Wilson loops and chiral correlators on squashed spheres
Received: July
Wilson loops and chiral correlators on squashed
F. Fucito 0 1
J.F. Morales 0 1
R. Poghossian 0 1 2
0 Alikhanian Br. 2, AM0036 Yerevan , Armenia
1 Via della Ricerca Scienti ca , I00133 Roma , Italy
2 Yerevan Physics Institute
We study chiral deformations of N = 2 and N = 4 supersymmetric gauge the N = 2 super eld. Using localization, we compute the deformed gauge theory partition function Z(~jq) and the expectation value of circular Wilson loops W on a squashed foursphere. In the case of the deformed N = 4 theory, exact formulas for Z and W are derived in terms of an underlying U(N ) interacting matrix model replacing the free Gaussian model describing the N = 4 theory. Using the AGT correspondence, the J deformations are related to the insertions of commuting integrals of motion in the fourpoint CFT correlator and chiral correlators are expressed as derivatives of the gauge theory partition function on a nite In the so called NekrasovShatashvili limit, the entire ring of chiral relations is extracted from the deformed SeibergWitten curve. As a byproduct of our analysis we show that SU(2) gauge theories on rational backgrounds are dual to CFT minimal models.
Nonperturbative E ects; Wilson; 't Hooft and Polyakov loops; Supersymmet

aI.N.F.N  Sezione di Roma 2,
Contents
1 Introduction and summary 2
The gauge partition function on R
The instanton moduli space and the equivariant charge
The deformed gauge partition function
Chiral correlators
The gauge theory on S4
Wilson loops
Perturbative expansion
The N = 4 deformed theory
Two point correlators htreC tr'J i
The CFT side
The conformal eld theory
Integrals of motion
The gauge/CFT dictionary
The gauge theory side
AGT duality: chiral correlators vs integrals of motion
Chiral relations: 1; 2 nite
AGT duals of minimal models
The minimal models
Degenerated states vs. critical masses
The partition function and the fourpoints conformal blocks
The gauge partition function on S4 vs the fourpoint correlator
Introduction and summary
extracted from the asymptotic fall o
of a Wilson loop made of two straight lines. The
correspondence.
involves both the gauge eld Am and its scalar superpartners 'i
W = htr eCi
C = i (Amx_ m + 'i y_i)ds
ent fractions of the original supersymmetry [1{3].
with Z(a; ) the gauge partition function on R4. Here Z is evaluated on an
such that 1 =
2 =
nite, so the gauge theory lives e ectively on a nontrivial
in the limit where the mass of the adjoint hypermultiplet is sent to zero.
GromovWitten invariants for certain complex manifolds.
Indeed in presence of an
background the zero mode solutions to the equations of motion
xm Am with
loop is given by the integral
tr eC =
dN a (a) tr e 1 e N V (~;a)
prediction S = ln W
e for the minimal area of a string worldsheet on the gravity
dual theory as a function of a 4dependent e ective t'Hooft coupling
e . As a bonus, in
this case, it will also be possible to
nd the correlators of the Wilson loop with tr 2 and
tr 4 and check our results against those of [20].1
1We thank the referee of the paper to point out this reference to us.
for a nite
background. We carry out the explicit computation in the SU(2) theory
partition functions are matched against the dual CFT correlators.
Witten curve in the limit in which one of the two parameters of the
background is sent
masses and rational backgrounds and CFT minimal models.
The gauge partition function on R
Sclass =
1 Z
Fclass( ) + h:c: + : : :
the scalar ' in the vector multiplet
has a nontrivial vacuum expectation value. In this
the classical prepotential of the general form
for some integer p and
Fclass( ) =
= ' +
Fclass( ) =
i tr 2 is recovered after setting J>2 to zero.
with gauge group U(N ) and fundamental or adjoint matter.
