Exact partition functions for the Ωdeformed \( \mathcal{N}={2}^{\ast } \) SU(2) gauge theory
HJE
SU(2) gauge theory
Matteo Beccaria 0
Guido Macorini 0
0 Via Arnesano , 73100 Lecce , Italy
We study the low energy e ective action of the gauge theory. It depends on the deformation parameters 1; 2, the scalar eld expectation value a, and the hypermultiplet mass m. We explore the plane ( m ; 2 ) looking for special features in the multiinstanton contributions to the prepotential, motivated by what happens in the NekrasovShatashvili limit 2 ! 0. We propose a simple condition on the structure of poles of the kinstanton prepotential and show that it is admissible at a nite set of points in the above plane. At these special points, the prepotential has poles at xed positions independent on the instanton number. Besides and remarkably, both the instanton partition function and the full prepotential, including the perturbative contribution, may be given in closed form as functions of the scalar expectation value a and the modular parameter q appearing in special combinations of Eisenstein series and Dedekind function. As a byproduct, the modular anomaly equation can be tested at all orders at these points. We discuss these special features from the point of view of the AGT correspondence and provide explicit toroidal 1blocks in nontrivial closed form. The full list of solutions with 1, 2, 3, and 4 poles is determined and described in details. ArXiv ePrint: 1606.00179
Extended Supersymmetry; Nonperturbative E ects; Supersymmetric E ec

2
INFN, Via Arnesano, 73100 Lecce, Italy Dipartimento di Matematica e Fisica \Ennio De Giorgi", Universita del Salento,
deformed N = 2 SU(2)
1
tive Theories
1 Introduction and results
2 Looking for simplicity beyond the NekrasovShatashvili limit
2.1
Back to the instanton partition functions
3 AGT interpretation
3.1 Explicit CFT computations
4 Perturbative part of the prepotential at the points Xi
5 Full prepotential and generalization to N poles points
5.1
Constraints from the modular anomaly equation
5.2 A worked out 3pole example
6 Predictions for the torus 1block
A 3 and 4poles Nekrasov functions data B On the structure of higher instanton Nekrasov functions 1 4
5
deformed N = 2 SU(2) gauge theory in four dimensions
and present novel closed expressions for its low energy e ective action at special values of
the deformation parameters. On general grounds, before deformation, the e ective action
of N = 2 theories is computed by the SeibergWitten (SW) curve [1, 2]. It is the sum
of a 1loop perturbative correction and an in nite series of nonperturbative instantonic
contributions that are weighted by the instanton counting parameter q = ei
where
is
the complexi ed gauge coupling constant at low energy. Due to N = 2 supersymmetry,
the full e ective action may be expressed in terms of the analytic prepotential F (a; m)
depending on the vacuum expectation value a of the scalar in the adjoint gauge multiplet
and on the mass m of the adjoint hypermultiplet [3].
Instead of applying the SW machinery, one may compute the e ective action by
topological twisting the theory and exploiting localization on the manyinstanton moduli
spaces [4{6]. Technically, this is made feasible by introducing the socalled
deformation
of the theory, i.e. a modi cation breaking 4d Poincare invariance and depending on two
parameters 1; 2 such that the initial theory is recovered when 1; 2 ! 0. The role of
{ 1 {
the deformation is that of a complete regulator for the instanton moduli space
integration [7{14]. In this approach, it is natural to introduce a well de ned partition function
Zinst( 1; 2; a; m) and its associated nonperturbative deformed prepotential by means of
F inst( 1; 2; a; m) =
1 2 log Zinst( 1; 2; a; m):
(1.1)
It is well established that the quantity in (1.1) is interesting at nite values of the
deformation parameters 1; 2, i.e. taking seriously the deformed theory. This is because the
amplitudes appearing in the expansion F pert + F inst = Pn1;g=0 F (n;g) ( 1 + 2)2n ( 1 2)g are
related to the genus g partition function of the N = 2 topological string [15{21] and
satisfy a powerful holomorphic anomaly equation [22{25]. Actually, understanding the exact
dependence on the deformation parameters is an interesting topic if one wants to resum
the above expansion in higher genus amplitudes. Clearly, this issue is closely related to the
AldayGaiottoTachikawa (AGT) correspondence [26] mapping deformed N = 2 instanton
partition functions to conformal blocks of a suitable CFT with assigned worldsheet genus
and operator insertions. AGT correspondence may be checked by working order by order
in the number of instantons [27{29]. For the N = 2
deformed SU(2) gauge theory the
relevant CFT quantity is the onepoint conformal block on the torus, a deceptively simple
object of great interest [27, 28, 30{37].
