#### Deformed SW curve and the null vector decoupling equation in Toda field theory

HJE
Deformed SW curve and the null vector decoupling
Rubik Poghossian 0 1
Yerevan Physics Institute 0 1
0 theory , Integrable Field Theories
1 Alikhanian Br. 2, AM-0036 Yerevan , Armenia
It is shown that the deformed Seiberg-Witten curve equation after Fourier transform is mapped into a di erential equation for the AGT dual 2d CFT cnformal block containing an extra completely degenerate eld. We carefully match parameters in two sides of duality thus providing not only a simple independent prove of the AGT correspondence in Nekrasov-Shatashvili limit, but also an extension of AGT to the case when a secondary eld is included in the CFT conformal block. Implications of our results in the study of monodromy problems for a large class of n'th order Fuchsian di erential equations are discussed.
Conformal and W Symmetry; Nonperturbative E ects; Supersymmetric gauge
1 Introduction
2.1
Exponents 2.1.1 2.1.2
3
Preliminaries on An 1 Toda CFT
Fusion with the completely degenerated eld V b!1
Derivation of null-vector decoupling equation in semiclassical limit
4
Comparison with the di erential equation derived from DSW
From the gauge theory di erential equation to the null-vector decoupling
equation
tal hypers
Matching S2(z) with w(
2
)(z): emergence of AGT
An 1-Toda 4-point functions versus SU(n) gauge theory with 2n
fundamen
Deformed Seiberg-Witten curve for Ar quiver
1
Introduction
Low energy behavior of N = 2 SYM theory admits an exact description including both
perturbative and non-perturbative layers [1, 2]. All relevant quantities, such as the
prepotential and chiral gauge invariant expectation values are nicely encoded in the geometry
of Riemann surfaces, called in this context the Seiberg-Witten curve. It was realized from
the very beginning that this curve is intimately related to the classical integrable
systems [3, 4]. Later development of this eld was triggered by the application of localization
method [5{8]. An earlier important reference is [9]. To make localization method e cient
one should
rst formulate the theory in a non-trivial background, commonly referred as
the -background [5, 8], which is parameterized by two numbers 1, 2 (these are rotation
angles in (x1; x2) and x3; x4 planes of Euclidean space time). The
-background brakes
the Poincare symmetry and e ectively regularizes the space-time volume, making the
partition function
nite. Using localization the instanton part of the partition function as
well as the chiral correlators of the theory can be represented as sum over arrays of Young
diagrams in such a way, that their total number of boxes coincides with the instanton
number. Sending the parameters 1;2 to zero one recovers the known results of the trivial
background. It appears, nevertheless, that the theory on
nite
-background has its own
signi cance. Namely, the recent developments recovered intriguing relations of these theory
{ 1 {
when only one of the parameters, say 1 ! 0. In this limit the classical integrable system
associated with SW curve gets quantized so that the remaining parameter 2 plays the role
of the Plank's constant. In [14{17] this limit has been investigated using Bohr-Sommerfeld
semiclassical method. Another approach initiated in [18] and further developed in [19{21]
is based on the careful analysis of the contributions of various arrays of Young diagrams.
It was shown in [18] that there is a single array of diagrams which dominates in the NS
limit. This approach leads to a generalization of the notion of Seiberg-Witten curve. The
limit corresponds to the classical (c ! 1) limit of 2d CFT conformal block of \heavy"
elds. The idea to apply DSW equation to investigate semiclassical limit of 2d CFT was
suggested in [18]. For the alternative approaches to the NS limit and the semiclassical CFT
see e.g. [16, 17, 24{26]. The case of irregular conformal blocks is considered in [27]. From
the AGT point of view the linear di erential equation discussed in previous paragraph
appears to be closely related to the null-vector decoupling equation in 2d CFT. Some
results in this direction has been already announced in [21]. For applications of CFT
degenerate elds in AGT context see also [28{32].
In this paper we systematically Investigate this relationship in a quite general setting
of Ar linear quiver theories with an arbitrary number (equal to r) of SU(n) gauge groups
corresponding in AGT dual CFT side to the r + 3-point conformal blocks in Wn Toda
eld theory.
The subsequent material is organized as follows. In section 2 we investigate DSW
equations for Ar quiver theories and establish explicit relations between curve parameters
and chiral expectation values. Then using Fourier transform we derive the
corresponding linear di erential equation and thoroughly investigate its singular points. In section 3
starting from the general structure of the fusion rule of the completely degenerated
eld
V b!1 and using Ward identities together with some general requirements necessary to get
{ 2 {
a0;u
U(n)
a1;u
U(n)
ar;u
1!1
2!1
P0;u P1;u
b
b
(b)
(the right edge) hypermultiplets; the lines connecting adjacent circles are the bi-fundamentals. (b)
The AGT dual conformal block of the Toda eld theory.
acceptable solutions, we derive the null-vector decoupling equation in the semiclassical
limit. Note that our approach here is somewhat heuristic and seems to be applicable only
in semiclassical case. To get exact di erential equation valid in full pledged quantum case
one should construct the null vector explicitly and make use of the complicated Wn-algebra
commutation relations and Ward identities. To my knowledge, at least in its full generality,
this goal has not been achieved yet. In section 4 we show that under a simple
transformation the two di erential equation of previous sections can be completely matched. Already
comparison of the rst non-trivial coe cient functions (in 2d CFT side this function is the
classical expectation value of the stress-energy tensor) readily recovers the celebrated AGT
correspondence. Thus our analysis provides a new, surprisingly elementary proof of the
AGT duality in semiclassical limit. Matching further coe cient functions (i.e. the higher
spin W -current expectation values in 2d CFT side and their gauge theory counterparts)
extends the scope of AGT correspondence: the conformal blocks including a descendant
eld get related to the higher power chiral expectation values in gauge theory. This new
relations are explicitly demonstrated in full details in the case r = 1 corresponding to
the four-point conformal blocks. Finally in Conclusion we emphasize the relevance of our
ndings in the context of the monodromy problems in a large class of Fuchsian di
erential equations.