The instanton moduli space and the equivariant charge
[N + 2k]
2k matrices
0; M characterizing the instanton gauge connection and the
satisfy the superADHM constraints
M + M
= 0
To localize the integral one rst introduces the equivariant charge on M
the moduli space via
degrees of freedom in
, M leaving a 4kN dimensional superspace Mk;N .Fundamental
hypermultiplets contribute an extra [k
Nf ] matrix K of fermionic moduli that can be
connection can be written in terms of the [N+2k] [N] matrix U de ned by
U = U
= 0
U U = 1l
and reads
Am = U @mU
Q = Q + d + i
= M
Q K = h
Q M = 0
Q h = 0
and i a contraction de ned by
i dxm =
i dMi =
an element of the symmetry group SO(4)
U(Nf ) rotating moduli
and spacetime coordinates. The combined action can then be written as
= M + d
Q K = h
We notice that Q2 =
, so Q is nilpotent at the xed points of . The charge Q can be
for localization, is the construction of an equivariant eld F de ned by
U + QU QU + [U QU; U QU ]
= r(U QU )
F = r(U dU )
components. Indeed, recalling that
rotates both the instanton moduli and the spacetime
variables xm one can write [33]
modU + QU QU + [U QU; U QU ] +
xm Am = ' +
In the limit ` to zero,
! 0, '~ reduces to ' and the equivariant correlator computes
operators are built out of '~ rather than '.
after replacing
m by dxm and evaluating the elds on the instanton background. Indeed,
background can be written in the explicit Q invariant form
Q F = (
+ gauge)(U Q U )
Sinst(~) =
2 J ! C2
level and instanton contributions
The oneloop partition function Zone loop is given by
with [17, 32]
Zognaeugeloop =
Zomnaetter
loop =
Z(~) = Zone loop Zinst+tree(~)
Zone loop = Zognaeugeloop Zomnaetter
QuN;v=1 2(av
m) 2(auv + m + )
mu) 2(au
mv+N + ) fund:
Here 2(x) is the Barnes double gamma function2 and
= 1 + 2
The instanton partition function is de ned by the moduli space integral
Zinst+tree( ) =
is given by the
localization formula [12]
= ( 2 ) D2 X
with D the complex dimension of the space M , x0 labelling the xed points of
take M = C2,
= P1
J=2 2 JJ! tr F
this action is the origin and the localization formula leads to
a rotation in the Cartan of the SO(4) Lorentz
Sinst(~) =
2 J ! C2
tr F J =
1 2 J=2
J tr '~(0)J
To perform the integral over the instanton moduli space we take
element of the Cartan subgroup of the symmetry group U(N )
= e Sinst(~) and
1 > 0; 2 > 0)
2(x) = Y
i;j=0 x + i 1 + j 2
or adjoint
I the Cartan of U(k) and 1;2 are Lorentz breaking parameters that
deform the R4 spacetime geometry. For a gauge theory on
at space 1;2 should be sent
to zero at the end of the computation. For
nite 's the integral describes the partition
function on a nontrivial gravitational background, the so called
bakcground. Fixed
the U(k) Cartan elements I . Explicitly
(i;j) = au + (i
1) 1 + (j
Zinst+tree( ) =
ZY =
= Zgauge ZYmatter
with ZY the inverse determinant of
and SY (~) the 0form of the instanton action (2.22)
evaluated at the xed point. For ZY one nds [11, 12, 32]
ZYmatter =
( QuN;v ZYu;Yv (auv + m)
QuN;v=1 Z;;Yv (mu
mv+N ) fund
ZYu;Yv (x) =
i) + 2 (1 + k~ui
(x + 1(1 + kuj
exchange mi $ mj can be obtained by replacing
mu+N ! mu+N +
SY (~) =
1 2 J=2
J tr '~0J Y =
1 2 J=2
with '~0 = U
U the zero form of the equivariant super eld F and OJ;Y the chiral primary
operator evaluated at the
xed point. To evaluate OJ;Y , one rst notices that for a
speci ed by the Young tableaux data Y one can
nd in nitely many solutions U (k;l) to the
de ning equation
U = 0. The
the chiral operators [14, 33].