The AGT interpretation emphasizes the importance of modular properties in the
deformed gauge theory. Indeed, it is known that SW methods can be extended to the case
of nonvanishing deformation parameters 1; 2 [38, 39] and modular properties have been
clari ed in the undeformed case [40, 41] as well as in presence of the deformation [42, 43].
The major outcome of these studies are explicit resummations of the instanton expansion
order by order in the large a regime. The coe cients of the 1=a powers are expressed in
terms of quasimodular functions of the torus nome q. This approach can be pursued in
the gauge theory [44{50], in CFT language by AGT correspondence [34{36, 51], and also
in the framework of the semiclassical WKB analysis [52, 52{57].
An important simpler setup where these problems may be addressed is the socalled
NekrasovShatashvili (NS) limit [58] where one of the two
parameters vanishes. In this
case, the deformed theory has an unbroken two dimensional N = 2 superPoincare
invariance and its supersymmetric vacua are related to the eigenstates of a quantum integrable
system. Under this Bethe/gauge map, the nonzero deformation parameter
plays the role
of ~ in the quantization of a classically integrable system. Saddle point methods allow to
derive a deformed SW curve [59, 60] that can also be analyzed by matrix model
methods [33, 52, 53, 61, 62]. In the speci c case of the N = 2 theory, the relevant integrable
system is the elliptic CalogeroMoser system [58] and the associated spectral problem
reduces to the study of the celebrated Lame equation. Besides, if the hypermultiplet mass
m is taken to be proportional to
with de nite special ratios m = n + 12 , where n 2 N, the
k
spectral problem is ngap. Remarkable simpli cations occur in the kinstanton
prepotential contributions F inst [63] that may be obtained by expanding the eigenvalues of a Lame
equation in terms of its Floquet exponent. As a byproduct of this approach, it is possible
to clarify the meaning of the poles that appear in the kinstanton prepotential at special
{ 2 {
values a = O( ) of the vacuum expectation value a. Indeed, the pole singularities turn out
to be an artifact of the instanton expansion.
In this paper, we inquire into similar problems when both the deformation parameters
are switched on, i.e. by going beyond the NekrasovShatashvili limit. In particular, we
explore the ( ; ) plane where ; are real parameters entering the scaling relation
m =
1
;
2 =
1
:
(1.2)
In other words, we keep the hypermultiplet mass to be proportional to one deformation
parameter with ratio , but 1; 2 are generic ( is just a convenient replacement of 2). By
dimensional scaling, the prepotential is a function Fe( ; )( ) of the combination
at the xed point ( ; ).1 The dependence on q is not written explicitly. After this stage
= 2 a= 1
preparation, the claim of this paper is the following
There exists a nite set of N poles points ( ; ) such that the kinstanton prepotential is
a rational function of
with poles at a xed set of positions
2 f 1; : : : ; N g independent
on k.
This claim is motivated by our previous analysis in the restricted NekrasovShatashvili
limit [63] and is far from obvious. Most important, it has far reaching consequences. At the
special N poles points, we show that the instanton partition function and the perturbative
part of the prepotential take the exact form
hm log
with coe cients depending on ; , and hm 2 N. The total prepotential is thus remarkably
simple and reads
Fe( ; )( ) =
hm log
N
n=1
log 1 + X
conformal dimension hm  the perturbative part providing interesting special instances of
the 3point DOZZ Liouville correlation function. These results are derived and tested by
giving a complete list of all the N
4 poles points. These turns out to be 4, 7, 12, and 11
(1.3)
(1.4)
at N = 1; 2; 3; 4 respectively.
of the variable .
