Toda 3point functions from topological strings II
JHE
Toda 3point functions from topological strings II
Mikhail Isachenkov 0 1 2 4
Vladimir Mitev 0 1 2
Elli Pomoni 0 1 2 3 4
0 15780 Zografou Campus , Athens , Greece
1 IRIS Haus , Zum Gro en Windkanal 6, 12489 Berlin , Germany
2 Notkestrasse 85 , D22607 Hamburg , Germany
3 Physics Division, National Technical University of Athens
4 DESY Hamburg, Theory Group
In [1] we proposed a formula for the 3point structure constants of generic primary elds in the Toda eld theory, derived using topological strings and the AGTW correspondence from the partition functions of the nonLagrangian TN theories on S4. In this article, we obtain from it the wellknown formula by Fateev and Litvinov and show that the degeneration on a rst level of one of the three primary elds on the Toda side corresponds to a particular Higgsing of the TN theories.
Topological Strings; Conformal and W Symmetry; Supersymmetry and Du

HJEP08(216)
1 Introduction
2 Toda CFT: a recap and a proposal
3 AGT dictionary
4 Semidegeneration from Higgsing the TN theories
4.1
4.2
Higgsing the TN  Review
The FateevLitvinov degeneration from Higgsing
4.3 The domain of the parameters restricts the contour
5 The semidegenerate W3 3point functions
6 The general WN case
7 Conclusions and outlook
A Notations, conventions and special functions
A.1 Parametrization of the TN junction
A.2 Conventions and notations for SU(N )
A.3 Special functions
A.4 Combinatorial special functions
B The sl(N ) KanekoMacdonaldWarnaar hypergeometric functions
B.1 The sl(N ) KMW hypergeometric functions and their qbinomial identity
B.2 The summation formula
C Higgsing and iterated integrals for the W4 case
(DOZZ) [3, 4] based on insightful and powerful consistency checks. This proposal was
rigorously derived by Teschner [5] who showed that the DOZZ 3point function is a solution
of the crossing symmetry equation.
The next natural step is to study multi eld nonrational CFTs, a prototype of which
is the Toda CFT. Obtaining the 3point functions of the Toda CFT is a longstanding
problem in mathematical physics. Attacking this problem purely by using 2D CFT
techniques is a notoriously di cult task and results exist only for particular specializations
of the external momenta. The state of the art can be found in the works of Fateev and
Litvinov [6{8], who obtained the 3point functions of primary operators if one of them is
appropriately degenerate.
In a previous publication [1], we presented a formula for the 3point functions of three
a very di erent nature than [6{8], namely topological stings, 5brane web physics and the
AGTW correspondence. The purpose of the present paper is to push forward the program
of further understanding and checking it. We begin with (2.16), specialize appropriately
one of the external momenta and obtain the formula of FateevLitvinov [6] after a direct
calculation, thus presenting a highly nontrivial check of our proposal. Specializing means
that the Verma module for the primary eld has a nullvector descendant at level one. In
the rest of the paper, we will refer to them as semidegenerate,1 as opposed to the
completely degenerate ones, containing N
1 linearly independent nullvectors. Furthermore,
we believe that the techniques of [1] will provide the solution not only for the 3points
functions of WN primaries, but also for those involving descendent elds. We leave this
for a future work.
The quirks of our formula for the 3point functions (2.16) stem from the strategy
employed in [1] to derive it. A key element was the AGTW correspondence [9, 10], which
is a relation between 4D N
= 2 SU(N ) quiver gauge theories and the 2D
WN Toda
CFT. Speci cally, upon an appropriate identi cation of the parameters, the correlation
functions of the 2D Toda CFT are equal to the partition functions of the corresponding 4D
N = 2 gauge theories. The conformal blocks of the 2D CFTs are given by the instanton
partition functions of Nekrasov [9, 10], while the 3point structure constants are obtained
by the partition functions of the TN superconformal theories [11, 12]. The TN theories
have no Lagrangian description and thus their partition functions were unknown until
recently [1, 12, 13]. The sole exception was the W2
Vir case, i.e. the Liouville case,
whose 3point structure constants are given by the famous DOZZ formula [3, 4] and equal
to the partition function of four free hypermultiplets [12, 14].
We were able to bypass the fact that the TN theories have no known Lagrangian
description by using a generalized version of AGTW: a relation between 5D gauge theories
compacti ed on S1 and 2D qdeformed Liouville/Toda CFT [12, 15{28], where the
circumference
of the S1 corresponds to the deformation parameter q = e
of the CFT. In
5D, the partition functions can be computed not only using localization, which requires
a Lagrangian, but also by using the powerful tool of topological strings [29]. Employing
1A representation of WN can contain a null vector at some level higher than one. Such
representarepresentation and two generic ones will not be considered in the present paper.
this technology, we calculated in [12] (see also [13]) the partition functions of the 5D TN
theories and suggested that they should be interpreted as the 3point structure constants
of the qdeformed Toda. Subsequently, we showed in [1] how to take the 4D limit,
corresponding to
! 0 or equivalently to q ! 1, thus obtaining the partition function (3.5) of
the 4D TN theories. We want to stress that taking this limit is a tricky business, as the
expression (2.16) includes nontrivial multiple sums and integrals. This is the reason why
we will always work with the qdeformed formulas and take the limit only at the end.
This article is organized as follows. After brie y reminding the reader of the essentials
of Toda CFTs, we recall the formula by Fateev and Litvinov for a special class of 3point
functions of Toda primaries, as well as its straightforward generalization to the conjectural
generic 3point functions of Toda primaries. To spell out the details of it, we will need some
basics of the AGT dictionary collected in section 3. In the next section 4, the discussion
temporarily deviates from the CFT matters focusing rather on the interplay between the
moduli spaces of the corresponding gauge theories and 5brane web physics. We argue
that the semidegeneration of a primary
eld on the (qdeformed) CFT side mirrors a
Higgsing of the TN theory on the 4D (5D) side. A more CFToriented reader can skip this
section, with the exception of 4.3. The AGT genesis of FateevLitvinov formula for W3
Toda 3point function, via pinching an integration contour by a particular residue of the
corresponding integrand and applying nontrivial summation theorems, is what section 5
focuses upon. With the details of W4 computation deferred to the appendix C, we then
proceed to a discussion of the general WN case in section 6. The conclusion and the
outlook follow, whereas the remaining appendices are devoted to overview of notations and
special functions, most importantly to describing and elaborating on the properties of the
KanekoMacdonaldWarnaar sl(N ) hypergeometric functions which play a major role in
our calculations.
2
Toda CFT: a recap and a proposal
In this section we brie y summarize some relevant facts about the Toda CFT, closely
following [6{8]. Furthermore, we spell out the FateevLitvinov formula for a special subset
of Toda structure constants and present our proposal for the Toda 3point functions of
generic primary elds.
The Lagrangian of the AN 1 Toda CFT is given by
1
8
L =
N 1
X eb(ek;');
k=1
(2.1)
where ' := PiN=11 'i!i, with ek, !k being the simple roots and the fundamental weights
of sl(N ) respectively. The de nition of the inner product ( ; ) along with other useful
Liealgebraic de nitions and notations are collected in appendix A.2 for the convenience of the
reader. The parameter
is called the cosmological constant, in analogy to the Liouville case
(N = 2) where it determines the constant curvature of a surface described by the classical
{ 3 {
~ (b 2) b !
=
1
(b2) b =) ~ =
(b2) 1=b2
(1=b2)
;
where (x) :=
(1(x)x) . As we mentioned in the introduction, the Toda CFT also has a
WN higher spin chiral symmetry generated by the elds W2
2; : : : ; N . The primaries under the full symmetry algebra WN T , W3; : : : ; WN of spins
WN are the exponential
elds of spin zero labeled by a weight of sl(N ):
In what follows, we will parametrize the fundamental weight decomposition of a weight
i as
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
! w
(2.8)
equation of motion. The normalization of the Lagrangian is chosen in such a way that
'i(z; z) 'j (0; 0) =
ij logjzj2 +
at z ! 0:
Following [7, 8], we consider the correlators on a twosphere, which prescribes putting a
background charge at the north pole in order to render the Toda action
nite:
'(z; z) =
Q logjzj +
at z ! 1;
V := e( ;'):
i = N
N 1
X
j=1
j
i !j :
Vw
= Rw( )V
{ 4 {
where Q := Q
= (b + b 1) with the Weyl vector
de ned in (A.8).
Analyzing the path integral of the theory (2.1), one can argue that the Toda CFT
must have an exchange symmetry b $ b 1 on a quantum level which simultaneously sends
the cosmological constant to its dual ~, de ned as
HJEP08(216)
2One should not confuse the a ne Weyl transformation, i.e. Weyl re ections accompanied by two
translations, with Weyl re ections belonging to the Weyl group of the a ne Lie algebra.
By looking at the corresponding OPEs, one reads o the central charge c of the Toda CFT
and the conformal dimensions
( ) of its primary elds:
c = N
1 + 12 (Q; Q) = (N
1) 1 + N (N + 1)Q2 ;
( ) =
(2Q
; )
;
(2.7)
2
with the antiholomorphic conformal dimensions of the primary elds being equal to the
holomorphic ones.
The conformal dimension, as well as the eigenvalues of all the other higher spin currents
Wk are invariant under the a ne2 Weyl transformations (A.13) of the weights
i, which
roughly means that several exponential elds correspond to the same `physical' eld. The
primary elds of Toda CFT transform under an a ne Weyl transformations
given in (A.13) as
(1
b (
Q; e))
in terms of the function
A( ) :=
(b2)
( bQ; ) Y
e>0
and by the normalization (2.5). They read
is an entire function de ned in appendix A.3.
Before presenting our formula for the 3point functions, we need to introduce the
qdeformed Toda theory. Albeit no Lagrangian description of the qdeformed version of
Toda eld theory has been found yet, many quantities of this conjectural deformation are
algebraically wellde ned, in full analogy to the Toda CFT (see [30] and references therein).
