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

0.48 MB·

by  Rubik Poghossian

Prepared for submission to JHEP YerPhI/2016/01
Deformed SW curve and the null vector
decoupling equation in Toda field theory
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y
Rubik Poghossian
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Yerevan Physics Institute,
0 Alikhanian Br. 2, AM-0036 Yerevan, Armenia
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E-mail: [email protected]

]
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-t Abstract: It is shown that the deformed Seiberg-Witten curve equation after
p
e Fourier transform is mapped into a differential equation for the AGT dual 2d CFT
h
cnformal block containing an extra completely degenerate field. We carefully match
[

parametersintwosidesofdualitythusprovidingnotonlyasimpleindependentprove
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v of the AGT correspondence in Nekrasov-Shatashvili limit, but also an extension of
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AGT to the case when a secondary field is included in the CFT conformal block.
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0 Implications of our results in the study of monodromy problems for a large class of
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n’th order Fuchsian differential equations are discussed.
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.
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0 Keywords: Deformed Seiberg-Witten equation, Toda field theory, AGT, Fuchsian
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1 differential equations, Accessory parameters
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i
X
r
a
Contents
1 Introduction 1
2 Deformed Seiberg-Witten curve for A quiver 3
r
2.1 Exponents 7
2.1.1 Points z = 0 and z = ∞ 7
r+2 0
2.1.2 Points z = 1, z = q , z = q q , ... , z = q q ···q 8
1 2 1 3 1 2 r+1 1 2 r
3 Toda CFT 9
3.1 Preliminaries on A Toda CFT 9
n−1
3.2 Fusion with the completely degenerated field V 10
−bω1
3.3 Derivation of null-vector decoupling equation in semiclassical limit 10
4 Comparison with the differential equation derived from DSW 16
4.1 From the gauge theory differential equation to the null-vector decou-
pling equation 16
4.2 Matching S (z) with w(2)(z): emergence of AGT 17
2
4.3 A -Toda 4-point functions versus SU(n) gauge theory with 2n fun-
n−1
damental hypers 20
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 field was triggered by
the application of localization method [5–8]. An earlier important reference is [9].
To make localization method efficient one should first formulate the theory in a non-
trivial background, commonly referred as the Ω-background [5, 8], which is parame-
terized by two numbers (cid:15) , (cid:15) (these are rotation angles in (x1,x2) and x3,x4 planes
1 2
of Euclidean space time). The Ω-background brakes the Poincar´e symmetry and
effectively regularizes the space-time volume, making the partition function finite.
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
– 1 –
in such a way, that their total number of boxes coincides with the instanton num-
ber. Sending the parameters (cid:15) to zero one recovers the known results of the trivial
1,2
background. It appears, nevertheless, that the theory on finite Ω-background has
its own significance. Namely, the recent developments recovered intriguing relations
of these theory with 2d CFT called the AGT correspondence [10–12]. According to
AGT correspondence the partition functions of gauge theories get identified with 2d
CFT conformal blocks.
AninterestingspecialcaseoftheΩ-backgroundistheNekrasov-Shatashvililimit
[13]whenonlyoneoftheparameters, say(cid:15) → 0. Inthislimittheclassicalintegrable
1
system associated with SW curve gets quantized so that the remaining parameter
(cid:15) plays the role of the Plank’s constant. In [14–17] this limit has been investigated
2
usingBohr-Sommerfeldsemiclassicalmethod. Anotherapproachinitiatedin[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 algebraic equations defining SW curve
get replaced by difference equations (referred as deformed Seiberg-Witten curve or
shortly DSW equations). It is worth noting that like the original SW curve, DSW
”curve” besides the prepotential encodes information about all chiral correlators.
Let me describe briefly how DSW equation emerges. One starts with an entire
function whose zeros encode the lengths of the rows of the dominant array of Young
diagrams mentioned above. The condition of giving the most important contribution
is translated then to a linear difference equation for this entire function [18]. The
DSW equation (no longer linear) emerges as a condition on the ratio of this entire
function with itself with a shifted argument. The initial linear difference equation is
closely related to the Baxter’s T −Q equation which plays an important role in the
context of exactly integrable statistical mechanics [22] and QFT models [23]. Fourier
transform of the linear difference leads to a linear differential equation. This is the
same equation which emerged as the Schro¨dinger equation in the already mentioned
alternative Bohr-Sommerfeld approach to the NS limit.