Y = V
ez (k;l) = V
u=1 (k;l)2=Yu
reproduce the classical result hez'
ez 2 ) in order to
of z one nds
OJ;Y =
O2;Y = tra2
O3;Y = tra3
For the rst few values of J one nds
u=1 (i;j)2Yu
u=1 (i;j)2Yu
O4;Y = tra4
(2 12 + 3 1 2 + 2 22 + 6
The deformed partition functions can then be written as
Zinst+tree(~) =
X ZY (~) =
1 2 J=2
energy associated to the deformed instanton partition function
Fe (~) =
1 2 ln Z(~)
Z( J =
J;2) = e
Zoneloop
with q = e2 i and Ztree = e
and Zinst = P
Y ZY qjY j the instanton part.
the tree level contribution to the partition function
function. To this aim, we rst introduce the Q invariant volume form
4 = dz1 dz1 dz2 dz2
i( 1jz1j2dz2dz2 + 2jz2j2dz1dz1)
and write the chiral operator in the equivariant form
d4x tr '~J =
htr '~J i =
or equivalently from (2.33)
J ! htr '~J i =
@ J ln Z(~)
derivative of the prepotential in the undeformed theory in
at space. On the other hand,
the general multitrace chiral correlators
htr '~J1 tr '~J2 : : :iundeformed
in the undeformed theory.
The gauge theory on S4
The gauge partition function on S4 is given by the integral [7]
ZS4 (~) = c
dN a jZone loop(a) Ztree+inst(a; ~)j2
vevs and masses taken in the domains
au =2 iR
mu =
mu =
nant Zone loop.
jZognaeugeloopj2 =
jZomnaetteloropj2 =
QuN;v=1 (av
(x) =
(x) =
is an entire function satisfying
x). It has an in nite number of single
zeros at x =
0 integers. Finally the
normalization c has been xed for later convenience to be4
Wilson loops
A supersymmetric Wilson loop is de ned by the line integral
with '1 = 12 ('
'y) and Am taken to be antihermitian matrices.
We use complex
where 1 2 = 1 and write 1 =
tion (2.18) is given by the same formula with
2(x) replaced by
x). We adopt units
one nds
c = q 21 m3(m3+ )+ 12 m4(m4+ )
C = i
(Am x_ m + jx_ j '1)ds
z`(s) = r` ei `s
L =
2 n1 =
and the Wilson loop can be written in the suggestive form
jx_ j2 = 12jr1j2 + 22jr2j2 = 1
C = i
xm + '1) ds =
precisely match.
C0 = i'~0 = iU
U is antihermitian and that correlators of '~(s) do not depend on s, so the
Wilson loop operator on the instanton background Y reduces to
= tr ei L '~0
Y =
S4 =
jZone loop(a)Ztree+inst(a; ~) j2
Plugging (3.16) into (3.15) one
nds, for the leading J correction to the Wilson loop
expectation value at weak coupling, the result
S4 =
dN a (a) e 12 2 Im tra2 tr aJ tr a2 + : : :
the correlators
tr eC tr '~J1 '~J2 : : : S4;undef:
tation value is computed by the correlator
S4 =
tr eC tr '~J
S4;undef:
Perturbative expansion
the weak coupling regime Im
! 1. In this limit, the integral (3.13) is dominated by the
one nds
(a) =
Zinst = 1 + : : :
The N = 4 deformed theory
at our disposal: the N = 4 theory.
of N = 2 theory where the N
and matter multiplets.
located at m =
1 or m =
2. Indeed, for any choice of J and instanton number the
1. Indeed
using the double Gamma function identity
V (a; ~) =
W =
J tr(ia)J + h:c:
dN a (a) tr e 1 e N V (a;~)
one nds
tion function reduces to the U(N ) matrix model integrals
jZoneloopj2 =
(a) =
u6=v
Z =
methods [5]. One de nes the resolvent
w(x) =
u=1 x
w(x) =
4fp 2(x)
where fp 2(x) a polynomial of order p
2 determined by the condition that w(x)
the square root, so we can write
with S the union of the cuts and (x) the density
w(x i0) =
(x) =
w(x) = 1 V 0(x)
x1 for large jxj and therefore w(x) is fully determined.