While the qdeformed Toda CFTs are vastly unexplored, for the qdeformed Liouville case
3We use a slightly di erent convention than [6]. One has to rescale { ! N{ to match the expressions.
{ 5 {
where \Weylre ections" stands for additional contributions that come from the eld
identi cations (2.8).
ture constant:
The coordinate dependence of 3point functions of primary elds (2.5) is xed by
conformal symmetry up to an overall coe cient C( 1
; 3) called the 3point
struch
V 1 (z1; z1)V 2 (z2; z2)V 3 (z3; z3)i =
C( 1
where zij := zi
zj and
i is the conformal dimension of the primary V i .
Up to now, the CFT machinery has produced expressions only for a restricted subset
of 3point functions, as well as for some interesting physical limits of those, see [6{8] for the
state of the art. The formula of Fateev and Litvinov [6] which we will quote in a moment
gives the Toda structure constants for the particular semidegenerate case when one of the
elds contains a nullvector at level one, implying that the corresponding weight becomes
proportional to the rst !1 or to the last !N 1 fundamental weight of sl(N ). Speci cally,
if one sets3
1 = N {!N 1, the structure constants read
C(N {!N 1
; 3) =
a bit more is known [12, 15{28]. The details of our working de nition for the qdeformed
Toda are presented in section 3.4 of [1]. The building blocks of our proposal are qdeformed
functions who reproduce the known limit as q := e
transformation properties as well as the poles and zeros4 of the undeformed ones. In the
Toda CFT, the dependence on the cosmological constant
is fully xed by a Ward identity
coming from the path integral formulation. The absence of a path integral formulation for
the qdeformed Toda implies that such quantities as structure constants of the theory are
ambiguous up to a function of , b and q. Due to this, we de ne the qdeformed structure
! 1, keep the same symmetries and
constants here up to the
symmetry b $ b 1:
(b2) term, having qdeformed only the part respecting the
the undeformed Toda structure constants in the limit q ! 1, one has to set, respectively:
then gives the undeformed one (2.13) upon taking the limit q ! 1 and reintroducing the
We nish this section with our proposal for the 3point function of of generic primary
lim
!0
2Q Pi3=1( i; ) I N 2 N 1 i " dA~i(j)
Y Y
topological string amplitude ZNtop, we require some notions and notations which will come
in the next section. The impatient reader may skip the explanations and proceed straight to
the formulae (3.7), (3.13), (3.15), (3.16), (3.17), (3.18) consulting also appendices A.3, A.4
for de nitions of the encountered special functions.
4To be more precise, the qdeformed functions have a whole tower of zeroes/poles for each zero/pole
of the undeformed function. The tower is generated by beginning with the undeformed zero/pole and
translating it by r l2ogiq =
r 2 i , where r is a positive integer.
{ 6 {
AGT dictionary
According to the AGTW correspondence [9, 10], the correlation functions of the 2D Toda
CFT are obtained from the partition functions of the corresponding 4D N
= 2 gauge
theories as
Z
S4 =
Z
2
[da] ZN4Dek(a; m; ; 1;2)
/ hV 1 (z1)
V n (zn)iToda ;
(3.1)
where the Omega deformation parameters are related to the Toda coupling constant5 via
1 = b and 2 = b 1. Moreover, a stands for the set of Coulomb moduli of the theory, m for
the masses of the hypermultiplets and
for the coupling constants. The correspondence
relates the masses m to the weights i and the couplings constants
to the insertion points
zi of the primary elds. In particular, the conformal blocks of the 2D CFTs are given by
the appropriate Nekrasov instanton partition functions [9, 10] and the 3point structure
constants by the partition functions of the TN superconformal theories on S4 [11, 12].
A similar relation between 5D gauge theories and 2D qCFT exists [12, 15{28], which
relates the 5D Nekrasov partition functions on S4
S1 to correlation functions the of
qdeformed Liouville/Toda eld theory:
Z
S4 S1 =
Z
2
[da] ZN5Dek(a; m; ; ; 1;2)
/ hV 1 (z1)
V n (zn)iqToda ;
(3.2)
(3.3)
(3.4)
! 0
(3.5)
where
=
parameters
q = e
1 ;
t = e 2 ;
are used in this case. The partition function on S4
S1 is the 5D superconformal index,
which as discussed in [29] can also be computed using topological string theory techniques
Z
S4 S1 =
Z
Z
[da] jZN5Dek(a)j2 /
[da] jZtop(a)j2 :
on S4
In [12] we computed the partition functions of the 5D TN theories on S4
S1 (see also [13])
and suggested that they should be interpreted as the 3point structure constants of
qdeformed Toda. We read them o from the toricweb diagrams of the TN junctions of [31]
by employing the re ned topological vertex formalism of [32, 33]. In a subsequent paper [1],
part one of the present series of papers, we showed how the 4D limit, corresponding to
or q ! 1, is to be taken. We thus obtained the partition function of the 4D TN theories
ZNS4 = const
lim
!0
N
1 2 ZN
S4 S1 ;
where by \const" we mean a function of 1, 2 that is independent of the mass parameters
of the theory. The degree of divergence was determined as proportional to the quadratic
5We also use the notation + = 1 + 2. When we specialize 1 = b and 2 = b 1 in order to connect the
topological string expressions to the Toda expressions, we have + = b + b 1 = Q.
log q is the circumference of the S1. The exponentiated Omega background
with the position of the avor branes on the TN side, here drawn for the case N = 5.
equality of (3.6), we have introduced the mass parameters mi, ni and li of the TN theory,
which, as shown in gure 1, are connected to the Toda theory parameters [1]
mi = ( 1
Q; hi) = N
ni =
li =
( 2
( 3
Q; hi) =
Q; hN+1 i) =
N
N 1
X
j=i
N
j
1
N 1
X
j=i
N 1
X j 1j
j=1
2j +
N + 1
2i
2
Q ;
N 1
X j 2j +
j=1
N + 1
2i
2
Q ;
N 1
X
j=N+1 i
3j +
N 1
X j 3j
j=1
N + 1
2i
2
Q :
It is important to note, that the mass parameters are not all independent, but obey
N
i=1
X mi =
X ni =
X li = 0 ;
N
i=1
N
i=1
which is re ected in the fact that the sum of the weights hi of the fundamental SU(N )
representation is zero. Then the structure constants of three primary operators in the
qToda theory are given by the TN partition functions on S4
S1 as
Cq( 1
2 3
4
Y Yq( j )5 (1
3
j=1
q)
ZN
N S4 S1 ;
where by \const" we mean a function of 1, 2 and
that is independent of the mass
parameters of the theory. We stress that the superconformal index ZN
the a ne Weyl transformations (A.12) and that all the nontrivial Weyl transformation
S4 S1 is invariant under
{ 8 {
(3.6)
(3.7)
(3.8)
(3.9)
denote 7branes by crossed circles. The left part of the
gure shows the original TN 5brane web
diagram, while the right one depicts the web diagram obtained by letting N
terminate on the same 7brane.
properties of the structure constants are captured by the following special functions:
Yq( ) :=
q de ned in (A.34) and the product taken over all positive roots e of
SU(N ). The partition function on S4
S1, or the superconformal index, for the TN theory
is given by an integral over the re ned topological string amplitude with an integration
measure containing the re ned MacMahon function6 M (t; q) [29]
ZN
S4 S1 :=
I N 2 N 1 i " dA~i(j)
Y Y
Here, we have removed the decoupled degrees of freedom, referred to as \nonfull spin
M(M~ iM~ j 1)M(t=qN~iN~j 1)M(L~iL~j 1) 2
q)N( k; k 2Q)
1
q
b 2b 1
1
q
b 1 2b ( k; )
Yq( k) ;
where the function M is de ned in (A.29). Interestingly enough, as noted in [1], these
degrees of freedom are responsible for the Weyl covariance of the Toda structure constants.
Here and elsewhere, we shall use the shorthand notation
jf (U1; : : : ; Ur; t; q)j2 := f (U1; : : : ; Ur; t; q)f (U1 1; : : : ; Ur 1; t 1; q 1) :
Inserting (3.11) into (3.9), we nd the nice expression
Cq( 1
I N 2 N 1 i " dA~i(j)
Y Y
6See (A.40) for the de nition of the re ned MacMahon function M (t; q).
re ned topological vertex formalism and reads
top obtained from the TN webdiagram by using the
ZN
top = ZN
pertZNinst ;
where the \perturbative" partition function7 is
q t A~(r)A~(r 1)
i j
q A~i(r)1A
~(r 1)
j+1
and the \instanton" one is
\interior" Coulomb moduli A~(i) = e
a(j)
j
given by
i
2
ai(j) , while the N
i(r), r = 1; : : : ; N
are given in (A.46). The
1, i = 1; : : : ; N
r. The
i are independent, while the \border" ones are
i
A~(0) =
Y M~ k ;
k=1
A~(0i) =
i
Y N~k ;
k=1
i
A~(N i) =
i
Y L~k ;
k=1
(3.18)
where M~ k := e
mk and similarly for N~k and L~k. See appendix A for more details on the
parametrization of the TN junction.
2
The formula (2.16) (correspondingly, (3.9)) for the structure constants of three primary
elds of (qdeformed) Toda CFT, has the correct symmetry properties, the zeros that it
should and, for N = 2, gives the known answer for the Liouville CFT [1]. However, it
is very implicit, requiring to perform N(N 1) sums over the partitions i(j), followed by a
2
(N 1)(N 2) dimensional8 integral over the Coulomb moduli A~(j) and nally to take the 4D
i
7We put the words \perturbative" and \instanton" inside quotation marks because for the TN there is
no notion of instanton expansion, since there is no coupling constant.
8It is the number of faces of the left diagram in gure 2.
unHiggsed TN with SU(N )3 global symmetry. On the right we show the sphere with two full
punctures and one Lshaped fN
1; 1g puncture. This particular Higgsing of TN leads to a theory
with with SU(N )
SU(N )
U(1) global symmetry. The partition function of this theory will lead to the Toda 3point function with one semidegenerate primary insertion.