For the purposes of this paper it is essential that from the AGT perpective the
NS limit corresponds to the classical (c → ∞) limit of 2d CFT conformal block of
”heavy” fields. 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 differential
equation discussed in previous paragraph appears to be closely related to the null-
vectordecouplingequationin2dCFT.Someresultsinthisdirectionhasbeenalready
announced in [21]. For applications of CFT degenerate fields in AGT context see
also [28–32].
In this paper we systematically Investigate this relationship in a quite general
– 2 –
setting of A linear quiver theories with an arbitrary number (equal to r) of SU(n)
r
gauge groups corresponding in AGT dual CFT side to the r + 3-point conformal
blocks in W Toda field theory.
n
The subsequent material is organized as follows. In Section 2 we investigate
DSW equations for A quiver theories and establish explicit relations between curve
r
parameters and chiral expectation values. Then using Fourier transform we derive
the corresponding linear differential equation and thoroughly investigate its singular
points. In Section 3 starting from the general structure of the fusion rule of the
completely degenerated field V and using Ward identities together with some
−bω1
general requirements necessary to get acceptable solutions, we derive the null-vector
decoupling equation in the semiclassical limit. Note that our approach here is some-
what heuristic and seems to be applicable only in semiclassical case. To get exact
differential equation valid in full pledged quantum case one should construct the null
vector explicitly and make use of the complicated W -algebra commutation relations
n
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 differential equation of previous chapters can be completely matched. Already
comparison of the first non-trivial coefficient functions (in 2d CFT side this func-
tion 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 coef-
ficient 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 field 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 findings in the context of
the monodromy problems in a large class of Fuchsian differential equations.
2 Deformed Seiberg-Witten curve for A quiver
r
Partition function and chiral correlators of N = 2 gauge theory can be represented
as sum over arrays of Young diagrams which label the fixed points of space time
rotations and global gauge transformations acting in the moduli space of instantons.
[5–8]. In the case of A quiver theory with fundamental and bi-fundamental matter
r
hypermultiplets and unitary U(n) gauge groups (see Fig.1a), there is an n-tuple of
Young diagrams associated to each of the r gauge groups (indicated by circles in
Fig.1a). It has been shown in [18] for the case of a single gauge group and later gen-
eralized further in [19–21] that among all fixed points in moduli space of instantons
there is a unique one giving a non-vanishing contribution in the Nekrasov Shatashvili
– 3 –
1 z2 zr zr+1
α1ω1 α2ω1 αrω1 αr+1ω1
U(n) U(n)
∞ 0
a a a a
0,u 1,u r,u r+1,u P0,u P1,u Pr,u Pr+1,u
b b b b
(a) (b)
Figure 1. (a) The quiver diagram for the conformal linear quiver U(n) gauge theory: r
circles stand for gauge multiplets; two squares represent n anti-fundamental (on the left
edge) and n fundamental (the right edge) hypermultiplets; the lines connecting adjacent
circlesarethebi-fundamentals. (b)TheAGTdualconformalblockoftheTodafieldtheory.
limit1. We will denote the (rescaled by (cid:15) ) lengths of the rows of this “critical” array
1
of Young diagrams Y by λ where α = 1,...r refers to the node of the quiver,
α,u α,u,i
u = 1,...,n is the index of the defining representation of the gauge group U(n)
associated with this node and i = 1,2,... specifies the row. The data λ can
α,u,i
be conveniently encoded in meromorphic functions y (x), which are endowed with
α
zeros located at x = a +i−1+λ and poles at x = a +i−2+λ where
α,u α,u,i α,u α,u,i
a are the Coulomb branch parameters. In addition we associate to the ”frozen”
α,u
nodes (indicated by squares in Fig.1a) the parameters a and a . In terms of
0,u r+1,u
these parameters the masses of fundamental and anti-fundamental hypermultiplets
are given by
n n
1 (cid:88) 1 (cid:88)
m = a − a and m¯ = a − a (2.1)
u r+1,u r,u u 0,u 1,v
n n
v=1 v=1
respectively. In terms of
n
1 (cid:88)
a¯ = a (2.2)
α α,u
n
u=1
the masses of the bifundamental hypermultiplets are simply
m = a¯ −a¯ . (2.3)
α,α+1 α+1 α
TheDeformedSeiberg-Witten(DSW)curveequationsarisefromtheconditiononthe
instanton configuration to give the most important contribution to the prepotential
in NS limit. In the case of our present interest of A quiver theory we get a system of
r
r (difference) equations for r functions y (x), α = 1,...,r. In addition we introduce
α
two polynomials
n n
(cid:89) (cid:89)
y (x) = (x−a ); y (x) = (x−a ) (2.4)
0 0,u r+1 r+1,u
u=1 u=1
1For simplicity in this paper we’ll set (cid:15) = 1. This is not a loss of generality since a generic (cid:15)
2 2
everywhere can be restored by simple scaling arguments.
