>0
for n > 0 so that
Elliptic DIM(gl1) algebra
Elliptic version of DIM algebra is generated by the same set of operators as the ordinary
DIM: x (z),
(z) and the central element . The relations are a copy of eq. (3.1), except
for the [x+; x ] relation, which changes to
q0 (q; q0) q0 (t 1; q0)
(q0; q0)31 q0 (q=t; q0)
( z=w)
where
p(z) = (p; p)1(z; p)1(p=z; p)1 is the thetafunction. Also, most importantly, the
structure function G (z) is now not trigonometric, but elliptic:
is exactly the same as in the trigonometric case, given by eqs. (3.2).
The essential di erence with the trigonometric case appears when one tries to build Fock
representation of elliptic DIM: one set of bosons turns out not to be enough. One needs at
least two sets of Heisenberg generators a^n and ^bn to reproduce the commutation relations
of the elliptic algebra. Concretely, we have for the level one representation:
u( +(z)) = '+(z) = exp
u( (z)) = ' (z) = exp
u( ) = (t=q)1=2
0
n>0
n>0
1
0
X (1 tn)z n
n6=0 n(1 q0jnj) ^anA exp @
n6=0
X (1 t n)q0jnjzn ^bnA :
n(1 q0jnj)
0
n6=0
X (1 tn)! jnjz n
n(1
q0jnj)
1
0
n6=0
X (1 t n)!jnjq0jnjzn ^bnA :
n(1
q0jnj)
!
X (1 tn)(! n
!n)! n=2
n(1 q0n)
X (1 t n)(! n
!n)! n=2
n(1 q0n)
z n^an
!nq0nzn^bn
zn^a n
!nq0nz n^b n
1
!
where the bosons a^n and b^n satisfy the following commutation relations:
1=2w)
(A.16)
(A.17)
1
(A.18)
(A.19)
[a^m; a^n] = m
[b^m; b^n] = m
[a^m; b^n] = 0:
(1
(1
q0jmj)(1
1
tjmj
q0jmj)(1
qjmj)
qjmj)
(pq0)jmj(1
tjmj)
m+n;0;
m+n;0;
The dressed current t(z) =
(z)x+(z) (z), corresponding to the stress energy tensor
is given by exactly the same expression (4.1), as in the ordinary DIM case. Moreover, the
dressing operators
(z) and
(z) are constructed from the
generators of the elliptic
DIM algebra using the same formulas (4.2) as give above. In the level two representation
(2)
u1;u2 the element t(z) produces the elliptic Virasoro stressenergy tensor
T (z) = : e ^(z)e ^(t 1z) : + t : e ^(tz=q)e^(z=q) :
(A.20)
where
^ (z) =
X
n6=0
z
n
n(1
q0jnj) p
a^ n
1 + !jnj
X
n6=0
n(1
z n
q0jnj) (!2q0)jnj=2b^ n
Let us also mention that the undressed elliptic DIM charge H x+(z)dz=z also leads to
several very interesting objects. In the level one representation it gives elliptic
Ruijsenaars Hamiltonian, while in the second level representation it is the di erence version of
the intermediate longwave Hamiltonian [197{202], which itself is a generalization of the
BenjaminOno system.
A.3
Rank > 1: DIM(gln) = quantum toroidal algebra of type gln
In complete parallel with the previous consideration, DIM(gln) emerge as a deformation of
the universal enveloping algebra of the Lie algebra An = Matn
C[z 1; D 1] with
i.e. of n
n matrices with entries being elements of the algebra of functions on the quantum
torus, zD = q1Dz. The deformation of A
N introduces another parameter, q2. Providing
this deformed algebra with twodimensional central extension, one arrives at DIM(gln).
The set of generators of DIM(gln) is Eik; Fik; Hir; Ki0; q c with k 2 Z, r 2 Z=f0g,
0
i
n
1. The generating functions (currents) are:
Ei(z) =
X Eikz k;
Fi(z) =
X Fikz k;
k2Z
k2Z
Ki (z) = Ki01 exp
q 1) X Hi; rz r
1
r=1
!
The two centers are qc and
= Qin=01 Ki0.