1We shall systematically add a tilde to quantities that are considered under (1.2) and expressed in terms
{ 3 {
Our results complement a similar analysis that has been performed in [64] on a special
class of N = 2 gauge theories on S4 with fundamental matter. In this reference, it has
been shown that, for a speci c choice of the masses, the partition function localizes around
a nite set of critical points where it can be explicitly evaluated and written in terms of
generalized hypergeometric functions. From the AGT perspective, these special points may
be viewed as a four point correlator involving the insertion of a degenerated eld.
The plan of the paper is the following. In section 2 we determine the 1pole points by
a direct inspection of the instanton prepotential contributions. In section 2.1 we discuss
the special features of the instanton partition function at the 1pole points. The AGT
interpretation is analyzed in section 3 where we also provide various explicit CFT tests
of the proposed partition functions. In section 4 we discuss the perturbative part of the
prepotential at the 1pole points. In section 5 the analysis is extended to N poles points
and the cases N = 2; 3 are fully classi ed. Finally, section 6 presents a list of special
toroidal blocks. Various appendices are devoted to additional comments.
2
Looking for simplicity beyond the NekrasovShatashvili limit
As discussed in the Introduction, we are interested in the scaling limit (1.2). The instanton
partition function is Zinst = Zinst( 1; 2; a; m) and it is convenient to introduce
Ze(ins;t )( ) = Zinst
1
;
1
; 1 ;
2
1
= Zinst 1; ; ;
;
2
where we used dimensional scaling independence to remove 1. Similarly, we de ne
F inst =
1 2 log Zinst;
with a polynomial Pk( ) and a single pole 1
0 in the variable j j. The Ansatz (2.3)
is a nontrivial requirement. It is motivated by the analysis in [63], but its admissibility
is actually one of the results of our investigation. To explore the constraints that (2.3)
imposes, we begin by looking at the simple oneinstanton case.2 For k = 1 we have the
explicit expression
e1
F inst( ) =
(2
2The functions Fekinst may be computed by the beautiful Nekrasov formula [5]. Alternatively, for a gauge
algebra g 2 fAr; Br; Cr; Drg one can also apply the methods described in [4, 5, 8, 11, 65{67].
{ 4 {
(2.1)
(2.2)
(2.3)
and there is a simple pole 1 = j + 1j. At the twoinstanton level, k = 2, the denominator
of Fe2( ) turns out to vanish at
j j =
+ 1(order 2);
Special cases occur when one of the poles coincides with those at 1 = j +1j. This happens
for
2
3
only nontrivial cases consistent with (2.3)3
is not in the set (2.6), one checks that Fe2 takes the form (2.3) if
Pushing the calculation up to 12 instantons, we con rm that the points in (2.7) and (2.8)
agree with the Ansatz (2.3). Thus, the 1pole condition (2.3) selects the following distinct
4 special points
2.1
Back to the instanton partition functions
We could analyze further the structure of the prepotential in (2.3) at the special points
Xi in (2.9) by looking for regularities in the polynomials Pk( ). However, it is much more
convenient to go back to the instanton partition function. To see why, let us consider X1
as a rst illustration. We nd indeed the simple expansion
2
After some educated trial and error, we recognize that (2.10) is the expansion of the
5
2
+
4
4
14
ZeXin1st( ) = 1
following expression
where
ZeXin1st( ) =
2
E2(q)
1
q 31 ( )4;
1
k=1
3Here, trivial means a constant Fek( ).
and E2 is an Eisenstein series.4 Similar expressions are found at the other three special
points. The detailed formulas are
The associated allinstanton Nekrasov functions are
ZeXin2st( ) =
ZeXin3st( ) =
ZeXin4st( ) =
FeXin1st( ) = 8 log[q 112 ( )] + 2 log
FeXin2st( ) = 3 log[q 112 ( )] +
FeXin3st( ) =
4
3
1
log[q 12 ( )] +
FeXin4st( ) = 2 log[q 112 ( )] +
3
2
2
3
1
2
log
log
log
2
2
Equations (2.13) and (2.14) are already remarkable because they are nontrivial closed
expressions for the instanton partition function, or prepotential, at all instanton numbers.