– 4 –
which encode fundamental hyper-multiplets attached to the first and the last nodes
of the quiver Fig.1a. The equations can be found using iterative procedure based on
so called iWeyl reflections (i stands for instanton) [33]
y (x−1)y (x)
α−1 α+1
y (x) → y (x)+ (2.5)
α α
y (x−1)
α
It appears that the result of this procedure is related to the q-Character of the
α’th fundamental representation of the group A . Explicitly for α = 1,2,...,r one
r
obtains2
 
χα(x) = y0(x−α) (cid:88) (cid:89)α y ykβ(x(x−−αα++ββ−) 1) qββ−αk(cid:89)β−1qγ,(2.6)
k −1
1≤k1<k2<···<kα≤r+1β=1 β γ=1
where q are the gauge couplings, χ (x) are n-th order polynomials in x with co-
α α
efficients related to the expectation values (cid:104)trφJ(cid:105) (φ are the scalars of the vector
α α
multiplet) in a way to be specified below. For later purposes we’ll set by definition
χ (x) ≡ y (x) and χ (x) ≡ y (x). The difference equations (2.6) are the de-
0 0 r+1 r+1
formed Seiberg-Witten equations [18] for the A quiver gauge theory [20, 21]. It is
r
assumed that all functions y (x) are normalized so that their large x expansions read
α
y (x) = xn(1−c x−1 +c x−2 −c x−3 +···). (2.7)
α α,1 α,2 α,3
The 1-forms dlogy (x) are the direct analogs of Seiberg-Witten differentials and
α
define the chiral correlators by the conventional contour integrals
(cid:73)
dx
(cid:104)trφJ(cid:105) = xJ∂ logy (x), (2.8)
α 2πi x α
γα
where γ are large contours surrounding all zeros and poles of y (x) in anti-clockwise
α α
direction. Comparison of (2.7), with (2.8) allows one to express the expansion coef-
ficients c in terms of chiral correlators (cid:104)trφJ(cid:105) with J ≤ k. Here are the first few
α,k α
relations
(cid:104)trφ (cid:105) = c
α α,1
(cid:104)trφ2(cid:105) = c2 −2c
α α,1 α,2
(cid:104)trφ3(cid:105) = c3 −3c c +3c (2.9)
α α,1 α,1 α,2 α,3
(cid:104)trφ4(cid:105) = c4 −4c2 c +4c c +2c2 −4c
α α,1 α,1 α,2 α,1 α,3 α,2 α,4
··· ··· ··· ··· ··· ···
2When comparing this formula with those of [20, 21] it should be taken into account that
we have shifted arguments in y (x), χ (x) appropriately to get rid of explicit appearance of the
α α
bifundamental masses.