`1; : : : ; `n 5branes each. On the right side of the gure, we depict the Young diagram f`01; `02; : : : ; `0ng
that gives the avor symmetry of the corresponding puncture. Having n bunches of 5branes, each
ending of a 7brane leads to a puncture in the Gaiotto curve with
avor symmetry S(U(k1)
U(kr)), where the widths ki of the boxes are equal to the numbers of stacks with the same number
of branes per stack.
(q ! 1) limit (3.5). In the subsequent parts of the paper we will show how to derive the
special case (2.13), known due to Fateev and Litvinov [6{8], from our formula (2.16). This
provides a strong check of our general proposal.
4
in
Semidegeneration from Higgsing the TN theories
In this section we argue that a particular way of Higgsing the TN theories, as depicted
gure 2, corresponds to the degeneration with one simple and two full punctures. On
the Toda side, this is equivalent to the semidegeneration of Fateev and Litvinov. On
the gauge theory side, the partition function of the theory with one simple and two full
punctures is the partition function of N 2 free hypermultiplets. Our discussion is based
on the physics of (p; q) 5brane webs and their symmetries. In particular, we identify
which Higgsing mechanism corresponds to the Fateev and Litvinov semidegeneration by
introducing 7branes on the 5brane web. Finally, in this section, we discuss the domain
in which the mass parameters, or Toda weights, take value, which will dictate the contour
for the integral (2.16).
In the next sections we will use the intuition acquired here to explicitly substitute
the values dictated by the web diagram, (4.10) and (4.6), in (3.9) so as to obtain the
formula (2.13) by Fateev and Litvinov.
4.1
The physics of the (p; q) 5brane webs that we will need in the context of this section is
studied in [13, 31, 34, 35]. We give a short review of their relevant results. A very useful
way of realizing 4D N = 2 quiver gauge theories in string theory is by using type IIA
string theory and the HananyWitten construction [36] of D4 branes suspended between
NS5 branes [37]. This con guration can be lifted to Mtheory, where both the D4 and
the NS5 branes become a single M5 brane with nontrivial topology, physically realizing
the SeibergWitten curve in which all the low energy data are encoded [37]. Similarly,
5D N = 1 gauge theories can be realized using type IIB string theory with D5 branes
suspended between NS5 branes forming (p; q) 5brane webs [38, 39]. A large class of N = 2
SCFTs, called class S, can be reformulated (from the realization in [37] with a single M5
brane with nontrivial topology) as a compacti cation of N M5 branes on a sphere [40].
This point of view is very useful since intersections of these N M5 branes with other M5
branes can be thought of as insertions of defect operators on the world volume of the M5
branes and thus punctures on the sphere. The name simple puncture is used for defects
that are obtained from the intersection of the original N M5 branes with a single M5 brane
(originating from D4's ending on an NS5 in the HananyWitten construction), while full or
maximal punctures stem from defects corresponding to intersections with N semiin nite
M5 branes (external avor semiin nite D4's in [37]).
More general punctures, naturally labeled by Young diagrams consisting of N boxes,
are also possible [40, 41]. In the (p; q) 5brane web language, they can be described when
additional 7branes are introduced [31]. Semiin nite (p; q) 5branes are equivalent to
(p; q) 5branes ending on (p; q) 7branes [42]. Consider N 5branes and let them end on n
7branes, as shown on the left of gure 4. The jth 7brane carries `j 5branes. We de ne
the numbers `0j as a permutation of the `j such that they are ordered
and arrange them as the columns of a Young diagram9 f`01; `02; : : : ; `0ng, see the right hand
side of gure 4. As we started with N 5branes, the `0j s must obey the condition Pjn=1 `0j =
N . The integers ka are de ned recursively
ka = f# `0j : `0j = `0k1+ ka 1+1g ;
and are equal to the number of columns of equal height. Since the diagonal U(
1
) of
the whole set of the N 5branes is not realized on the low energy theory [42], the avor
symmetry of the corresponding puncture in the Gaiotto curve is S(U(k1)
U(kr)) [40].
The Coulomb branch of the TN theories, corresponding to normalizable deformations
of the web which do not change its shape at in nity, has dimension equal to the number of
faces in the TN web diagram, see the left part of gure 2, and has dimension (N 1)(N
2
2) ,
as it should [41]. Moreover, the dimension of the Higgs branch of the TN theories, known
9In this article, we draw the Young diagrams in the English notation. By fc1; : : : ; crg we mean a Young
diagram with r columns for which the jth column has cj boxes, j = 1; : : : ; r. Furthermore, we use the
notation fabg for the partition fa; : : : ; ag with b columns.
(4.1)
(4.2)
the left we have the unHiggsed dot diagram with three full punctures, SU(4)3 global symmetry
and three Coulomb moduli. In the middle, the four D5 branes end on two D7 branes with two D5
branes on each, which corresponds to the Young diagram f2; 2g. This theory has apparent global
symmetry SU(4)2
SU(2) and one closed polygon corresponding to one leftover Coulomb modulus.
Finally, on the right we have the fullyHiggsed theory with three D5 branes on the rst D7 brane
and one D5 brane on the second D7. This theory has no Coulomb moduli left.
2
to be 3N2 N 2 [41], was obtained by terminating all the external semiin nite 5branes on
7branes and counting the independent degrees of freedom for moving them around on the
webplane [31]. Finally, the global symmetry SU(N )3 of the TN theories is realized on the
Higgsed TN theories can also be understood in this way [31]. Beginning with the TN
5brane webs which correspond to the sphere with three full punctures (labeled by the Young
diagrams f1N g) and grouping the N parallel 5 branes of the punctures into smaller bunches
(labeled by the Young diagrams f`01; `02; : : : ; `0ng), 5brane con gurations which realize 5D
theories with E6;7;8
avor symmetry were obtained. These theories have Coulomb and
Higgs branches of smaller dimension than the original TN which can be counted using a
generalization of the srule [43{45] from the socalled dot diagrams,10 see also [13, 34, 35].
For us, the important result from [31] is that the dimension of the Higgs moduli space of
a puncture corresponding to the Young diagram depicted in gure 4 is
dimHMpH =
1) `j ;
n
X (j
j=1
(4.3)
and that the Coulomb branch is the number of closed dual polygons in the dot diagram.
4.2
The FateevLitvinov degeneration from Higgsing
We need to decide which puncture (Young diagram f`01; `02; : : : ; `0ng) corresponds to the
FateevLitvinov semidegenerate primary operator. This puncture should have only U(
1
)
symmetry (for N > 2). Thus, it can be obtained by grouping the N 5branes in two
bunches of unequal number of 5branes, N
1 and 1 respectively, forming the Lshaped
Young diagram fN
symmetry, while for N
1; 1g shown in gure 3. For N = 2, the puncture has an SU(2) avor
3 the avor symmetry gets reduced to U(1), as required for the
semidegenerate eld. This Young diagram fN
1; 1g corresponds to the simple punctures
10The dot diagrams are the dual graphs of the web diagrams with the additional information about the
7branes encoded in white and black dots.
discussed before. The Higgs moduli space of this con guration has dimHMH
which is consistent with the fact that we have only one parameter { in the CFT side.
Finally, the dot diagrams tell us that the dimension of the Coulomb branch in this case is
zero, which, as we will see later, is consistent with what one gets by just substituting (4.7)
semideg = 1
Now, let us discuss what happens with the Kahler moduli that parametrize the TN
partition functions as we bring together N
1 parallel horizontal external D5 branes on a
single D7 brane. These we will then translate in the language of mass parameters mi; ni; li
(i = 1; : : : ; N ) and Coulomb moduli ar (r = 1; : : : ; (N 1)(N 2)=2) using the dictionary of
appendix A.1 and in particular equation (A.4) and,
nally, to the Toda weights
1;2;3
using (3.7). We follow closely the discussion in [34]. For simplicity, we begin with two
parallel D5 branes that originally end on di erent D7 branes. This process is depicted in
gure 6. First we need to shrink u2 of U2 = e
In the process of sending the u1 of U1 = e
u2 to zero while still having two 7branes.
u1 to zero, one of the two D7 branes will meet
a D5 brane and the two parallel D5 branes will fractionate on the D7 branes. After moving
the cut piece to in nity it e ectively decouple from the rest of the web.
U = pt=q.
For the unre ned topological strings, i.e. for 2 =
1, shrinking the length of a
5brane that is parametrized11 by U = e
u corresponds to setting U = 1. This is not true
any more in the case of the re ned topological string where zero size will correspond either
to U = pt=q or U = pq=t [46{49]. It turns out that both choices are equivalent as is
extensively discussed in [34]. In this paper we wish to consider only the parameter space
that corresponds to Toda CFT with Q = 1 + 2 > 0, i.e. t=q > 1, and thus we have to pick
For the T3 case the situation is exactly the same as the simple example depicted in
gure 6. The following two Kahler parameters
HJEP08(216)
Q(m1;)1 = Ql(;11) =
r t
q
:
Q(m1;)1 = A 1M~ 1N~1
and
Ql(;11) = AM~ 2 1N~ 1
1
are the ones we have to shrink, where A
appendix A.1 for notations and gure 9 for the web diagram of T3. Thus, we have to set
A~(
1
) is the Coulomb modulus of T3. See
1
11The parameter u in the exponent is the length of the 5brane segment.
(4.4)
(4.5)
In general for TN as depicted in gure 13 we must tune
Q(mj);i = Ql(;ji) =
r t
q
1 = N {!N 1
()
mi =
with
Going back to the Toda side, we wish to semidegenerate the weight
q qt ; t; q
1
q t
q
e
1
2
q(m
2+ ) :
mij
2
:
8
>< {
>
where the implications from (3.7) of the semidegeneration on the mass parameters are
written on the right. For the T3 case that implies for the exponentiated mass
parameters that
and
M~ 2 = K~
which is consistent with (4.4) and (4.5) when the Coulomb moduli is tuned to the value
where i; j = 1; : : : ; N
parameters obey (4.6).