It is clear that it would be nice to provide some clarifying interpretation for this features
at the special points Xi. In the next section, we shall examine the clues coming from AGT
correspondence.
3
AGT interpretation
According to the AGT correspondence, the instanton partition function of N = 2 SU(2)
gauge theory is [26{28]
Zinst(q; a; m) =
q2k)
" 1
Y(1
k=1
# 1+2 hm
Fhhm (q);
where F m(q) is the 1point toroidal block of the Virasoro algebra of central charge c =
1 + 6 Q2 on a torus whose modulus is q, with one operator of dimension hm inserted and
a primary of dimension h in the intermediate channel. The precise dictionary in terms of
the deformation parameters is
b = p 2= 1;
hm =
Q2
4
m2
{ 6 {
Assuming the scaling relations (1.2), the expressions in (3.2) read
In particular, at the four points Xi we obtain the following values for (c; hm)
Of course, points appear in pairs with the same central charge and
values related by
!
1
. More remarkably, the associated values of the parameter
is always such that hm
is a positive integer. The toroidal block has a universal prefactor q 12 = ( ) that is its value at
h ! 1. Comparing (3.1) with (2.13) we can write the general form for all four Xi points as
log
1
1
2
2
1 = j + 1j;
( )
1 +
c
1
24 h
(E2(q)
1) ;
(c; hm) = (0; 2) or ( 2; 3):
(3.7)
We remark that the above (c; hm) may well be pathological for a physical CFT.
Nevertheless, the toroidal block is de ned by the Virasoro algebra for abritrary values of c; hm, and
h. Eq. (3.7) must be taken in this sense. We checked (3.7) against Zamolodchikov recursive
determination of the toroidal block [69{71] with perfect agreement. Of course, by AGT,
this is same as Nekrasov calculation. The remarkably simple form (3.7) is clearly consistent
with general results for the torus block. For instance, at leading and nexttoleading order
and generic operator dimensions we have [35]
Fhhm (c; q) = 1 + F1(h; hm; c) q2 + F2(h; hm; c) q4 + : : : ;
where
F1(h; hm; c) = 1 +
hm (hm
2 h
1)
;
F2(h; hm; c) = [4 h (2 c h + c + 16 h2
10 h)] 1
{ 7 {
Thus,
in full agreement with (3.7) for c = 0; 2. Notice also that (3.7) may be written
1
( )
1 +
c
1
( )
=
1
X
k=0
1 +
1
h
c
k
Pk q2k; (3.11)
where Pk are the coe cients of the expansion of Qk1=1(1
unrestricted partitions of k (nm means m copies of n)
q2k) 1, i.e. the number of
P1 = #f(1)g = 1; P2 = #f(2); (12)g = 2; P3 = #f(3); (2; 1); (13)g = 3;
P4 = #f(4); (3; 1); (22); (2; 12); (14)g = 5;
P5 = #f(5); (4; 1); (3; 2); (3; 12); (22; 1); (2; 13); (15)g = 7; and so on.
3.1
Explicit CFT computations
The (c; hm) = (0; 2) conformal block. It is instructive to derive the result (3.7) at
c = 0 from a direct CFT calculation.5 In other words, we want to show that
h
Fhm=2(q; c = 0) =
1
( )
1
1
24 h
(E2(q)
1) =
1 +
k
Pk q2k;
1
X
k=0
1
h
where we used (3.11). The toroidal block is obtained as
Fhhm (q; c) = q h+ 1c2 Trh qL0 1c2 'hm (1) ;
where the trace is over the descendants of 'h. The starting point is thus conformal
descendant decomposition of the diagonal part of the OPE
'hm (x) 'h(0) =
X x hm+jY j Y L Y 'h(0)
Y
= x hm (1 + x (1) L 1 + x2 ( (2) L 2 +
(1;1) L2 1) + : : : ) 'h(0);
5CFT at c = 0 is obviously quite special since the 2point function of the stress energy tensor is then
zero, so strictly speaking, the theory is not conformal any more (since the stresstensor vanishes identically).