2
Deformed Seiberg-Witten curve for Ar quiver
Partition function and chiral correlators of N = 2 gauge theory can be represented as
sum over arrays of Young diagrams which label the xed points of space time rotations
and global gauge transformations acting in the moduli space of instantons. [5{8]. In the
case of Ar quiver theory with fundamental and bi-fundamental matter hypermultiplets and
unitary U(n) gauge groups (see gure 1a), there is an n-tuple of Young diagrams associated
to each of the r gauge groups (indicated by circles in
gure 1a). It has been shown in [18]
for the case of a single gauge group and later generalized further in [19{21] that among
all xed points in moduli space of instantons there is a unique one giving a non-vanishing
{ 3 {
contribution in the Nekrasov Shatashvili limit.1 We will denote the (rescaled by 1) lengths
of the rows of this \critical" array of Young diagrams Y ;u by
;u;i where
= 1; : : : r refers
to the node of the quiver, u = 1; : : : ; n is the index of the de ning representation of the
gauge group U(n) associated with this node and i = 1; 2; : : : speci es the row. The data
;u;i can be conveniently encoded in meromorphic functions y (x), which are endowed
with zeros located at x = a ;u + i
1 +
;u;i and poles at x = a ;u + i
2 +
;u;i
where a ;u are the Coulomb branch parameters. In addition we associate to the \frozen"
nodes (indicated by squares in gure 1a) the parameters a0;u and ar+1;u. In terms of these
parameters the masses of fundamental and anti-fundamental hypermultiplets are given by
HJEP04(216)7
respectively. In terms of
the masses of the bifundamental hypermultiplets are simply
n
v=1
n
1 X ar;u
mu = ar+1;u
and
mu = a0;u
n
v=1
n
1 X a1;v
a =
n
u=1
n
1 X a ;u
m ; +1 = a +1
a :
The Deformed Seiberg-Witten (DSW) curve equations arise from the condition on the
instanton con guration to give the most important contribution to the prepotential in NS
limit. In the case of our present interest of Ar quiver theory we get a system of r (di erence)
equations for r functions y (x),
= 1; : : : ; r. In addition we introduce two polynomials
n
Y(x
u=1
n
Y(x
u=1
y0(x) =
a0;u);
yr+1(x) =
ar+1;u)
which encode fundamental hyper-multiplets attached to the rst and the last nodes of the
quiver gure 1a. The equations can be found using iterative procedure based on so called
iWeyl re ections (i stands for instanton) [33]
{ 4 {
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
y (x) ! y (x) +
y
1(x
1)y +1(x)
y (x
1)
It appears that the result of this procedure is related to the q-Character of the 'th
fundamental representation of the group Ar. Explicitly for
= 1; 2; : : : ; r one obtains2
(x) = y0(x
)
X
Y
1 k1<k2< <k r+1 =1
0
yk (x
+ )
+
1)
q
k 1
Y q A ;
1
=1
1For simplicity in this paper we'll set 2 = 1. This is not a loss of generality since a generic 2 everywhere
can be restored by simple scaling arguments.
2When comparing this formula with those of [20, 21] it should be taken into account that we have shifted
arguments in y (x),
(x) appropriately to get rid of explicit appearance of the bifundamental masses.
The 1-forms d log y (x) are the direct analogs of Seiberg-Witten di erentials and de ne the
chiral correlators by the conventional contour integrals
htr J i =
I
2 i
where
are large contours surrounding all zeros and poles of y (x) in anti-clockwise
direction. Comparison of (2.7), with (2.8) allows one to express the expansion coe cients
c ;k in terms of chiral correlators htr J
i with J
k. Here are the rst few relations
htr
htr 2
htr 3
htr 4
i = c ;1
i = c2 ;1
i = c3 ;1
i = c4 ;1
2c ;2
3c ;1c ;2 + 3c ;3
4c2 ;1c ;2 + 4c ;1c ;3 + 2c2 ;2
4c ;4
where q are the gauge couplings,
(x) are n-th order polynomials in x with coe cients
related to the expectation values htr J i (
are the scalars of the vector multiplet) in
a way to be speci ed below. For later purposes we'll set by de nition
0(x)
y0(x)
yr+1(x). The di erence equations (2.6) are the deformed Seiberg-Witten
equations [18] for the Ar quiver gauge theory [20, 21]. It is assumed that all functions
y (x) are normalized so that their large x expansions read
y (x) = xn(1
On the other hand, inserting the expansion (2.7) into (2.6) and comparing left and right
hand sides one can express the coe cients c ;k (and due to (2.9) also htr J i ) in terms of
coe cients of the polynomials
(x). In fact the rst n of these relations can be inverted to
get the coe cients of the polynomials
(x) in terms of c ;1; : : : ; c ;n (or, equivalently, in
terms of htr
i; : : : ; htr ni). Then the remaining in nite number of relations are nothing
but the deformation of the celebrated chiral ring relations [34] expressing higher power
(J > n) chiral expectation values htr J i in terms of lower, up to the n'th power expectation
values.3 For our later purposes let us display explicitly the relations for the rst three
coe cients of the polynomials
3For the generalization of chiral ring relation for the generic -background see [35].
(x) =
n
i=0
X( )
i
;ixn i
:
{ 5 {
Expanding l.h.s. of (2.6) up to the order
xn 3 we get
2
4
1
X
Clearly with more e orts it should be possible to write down expressions for further
coe cients but unfortunately these expressions soon become quite intractable. In section 4
we'll do one more step giving an explicit expression for the next coe cient in the special
case when r = 1.