– 5 –
On the other hand, inserting the expansion (2.7) into (2.6) and comparing left and
right hand sides one can express the coefficients c (and due to (2.9) also (cid:104)trφJ(cid:105) ) in
α,k α
termsofcoefficientsofthepolynomialsχ (x). Infactthefirstnoftheserelationscan
α
be inverted to get the coefficients of the polynomials χ (x) in terms of c ,...,c
α α,1 α,n
(or, equivalently, intermsof(cid:104)trφ (cid:105),...,(cid:104)trφn(cid:105)). Thentheremaininginfinitenumber
α α
of relations are nothing but the deformation of the celebrated chiral ring relations
[34] expressing higher power (J > n) chiral expectation values (cid:104)trφJ(cid:105) in terms of
α
lower, up to the n’th power expectation values3. For our later purposes let us display
explicitly the relations for the first three coefficients of the polynomials
n
(cid:88)
χ (x) = (−)iχ xn−i. (2.10)
α α,i
i=0
Expanding l.h.s. of (2.6) up to the order ∼ xn−3 we get
α kβ−1
(cid:88) (cid:89) (cid:89)
χ = q ; (2.11)
α,0 γ
1≤k1<···<kα≤r+1β=1 γ=β
(cid:32) α (cid:33) α kβ−1
(cid:88) (cid:88)(cid:0) (cid:1) (cid:89) (cid:89)
χ = c − c −c q ; (2.12)
α,1 0,1 k −1,1 k ,1 γ
β β
1≤k1<···<kα≤r+1 β=1 β=1 γ=β
(cid:34)
α−1 α
(cid:88) (cid:88) (cid:88) (cid:0) (cid:1)(cid:0) (cid:1)
χ = c −c +2 c −c +2
α,2 kβ−1,1 kβ,1 kγ−1,1 kγ,1
1≤k1<···<kα≤r+1 β=1γ=β+1
α
(cid:88)(cid:0) (cid:0) (cid:1) (cid:0) (cid:1)
+ c −c +α−β +3 −(c +2α) c −c +2
k −1,1 k ,1 0,1 k −1,1 k ,1
β β β β
β=1
(cid:17)
+(−α+β −2)c +c2 −c +c +2α−2β +3
kβ,1 kβ−1,1 kβ−1,2 kβ,2
(cid:21) α kβ−1
(cid:89) (cid:89)
+α(c +α)+c q . (2.13)
0,1 0,2 γ
β=1 γ=β
Clearly with more efforts it should be possible to write down expressions for further
coefficients but unfortunately these expressions soon become quite intractable. In
Section 4 we’ll do one more step giving an explicit expression for the next coefficient
in the special case when r = 1.
Quite remarkably it is possible to eliminate the functions y (x),...,y (x) from
2 r
eq. (2.6) and find a single equation for y (x). Here is the result4:
1
r+1 i−1
(cid:88) χ (x) (cid:89) y (x−j)
1+ (−)i i 0 qi−j = 0, (2.14)
y (x) y (x−j) j
1 1
i=1 j=1
3For the generalization of chiral ring relation for the generic Ω-background see [35].
4Of course, the same can be done also for y (x).
r
– 6 –
It is useful to represent the meromorphic functions y (x) as a ratio:
1
Y(x)
y (x) = y (x) , (2.15)
1 0
Y(x−1)
where Y(x) is an entire function with zeros located at x = a + (i − 1) + λ
1,u 1,u,i
(remind that λ is the appropriately rescaled length of the i’th row of the Young
α,u,i
diagram Y ). In terms of Y(x) the eq. (2.14) can be rewritten as
α,u
(cid:32) (cid:33)
r+1 α
(cid:88) (cid:89)
(−)α qα−β χ (x)Y(x−α) = 0. (2.16)
β α
α=0 β=1
Since for small values of the gauge couplings q (cid:28) 1 the i’th row length
α
λ → 0 when i → ∞, it is reasonable to expect that the sum
α,u,i
(cid:88)
ψ(z) = Y(x)z−x (2.17)
x∈Z+a1,u
will converge in some ring with the center located at 0. Then the difference equation
for Y(x) can be easily “translated” into a linear differential equation for ψ(x) [19, 20]
(cid:32) (cid:33)
r+1 α
(cid:88) (cid:89) d
(−)α qα−β χ (−z )z−αψ(z) = 0 (2.18)
β α dz
α=0 β=1
It is not difficult to find the coefficient in front of the highest derivative dn/dzn in
(2.18). Using (2.11) one can show that this coefficient has a nice factorized form
(cid:32) (cid:33) (cid:32) (cid:33)
r+1 α r α
(cid:88) (cid:89) (cid:89) (cid:89)
(−)n+α qα−β χ zn−α = (−)nzn−r−1 z − q (2.19)
β α,0 β
α=0 β=1 α=0 β=1