However, as we remarked, we are considering the toroidal block as a wellde ned function of (c; hm; h) that
may be regarded as the solution to the Zamolodchikov recursion relations. It would be interesting to revisit
our calculation in the language of logarithmic CFT, see for instance [72, 73].
{ 8 {
(n) =
1
h
;
(1) =
Computing them at (c; hm) = (0; 2) we see that indeed
Hence, if we apply (3.15) to the vacuum, we get6
(1) =
(2) =
where Y denotes a unrestricted partition of jY j and Ln are Virasoro generators
they are all equal
The coe cients
in (3.15) are determined by conformal symmetry and are functions of
h; hm; c. As a consequence of unbroken conformal symmetry (c = 0) and of the fact that
hm = 2 is the same dimension as that of the energy momentum tensor, one nds that the
coe cients are those associated with simple L n descendants. Besides
Using we obtain with one index
Now, to get the torus block, we need to evaluate the diagonal matrix elements of '2(x).
[Ln; '2(x)] = xn (x @ + 2 (n + 1)) '2(x);
1
h
;
1
h
1
X x
n=0
{ 9 {
(3.16)
(3.17)
(3.18)
(3.19)
(3.20)
(3.21)
(3.22)
(3.23)
HJEP07(216)
'2(x) L k jhi =
[L k; '2(x)] jhi + L k '2(x) jhi
=
=
h
1 X1 (2k
n=0
+
1
x2
n) xn k 2 L njhi +
n 2 L kL n jhi
1
h
1
X x
n=0
1 +
k
h
L kjhi + : : : ;
where we have shown only the diagonal entry. Adding one index each time, a similar
calculation shows that for any number of indices
'2(x) L Y jhi =
+
1
x2
1 + jY j
h
L Y jhi + : : : :
Thus the diagonal matrix element of '2(x) associated with the Y descendent depends only
on jY j. The number of Y with
xed jY j is the number PjY j of unrestricted partitions of
jY j. Summing over Y with jY j = k we prove (3.13).
6Notice that hm = 2 is not enough to achieve such simpli cation. A vanishing central charge is also
needed to remove descendants with multiple applications of Virasoro operators.
3
2h
;
(2) =
12
8h + 1
;
for certain coe cients MY functions of h. At level 1, we have only
'3(x) L Y jhi =
1
+
x3 MY L Y jhi + : : : ;
hm = 3 is apparently quite less trivial. The main reason is that the
coe cients in the
conformal decomposition (3.15) do not trivialize in this case. This complication forbids
us to prove the wanted result in general. Nevertheless, we provide an explicit check at
level 4. Of course, one could simply use the recursive de nition of the toroidal block,
but our brute force calculation is perhaps more transparent. Besides, it emphasizes the
di erence compared with the previous (c; hm) = (0; 2) case. The starting point is again the
OPE (3.15) that now takes the following form up to level 4 descendants
with the simple but non trivial
coe cients
(1) =
(2;1) =
(3;1) =
(2;1;1) =
As in (3.23), we write
At level 2,
At level 3,
M4 =
M2;2 =
MY = 5 +
60
h
X
jY j=4
2. It would be nice to prove the agreement
at all levels, possibly working in a de nite c =
2 CFT like the triplet model considered
4
Perturbative part of the prepotential at the points Xi
The prepotential has also a perturbative part, related by AGT to the DOZZ 3point
function in the Liouville theory [70, 76, 77]. The general expression for the perturbative part
of the prepotential is ( me = m + 1+2 2 ) [4, 5]
F pert = 1 2
1; 2 (2 a) +
1; 2 ( 2 a)
1; 2 (2 a + me)
1; 2 ( 2 a + me) ;
1; 2 (x) =
s Z 1 dt
(s) 0
ts e t x
t (e 1 t
1) (e 2 t
1)
and
is a renormalization scale. Evaluating Fpert by expanding at small 1 and resumming,
we nd that at all Xi points it is possible to write
d
ds
4
F pert =
Again, this appears to be a special feature of the Xi points because it is not possible to give
such a simple expression for F pert at generic 1; 2 from (4.1). With a rede nition of the UV
cuto , this may be written in the following suggestive form that we shall generalize later
e
F pert( ) =
Full prepotential and generalization to N poles points
If we combine the perturbative (4.4) and instanton (3.6) parts of the prepotential, we
obtain the remarkably simple expression
Fe = Fepert + Feinst =
4 a2 :
(4.1)
(4.2)
(4.3)
(4.4)
(5.1)
with a certain coe cient . This suggests that it is convenient to organize the total
prepotential in the form
Fe =
where M2n(q) is a polynomial in E2;4;6 of (quasi) modular degree 2n. We emphasize again