Quite remarkably it is possible to eliminate the functions y2(x); : : : ; yr(x) from eq. (2.6)
and nd a single equation for y1(x). Here is the result:4
It is useful to represent the meromorphic functions y1(x) as a ratio:
where Y (x) is an entire function with zeros located at x = a1;u + (i
1) + 1;u;i (remind
that
;u;i is the appropriately rescaled length of the i'th row of the Young diagram Y ;u).
In terms of Y (x) the eq. (2.14) can be rewritten as
r+1
1 + X( )
i=1
i i(x) Yi1 y0(x
y1(x) j=1 y1(x
j) j
j) qi j = 0;
y1(x) = y0(x)
Y (x)
Y (x
1)
;
r+1
Y q
=1
1
A
(x)Y (x
) = 0 :
Since for small values of the gauge couplings q
1 the i'th row length
;u;i ! 0 when i ! 1, it is reasonable to expect that the sum
(z) =
X
x2Z+a1;u
Y (x)z x
4Of course, the same can be done also for yr(x).
{ 6 {
(2.11)
(2.12)
(2.13)
(2.14)
(2.15)
(2.16)
(2.17)
will converge in some ring with the center located at 0. Then the di erence equation for
Y (x) can be easily \translated" into a linear di erential equation for (x) [19, 20]
r+1
A
d
dz
( z
)z
(z) = 0
It is not di cult to nd the coe cient in front of the highest derivative dn=dzn in (2.18).
Using (2.11) one can show that this coe cient has a nice factorized form
r+1
X( )n+
=0
0
Y q
=1
1
A
;0z
n
= ( ) z
n n r 1 Y
1
Y q A
=1
Further investigation con rms that indeed (2.18) is an n-th order Fuchsian di erential
equation with r + 3 regular singular points located at
z0 = 1; z1 = 1; z2 = q1; z3 = q1q2; : : : ; zr+1 = q1q2
qr; zr+2 = 0;
where for later use we have introduced new parameters z related to the gauge couplings
through conditions
2.1
Exponents 2.1.1
Points zr+2 = 0 and z0 = 1
First let's look after a solution of the form
q =
z +1 :
z
(z) = zs(1 + O(z))
yr+1(r + 1
s) = 0
y0( s) = 0
(2.18)
(2.19)
(2.20)
(2.21)
(2.22)
(2.23)
(2.24)
HJEP04(216)7
Inserting this in (2.18) we see that when z ! 0 the term with
= r +1 of (2.18) is the most
singular one. So, for yet unknown constant s we get the characteristic equation (sometimes
called indicial equation)
Similarly the characteristic equation for the in nity reads
2.1.2
Investigation of these points is slightly more subtle. Consider the ansatz
(z) = (z
z )s (1 + O(z
z )))
{ 7 {
for some xed
get the equation
0 = ( ) z
2 f1; : : : ; r + 1g. Taking into account (2.19) and (2.12) for the index s we
6= ; =1
(z
1
X
=1
The rst two lines come from the terms proportional to (zd=dz)n. The rst (second) line
includes part with n \hits" (n
1 hits) on
(z) by the operator d=dz. The third line is
coming from the terms
(zd=dz)n 1 with all n operators d=dz hitting
(z). Though the
second and especially third lines of this equation look quite complicated, fortunately they
can be simpli ed drastically. Indeed it can be shown that the second line is equal to
( )n nzn r 1
r+1
Y
6= ; =1
(z
while the third line can be rewritten as
( ) z
n n r 1 (c
1;1
c ;1)
r+1
Y
6= ; =1
(z
These are 2d CFT theories which, besides the spin 2 holomorphic energy momentum
current W (
2
)(z)
T (z) are endowed with additional higher spin s = 3 : : : ; n currents W (3),
. . . W (n) [36{38]. The Virasoro central charge is conventionally parameterised as
c = n
1 + 12(Q; Q) ;
{ 8 {
where the \background charge" Q is given by
is the Weyl vector of the algebra An 1 and b is the dimensionless coupling
constant of Toda theory. In what follows it would be convenient to represent roots, weights
and Cartan elements of An 1 as n-component vectors subject to condition that sum of
components is zero and endowed with the usual Kronecker scalar product. Obviously this
is equivalent to a more conventional representation of these quantities as diagonal
tracen matrices with pairing given by trace. In this representation the Weyl vector is
and for the central charge we'll get
where for the later use we have introduced the parameter
(3.1)
(3.2)
(3.3)
The primary elds V (in this paper we concentrate only on the left moving holomorphic
parts) are parameterized by vectors
with vanishing center of mass. Their conformal
wights are given by
In what follows a special role is played by the elds V !1 with dimensions
For further reference let us quote here explicit expressions for the highest weight !1 of the
rst fundamental representation and for its complete set of weights h1; : : : ; hn (h1 = !1)
For generic these elds admit a single null vector at the rst level.
3.2
Fusion with the completely degenerated
eld V b!1
The eld V b!1 plays a special role in Toda theory. Fusion rules with this eld are especially
simple (see e.g. [39])
=
n
2
1 n
;
3
; : : : ;
2
1
n
2
c = (n
1)(1 + n(n + 1)q2)
q = b +
1
b
where hk are the weights of the rst fundamental representation and [V ] denotes the W
class of the primary eld V . In the case when V is partially degenerated (i.e.
=
!1 for
some scalar
), then in (3.4) only the rst two terms contribute, all other classes drop out
due to vanishing of the relevant structure constants. The remaining exponents are equal
to respectively
In classical limit (q
1=b) the eld V b!1 satis es the n-th order di erential equation [39, 40]
b(n
1)
n
n
k=0
and
b q
n
0 and the other coe cients w(n k)(z) are the expectation values of
the currents bn kW (n k)(z) in the classical limit b ! 0 (the prefactor bn k is included to
secure a nite limit).