At the level of partition functions, the FateevLitvinov formula for the special 3point
functions can be identi ed with the partition function of N 2 free hypermultiplets, after
removal of the decoupled degrees of freedom (3.12).
We know from [12, 14], that the partition function of a single free hypermultiplet is given by 2, i + j
N
1 and K~ = e
{. This implies that the Kahler
This is compatible with the statement that after Higgsing, the T3 the dimension of the
Coulomb branch is zero, and also with the fact that we will discuss in next section, the
contour integral gets pinched once one substitutes (4.7) in (3.9). In the general TN case,
Higgsing forces the Coulomb parameters to become A =
r t K~ N~1 :
q
A~(j) =
i
t
q
i(N i j)
2
j
k=1
K~ i Y N~k ;
(4.7)
(4.8)
(4.9)
(4.10)
(4.11)
(4.12)
Thus, the 5D superconformal index of N 2 free hypermultiplets is the product of N 2 such
partition functions
Up to factors that for now we drop and using (3.12), we can identify
From this knowledge, one could go ahead and guess some of the complicated summation
formulas like (5.20), as was done by [34] for the T3 case.
The domain of the parameters restricts the contour
An important step we will have to take is to perform the contour integral in (2.16). For
that we need to carefully discuss the domain in which our parameters take values. On the
Toda side, this type of conditions is obtained by considering the physicality of the WN
Toda weights is in order. Denoting by ( ) the conformal dimension of the primary eld V , the formula for the 2point functions
V 0 (z0; z0)V (z; z) =
(2 )
N 1 (
+ 0
j
z
;
tells us that requiring that V 0 be the conjugate eld to V leads to the following reality
condition12
<( ) = Q
()
mi; ni; li 2 iR :
The physicality condition for the Toda weights (4.15) implies through the dictionary (3.7)
that the mass parameters are purely imaginary. On the (p; q) 5brane web diagram side,
distances are measured by the real part of the mass parameters, see equations (2.72.12)
of [20] for a review of our conventions. When the 5branes are on top of each other, i.e.
when their distance is zero,13 TN has SU(N )3 symmetry [31] and we can have physical
Toda theory states. Since Q = Q PiN=11 !i and since semidegeneration requires that
= N {!N 1, we see that semidegeneration/Higgsing is incompatible with the physicality
condition (4.15). This agrees with a CFT intuition [5].
We wish to conclude this section by stressing that the formulas we are dealing with
have di erent domains with di erent convergent expansions depending on the values of the
masses, just like in (A.29). In the topological string language they correspond to di erent
12See section 4 and 11 of [5] for a detailed discussion of the physicality condition in the Liouville case.
13In the re ned topological vertex, the SeibergWitten curve is replaced by its quantum version in which
zero distance is understood as integer multiples of +.
(4.14)
(4.15)
the contour integral diverges, which is why we regulate it by multiplying with a
b. In the limit
a ! b, the integral is given by a single residue.
geometries that are related to each other by opping. For each Kahler parameter U , we
distinguish between the region jU j > 1 and the one with jU j < 1; to each we associate a
di erent (p; q) 5brane web diagram. Going from one region to the other involves \ opping"
which transforms the Kahler parameters as depicted in
gure 7. See [50] for a recent
discussion of the topic. In the next section, we explain how the contour in (3.9) is to be
chosen and we argue that the contour is dictated by the choice of the opping frame.
function in a domain D
that it can be written as
5
The semidegenerate W3 3point functions
In this section we explicitly derive the FateevLitvinov result for the semidegenerate
3point functions of the sl(3) Toda theory from our general formula. To succeed in this
calculation, we need to do two things: to evaluate the contour integral in (3.14) and to
perform the sum in (3.17). For general values of the parameters, in nitely many poles
contribute to the contour integral, but luckily in the semidegeneration limit only two of
them do for the sl(3) case. This is due to a phenomenon known as \pinching", which we
illustrate in the beginning of the section with a very simple example. Then, we show that in
the sl(3) case, there are two possible poles where the contour can be pinched, each of them
corresponding to a di erent opping frame of the T3 geometry. From this observation, we
infer three di erent possible choices for the contour in (3.14). We compute the integral for
each of them and
nd the same result. Finally, we show that for the particular residues
that contribute it is possible to compute the sum in the \instanton" factor.
Let us rst make a simple example to illustrate pinching. Let g be a meromorphic
C that has only simple poles at the points a, b and pi, meaning
where f is a holomorphic function in D. Let C be a closed contour in D that encircles a
as well as the pi but not b. We write a = p +
and b = p
and take the limit
thus letting the two points a and b collide on the contour C on both sides, as depicted in
gure 8. If we now compute the contour integral of g around C and multiply it by a
b,
(5.1)
! 0,
from the left one by applying two opping moves, see gure 7, to the encircled segments.
Qi(a
f (a)
f (p)
!0
! Qi(p
pi)
pi)
+ X
i
= lim [(a
a!b
(pi
a)(pi
(a
b)f (pi)
b) Qj6=i(pi
pj )
b)Res(g(z); a)] :
(5.2)
Thus, in the limit a ! b, the contour gets pinched at the point a = b = p and the integral
is given by a single residue. This is essentially the contour integral version of the identity
"
lim"!0 (x+i")(x i") =
which g has not only simple poles, but we will not need it.
(x). This example can also be easily generalized to the case in
We now want to explain how this simple example applies to our integral formulas for
the correlation functions of sl(3). In the sl(3) case, our contour integral formula (3.14) for
the structure constants reads
Cq( 1
; 2
with the sum going over all partitions
= f 1(
1
); 2(
1
); 1(2)g. Since we wish to evaluate the
contour integral (5.3) in the semidegenerate limit
1 = 3{!2, we introduce a regulator
and parametrize the three masses labeling the positions of the branes on the left as
m1 = { +
Q ;
m2 = {
;
m3 =
2{ + Q ;
which implies that the exponentiated masses M~ i = e
mi are
t K~ e
q
;
M~ 2 = K~ e ;
{. The semidegenerate limit then corresponds to
the masses, the numerator of jZ3topj2 in (5.4) goes to zero, just like the term a
b in the
! 0. For these values of
simple example (5.2) above, since
HJEP08(216)
(5.5)
(5.6)
(5.7)
jM(M~ 1M~ 2 1)j2 = (1
e 2 )
reg.
reg. ;
where \reg" are terms that don't vanish for
! 0.
The next step is to analyze the poles in the integrand of (5.4) and determine which ones
will contribute in the semidegenerate limit. We make the assumption14 that only poles
from the \perturbative" part, i.e. the rst line of (5.4), are relevant for this computation,
which will be justi ed by the nal result.
Due to the vanishing of the numerator (5.7), we need to have pinching in order to
get a nonzero answer. As we learned from the simple example at the beginning of the
section, we need to
nd poles that lie on di erent sides of the contour and that collide
when the regulator is removed. The poles in the integrand come from the zeroes of the
functions jM(U )j2 in the rst line of (5.4). Since, in order to obtain the Toda theory from
topological strings we wish to have b > 0, so that jqj < 1 and jtj > 1, we get from (A.29)
the expression
jM(U ; t; q)j2 = M(U ; t; q)M(U 1; t 1; q 1) =
U t iqj )(1
U 1t1 iqj 1) : (5.8)
1
Y (1
i;j=1
Thus, the zeroes of jM(U )j2 are to be found on the points
U = t mqn ;
U = tm+1q n 1
;
(5.9)
for m; n 2 N0 = f0; 1; 2; : : :g. We see that there are two classes of poles of jZtopj2, namely
those that condense around zero in the A complex plane and those that condense around
in nity.
14This can be supported by a following simple observation. The integral in our formula (2.16) for the
Toda threepoint function should be regarded as a complicated deformation of a conventional MellinBarnes
contour integral of ratio of gamma functions multiplying a hypergeometric function. The \perturbative"
part of the integrand corresponds to the deformed gamma functions, whereas the \instanton" part is the
analogue of the hypergeometric function. As the usual hypergeometric function is an entire function of its
parameters, it cannot give residue contributions to the value of the MellinBarnes integral. It is natural to
expect the same property for its deformation.
Reproducing the FateevLitvinov formula is a powerful test in support of our proposal
for 3point functions of generic Toda exponential elds.
We would, of course, like to
obtain further checks of (3.9) which is currently the work in progress. There are two
natural steps to take here. The rst one involves placing a more general semidegenerate
eld to the 3point function. Speci cally for W3, if a semidegenerate condition reads
1 = N {!2
mb!1, where m is a positive integer, it corresponds to a primary eld having
a nullvector on a level m + 1 > 1. The Toda 3point functions containing such a
eld
are also known from [8]. In fact, these are the best of the CFT knowledge for the 3point
functions of generic primaries. The corresponding formula (see (3.11) and appendix B
of [8]) involves two very di erent pieces: a straightforward generalization of (2.13) and
a 4mdimensional Coulomb integral. This intriguing factorization indeed looks like to be
reproducible from our general perspective.
The second natural step is matching the known semiclassical asymptotics [7]. We
observe that in such a limit the combinatorial functions N
factorize as
(7.1)
The sums over partitions thus disentangle, and proper generalizations of hypergeometric
identities for the case of sl(2) KM hypergeometric functions can be found to perform
them. In fact, this step could then serve as a launch pad for a more ambitious goal of
guessing a still unknown `Lagrangian' for the qdeformed Toda theory. One would have to
begin here by looking for the Lagrangian description of the qdeformed Liouville theory,
returning to the work of [21, 22]. It could well be that the 2D space has to be made
noncommutative [51{53].