that the Ansatz (5.2) is non trivial because Fetot is a combination of the perturbative and
instanton contributions. Our claim is that (5.2) can be truncated at maximum degree
n = N for a special set of points ( ; ). At such points, the instanton partition function
takes the special form, see (1.3)
(5.2)
(5.3)
(5.4)
(5.5)
(5.6)
(5.7)
(5.8)
4 =
2;2 =
1
23040
1
To see how this works in practice, let us parametrize
Replacing these combinations in (5.3) and comparing with the Nekrasov function at 5
instanton we obtain explicit expressions for the
coe cients in terms of ; . The rst
coe cients are
The non trivial solutions of 4 = 2;2 = 0 (as always, up to the choice
> 0) are
that is precisely the four points Xi. The coe cient 2 is also an output of the calculation
and its general expression is
2 =
( 2 +
1)( 2 +
Finally, comparing (5.2) and (5.3) we determine the perturbative part of the prepotential
to be, see (1.3)
that is a nice generalization of (4.4). In all cases, we have con rmed that this is in full
agreement with the explicit evaluation of the general expression (4.1). We remark once
again that it is remarkable that at the special npoles points, it is possible to get such a
simple expression.
Performing the above analysis for 2 and 3 poles Nekrasov functions, we
nd the
following complete results. We have four 1 and seven 2poles Nekrasov functions that are
fully characterized by the following tables where we have separated by a horizontal line
points with di erents (c; hm)
5
2
5
4
7
4
7
6
2
1
2
Looking for 3poles points, we identify 12 cases whose full data is collected in the table
in (A.1). With 4poles, we found 11 solutions collected in the two tables (A.2) and (A.3).
Notice that in all presented cases, the parameter
is always rational negative. This implies
that central charge takes the form of extended minimal models
with coprime integers p, q. Here, the extension is due to the fact that the minimum value
of p; q is one instead of 2, see also gure 1.
5.1
Constraints from the modular anomaly equation
An important test of the expression (5.2) is the validity of the modular anomaly equation
expressing Sduality [44{49]. This is a nontrivial constraint capturing the dependence on
the quasimodular series E2 and reads [45]
An alternative form is obtained by expanding at large
and identifying the coe cients h` in
Then, (5.11) may be written in the equivalent form
Fe = h0 log
1
X
`=1
h`
21 ` ` 2`
1
;
h0 =
hm:
n=0
X hn h` 1 n +
` (2 `
1)
12
h` 1:
HJEP07(216)
(5.10)
(5.11)
(5.12)
(5.13)
 20
 15
 10
 5
8
6
4
2
0
c
2;2
2
2;2;2
2;2
2;4
4
Just to give a simple example, let us consider a 1pole function and write the dependent
part of Fe as
F ( ) = h0 log
e
Imposing (5.11) or (5.13), we recover the expression (5.7) for 2. Besides, we also get the
following constraint between
and
(excluding trivial solutions with constant prepotential)
have presented.
ratio 2;2= 2:
The condition (5.15) is indeed satis ed by the values in the rst table of (5.9). However,
there are in nite other pairs ( ; ) that make (5.14) a solution of (5.11) that is not realized
in the gauge theory. Of course, this is because (5.14) predicts all the higher order terms in
the large
only for a
expansion and this is correct in comparison with the actual Nekrasov formulas
nite set of values of , and . Actually, the admissibility of the Ansatz (5.3)
is de nitely non trivial. Anyway, we checked the validity of (5.11) for all the solutions we
As a
nal comment, we notice that using the explicit expressions for 2;2 and 2 it
turns out that the constraint (5.15), up to an multiplicative factor, can be expressed as the
2;2 = 0 is of course a necessary condition in the Ansatz (5.3) to truncate the sum at
N = 1. Looking at higher values of N , one gets similar constraints for any N . In fact,
since the modular anomaly equation controls the dependence of Fe on E2, the consistency
of our Ansatz with (5.11) imposes relations between all the coe cients of the form
2;X
with X . For example, in the 2poles case we have that
(5.14)
(5.15)
HJEP07(216)
Similarly for 3poles we get
2;2;2;2
2;2;2
2;2;4
2;4
and
2
2;6
6
and the constraint for
ensuring that 2;2;2 = 2;4 = 0 is
To appreciate the result of our analysis, let us consider in some details the rst line of (A.1).