3.3
Derivation of null-vector decoupling equation in semiclassical limit
Let us consider the semi-classical limit of the correlator
hV (0) (
1
)V b!1 (z)V (
1
) (z1)
V (r+1) (zr+1)V (r+2) (0)i ;
where we'll assume that all the elds besides V b!1 (z) are \heavy", namely
(0) = (0)=b;
(r+2) = (r+2)=b;
(k) = (k)!1=b
(the parameters (0) and (r+2) are n-component vectors while (k) are scalars all of them
remaining
nite in the b ! 0 limit). We have chosen all the parameters, besides the rst
and the last ones, to be proportional to the rst fundamental weight !1 since this is the
case when the AGT relation between correlation functions and the partition function of
quiver gauge theory holds. As we'll see later, the AGT correspondence in the b ! 0 limit
emerges as a special case of a more general construction presented in the remaining part
of this paper. In semiclassical limit the correlator (3.7) factorizes into a product of the
classical (normalized) expectation value of the light eld V b!1 (z) with the correlator of
the remaining heavy operators, hence, with respect to the variable z it must satisfy the
same di erential equation (3.6):
n
k=2
d
n
dzn + X w(k)(z)
d
n k !
dzn k
G(z) = 0 :
(3.9)
As already mentioned the coe cient functions w(n k)(z) are the classical expectation values
of the holomorphic currents bn kW (n k)(z) in the background of heavy operators. Due to
the form of OPE of W (k)-current with primary
elds, this classical expectation values
should be rational functions of z, with k-th order poles located at the insertion points of
heavy primary elds. The OPE (3.4) completely xes the indices at the singular points
as follows:
( )=n
There is a small puzzle to be understood here. The indicial equation at z = z is a degree
n algebraic equation while on the second line of eq. (3.10) we quoted only two indices.
Multiple roots are not admissible, both eld appearing on the r.h.s. of the OPE should
nd the missing indices, let us slightly change the charge
parameters of the
eld at the point z . We'll immediately see that besides two indices
close to those given on the second line of (3.10), there are n
2 additional indices, located
close to the points
Thus it is natural to assume that besides (n
1) ( )=n we have sequence of n
1 indices
2
1
( )=n; 3
( )=n; : : : ; n
( )=n; 2
( )=n; : : : ; n
1
1
( )=n :
( )=n :
The extra indices we got are naturally attributed to the contribution of descendant elds.
This is not the end of story yet. It is well known that if there are indices at a singular point
which di er from each other by integers, generally speaking logarithmic solutions emerge,
something, which is not acceptable in a standard CFT such as Toda theory. The condition
that logarithmic solutions are actually absent, imposes further restrictions on the rational
coe cient functions w(k)(z). We'll explicitly parameterize these rational functions as
Consider a solution of the di erential equation (3.9) around z = z represented as a series
w(k)(z) =
Let us insert this expansion into (3.6) and read o the rst n
1 constraints imposed by the di erential equation. We immediately get the relations
bm(s + m
n + 1)n + X
n
m
X
k=2 l=m k+1
valid for m 2 f0; 1; : : : ; n
2g, where
blwm(k; l)(s + l
n + k + 1)n k = 0
(x)l = x(x + 1)
(x + l
1)
is the Pochhammer's symbol. According to what has been discussed above, the coe
cients w(n k)(z) of the di erential equation should be chosen so that these constraints be
satis ed for
s = 1
( )
n
k
!
=
n
k
!
k!(n
k)!
(k
n!
2
valid for each value of m 2 f1; 2; : : : ; n
quantities w(km; ) with k 2 f2; 3; : : : ; ng and m 2 f1; 2; : : : ; k
g
Here is the explicit expression:
2 . These equations allow one to express all
1g in terms of w(mm+1; ).
n
m
n
k
1
!
w(km; ) = (
1
)k m+1
(s )k m 1w(mm+1; ) :
(3.22)
where the standard notation
for the binomial coe cient is used. It is rewarding to see that this formula with parameter
s speci ed in (3.17) gives correct zero mode eigenvalues of W-currents on the eld V!1 =b
in semiclassical limit b ! 0. In particular for the (rescaled) conformal dimension (i.e. for
k = 2) we get
0
w(2; ) =
(n
1) ( ) 1
( )=n
in complete agreement with (3.3). I have checked that also the other eigenvalues agree
with data available in the literature.
The remaining constraints that follow from eq. (3.15) can be represented as
n
X
k=m+1
w(km; )(s
n + k + l)n k = 0;
for l = 1; 2; : : : ; n
m
1
(3.21)
and for arbitrary constants b0; : : : ; bn 2. In particular choosing bl = l;m we arrive at
(s + m
n + 1)n + X w(k; )(s + m
0
n + k + 1)n k = 0 ;
(3.18)
satis ed for each m = 0; 1; : : : ; n
2. Solving this system of equations with respect to
0
w(k; ) we get
0
w(k; ) = (
1
)k+1
1)(s
1)k ;
(3.19)
Thus we managed to express all the coe cients of higher order poles at z = z ,
1; 2; : : : ; r + 1 (but not those corresponding to z = 0) in terms of the residues w(mm+1; ).
=
Recall now that there should be one more independent index at z = z (see middle
line of (3.10) and (3.17)) equal to
s~ = (n
1) ( )=n = (n
1)(1
s ) :
The indicial equation to be satis ed is (cf. (3.15)):
n
k=2
(s~
n + 1)n + X w(k; )(s~
0
n + k + 1)n k = 0 :
It can be checked that quite remarkably this equality under substitutions (3.19), (3.23) is
indeed satis ed automatically.