As an example, let us consider the quartic potential
with Qp 2(x) a polynomial of order p
2. Indeed the number of unknown variables in
b2 =
V (a) =
g4 =
4 Im( 4)
x1 for large jxj one nds
w(x) =
(1 + 4 g4(x2 + 2b2))px2
4
2
line where Wilson loop operators C( ) are inserted.
Plugging (3.31) into (3.30) and expanding for large x and small
one nds [5]
x w(x) =
N n=0
= 1 +
with 2n external lines, p propagators and k fourpoint vertices. In
gure 1 we show the
the analytic form [5]
x w(x) = 1 + X
X1 ( 12g4 2)k n (2n)! (2k + n
x2n n! (n
1)! k! (k + n + 1)!
For the Wilson loop one then nds
W =
N n=0
= 1 + X
= p
n=1 k=0
n=1 k=0
e =
= gY2 M N n1n2
0 and n1n2 = 1 one nds the
familiar N = 4 formula [1, 4, 6]
YM = gY2MN
On the other hand in the limit of large
with g4 2 kept nite one
nds the same
ending on the loop is given by ln W so one expects
S = ln W
providing a precise test of the duality.
Two point correlators htreC tr'J i
the expansion of the exact formula (3.34) in the limit where 2
0 i.e. the
undeformed theory.
correlator are traceless, so the diagrams in
gure 1 and
gure 2 with loops starting and
YM
from the quartic vertex contract among themselves.
The correlators hW tr'J i can be extracted from the small g4expansion of (3.34)
W0 =
= 1 +
W = W0
g4 W1 + : : :
W1 = 12 2YM I4
the number of diagrams with zero and one g4vertices in
gure 1. The insertion of tr'2
respect to . Indeed using (3.22) and (3.28) one nds
htreC tr'2i = 2 2YM @ YM
W0 = 2 YM I2
YM
legs. These diagrams are counted by
W0 = 8 YM I2
Subtracting this contribution from W1 one nds
htreC tr'4i =
YM) =
localization and in [1, 4, 6] from perturbation theory.
AGT duality: chiral correlators vs integrals of motion
The CFT side
with the insertion of the integrals of motion In introduced in [35].
The conformal eld theory
algebra with commutation relations
[Lm; Ln] = (m
n) Lm+n + m) m+n;0 [am; an] =
[Lm; an] = 0
The central charge c is parametrized by
The primary elds V are de ned as
c = 1 + 6 Q2
Q = b +
V (z) = V
vir a primary eld of the Virasoro algebra with dimension
( ) = (Q
heis(z) = e2i(
Q) Pn<0 ann z n e2i Pn>0 ann z n
The commutation relations of the eld V and the generators Lm, an are
[Lm; V (z)] = V
[an; V (z)] =
heis(z) zm+1@z + (m + 1) ( ) zm
) zn V (z) for
Lmj0i = anj0i = 0
at zero and in nity respectively
j i = V (0) j0i
h j = lim z
Consider the remaining Virasoro part of the fourpoint correlator
Gvir ( i; jzi) = z1
hhVvi1r(z1) Vvi2r(z2) Vvi3r(z3) Vvi4r(z4) ii
of conformal dimension
and the factor z12 ( 1) is included to guarantee a nite limit at z1 !