Having checked the known cases, it is very interesting to go beyond them, the ultimate
goal being to compute the contour integral in (3.14) exactly for generic values of the
parameters. This will mean a considerable simpli cation of our general formula for the
3point functions of Toda primaries. Doing so requires
nding a closed form expression
for the \instanton" sum of (3.17), meaning that a suitable generalization of the KMW
sl(n) hypergeometric functions, as well as corresponding summation identities for them,
have to be found. As an exercise to do before going for this serious problem, one could
like to compute the corresponding sums for the cases with E6;7;8
avor symmetry studied
in [13, 31, 34, 35] which are obtained from the general TN by a less severe Higgsing than
the one we perform here.
Putting the above into the perspective of a full solution of the Toda theory, let us
mention the remaining ingredients of it. First, a wellknown fact is that, unlike the Virasoro
case, the WN symmetry is not restrictive enough to constrain the 3point functions of
descendent elds from those of primaries [54]. The number of corresponding Ward identities
is simply too small to
nd from them the descendent structure constants. This means
that in order to
nd all the 3point correlators, one needs to calculate independently the
3point structure constants of two primaries and one descendent. It is however rather
straightforward from the topological strings point of view.
The second remaining ingredient of a complete solution of Toda CFT are the conformal
blocks. The paper [55] describes the particular family of blocks which can be obtained
by gluing the FateevLitvinov 3point functions (2.13). Gluing the general (qdeformed)
Toda 3point functions in the same way would give the general conformal blocks of the
(qdeformed) Toda CFT. Addressing this problem for qLiouville, that is a starting point
in such an investigation, is work in progress [56]. Due to many uncertainties in properly
de ning a qdeformed Liouville (Toda) theory, such a
nding would then as well work in
opposite direction, allowing to know more about the qdeformed AGTW correspondence
and its relation to topological strings (see [57]). The novel identities for
KanekoMacdonaldWarnaar sl(n) hypergeometric functions could probably be as helpful here as they were in
the present note, to sum up known and new expressions for conformal blocks.
We nish with two remarks on the gauge theory side. The degeneration we study in
this paper, and in general Higgsing, should also be understood on the 4D/5D gauge theory
side using a generalization of the AGT correspondence with additional codimension two
halfBPS surface defects [58] as in [46, 59{61]. See also [62, 63]. The partition functions
with halfBPS surface operators can be obtained form certain 2D partition functions [64].
This 2D/4D relation has its qdeformation to a 3D/5D relation that was initiated by [21]
and further studied by [22{24]. See [65] for the latest advancements on the subject.
Lastly, by observing that the Higgsed geometry corresponding to the degeneration, see
the right side of gure 2, is related to the strip geometry T~N , see gure 7 in [11], by the
HananyWitten e ect. We refer the interested reader to [11, 66] for a nice discussion on
the subject. The invariance of the topological string amplitude under the HananyWitten
transition is nontrivial. It would be important to see how one can relate formula (2.14)
for the qdeformed structure constants to the topological string amplitude for the strip, see
equation (4.66) of [12].
Acknowledgments
We would like to thank rst our collaborators on closely related projects Masato Taki and
Futoshi Yagi. We are indebted to Volker Schomerus and Futoshi Yagi for reading the draft
of this paper and making helpful comments. In addition, we are thankful to Can Kozcaz,
Fabrizio Nieri, Jorg Teschner and Dan Xie for insightful comments and discussions. We
furthermore gratefully acknowledge support from the Simons Center for Geometry and
Physics, Stony Brook University, as well as of the C.N. Yang Institute for Theoretical
Physics, where some of the research for this paper was performed. V.M. acknowledges the
support of the Marie Curie International Research Sta
Exchange Network UNIFY of the
European Union's Seventh Framework Programme [FP7People2010IRSES] under grant
agreement n269217, which allowed him to visit Stony Brook University. E.P. is partially
supported by the Marie Curie action FP7PEOPLE2010RG. M.I. thanks the Research
Training Association RTG1670 for partial support. The research leading to these results
has received funding from the People Programme (Marie Curie Actions) of the European
Union's Seventh Framework Programme FP7/20072013/ under REA Grant Agreement
the horizontal lines as Q(nj;)i, to the vertical lines as Ql(;ji), and to tilted lines as Q(mj);i. We denote the
breathing modes as A~i(j). The index j labels the strips in which the diagram can be decomposed.
A
In this appendix, we summarize our conventions and the main properties of the special
functions that we use the most.
A.1
Parametrization of the TN junction
We gather in this appendix all necessary formulas for the parametrizations of the Kahler
moduli of the TN . First, the \interior" Coulomb moduli A~(i) = e
j
while the \border" ones are given by
a(j)
i
are independent,
Therefore, A~(0) = A~(N0) = A~(N) = 1 and we can invert relation (3.18) as
0 0
M~ i =
A~(0)
A~(i0) ;
i 1
N~i =
A~(i)
0
A~(i 1)
0
;
L~i =
i
A~(N i)
i 1
A~(N i+1)
:
j
k=1
k=1
N
k=1
(A.1)
(A.2)
(A.3)
The parameters labeling the positions of the avors branes obey the relations
N
k=1
Y M~ k = Y N~k = Y L~k = 1 ()
X mk =
X nk =
X lk = 0:
N
k=1
N
k=1
N
k=1
i
k=1
N
k=1
A~(0) = Y M~ k;
i
A~(j) = Y N~k;
0
A~(N i) = Y L~k:
i
All placements are illustrated in gure 13. The Kahler parameters associated to the edges
of the TN junction are related to the A~(j) as follows
i
Q(nj;)i =
A~(j)A~(j)
i
A~(j 1)A~(j+1)
Ql(;ji) =
A~(j)A~(j 1)
A~(j) A~i(+j11)
i 1
;
Q(mj);i =
i
A~(j 1)A~(j)
A~(j)A~(j 1) :
For each inner hexagon of (13), the following two constraints are satis ed
Ql(;ji)Q(mj);i+1 = Q(mj+;i 1)Ql(;ji+1);
Q(nj;)iQ(mj+;i 1) = Q(mj);i+1Q(nj;)i+1:
A.2
Conventions and notations for SU(N )
scalar product is de ned via (hi; hj ) = ij
N
1 . The simple roots are
For the convenience of the reader we summarize here our SU(N ) conventions. The weights
of the fundamental representation of SU(N ) are hi with PiN=1 hi = 0. We remind that the
and the positive roots e > 0 are contained in the set
+ := fhi
hj giN<j=1 = feigiN=11 [ fei + ei+1giN=12 [
N
X (hi
N
X N + 1
i=1
2
2i
hi = !1 +
+ !N 1
;
and it obeys ( ; ei) = 1 for all i. The N
1 fundamental weights !i of SU(N ) are given by hj ) =
i
X hk ;
k=1
i = 1; : : : ; N
1
and the corresponding nite dimensional representations are the ifold antisymmetric tensor
product of the fundamental representation. They obey the scalar products (ei; !j ) = ij ,
i.e. they are a dual basis. Furthermore, we nd the following scalar products useful
N + 1
2
( ; hj ) =
N
i
N
12
j
i
j > i
1)
:
(!i; !j ) =
min(i; j) (N
max(i; j))
N
;
( ; ) =
The Weyl group of SU(N ) is isomorphic to SN and is generated by the N
re ections associated to the simple roots. If
is a weight, we de ne the Weyl re ections
with respect to the simple root ei
wi
:=
2
(ei; )
(ei; ei) ei =
(ei; ) ei :
Furthermore, we de ne the a ne Weyl re ections with respect to ei as follows
wi
:= Q + wi (
Q) = wi
+ Qei =
(
Q; ei) ei ;
One can show from the alternative de nition below that the following shift identities are
obeyed
(x + b) = (xb) b1 2bx (x);
(x + b 1) = (xb 1) b2xb 1 1 (x):
where (x) := (1(x)x) . An useful implication is
(x + Q) = b2(b 1 b)x
1 + bx
1 bx
b 1x
b 1x
(x);
log (x) :=
e t
sinh2 h Q
2
x 2t i 3
sinh b2t sinh 2tb
Q
2
= 1:
which is used in the derivation of the re ection amplitude (2.9). It follows from (A.16)
that
is an entire function with zeroes at
x =
n1b
n2b 1;
or
x = (n1 + 1)b + (n2 + 1)b 1;
where ni 2 N0.
The function
<(t) > 2) of
can be connected to the Barnes Double Gamma function 2(xj!; !2).
First, we de ne 2(xj!1; !2) via the analytic continuation (the sum is only wellde ned if
Special functions
main text.
the integral
It is clear from the de nition that
In this section we gather the de nitions and properties of the special functions used in the
We begin with the function (x) which is de ned for 0 < <(x) < Q = b + b 1 as
(A.14)
(A.15)
(A.16)
(A.17)
(A.18)
(A.19)
(A.21)
:
(A.20)
HJEP08(216)
From this de nition, one can prove (see A.54 of [67]) the di erence property
2(s + !1j!1; !2) =
2(sj!1; !2)
p
2
s 1
! !2 2
2
s
!2
;
(s + n1!1 + n2!2) t5
2(s + !2j!1; !2) =
2(sj!1; !2)
3
:
t=0
p
2
s 1
! !1 2
1
s
!1
In order to express the
function using the Barnes double Gamma function, we have to
rst de ne the normalized function
b(x) :=
2(xjb; b 1)
2( Q2 jb; b 1)
:
The log of the function b(x) has an integral representation as log b(x) =
e xt
e tb)(1
e 2
Qt
e tb 1 )
e t
Q
2
t
C :
A
Then, using (A.21) we can express the (x) as
This, together with the di erence properties of 2 proves the shift identities (A.16).