The commutation relations are
dij Gij (z; w) Ki
q(1 1)c=2z Ej (w) + Gji(w; z) Ej (w)Ki
q(1 c)=2z
= 0;
dij Gij (z; w) Ei(z)Ej (w) + Gji(w; z) Ej (w)Ei(z) = 0;
djiGji(z; w) Fi(z)Fj (w) + Gij (w; z) Fj (w)Fi(z) = 0;
djiGji(z; w) Ki
q(1 1)c=2z Fj (w) + Gij (w; z) Fj (w)Ki
q(1 c)=2z
= 0;
Gij (q cz; w)
Gij (qcz; w)
h
Ei(z); Fj (w)i =
Ki (z)Kj+(w) =
h
Ki (z); Kj (w)i = 0
q
ij
q 1
Gji(w; q cz)
Gji(w; qcz)
qcw
z
Ki+(z)
Ki (z)Kj+(w)
qcz
w
Ki (w)
(A.21)
(A.22)
(A.23)
(A.24)
HJEP07(216)3
where, in variance with the DIM(gl1)case,
and powers of q are made from entries of the Cartan matrix. The commutation relations
can be added with the Serre relations
q1 = tq 1;
q2 = q2;
q3 = t 1q 1
(A.25)
for n
3
for n = 2
symz1;z2
Ei(z1); hEi(z2); Ei 1(w)i
Ei(z1); Ei(z2); hEi(z3); Ei 1(w)i
(A.26)
and similarly for F . The qcommutator is [A; B]q = AB
qBA.
The comultiplication is the same as for DIM(gl1).
The structure functions are build from the a ne Dynkin diagrams and for glncase
are de ned as follows:
for the simply laced case n
3
Abn
Ab1
Gij (z; w) =
>
8> (z
>< (z
> (z
>
>: (z
(
1
q1w) for i = j
q2w) for
i = j
q3w) for i = j + 1
w) for i 6= j; j
1
1
3
dij =
t 1 for i = j
1; n
otherwise
(A.27)
Gg0l02 (z; w) = Gg1l12 (z; w) = (z
Gg0l12 (z; w) = Gg1l02 (z; w) = (z
q2w)
q1w)(z
q3w)
d00 = d11 = 1;
d01 = d10 =
1
(A.28)
The a ne Dynkin diagram for n = 2 is not simply laced, and in this case
For n = 1 we return to section 3.1, i.e.
Gg0l01 (z; w) = (z
q1w)(z
q2w)(z
q3w);
d00 = 1 (A.29)
One expects in the Pagoda (triplea ne) case DIM(gcl1) (or Uq;t;et
(gbbbl1), hence, the
name Pagoda) the Dynkin diagram of the form:
A.4
A
ne Yangian of gl1 [139]
One can consider a \quasiclassical" limit of the DIM(gl1) algebra, q = e~h1 , t 1 = e~h2 ,
t=q = e~h3 with properly rescaled generators. We also use another parameterizations:
1 = h1 + h2 + h3 = 0;
(A.30)
3 = h1h2h3
In the limit of ~ ! 0, one obtains the a ne Yangian, which, on the gauge theory side,
describes the 4d theories/Nekrasov functions. It is given by the commutation relations:
and two more relations similar to (A.34) with ei substituted by fi and 3 substituted by
3. These commutation relations should be added by the Serre relations
symi1;i2;i3 ei1; [ei2; ei3+1] = 0
i
and similarly for fi.
The commutation relations should be supplemented with the \initial conditions":
0;1 are the central elements, i.e. commute with everything all generators
2 is the grading element, i.e.
[ 2; ej] = 2ej;
2fj;
(A.36)
(A.31)
(A.32)
(A.33)
(A.34)
(A.35)
(A.37)
(A.38)
Note that, introducing the generator functions
one can rewrite the commutation relations as
with (u) = ((uu+hh11))((uu+h2)(u+h3) .
h2)(u h3)
h
1
i=0
1
i=0
e(u)e(v)
f (u)f (v)
(u)f (v)
e(u)f (v) f (v)e(u)
e(u) = X eiu i 1;
f (u) = X fiu i 1;
(u) (v)
(v) (u)
1
i=0
v) e(v)e(u);
u) f (v)f (u);
v) e(v) (u);
u) f (v) (u);
v
Heisenberg subalgebra
The commutation relations of the Virasoro algebra with extended U[(
1
)algebra,
[Jm; Jn] = km m+n
nJm+n
n)Lm+n +
(A.39)
can be realized with identi cation:
HJEP07(216)3
c
12
1
2
J 1 = e0;
J1 =
f0;
L 1 = e1 + e0;
L1 = f1
L 2 =
1
L0 =
2 + 2
1 +
From the rst line it follows that k =
. The other current mode are constructed by
repeated commutators: J 2 = [e1; e0], J2 =
[f1; f0] etc. Consistency conditions (e.g.
J 3
[L 1; J 2
]
[L 2; J 1]) require 2
= (
1
) 3 0 (the dependence on hparameters
comes from relation with [e0; 3], which does not involve e3, because [e3; 0] = 0). Thus,
there remains a free parameter .