1. The fact that
to z. For the choice above one nds [32]
@z1 Gvir = 0
@z3 Gvir = @z Gvir
@z2 Gvir = ( z@z + 2 1
@z4 Gvir = ((z
1)@z +
= Pi4=1 i and
sector one nds the conformal block
i =
( i). Including also the contribution of the Heisenberg
hh 1jV 2 (1) V 3 (z) j 4ii = (1
conformal block (4.12)
G( ijz) =
where C 1 2 are the Liouville structure constants [36, 37].
integrals are [35]
I2 = L0
I3 =
I4 = 2 X L k Lk + L
k= 1;k6=0
k= 1;k6=0
a k Lk + 2 i Q X k a k ak +
3 i+j+k=0
k= 1;k6=0 i+j=k
L k ai aj + 12 L0
jkja k Lk + 2(1
5 Q2) X k2 a k ak + 6iQ
formal blocks. We de ne
hh 1j[V 2 (1); a k] [ak; V 3 (z)] j 4ii +
= z@z +
Similarly for G3 one nds
G3 = i X z
the Virasoro part of the composite eld V .
+(k +1) 2 3 +(k 1)(Q
5 G
3) z@z +(Q
di erential operators acting on G. We write
Gn( i; jz) = Ln G( i; jz)
L2 = z@z +
L3 =
3) z @z + (Q
(see eq. (4.26)).
The gauge/CFT dictionary
to the partition function of the N
characterizing the
gravitational background parametrizes the central charge of the CFT.
The full dictionary is given by [17]
G( i; q) = ZSU4(2)(mi; q)
1 =
3 =
(m3 + m4)
2 =
4 =
1 = b
(m1 + m2)
2 = b 1
= 1 + 2 = Q
z = q
block G( i; jq) via
with Zinst
block is related to the SU(2) partition function via
4 for small q. On the other hand, the Virasoro conformal
Gvir( ; i; q) = q
4 ZiSnUst(2)(a; miq) = q
3) ZiUns(t2)(a; miq)
function on the sphere via
valid at allinstanton orders for a
background. The results will be checked against
Witten curve obtained in [25, 26].
Chiral relations: 1; 2
Using the AGT dictionary (4.20), (4.21) leads to
2 G2(q)
htr'~2i =
htr'~3i = 6 i G3(q)
htr'~4i = 2h4 G4(q)
= 1 + 2
h2 = 1 2
2 M3 + 2M4 + 2 q (M1 M3
M4) + h4 q(1
2q( M1 + M2) + q2( h2 + 2M2
htr'~2i =
htr'~3i =
htr'~4i =
with Z = ZoUn(e2) loopZinst+tree and
U(2)
htr'~3i =
htr'~4i =
M2 =
(1 + q)
q@qhtr'~2i +
M1 =
M3 =
mimj mk
M4 = m1m2m3m4
We notice that the last two equations of (4.26) can be rewritten in the form
q) + qM12 + M1 htr'~2i
q) + qM1M3 + M3)
which shows that in a nite
background, chiral correlators can be written in terms of
instanton computation. Using (2.26){(2.39) one nds
Zinst+tree = q h2 1
M1 + 2M2)
M3 + 2 M4) + : : : : (4.30)
The explicit form of Zone loop is irrelevant to our purposes since it is
independent. It is
to order q. We have checked this up to order q4.
h2 ln Zinst nite. For
the di erence equation [25, 26]
) y(z) y(z
) + (1 + q) P (z)y(z)
1 = 0
P (z) = z2
u1z + u2
Q(z) = 1 + X
The chiral correlators can be extracted from the expansion at large z of
tr '~2 = 2 a2 +
tr '~4 = 2 a4 +
tr '~3 = q 3 a2 M1 + 3 M3 + : : :
M1 + 2 M2)
M3 + 2 M4 + : : :
a2 M1 +M3
@z log y(z) =
y(z) =
and using (4.33) one nds the chiral correlators as functions of the yi's.
y34 + 4 y32 y4
3 y3 y4 + 3 y5
determines u1 while u2 can be solved in terms of htr'~2i. The results are
u1 =
u2 =
( 1 + q)htr'~2i + 2qM2 + 2q (
2(1 + q)
ring type. Explicitly
htr'~3i =
htr'~4i =
(1 + q)
q) + qM1M3 + M3)
q) + qM12 + M1 htr'~2i
for chiral correlators involving higher powers of the scalar eld.