We proceed by de ning some qdeformed special functions we need in the main text,
such as shifted factorials15
for positive p, which is continued to negative p according to
Q
2
2
x)
:
(x) =
1
b(x) b(Q
i=1
(U ; q)p := Y(1
U qi 1)
(U ; q)p =
1
(U qp; q) p
:
(A.22)
(A.23)
(A.24)
(A.25)
(A.26)
(A.27)
(A.28)
(A.30)
;
(A.29)
In particular for p ! 1, and for arbitrary number of q's, we have (we require for
convergence that jqij < 1 for all i)
(U ; q1; : : : ; qr)1 :=
(1
U q1i1
qrir ) :
1
Y
i1=0;:::;ir=0
We can extend the de nition of the shifted factorial for all values of qi by imposing
the relations
(U ; q1; : : : ; qi 1; : : : ; qr)1 =
1
(U qi; q1; : : : ; qr)1
:
Furthermore, they obey the following shifting properties
(qj U ; q1; : : : ; qr)1 =
(U ; q1; : : : ; qr)1
(U ; q1; : : : ; qj 1; qj+1; : : : ; qr)1
:
We then de ne the function M(U ; t; q) as
M(U ; t; q) := (U q; t; q)11 = <>> Q1
>
>
>
8 Qi1;j=1(1
> Qi1;j=1(1
>
>
>
>: Q1
i;j=1(1
U ti 1qj ) 1
U ti 1q1 j )
U t iqj )
for jtj < 1; jqj < 1
for jtj < 1; jqj > 1
for jtj > 1; jqj < 1
U t iq1 j ) 1 for jtj > 1; jqj > 1
converging for all U . This function can be written as a plethystic exponential
15A good source for the properties of the shifted factorials is [68].
M(U ; t; q) = exp
" 1
X U m
m=1
m (1
qm
tm)(1
qm)
#
;
which converges for all t and all q provided that jU j < q 1+ (jqj 1)t (jtj 1). Here and
elsewhere (x) = 1 if x > 0 and is zero otherwise. The following identity is obvious from
the de nition
M(U ; q; t) = M(U t=q; t; q) :
From the analytic properties of the shifted factorials (A.27), we read the identities
M(U ; t 1; q) =
1
M(U t; t; q)
;
M(U ; t; q 1) =
1
M(U q 1; t; q)
;
while from (A.28) we take the following shifting identities
M(U t; t; q) = (U q; q)1M(U ; t; q);
M(U q; t; q) = (U q; t)1M(U ; t; q) :
1
Y
n1;n2=0
M(q x; t; q)
M(q qt ; t; q)
2
(1
;
:=
M
r t
q
; t; q
2
:
(1
qx+n1 1+n2 2)(1
q + x+n1 1+n2 2)
q +=2+n1 1+n2 2)2
We de ne the qdeformed
function as
q(xj 1; 2) = (1
q) 11 2 (x 2+ )2
= (1
q) 11 2 (x 2+ )2
where we have used the de nition (3.13) for the norm squared. From time to time we will
use the shorthand notation
If follows from the de nition (A.34) that q( +=2j 1; 2) = 1, that
q(xj 1; 2) =
xj 1; 2) and that
q(xj 1; 2) =
q(xj 2; 1). Furthermore, from the shifting identities for
M, we can easily prove that
q(x + 1j 1; 2) =
1
1
q 2
1 2 2 1x
q 2 (x 2 1) q(xj 1; 2) ;
together with a similar equation for the shift with 2. Here, we have used the de nition of
the qdeformed
and
functions
q(x) := (1
q)1 x (q; q)1 ;
(qx; q)1
q(x) :=
= (1
q)1 2x (q1 x; q)1 ; (A.37)
(qx; q)1
valid for jqj < 1. They obey q(x+1) = 11 qqx q(x), implying q(x+1) = (1 qx)(1 q x) q(x).
(1 q)2
Because of the normalization of
q(xj 1; 2) and since the factors of the right hand side
of (A.36) have a well de ned limit for q ! 1, we nd by comparing functional identities that
q(xj 1; 2) q!!1
(xj 1; 2) :=
2 2+ j 1; 2
2
2 xj 1; 2 2 +
xj 1; 2
:
(A.38)
where 2 is the Barnes Double Gamma function. In particular, the usual function (x)
introduced in [4] is equal to
(xjb; b 1). We shall often just write
q(x) instead of q(xj 1; 2)
and indicate in the text whether the i parameters are arbitrary or whether b = 1 = 2 1.
(A.31)
(A.32)
(A.33)
(A.34)
(A.35)
q( +
(A.36)
HJEP08(216)
at x = 0 is due to the factor (1
piece of the derivative that survives is
We will also need to evaluate the derivative of q(x) at x = 0. Since the zero of q(x)
qx) in the numerator of (A.34), we nd that the only
0q(0) =
1
q(b) :
From this formula we can then obtain an identity useful for the calculations of the main
text. Let us de ne the norm squared of the re ned McMahon function following [29]:
U!1
1
U 1
jM (t; q)j2 := lim jM(U ; t; q)j2 = jM(q 1; t; q)j2 = (1
Then, from (A.35) and (A.39) we get for 1 = b and 2 = b 1
( 1 2)
q) 4 1 2
2
q(
1
) :
We shall use in the following
`( )
X
i=1
i
jj jj2 :=
`( )
X
i=1
2
i
;
n( ) :=
`( )
X(i
i=1
1) i = jj tjj2
2
j j
;
(A.42)
where `( ) is the number of rows of the partition . We also de ne the relative armlength
a (s), armcolength a0 (s), leglength l (s) and legcolength l0 (s) of a given box s of the
partition
with respect to another partition
as:
a (s) := i
j ;
a0 (s) := j
1 ;
t
l (s) := j
i ; l0 (s) := i
1 :
(A.43)
It is of course also possible to have
= . The (q; t)deformed factorial of U depending on
a partition
is then given as a following product over its boxes:
(U ; q; t) := Y(U t1 i; q) i = Y(1
U qa0(s)t l0(s)) :
(A.44)
The next piece of notation that we need are the (q; t)deformations of the hook product
of a Young diagram
. There are two inequivalent ways for this number to be deformed to
a twovariable polynomial, namely:
h (q; t) := Y(1
qa(s)tl(s)+1) ;
h0 (q; t) := Y(1
qa(s)+1tl(s)) :
(A.45)
Our last de nition is that of the 5D uplift of Nekrasov functions, which we write as
s2
s2
N (u; 1; 2) :=
2 sinh
u + 1( i
j + 1) + 2(i
t
j
Y 2 sinh
s2
2
[u
1a (s) + 2 (l (s) + 1)]
2
2 sinh
u + 1(j
i) + 2( tj
i + 1)
(A.46)
=
Y 2 sinh
[u + 1 (a (s) + 1)
2l (s)]
s2
Y
Y
where the products are taken over boxes of partitions
and , respectively. By pulling
some factors out of the products, the de nition can also be rewritten as
r t 1
q U
Y
j j+2j j
jj tjj2 4 jj tjj2 q jj jj2 4 jj jj2
1
U t tj+i 1
q i+j ;
Y
u. For particular values of the parameter u, the introduced functions behave
where + = 1 + 2. Furthermore, they obey the exchange identities
(A.47)
(A.48)
(A.49)
(A.50)
(A.51)
Finally, there are two relations involving the functions we just de ned, namely
N (
+) = N (0) =
N (u;
2
1) = N t t (u
+; 1; 2);
N (u; 2; 1) = N t t (u; 1; 2):
1
h (q; t)h0 (q; t)
(
1
)j jt jj 2tjj2 q jj jj2
2
N
(0)
(U )
(U ; q; t) =
r t
U
j j
2
t jj 4tjj2 q jj 4jj2 N
;
as well as
where U = e
u
.
B
The sl(N ) KanekoMacdonaldWarnaar hypergeometric functions
This appendix contains the derivation of the summation formula (6.5) used in the main
text. It exploits a binomial identity for the KanekoMacdonaldWarnaar extension of
basic hypergeometric functions [69] which generalizes the KanekoMacdonald sl(2) identity
of [70{72].
B.1
The sl(N ) KMW hypergeometric functions and their qbinomial identity
The Macdonald polynomials P (x; q; t) (in the case of in nite alphabet x referred as the
Macdonald symmetric functions) are labeled by a number partition
= ( 1; : : : ; `( )) and
form an especially convenient basis in the ring of symmetric functions of x = (x1; x2; : : : )
over the eld F = Q(q; t) of rational functions in two variables q and t [73].
0
X
(A1; : : : ; Ar+1; q; t) (N 1)
(
1
);:::; (N 1) (qtkN 1 1; B1; : : : ; Br; q; t) (N 1)
Y
s=1
Y
Y
Y
s=1 i=1 j=1
h0 (s) (q; t)
tn( (s)) (qtks 1; q; t) (s) P (s) (x(s); q; t)
N 2 ks ks+1 (qtj i 1+ks ks+1 ; q) (s) (s+1)
(qtj i+ks ks+1 ; q) (s) (s+1)
i
i
j
j
;
where the integer parameters ks are such that 0
summations are performed over partitions (s), 1
k0 < k1 < k2 <
< kN 1 and the
s
N
1 satisfying ks
`( (s)). We
have used here the de nitions (A.24), (A.42), (A.44), (A.45). The prime symbol above
marks the fact that entries of the partitions giving a nonzero contribution to the sum all
satisfy an additional condition
i ks+ks+1 for 1
i
ks. It provides a convenient
visualization of the multiple sum as running over single skew plane partitions of shape
, where
= (kNN 11) is a rectangle and
= (kN 1
k1; : : : ; kN 1
kN 2).