(3.20)
(3.23)
(3.24)
are consequences of5
and the commutation relations
where
h
l
k
l=1
X zk+m l
I
z
(
It remains to take into account constraints imposed by Ward identities. These identities
w(k) being the zero mode eigenvalue corresponding to the eld V . For m > 0
combining (3.25) and (3.26) we easily get
or, in view of (3.22)
w(km;r+2) =
X
r+1 k m
X z
Similarly analysing m = 0 case of (3.25) and (3.26) we get the relation
k
1
0
(
1
)k+1w(k;0) + X w(k; )
0
r+2
=1
Mi;j =
(s)i j
((i
j)!)2
;
(3.25)
(3.26)
(3.27)
(3.28)
(3.29)
(3.30)
(3.32)
r+1 k 1
+ X
X( )l+1zk l k
=1 l=1
l
1
1
n + l
l
k
1
1
(s )l 1wl(k k l+1; ) = 0 :
(3.31)
These equations allow one to express all coe cients related to the point z1 = 1 (i.e. w1(k;1k))
in terms of coe cients w(k; ) with
1
k
respect to the variables w1(k;1k) has a Gaussian triangular form and can be solved. As a
side remark note that this problem reduces to the inversion of the (n
1)
(n
1) lower
= 2; 3; : : : ; r + 1. Indeed the system (3.31) with
triangular matrix
5The sign factor (
1
)k+1 re ects the fact that a two-point function of primaries is non-zero if the zero
modes of even spin currents (e.g. dimensions) coincide while those of odd currents have opposite signs (see
e.g. [40]).
where i; j 2 f1; 2; : : : ; n
1g. It's possible to show that the inverse matrix can be
represented as
where the polynomials Pl(s) are conveniently given by means of a generating function
M 1
i;j =
X Pl(s) i j;l ;
1
l=0
X Pl(s)ql =
1
1F1(s; 1; q)
1F1(a; b; q) =
l=0
X1 (a)l l
q :
(b)ll!
with
HJEP04(216)7
(3.33)
(3.34)
(3.35)
(3.36)
For the later reference let us write down w(
2
)(z) (see eq. (3.13)) in terms of dimensions
and the parameters w(21; ),
= 2; 3; : : : ; r + 1, explicitly
w(
2
)(z) =
0
w(2;r+2)
z2
To conclude we succeeded to express all the parameters of the di erential equation (3.9)
in terms of (see eqs. (3.19), (3.22), (3.30) and (3.31)):
zero-mode eigenvalues of the W -currents corresponding to the initial state, insertion
elds and the nal state (i.e. w0(k; ) for k 2 f2; 3; : : : ; ng,
2 f0; 1; : : : ; r + 2g)
the coe cients w1(k;k) fork 2 f2; 3; : : : ; ng,
rational functions w(k)(z) at the points z = z
2 f2; 3; : : : ; r + 1g, which are residues of
The parameters of the second list will be referred as the accessory parameters, since these
are direct generalizations of the accessory parameters of the Liouville theory (the
particular n = 2 case of Toda theory) [41]. To avoid confusion notice that we have in mind
not the \real" monodromy problem of [41], but a generalization of the complex SL(2; C)
monodromy problem, discussed in [42, 43], to the SL(n; C) case.
It is essential to note that there are exactly (n
1)
r accessory parameters, as many
as the number of parameters necessary to
x r intermediate W -families of the conformal
block. We will see soon, that the DSW \curve" is the appropriate tool to solve the related
monodromy problem, namely, to nd such set of accessory parameters, which corresponds
to a given set of intermediate W -families. In particular concentrating on the accessory
parameters w(21; ), we will easily reestablish the famous AGT relations (in the semiclassical
limit). Consideration of the remaining accessory parameters lead to a generalization of
AGT, relating conformal blocks including non-primary
elds on 2d CFT side with gauge
invariant expectation values on the N = 2 gauge theory side. Investigation of this new
relations in general quantum Toda case (e.g. along the lines of [44, 45], seems to be quite
an interesting task.
4.1
From the gauge theory di erential equation to the null-vector decoupling
We'll argue here that the di erential equation obtained from (2.18) by substitution
(z) = G(z) Y (z
z )t
with suitably chosen parameters t coincides with the di erential equation (3.9) satis ed
by Toda CFT semi-classical conformal block (3.7). The idea is to choose parameters t so
that the term with derivative of order n
1 disappears. Straightforward calculations show
that this task is achievable with the unique choice
The resulting di erential equation for G(z) can be represented as
tr+2 = r + 1 +
t =
c
n
1
n
2
1;1
= 1; 2; 3; : : : ; r + 1 :
+ Sn(z) G(z) = 0 ;
2
r !
r+1
(c
1;1
c ;1) (c
1;1
n (z
z ) (z
z )
:
In general Su(z) for u = 2; 3; : : : ; n, are rational functions with poles of order u located at
z 2 fz1 = 1; z2; : : : ; zr+1; zr+2 = 0g.
Note that the substitution (4.1) shifts all exponents described in section 2.1 in an
obvious manner. Namely the exponents at z get shifted by t for
= 1; 2; : : : ; r + 2 and
the exponent at z0 = 1 by
so that the resulting indices become
and
n
tr+2 +
+ tr+1 =
1
n
2
u = 1; 2; : : : ; n
(n
1)(c ;1
c
n
u = 1; 2; : : : ; n :
1;1) ;
u = 1; 2; : : : ; n
1
(4.1)
(4.2)
(4.3)
(4.4)
(4.5)
Remind that a0;u and ar+1;u are the roots of the polynomials y0(x) and yr+1(x) respectively
(see eqs. (2.4)). In particular
Now we see that the simple identi cation
n
u=1
and
n
c
c0;1
n
provides a perfect matching of indices (4.5) with (3.10).