Furtherinvestigationconfirmsthatindeed(2.18)isann-thorderFuchsiandifferential
equation with r+3 regular singular points located at
z = ∞, z = 1, z = q , z = q q , ..., z = q q ···q , z = 0,
0 1 2 1 3 1 2 r+1 1 2 r r+2
where for later use we have introduced new parameters z related to the gauge
α
couplings through conditions
z
α+1
q = . (2.20)
α
z
α
2.1 Exponents
2.1.1 Points z = 0 and z = ∞
r+2 0
First let’s look after a solution of the form
ψ(z) = zs(1+O(z)) (2.21)
– 7 –
Inserting this in (2.18) we see that when z → 0 the term with α = 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)
y (r+1−s) = 0 (2.22)
r+1
Similarly the characteristic equation for the infinity reads
y (−s) = 0 (2.23)
0
2.1.2 Points z = 1, z = q , z = q q , ... , z = q q ···q
1 2 1 3 1 2 r+1 1 2 r
Investigation of these points is slightly more subtle. Consider the ansatz
ψ(z) = (z −z )s(1+O(z −z ))) (2.24)
α α
for some fixed α ∈ {1,...,r+1}. Taking into account (2.19) and (2.12) for the index
s we get the equation
r+1 n−1
(cid:89) (cid:89)
0 = (−)nzn−r−1 (z −z ) (s−i) (2.25)
α α β
β(cid:54)=α,β=1 i=0
 
r+1 β n−2
(cid:88) (cid:88) (cid:89) (cid:89)
+ nβ(−zα)n−β−1 zkγ (s−i)
β=0 1≤k1<···<kβ≤r+1γ=1 i=0
 
(cid:32) (cid:33)
r+1 β β n−2
(cid:88) (cid:88) (cid:88)(cid:0) (cid:1) (cid:89) (cid:89)
+ (−zα)n−β−1 c0,1 − ckγ−1,1 −ckγ,1 zkγ (s−i).
β=0 1≤k1<···<kβ≤r+1 γ=1 γ=1 i=0
The first two lines come from the terms proportional to (zd/dz)n. The first (second)
line includes part with n ”hits” (n−1 hits) on ψ(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 ψ(z).
Though the second and especially third lines of this equation look quite complicated,
fortunately they can be simplified drastically. Indeed it can be shown that the second
line is equal to
r+1 n−2
(cid:89) (cid:89)
(−)nnzn−r−1 (z −z ) (s−i) (2.26)
α α β
β(cid:54)=α,β=1 i=0
while the third line can be rewritten as
r+1 n−2
(cid:89) (cid:89)
(−)nzn−r−1(c −c ) (z −z ) (s−i). (2.27)
α α−1,1 α,1 α β
β(cid:54)=α,β=1 i=0
Thus for the allowed exponents we get
s ∈ {0,1,...n−2,c −c −1}. (2.28)
α,1 α−1,1
– 8 –
It is known in general that if indices differ by integers one might have logarithmic
solutions. This is not the case however in the example at hand. A closer look ensures
that our differential equation around z = z , α = 1,2,...,r admits n independent
α
solutions of the types
(z −z )m +O((z −z )n−1); m ∈ {0,1,2,...,n−2} and
α α
(z −z )cα,1−cα−1,1−1(1+O(z −z )). (2.29)
α α
3 Toda CFT
3.1 Preliminaries on A Toda CFT
n−1
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),
where the ”background charge” Q is given by
Q = ρ(b+1/b),
where ρ is the Weyl vector of the algebra A and b is the dimensionless coupling
n−1
constant of Toda theory. In what follows it would be convenient to represent roots,
weights and Cartan elements of A as n-component vectors subject to condition
n−1
thatsumofcomponentsiszeroandendowedwiththeusualKroneckerscalarproduct.
Obviously this is equivalent to a more conventional representation of these quantities
asdiagonaltracelessn×nmatriceswithpairinggivenbytrace. Inthisrepresentation
the Weyl vector is given by
(cid:18) (cid:19)
n−1 n−3 1−n
ρ = , ,..., (3.1)
2 2 2
and for the central charge we’ll get
c = (n−1)(1+n(n+1)q2)
where for the later use we have introduced the parameter
1
q = b+ .
b
For further reference let us quote here explicit expressions for the highest weight ω
1
of the first fundamental representation and for its complete set of weights h ,...,h
1 n
(h = ω )
1 1
(ω ) = δ −1/n,
1 k 1,k
(h ) = δ −1/n. (3.2)
l k l,k
– 9 –