In the following, it will be enough to restrict ourselves to a socalled principal
specialization of a Macdonald polynomial, for which the string of arguments x is set to
~x := z(1; t; : : : ; tk 1):
P (~x; q; t) = zj jtn( ) (tk; q; t)
:
h (q; t)
The corresponding specialization of the sl(N ) multiple qbinomial theorem is then
writ(B.1)
(B.2)
(B.3)
N 1 ks ks 1 (Azs
Y Y
s=1
Y
i=1
(zs
ks ks 1 (qzs
Y
1 s r N 2 i=1
zN 1ti+s+ks 1+ +kN 2 N ; q)1
zN 1ti+s+ks 1+ +kN 2 N ; q)1
(zs
zrti+s r+ks 1+ +kr kr+1 2; q)1 ;
zrti+s r+ks 1+ +kr 1 1; q)1
s
N
1 and \ " indicates the absence of the Having many nice properties, the Macdonald polynomials are applied in various areas of contemporary mathematics. One of them is the theory of sl(N ) KanekoMacdonald
Warnaar analogues of basic hypergeometric functions. These functions, of type (r + 1; r),
are de ned as
r+1 r
B1; : : : ; Br
A1; : : : ; Ar+1 ; q; t; x(
1
); : : : ; x(N 1)
ten as:
Theorem: [See [69], Cor. 3.1]
1 0
A
; q; t; ~x(
1
); : : : ; ~x(N 1) =
where ~x(s) := zs(1; t; : : : ; tks 1) for 1
parameters Bi in the de nition (B.1).
B.2
The summation formula
It will be convenient for the subsequent argument to rewrite the above formula (B.3)
in the topological string conventions. This turns out to be possible due to the
identities (A.29), (A.50), (A.51) and the following lemma:
k1 k2 (Atj i) 1;i 2;j
i=1 j=1 (Atj i+1) 1;i 2;j
= t k1j 2j 2k2j 1j
.
N
( a
2;
( a)
k1 2) N
; 1
( a + k2 2)
;
(B.4)
Proof. Let us
rst notice that by using de nition (A.47) as well as exchange
identities (A.49), the righthand side of the above formula can be written as a following product:
t k1j 2j 2k2j 1j
k1 2) N
( a)
N
Y
A t t t2;j iq 1;i j+1
Y
q
A t t t1;j+i 1q 2;i+j
A t t k1+i 1q 2;i+j : (B.5)
HJEP08(216)
In proving the lemma, we will deal with formal power series in variables t and q, so that we
will not be concerned with issues of convergence of the intermediate expressions, requiring
only that t; q 6= 1. We also extend the entries of partitions 1 and 2, such that
1;i := 0; i > `(
1
);
2;i := 0; i > `( 2)
and for now assume `(
1
) = k1, `( 2) = k2.
So, let us start with the following obvious identity:
1
k1 k2
X X +
1
X
k2
X + X
k1
1
X
i=1 j=1 i=k1+1 j=1 i=1 j=k2+1
q 1;i 2;j :
k1 1
k1 k2
k1
i=1
k2
j=1
q 2;j + X t i+k2 1
q 1;i :
k1
i=1
Taking the last two sums of the righthand side, shifting their summation indices and using
convention (B.6), one gets:
1
X
k2
X + X
k1
1
X
i=k1+1 j=1
i=1 j=k2+1
q 1;i 2;j
= X X tj i k1 1
q 2;j + X X tj i+k2 1
q 1;i
k2
j=1
X tj 1 k1 1
q 2;j + X t i+k2 1
q 1;i
;
(B.8)
where in the last step we used the sum of an in nite geometric progression. Substituting
this back and multiplying the whole expression by t 1
1, we obtain:
(t 1
q 1;i 2;j = (t 1
1) X X tj i 1
1 k2
1
t 1
1
1
i;j=1
(B.6)
(B.7)
(B.9)
Now we will use the following identity which the reader can nd for instance in [32]:
1
1) X q 1;it1 i = (q 1
1) X t t1;iqi:
(B.10)
1
i;j=1
X tj 1qi t 1;iq 2;j
t
1 =
Multiplying it by Pj1=1 tj 1q 2;j and subtracting from the result the same with 1, 2 set
(t 1
q 1;i 2;j = (q 1
1) X tj 1qi t 1;iq 2;j
t
1 :
(B.11)
Substituting this back as a lefthand side of (B.9) and dividing everything by q 1
1, we
k1
i=1
k1 1;i
s2 2
X
t t2;j iq 1;i j+1 +
ti t1;j 1qj 2;i: (B.14)
= X X
X
k1 k2 1;i 2;j
i=1 j=1 l=1
tj i ql:
(B.15)
X X q tj i 1
k2
j=1
+ X q1 2;j tj 1 k1 1
1
q 2;j
+ X qt i+k2 1
1
q 1;i
(B.12)
s2 1
X
(i;j)2 1
X
(i;j)2 2
For clarity, the upper bound of the rst summation on the right is written schematically,
implying that for terms having 1;i
2;j < 0 the sum should be replaced by an equivalent
corresponding to a negative Pochhammer symbol.
For the lefthand side one now should employ an identity from [74] (our t and q are
interchanged with respect to the formula there):
tj 1 t1;iqi 2;j
tj 1qi =
X tl 2(s)qa 1(s)+1 + X t l 1(s) 1q a 2(s)
Interchanging the indices in the second summand of the righthand side of (B.13), changing
the summation order in the third summand and moving them to the left, one nally obtains:
t t2;j i
tk2 i q 1;i j+1 +
t t1;j+i 1
t k1+i 1 q 2;i+j
1
i;j=1
1
X
i;j=1
1
X
i;j=1
X
(i;j)2 1
where one can now use the formula for nite geometric progression to get rid of the fractions
in the righthand side:
tj 1 t1;iqi 2;j
tj 1qi =
+ X X tj 1 k1qi 2;j + X X t i+k2qj:
(B.13)
Substituting here t; q
logarithm, we get
! tr; qr, multiplying by A qt r=r and using a series expansion of the
X
ln
A t t t2;j iq 1;i j+1 !
X
ln
A t t t1;j+i 1
q 2;i+j !
A t t k1+i 1q 2;i+j
=
X
k1 k2
X ln
1
Atj iql 1 !
Atj i+1ql 1 :
(B.16)
HJEP08(216)
Exponentiation concludes the proof.
Remark. Tracing the above argument, one can see that it can be literally extended to
the case `(
1
)
k2. This will be crucial for what follows.
Having the lemma, we now can show that (B.3) is equivalent to:
(
1
);:::; (N 1) i=1
0
X
"N 2
Y
Y
i=1
Y
N 1
Y
Theorem:
"N 1 N (i) (i 1) ((ki 1
ki) 2
N (i) (i) (0)
t
t zN 1 kN 2+kN 1 j (N 1)
j
2
r
+) #
A
N; (N 1) ( a)
(B.17)
M ti (j+1)+ki kj+1 Qjs=i(zstks) M qt t(i 1) j+ki 1 kj Qjs=i(zstks)
1 i j N 2 M t t(i 1) (j+1)+ki 1 kj+1 Qjs=i(zstks) M 1q ti j+ki kj Qjs=i(zstks)
M Aq ti (N 1)+ki kN 1 QsN=i1(zstks) M qt t(i 1) (N 1)+ki 1 kN 1 QsN=i1(zstks)
i=1 M Aqt t(i 1) (N 1)+ki 1 kN 1 QsN=i1(zstks) M 1q ti (N 1)+ki kN 1 QsN=i1(zstks)
:
Finally, we are in position to prove the required summation formula:
X
Y
"N 1 VipUiUi+1
j (i)j #
(
1
);:::; (N 1) i=1
N (i) (i) (0)
i=1
Y N (i+1) (i) (ui+1
+=2)
#
+=2) N; (N 1) (uN
+=2)
(B.18)
N 1 N i
Y
Y
i=1 j=1 M
M
q qt Ui+j
Qis+=jj 1(VsUs) M q Uj
t Ui+j
Qis+=jj 1(VsUs)
Qis+=jj 1(VsUs) M
q t 1
q Uj
Qis+=jj 1(VsUs)
;
(multiplied by q qt , single site parameters are excluded).
with N site parameters Ui = e
ui and N
1 link parameters Vj . One can visualize the
righthand side of this formula by noticing that the arguments of numerator are precisely
all the simplyconnected combinations of even number of site and link parameters
(multiplied by qt when starting with a link parameter), whereas the arguments of denominator
represent all the simplyconnected combinations of odd number of site and link parameters
Proof. We use a socalled specialization technique [73]. Let us group all terms on the left
having the same powers of Vi, i = 1; : : : ; N
1, i.e. grade our in nite sum with respect to a number of boxes of partitions we sum over. The coe cient of each combination of
V i1
1
V iN 11 is a polynomial in variables Ui, i = 1; : : : ; N of degree 2(i1 +
N
+ iN 1),
having its coe cients in F. Similarly, expanding the righthand side as a series in Vi
and resumming geometric progressions in q; t into rational functions, we learn that the
corresponding coe cients are as well polynomial in variables Ui with coe cients in F.
Let us now take any ordered combination of positive integers ki, k1 <
< kN 1
,
such that
ki+1
(i)
ki
(i+1)
`( (i+1)):
One can see that the condition s
s ki+ki+1 is trivially satis ed in this way, turning
the corresponding skew plane partition into a horizontal strip plane partition. Making the
following specialization of Ui (remember that k0
0):
and reparametrizing the remaining variables as
as well as
Ui =
r t tki ki 1 ;
t t
Vj =
r q zj tkj 1+kj kj+1 ;
j = 1; : : : ; N
2
UN =
r q 1
t A
VN 1 =
r t
A
zN 1 tkN 2
(B.19)
HJEP08(216)
(B.20)
(B.21)
(B.22)
one can readily check that formula (B.18) then degenerates to the established sl(N )
qbinomial identity (B.17). Correspondingly, the above statement on equality of two
polynomial coe cients translates into a statement on equality of corresponding polynomial
coe cients of z1i1
N
1
ziN 1 , which holds true.