Matching S2(z) with w(
2
)(z): emergence of AGT
Now let us compare S2(z) (eq. (4.4)) with w(
2
)(z) (eq. (3.36)). Clearly the coe cients
at the double poles under the map (4.7) become identical (not a surprise, since we have
already checked that indices coincide).
Comparison of residues of poles at z = z for
2 f2; : : : ; r + 1g leads to identi cation
w(21; ) =
c
1;2
2
r
z
z )
+
X
r+1 (c
= +1
1;1
c ;1)(c
n (z
1;1
z )
Matching the residue at the pole z = z1
1 requires
c0;2
c1;2 + (c0;1
c1;1)
c1;1 +
n
c1;1) (c
n (1
1;1
z )
r+1
X z w(21; ) =
=2
n
2
1
A careful calculation shows that upon replacement of dimensions w(2; ) by their expressions
in terms of parameters ( ) speci ed in (4.7) and substitution (4.8) for w(21; ), the eq. (4.9)
0
becomes an identity. To prove this statement we have used the useful identity (for our
purpose one should take d = c ;1
C 1
X
1
< r+1
1
X d (n
n (z
=1
X
< r+1 (z
d (n
d )
zd (n
d )
z )
+
d )
r+1
X
= +1
d (n
n (z
dz
z )(z
z ) 2 i
Xr+1 d1(n
=2
z1
d )
z
1
d )
z ) A
n
u=1
X a ;u = c ;1 = htr
i
:
1 2 =
1 = b2 ;
2
P ;u = a ;u
c ;1
n
r+1
X
= +1
where Zinst is the instanton part of the partition function and a ;u are the Coulomb branch
parameters which are related to the parameters c ;1 by
Notice also that in (4.12) taking into account our convention 2 = 1 we have replaced
where the last equality, as we'll see soon, is necessary to match the gauge theory and Toda
theory sides. It will be convenient to separate the \center of mass" of the quantities a ;u
introducing new parameters (these are parameters indicated in gure 1b)
Let me emphasize that the quantities a ;u are genuine parameters of our gauge theory and
as such, they do not depend on the gauge couplings q (or, equivalently, on z ). Thus, on
r.h.s. of eq. (4.8), besides their explicit appearance, the variables z are hidden only in the
di erences c
1;2
c ;2. Due to (2.20) we have
where the contour C encloses the points z2; : : : ; zr+1 but not z1. To pass from the second
line to the third, one should notice that the same integral alternatively can be computed
as sum of residues at in nity and at z1 taken with negative signs.
There is no need to compare residues at the remaining pole at z = zr+2 = 0 since all
residues sum to zero (both w(
2
)(z) and S2(z) vanish at large values of z as 1=z2).
Now let us look on identi cation (4.8) more closely. The gauge theory expectation value
is related to the partition function through Matone relation [46] which is valid also in
presence of a nontrivial -background [47]:
(4.10)
HJEP04(216)7
(4.11)
(4.12)
(4.13)
(4.14)
(4.15)
(4.16)
Hence, combining (4.11), (4.12), (4.13) and (4.15) we get
c
1;2
c ;2 =
P 2;u
P 2
1;u +
c
2
1;1
n
1
2n
Inserting this expression into (4.8) after few simple manipulations, for
2 f2; : : : ; r + 1g we obtain
1;u)
h(P ;u)
HJEP04(216)7
1
X
=1
( ) 1
( ) !
n
z
z
r+1
X
= +1
( ) !
n
( ) 1
(4.17)
;
z
z
(4.18)
(4.19)
(4.20)
(4.21)
where
h(P ;u) =
n n
2
1
24
are the (rescaled) dimensions with Toda momenta P ;u (Toda momenta are related to the
charge parameters via
=
P ) and dimensions w(2; ) are given in (3.20). Remind now
that w(21; ) is related to the Virasoro generator b 2L 1 acting on the eld V (z ). Thus
2
1 Xn P 2;u
u=1
0
where FCFT is the r + 3 point conformal block of Toda theory depicted in
gure 1b.
Comparing (4.18) with (4.20) it is not hard to recognize the celebrated AGT relation.
The same technique, in principle, can be applied to investigate the remaining accessory
parameters w(k; ) with k 2 f3; : : : ; ng. The result would be an extension of the AGT
1 k
correspondence to the case of conformal blocks including a descendant eld on CFT side
and higher power expectation values in gauge theory side. Unfortunately in general case
the next steps along this line seem to be quite complicated, one soon encounters awkward
expressions. Instead in the next section we will give few more details for the simpler case
of four point functions (i.e. for r = 1).
4.3
An 1-Toda 4-point functions versus SU(n) gauge theory with 2n
fundamental hypers
In this case we have a single DSW curve equation
1(x) =
q1y0(x
1)y2(x)
y1(x
1)
+ y1(x)
The rst three coe cients of the n'th order polynomial 1(x) can be read o from the
general formulae (2.11){(2.13). We'll need also the forth coe cient which in our case r = 1
is easy to obtain. Without loss of generality we can set6 c1;1 = 0. As a result we get
1;0 = 1 + q1;
1;1 = q1 (c0;1 + c2;1) ;
1;2 = c1;2 + q1 (c0;2
c1;2 + c0;1 (c2;1
1) + c2;2) ;
Now starting from the eq. (2.18) and following the steps described in previous sections
we'll try to recast the di erential equation in the form (4.3). The coe cient function
S2(z) of (4.3) we have already calculated for the general case, so we'll simply specify the
expression (4.4) to the case r = 1.