AGT duals of minimal models
The minimal models
is characterized by its Virasoro central charge
There are (p
1g with conformal dimensions
1)=2 primary elds denoted as
m;n, m 2 f1; 2; : : : ; p
c = 1
m;n =
where Q = b + 1=b, b = ipp=q and
m;n =
whose dimensions are
Note the identi cation m;n
p m;q n, which re ects the symmetry of the dimension (5.2)
with respect to
Since a primary eld
m;n is degenerated at the level mn, any correlation function
2;1 satisfy a second order di erential equation and can be explicitly
of squares of the corresponding conformal blocks.
o = 1;1
= 1;2
" = 2;1
The nontrivial threepoint structure constants are
leading to the Ising fusion rules
o = 0;
" =
Cooo = C
o = C""o = 1
C " =
[o] + ["]
io = p
h""""io =
io =
2 z1=8(1
z)1=8
2z1=8 p
i" =
i =
""io =
z1=8(1
z)1=8
one nds
z)j1=4
h""""i = j
z=2j2
jzj1=4j1
1 = 2
1 = b = i
m3 + m4 =
3 =
O3 =
O3 = "
O3 =
O3 = "
The masses, mi, are chosen such that the i's in (4.20) belong to the set
ZiUns(t2;)+ = 2F1
ZiUns(t2;) = 2F1
1 B+A1;1 B+A2 q
; "g =
Q; Q + ; Q +
associated to the primary
elds (5.4).
Without any loss in generality we can discard
expectation value a = a1 =
a2. This integration can be formally evaluated with the
a+ =
Using (4.20), (5.13), (5.14) one nds
+ =
O3 =
O3 = "
Young tableaux with a single row(column) at a+ and no boxes at
a+ for O3 =
Similarly, for a = a
the relevant singlerow(column) tableaux are centered at a . The
whole instanton sum adds to a hypergeometric function [32]
are the operators associated to
O3 =
: A1 = (
+)b = (a+
O3 = " :
A2 = ( 1 + 2 +
Q)b = (a+
B = 2 +b = (2a+ + ) 1
A1 =
A2 =
B =
+ =
The partition function and the fourpoints conformal blocks
model are listed in table 1. Using (5.16){(5.18) one
nds the U(2) instanton partition
ZhU"("2")";iionst = 2F1
ZhU("2)";iinst = 2F1
ZU(2);inst = ZhU"("2);inst = 2F1
h ""io io
are reproduced from (5.20){(5.23) via (4.22)
hO1O2O3O4iO
= q
4 ZU(2);inst
hO1O2O3O4iO
= (1
= 1
O3 =
O3 = "
function vanishes at the corresponding critical values.
The gauge partition function on S4 vs the fourpoint correlator
hO1O2O3O4i
constants C
are de ned by
= C+jq +
4 ZU(2)
4 ZU(2)
4 to rewrite c q a2 = q
with (x) =
(1(x)x) . Plugging the special values of i from table 1 into (5.28) one nds
C =C+
get exchanged, so the
involved in the corresponding conformal block
The ratio between the two is given by [32]
= Resa jZhoOne1Ol2oOop3O4ia j
= CO1O2O C
O O3O4
(A1) (A2) (B
(B) (B
A1) (B
ZhU"("2")"i = j
= j
ZhU("2)"i = j
q=2j2
jqj1=4
= j1 + p
2jqj1=4j1
hO1O2O3O4i = ZU(2)
hO1O2O3O4i
O3 =
O3 = "
with results in perfect agreement with (5.9).
Acknowledgments
mon projects in Fundamental Scienti c Research"2013.
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