We see that two polynomials in N
1 variables16 coincide on an (N
1)dimensional semilattice, meaning they just coincide. Term by term, this proves the theorem. Finally, let us remark that the summation formula (B.18) for N = 2
X
(
1
)
V1pU1U2
j (
1
)j N (
1
); (u1
+=2) N; (
1
) (u2
N (
1
) (
1
) (0)
+=2)
=
M U1V1 M qt V1U2
q qt U1V1U2
(B.23)
{ 40 {
reproduces the nontrivial part of (5.3) of [11], whereas, taken for N = 3
X
+=2) N (2) (
1
) (u2
+=2) N; (2) (u3
+=2)
N (
1
) (
1
) (0) N (2) (2) (0)
(B.24)
M U1V1 M qt V1U2 M U2V2 M qt V2U3 M U1V1U2V2 M qt V1U2V2U3
M
q qt V2 M
q qt U1V1U2 M
q qt V1U2V2 M
q qt U2V2U3 M
q qt U1V1U2V2U3
;
HJEP08(216)
it is equivalent to the formula (6.7) conjectured in [13].
C
We saw in section 5 how for T3 the semidegeneration of the mass parameters mi pinches
the integral contour, so that the W3 structure constants are given by a
nite number of
residues  one or two depending on the choice of contour in
gure 10. The purpose of
this section is to show a similar computation in the T4 case, in order to illustrate some of
the complexities that arise when we are confronted with iterated contour integrals. For
simplicity of notation, we set A1
A(11), A2
A(
1
) and A3
2
A(12). From (3.16), we read
the \perturbative" part of the the topological string partition function
Z4
Q4k=1 M
M
M
q t A1
q M~kN~1
A21
A2N~1
M
q t A1A3
q A2N~1N~2
M
q qt AA1 N3~2
A22L~4
A1
M
M
M
Q
M
M
1 i<j 4 M M~j
~
Mi
q t A1M~k
q A2
A2N~1
A21
q t A2A3
q A1L~1L2
~
A1
A22L~4
q A3L~3
M
M
M
A23
N~1N~2L~1L~2
M
A1A2L~4
~
N1
M
q qt A2M~ kL~4
q t A3N~3
q L~1
M
M
~
N1
A1A2L~4
M
N~1N~2L~1L~2
A23
q t A1A3
q N~1L~1L2
~
M
q t A2A3L~4
q N~1N~2
q t A3N~4
q L~2
2
(C.1)
2
2
:
In addition, the \instanton" part (3.17) takes for N = 4 the form
Z4
inst = X
N~1L~3 ! j 1(
1
)j+j 2(
1
)j+j 3(
1
)j
2
N~2L~2 ! j 1(2)j+2j 2(2)j
N~3L~1 ! j 1(23)j
N (
1
) (2)
2 2
N (
1
) (
1
)(0)N (2) (2)(0)N (3) (3)(0)N (
1
) (
1
)(0)
1 1
1 1
1
1
2 2
N (2) (2)(0)N (
1
) (
1
)(0)N (
1
) (
1
)( a1 + 2a2 + l4)
2 2
3 3
2 3
N 3(
1
);
2 3
1 2
1 1
1 1
Q
2
Q
2
(C.2)
N (
1
) (
1
)( a1 + 2a2
Q + l4)N (
1
) (
1
)(2a1 a2 n1)N (
1
) (
1
)(2a1 a2
Q
a2 l
Q2 N (
1
) (2) a1 + a3 l
m2 n1
Q
2
1 3
N 1(
1
); a1
m3 n1
Q2 N 1(
1
); a1
N 1(2); 2(2)(2a3 l
1 l
2 n1 n2)N 1(2); 2(2)(2a3
Q l
1 l
2 n1 n2)
1 2
Q2 N 1(
1
); a1
Q + l4 n1)
a1 + m1 + n1
N (2) (
1
) a1 a2 +a3 n1 n2
N (2) (
1
) a2 +a3 +l4 n1 n2
1 2
1
2
Q
2
Q
2
Q
2
1 2
1 3
m4 n1
Q2 N; 1(
1
)
1 3
Q
2
N (
1
) (2) a1 a3 +n2
N (3) (2) a3 l2 +n4
N (2) (3) a3 l1 +n3
where the summation goes over partitions
= f 1(
1
); 2(
1
); 3(
1
); 1(2); 2(2); 1(3)g. Let us
perform the contour integrals over the Coulomb moduli Ai's. As demonstrated in 5, there are
multiple ways to choose the contour in such a way that the contours gets pinched in the
semidegeneration limit. We will in this appendix just show the computation for a contour
that leads to a single residue contributing. We have also performed the computation for
other contours and, up to an irrelevant multiplicity, have obtained the same results.
Let us start by looking at the mass parameters. Using the T4 parametrization of (A.4),
we nd the expressions for the Kahler parameters Q(mj);i and Ql(;ji). The mass parameters for
the 5branes on the left side of the T4 junction are parametrized as follows
t 2 K~ d1 ;
M~ 2 =
1
t 2 K~ d2 ;
M~ 3 =
q
1
2 K~ d3 ;
M~ 4 =
q
3
2 K~ 3
;
(C.3)
the regulators i are
by the zeroes coming from the
M
M~ iMj
~ 1 2 in the numerator.
with Qi3=1 di = 1. We set di = e
i with Pi3=1 i = 0. We will compute the integrals in
the order A1, A2 and A3 and are interested in the result in the limit a ! 0. Thus, in the
calculation of the contour integrals, we will only keep the residues that will diverge when
nally all set to zero. Their divergences will be canceled in the limit
Let us now consider the contour integral over A1. The possible contributing poles
come from the following terms in the denominator of (C.1)
3
Y
j=1
M
r t
A1
q M~ j N~1
! 3
Y
k=1
M
r t A1M~ k ! 2
A2
We number the terms with j = 1; 2; 3 as 1 to 3 and those with k = 1; 2; 3 as 4 to 6 and
we need to investigate which of them might pinch the integral contour. The situation for
imaginary a is depicted in
gure 14. We see that for jKj > 1 and imaginary masses ni
and li the contour for A1 can be chosen in such a way that in the limit a ! 0 only one
~
1.5
1.0
lim
a!0
I
a!0
a!0
3
Y
k=1
I
I
3
Y
3
Y
dAk
2 iAk jM (t; q)j2
Z4
top 2
dAk
2 iAk jM (t; q)j2 Res
Z4
top 2
; A1 = K~ N~1d1
q
dAk
2 iAk jM (t; q)j2
M q d3 M qt22 K~ 4d2 M q
t d2
t K~ 4d3
2
M K~ 2N~1 M t K~ 2N~1 M qt A2K~ L~4d2 M A2K~ L~4d3
A2d2 q A2d3
2
q A2 q A22L~4 A22L~4
M t K~ 2N~1d12 M qt A2K~ L~4d1 M t K~ N~1d1 M K~ N~1d1
2
M
A23
3
A23
q q A2N~2
t A3K~ d1 M
2
M qt AK~2 L3~4
M qt AN~21Kd~12
q A3L~3 M
q q
A2A3
t K~ N~1L1L~2d1 M
~
q qt AN2~A1N~3L2~4 2
M N~1N~2L~1L~2 M q N~1N~2L~1L~2
M
q qt AL~3N2~4
M
q qt AL~3N1~3
M
t 2 A3K~ d1
L~1L2
~
M
t 23 K~ N~1N~2d1
A3
inst 2
A1= qt K~ N~1d1
are taken to zero, the integral is given by just one residue whose position is indicated by a small
circle.
residue contributes, namely the one for
A1 =
t K~ N~1d1 :
Thus, we can compute the integral over A1 just as in the T3 case and, after some simpli
cations, obtain the integral expression
where we have used (5.13).
denominator of the integrand in (C.6) are
M
A2d2
K~ 2N~1
M
q A2d3
t K~ 2N~1
We must now perform the integration over A2. We nd that the relevant terms in the
q
From the above, we read that there are two poles that are potentially relevant for the
semidegenerate limit, namely those for
A2 =
t K~ 2N~1d3 1;
A2 = K~ 1L~ 1
4 d 1
3 :
These are the two residues that could contribute due to pinching. We need now to set the
exact integral contour for A2 to see which one of them actually contributes. The contour
can be chosen in such a way as to have the residue at A2 = qt K~ 2N~1d3 1, but not the one at
A2 = K~ 1L~ 1
4 d3 1. Finally, we have to compute the integral over A3. Arguments similar
to the ones used for A2 tell us that the contour can be chosen such as to have a pinching
when the regulators are removed at the pole
A3 =
r t K~ N~1N~2d1:
Performing the same kind of computation that led to (C.6), we obtain the integral in the
2
Z4
inst 2
A~(j)
i ! qt
i(4 2i j)
K~ i Qjk=1 N~k
: (C.10)
semidegenerate limit
lim
a!0
I
3
Y
k=1
dAk
2 iAk jM (t; q)j2
Z4
top 2
K~ 4
M
Qi4=1 M
N~5 iL~i
~
K
Computing the \instanton" contribution to residues, we nd that inserting the values of he
Coulomb moduli, namely (C.5), the left part of (C.8) as well as (C.9) into (C.2) immediately
gets rid of the sums over 1(
1
), 1
(2) and 2(
1
) due to (A.48). Thus, we obtain the \instanton"
contribution to the contour integral in the semidegenerate limit:
Z4
inst
A1= qt K~ N~1d1;A2= qt K~ 2N~1;A3=q qt K~ N~1N~2
X
1(3); 2(2); 3(
1
)
N 1(3);(n4 + l1
N~3L~1 ! 2
(3)
1
2 1
N~2L~2 ! 2
N~3L~3
2
N~1L~3 ! 2
N~2L~4
(
1
)
3
{)N (
1
) (2)(n2 + l3
3
2
1 1
2 2
3 3
{)N; 3(
1
)(n1 + l4
: (C.11)
We can now plug the summation formula (6.5) in (C.11) and inserting the result in (C.10)
we get the nal result:
lim
a!0
I
3
Y
k=1
dAk
2 iAk jM (t; q)j2
Z4
top 2
M(K~ 4) Q
1 i<j 4 M
N~j=N~i M
L~i=L~j
Qi4;j=1 M(N~iL~j K~ 1) 2
Thus, we obtain our general formula (6.4), specialized for N = 4.
(C.8)
(C.9)
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