The data (4.22) are su cient for calculation of the next coe cient function S3(z)
of (4.3). The calculation is rather lengthy but quite straightforward. Here we'll not give all
the details. Those interested readers who might have desire to recover the results presented
below could bene t from the useful identity
z
d
dz
n
=
X dn;lzl
n
l=1
d
dz
l
;
dl+k;l =
l
X
i1
X
i1=1 i2=1
il 1
X i1i2
il=1
il :
dn;1 = 1 ;
dn;2 =
dn;3 =
dn;4 =
n(n
n(n
n(n
2
1)
;
1)(n
1)(n
24
48
2)(3n
2)2(n
5)
3)2
;
:
where the expansion coe cients admit a nice representation as
In fact, to calculate S3(z) one needs only the rst four coe cients which can be easily
deduced from (4.24)
It is clear from discussions in previous sections that knowing the residue of the rational
function S3(z) at z = z2 (i.e. w(32;2)) is su cient to fully recover the function itself. Here is
the nal result for this residue
w(32;2) =
(z2
A
1)2 +
B
z
2
2 +
C
z2(z2
1)
;
6In fact a uniform shift of all parameters a ;u indicated in gure 1a is immaterial and can be compensated
by the shift of same amount of the parameter x of DSW equations (2.6).
(4.23)
(4.24)
(4.25)
(4.26)
where
2
n2
n
1
C = 2 1
A =
c0;1 (c0;1 + c2;1) (n
c2;1
n
6n2 c2;1 12c22;1
2
n
2
n
+c0;1 n
Recall now that the parameters c0;1; c0;2 (c2;1; c2;2) are related to the fundamental
(antifundamental) hyper-multiplet masses mu, (mu) as
n
Due to eq. (2.9)
Conclusion.
shown that:
Thus (4.26), (4.27) go beyond the standard AGT correspondance explicitly relating the
semiclassical four-point function including a descendant
theory expectation values htr 2i and htr 3i. The latter quantities can be calculated either
with direct instanton calculus or DSW curve methods thus leading to the evaluation of the
eld W (32)V (z2) with the gauge
Toda descendant including four point conformal block.7
As a nal remark note that from the mathematical point of view we have
form (2.18), where
by (2.11), (2.20).
Any n'th order Fuchsian di erential equation with generic ordinary singular points
at 0 and 1, additional r + 1 ordinary points at z1 = 1; z2; : : : ; zr+1 each of them
obeying a set of indices of the form f0; 1; : : : ; n
2;
g (
are arbitrary), such
that the equation doesn't accept any logarithmic solution can be represented in the
(x) are n'th order polynomials with highest coe cient given
The monodromy problem of such di erential equation is intimately related to the
system of di erence equations (2.6) in the sense that the eigenvalues of the monodromy
matrices M
2 SL(2; C) are given in terms of the period integrals of the meromorphic
di erentials xd log(y (x)), where y (x) are the solutions of di erence equations (2.6).
7Notice that this conformal block can not be related to the respective block of primaries via Ward
identities, so that a straightforward CFT calculation is not available.
Acknowledgments
I thank F. Fucito, J. F. Morales and G. Sarkissian for many interesting discussions. I am
grateful to S. Theisen for hospitality at the AEI Potsdam-Golm where the bulk of this
work has been completed and for a discussion. A discussion with S. Fredenhagen is also
gratefully acknowledged. This work was supported by the Armenian State Committee
of Science in the framework of the research project 15T-1C308 and by the Volkswagen
Foundation of Germany.
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
HJEP04(216)7
any medium, provided the original author(s) and source are credited.
[Erratum ibid. B 430 (1994) 485] [hep-th/9407087] [INSPIRE].
[3] A. Gorsky, I. Krichever, A. Marshakov, A. Mironov and A. Morozov, Integrability and
Seiberg-Witten exact solution, Phys. Lett. B 355 (1995) 466 [hep-th/9505035] [INSPIRE].
[4] E.J. Martinec and N.P. Warner, Integrable systems and supersymmetric gauge theory, Nucl.
Phys. B 459 (1996) 97 [hep-th/9509161] [INSPIRE].
[5] N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math.
Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].
[6] R. Flume and R. Poghossian, An Algorithm for the microscopic evaluation of the coe cients
of the Seiberg-Witten prepotential, Int. J. Mod. Phys. A 18 (2003) 2541 [hep-th/0208176]
[7] U. Bruzzo, F. Fucito, J.F. Morales and A. Tanzini, Multiinstanton calculus and equivariant
cohomology, JHEP 05 (2003) 054 [hep-th/0211108] [INSPIRE].
244 (2006) 525 [hep-th/0306238] [INSPIRE].
[8] N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, Prog. Math.
[9] A. Lossev, N. Nekrasov and S.L. Shatashvili, Testing Seiberg-Witten solution, in Strings,
branes and dualities. Proceedings, NATO Advanced Study Institute, Cargese, France, May 26
{ June 14, 1997 [hep-th/9801061] [INSPIRE].
[10] L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville Correlation Functions from
Four-dimensional Gauge Theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219]
[11] N. Wyllard, AN 1 conformal Toda eld theory correlation functions from conformal N = 2
SU(N ) quiver gauge theories, JHEP 11 (2009) 002 [arXiv:0907.2189] [INSPIRE].
[12] V.A. Fateev and A.V. Litvinov, Integrable structure, W-symmetry and AGT relation, JHEP
01 (2012) 051 [arXiv:1109.4042] [INSPIRE].
[13] N.A. Nekrasov and S.L. Shatashvili, Quantization of Integrable Systems and Four
Dimensional Gauge Theories, in Proceedings, 16th International Congress on Mathematical
Physics (ICMP09) (2009) [arXiv:0908.4052] [INSPIRE].