The central charge is c =
2 0
3 03 = 1
(1
1)(1
2)(1
3), where a =
0hbhc with (abc) is a cyclic permutation of (123).
Representations: plane partitions
The basis of a quasi nite representation of this a ne Yangian6 can be described by plane
partitions (3d Young diagrams). The generators of algebra act on the plane partition as
follows:
e(u)
f (u)
(u)
adding a box to 3d Young diagram
removing a box to 3d Young diagram
diagonal action
More precisely,
the diagonal action is
> =
(u) =
;(u) Y
>
2
u
u0
h( )
where h( ) = xh1 + yh2 + zh3 and (x; y; z) are the coordinates of the box within the
plane partition;
6Such representations are labeled by a triple of ordinary Young diagrams: \minimal" plane partitions
are labeled by boundary conditions, [65, 139].
(A.41)
(A.42)
the raising (lowering) action is
e(u)j
f (u)j
> =
> =
X
2 +n
X
2 n
E(
F (
u
u0
u0
h( )
h( )
j
j + >;
A (
1
) +
A (2) E
A (
1
) +
A (2) +
A (
3
)
= 0 (A.48)
(A.43)
(A.45)
(A.47)
h B
h A
(A.46)
n
n
(u) u h( A) o
(u) u h( B) o =
where + (
box as compared to .
) denotes arbitrary plane partition with one additional (one subtracted)
Here F and E are coe cients which have to be de ned from the commutation relations
of the algebra and are some residues of
inhomogeneity in the standard spin chain.
(u), u0 is a constant shift, a counterpart of
and F (
generating functions it looks like
Formula (A.42) is derived by acting with the both sides of the commutation relation
(u v) e(v) (u) on j
>, using (A.43) and then taking the residue at v = h( )
Constraints on the coe cients E and F . Constraints on functions E(
) can be derived from the commutation relations [ei; fj ] =
i+j . For the
(u) = 1 + 3 X E(
h( )
h( )
(A.44)
where the secondorder pole does not contribute. This relation does not x E and F
completely. Imposing an additional requirement of unitarity E(
one immediately obtains [139]
+) = F ( +
),
3E(
3E(
resu !h( )
)2 = resu !h( )
(u)
(u)
The commutation relation e(u)e(v)
v)e(v)e(u) relates adding two boxes in
One still has to x the sign (after taking the square root).
di erent order:
E(
E(
A)E( +
B)E( +
A
B
A +
A +
B)
B)
To check that it is satis ed, calculate the square of the l.h.s.:
resu !h( A) (u) resu !h( B) + A (u) resu !h( A) (u) resu !h( B)
resu !h( B) (u) resu !h( A) + B (u) = resu !h( B) (u) resu !h( A)
h( B) h( A)
h( A) h( B)
h( B) h( A)
2
Similarly one can check the Serre relations by adding three boxes:
X hh( A (
1
) )
2S3
E
2h( A (2) ) + h( A (
3
) ) E
A (
1
)
i
{ 54 {
B (ei; ej ) <
jfiej j
>= i+j; <
should have only r
1 independent lines, i.e. there is a relation
Then, the generating function of eigenvalues
r 1
X
i=0
i i+k; = 0;
k
0
j; u j 1 =
1
j=0
f (u)
where f (u) and g(u) are polynomials of degree r
and g(u):
Consider the case of r = 2, i.e. a single state at the level one and linear functions f (u)
(u) =
u + 3 0;
= 1 +
3 0;
A simplest example of the highestweight representation.
tion with the highest weight j
>:
j j
j; j
fj j
>= 0
Since we consider the quasi nite representations, there should be linear relations among
eij
>. Consider vectors in the representation at the rst level with
nitely many, r
independent vectors. This means that the Shapovalov matrix at the level one, which is
(A.49)
(A.51)
(A.52)
(A.53)
(A.54)
(A.55)
(A.56)
Then, the commutation relations and the Serre relations implies that there are 3 states
at the second level (this is since the function
(u) is a ratio of cubic polynomials) and 6
states at the third level. These particular numbers are equal to the number of 3d Young
diagrams with a given number of boxes. This means that the highest weight is associated
with the trivial plane partition j >= ;, and the single rst level vector is associated with
the only one box plane partition j
>:
eij; >= 0; i > 0;
e0j; >
ij; >= 0; i > 0
Since 1 is a center and [ 2; e0] = 2e0 one immediately obtains
Using these formulas, from the Serre relations that involve j and e0;1;2;3, one gets
2j
1j
>= 0;
>= 2j
(u)j
> =
u
u + 3 0;; '(u)j
>
>
Open Access.
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any medium, provided the original author(s) and source are credited.
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