The instanton partition function for X = ( ; ) = ( 11 ; 4) is up to 12 instantons
2
This is far more involved than (2.10) and a brute force guess would not be possible.
Nevertheless, it is a straightforward calculation to check that this is the expansion of
ZeXinst( ) = [q 112 ( )]12
6
14 E2 4 + 1430 E22 + 73 E4
( 2
9) ( 2
2
4) ( 2
in agreement with the data in (A.1). The perturbative part of the prepotential is computed
from (4.1) by expanding at large a. This gives, in the
variable
eX
F pert( ) = 28 log
56
2
196
4
3176
and this is indeed the large
expansion of
eX
F pert( ) = 28 log
+ 4 log
1
1
2
4
2
1
9
2
;
(5.23)
where 28 =
hm =
( 4)
7. Hence, the full quantum prepotential is in this case
Fe = 28 log
+ 4 log 1
14
2 E2 +
7
3 4 (20 E22 + E4)
9 6 (140 E23 + 21 E2 E4 + E6) ; (5.24)
and one can check that (5.11) is satis ed.
1
2
(5.20)
+: : : :
;
(5.21)
Predictions for the torus 1block
We have already seen that AGT implies explicit expressions for special torus blocks that
we write stripping o the large h dominant term
From the 1pole partition functions, we have obtained
From the 2poles partition functions, we get similar expressions
Notice that there are multiple entries in the tables (5.9) and (A.1) with the same value of
(c; hm), but di erent ( ; ). Nevertheless, the associated block is consistently the same as
soon as
is expressed in terms of h. From the 3poles partition functions, we get
H5h(q; 7) = 1 +
H4h q;
22
5
= 1 +
H4h(q; 1) = 1 +
H5h(q; 0) = 1 +
1
24h
E2
;
15E22 + E4
25E22 + 2E4
E
2
2
3 5E22
H3h(q; 2) = 1 +
1
8h
E2
:
80E2(3 h + 1) + 16(15 h + 4)
144 h(4 h + 1)
30E2(40 h + 9) + 3(400 h + 81)
960 h(5 h + 1)
E4
48E2 h + 48 h
48 h(4 h
1)
;
2E4
10E2(24 h + 1) + 240 h + 1
;
5 E23 + 3 E4 E2 + 2 E6 45 E2(8 h + 1)2
+ 9(5 E22
30 E2(48 h + 1)2 + 3(60 E22 17 E4)(48 h + 1) 3312 h + 1 ;
126 E2(16 h + 9)2 + 21(20 E22 + E4)(16 h + 9) ;
2(140 E23 + 21 E4 E2 + E6) + 9(3584 h2 + 3248 h + 729)
32256 h(7 h + 2)(7 h + 3)
23040 h(5 h 2)(5 h + 1)
While the structure is extremely similar, the selection rule is no longer true here. At the
point X1, the Laurent expansion of FeXin1s;tk( ) around
= 1 has the generic form
FeXin1s;tk( ) =
(
d(k)
0
1)k +
(
d(k)
where all the coe cients are non vanishing. The same is true for all the other three 1pole
cases X2; X3; X4. As in [63], we can rewrite the kinstanton functions in terms of the d(pk)
in the exact form
FeXin1s;tk( ) = ck +
independent part (i.e. the term proportional to the
logarithms of the Dedekind function. The coe cients d(pk) can be obtained from the expansion
of the exact expressions of FeXinis;tk( ). For X1 they read
1)
1
(B.5)
Open Access.
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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