[14] A. Mironov and A. Morozov, Nekrasov Functions and Exact Bohr-Zommerfeld Integrals,
JHEP 04 (2010) 040 [arXiv:0910.5670] [INSPIRE].
[15] A. Mironov and A. Morozov, Nekrasov Functions from Exact BS Periods: The Case of
SU(N ), J. Phys. A 43 (2010) 195401 [arXiv:0911.2396] [INSPIRE].
[16] K. Maruyoshi and M. Taki, Deformed Prepotential, Quantum Integrable System and Liouville
Field Theory, Nucl. Phys. B 841 (2010) 388 [arXiv:1006.4505] [INSPIRE].
HJEP04(216)7
[17] A. Marshakov, A. Mironov and A. Morozov, On AGT Relations with Surface Operator
Insertion and Stationary Limit of Beta-Ensembles, J. Geom. Phys. 61 (2011) 1203
[18] R. Poghossian, Deforming SW curve, JHEP 04 (2011) 033 [arXiv:1006.4822] [INSPIRE].
[19] F. Fucito, J.F. Morales, D.R. Paci ci and R. Poghossian, Gauge theories on
-backgrounds
from non commutative Seiberg-Witten curves, JHEP 05 (2011) 098 [arXiv:1103.4495]
[20] F. Fucito, J.F. Morales and D.R. Paci ci, Deformed Seiberg-Witten Curves for ADE Quivers,
[21] N. Nekrasov, V. Pestun and S. Shatashvili, Quantum geometry and quiver gauge theories,
[22] R. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, London (1982).
[23] V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Integrable structure of conformal
eld theory. 3. The Yang-Baxter relation, Commun. Math. Phys. 200 (1999) 297
[hep-th/9805008] [INSPIRE].
JHEP 06 (2011) 050 [arXiv:1102.5403] [INSPIRE].
[24] M. Piatek, Classical conformal blocks from TBA for the elliptic Calogero-Moser system,
[25] K. Bulycheva, H.-Y. Chen, A. Gorsky and P. Koroteev, BPS States in Omega Background
and Integrability, JHEP 10 (2012) 116 [arXiv:1207.0460] [INSPIRE].
[26] M. Piatek, Classical torus conformal block, N = 2 twisted superpotential and the accessory
parameter of Lame equation, JHEP 03 (2014) 124 [arXiv:1309.7672] [INSPIRE].
[27] S.K. Choi, C. Rim and H. Zhang, Irregular conformal block, spectral curve and ow
equations, JHEP 03 (2016) 118 [arXiv:1510.09060] [INSPIRE].
[28] G. Bonelli, A. Tanzini and J. Zhao, Vertices, Vortices and Interacting Surface Operators,
JHEP 06 (2012) 178 [arXiv:1102.0184] [INSPIRE].
[29] G. Bonelli, A. Tanzini and J. Zhao, The Liouville side of the Vortex, JHEP 09 (2011) 096
[arXiv:1107.2787] [INSPIRE].
[30] F. Fucito, J.F. Morales, R. Poghossian and D. Ricci Paci ci, Exact results in N = 2 gauge
theories, JHEP 10 (2013) 178 [arXiv:1307.6612] [INSPIRE].
[31] M. Bershtein and O. Foda, AGT, Burge pairs and minimal models, JHEP 06 (2014) 177
[arXiv:1404.7075] [INSPIRE].
gauge theories, arXiv:1211.2240 [INSPIRE].
supersymmetric gauge theory, JHEP 12 (2002) 071 [hep-th/0211170] [INSPIRE].
spheres, JHEP 11 (2015) 064 [arXiv:1507.05426] [INSPIRE].
JHEP 11 (2007) 002 [arXiv:0709.3806] [INSPIRE].
the accessory parameters and as a potential for the Weil-Petersson metric on the Teichmuller
geometry of quiver gauge theories, arXiv:1512.02492 [INSPIRE].
[1] N. Seiberg and E. Witten , Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl . Phys. B 431 ( 1994 ) 484 [ hep -th/9408099] [INSPIRE].
[2] N. Seiberg and E. Witten , Electric-magnetic duality, monopole condensation and con nement in N = 2 supersymmetric Yang-Mills theory , Nucl. Phys. B 426 ( 1994 ) 19 [32] S.K. Ashok , M. Billo , E. Dell'Aquila , M. Frau , R.R. John and A. Lerda , Non-perturbative studies of N = 2 conformal quiver gauge theories , Fortsch. Phys . 63 ( 2015 ) 259 [33] N. Nekrasov and V. Pestun , Seiberg-Witten geometry of four dimensional N = 2 quiver [34] F. Cachazo , M.R. Douglas , N. Seiberg and E. Witten , Chiral rings and anomalies in [35] F. Fucito , J.F. Morales and R. Poghossian , Wilson loops and chiral correlators on squashed [36] A.B. Zamolodchikov , In nite Additional Symmetries in Two-Dimensional Conformal Quantum Field Theory, Theor . Math. Phys. 65 ( 1985 ) 1205 [Teor . Mat. Fiz . 65 ( 1985 ) 347] [37] V.A. Fateev and S.L. Lukyanov , The Models of Two-Dimensional Conformal Quantum Field Theory with Z(n) Symmetry, Int . J. Mod. Phys. A 3 ( 1988 ) 507 [INSPIRE].
[38] A. Bilal and J.-L. Gervais , Systematic Approach to Conformal Systems with Extended Virasoro Symmetries , Phys. Lett. B 206 ( 1988 ) 412 [INSPIRE].
[39] V.A. Fateev and A.V. Litvinov , On di erential equation on four-point correlation function in the Conformal Toda Field Theory , JETP Lett. 81 ( 2005 ) 594 [Pisma Zh . Eksp. Teor. Fiz. 81