Classical conformal blocks, twisted superpotentials

Transkrypt

Classical conformal blocks, twisted superpotentials
and
accessory parameters
Marcin Pia̧tek
Institute of Physics, University of Szczecin
stringtheory.pl/2013
Kraków, 5-7.04.2013
()M. Pia̧tek, University of Szczecin
1 / 22
Papers
in collaboration with Franco Ferrari (University of Szczecin)
F. Ferrari, M. Pia̧tek, On a singular Fredholm-type integral equation
arising in N = 2 super-Yang-Mills theories, Phys. Lett. B 718
(2013)
F. Ferrari, M. Pia̧tek, Liouville theory, N = 2 gauge theories and
accessory parameters, JHEP 05 (2012) 025
M. Pia̧tek, Classical conformal blocks from TBA for the elliptic
Calogero-Moser system, JHEP 06 (2011) 050
F. Ferrari, M. Pia̧tek, On a path integral representation of the
Nekrasov instanton partition function and its NS limit,
arXiv:1212.6787 [hep-th]
2 / 22
Scope and aims
gauge/Bethe correspondence
''Classical version'' of AGT correspondence
Nekrasov-Shatashvili (2009)
classical conformal blocks,
2d N=2 (Ω-deformed)
SYM theories:
Quantum Integrable
Systems:
classical Liouville actions
twisted superpotentials
Yang-Yang functions
Classical Liouville theory:
uniformization and the problem
of accessory parameters
Nekrasov-Shatashvili limit:
classical
limit
Quantum Liouville theory
AGT (2009)
(2d CFT)
quantum conformal blocks,
correspondence
4d N=2 (Ω-deformed)
SYM theories:
Nekrasov partition functions
correlation functions
Nekrasov-Okounkov limit:
low energies
4d N=2 SYM theories
Seiberg-Witten theory:
Coulomb branch
prepotentials
3 / 22
The problem of accessory parameters (Riemann sphere)
The Fuchs equation:
∂z2 ψ(z) + T cl (z) ψ(z) = 0.
The accessory parameters:
n−1
X
ck = 0,
k=1
n−1
X
k=1
n−1
X
k=1
T cl (z) is a meromorphic function on
C = C ∪ {∞} of the form:
n−1 X
δk
ck
cl
T (z) =
+
,
(z − zk )2 z − zk
k=1
T cl (z)
z→∞
=
δn
cn
+ 3 + O z −4 ,
2
z
z
δi = 14 (1 − ξi2 ), ξi ∈ R≥0 , i = 1, . . . , n.
(δk + ck zk ) = δn ,
2δk zk + ck zk2 = cn .
The problem is to tune c1 , . . . , cn in
such a way that the Fuchs eq. admits a
fundamental system of solutions
(ψ1 , ψ2 ) with monodromy in PSL(2, R).
4 / 22
Uniformization of C (0, n) = C \ {z1 , . . . , zn = ∞}, n ≥ 3
UNIFORMIZATION THEOREM (Poincaré, Koebe, 1907): there exists for
n ≥ 3 a universal covering map λ : H −→ C (0, n) ∼
= HG ;
H = {τ ∈ C : Im τ > 0} and G ⊂ PSL(2, R) = SL(2, R)/{±I} is a
discrete subgroup acting as Möbius transformations.
Up to a Möbius transformation the (locally defined) inverse map
λ−1 : C (0, n) 3 z −→ τ (z) ∈ H is λ−1 = ψ1 /ψ2 , where (ψ1 , ψ2 ) are
fundamental solutions of the equation:
"
#
n−1
1
X
c
k
4
∂z2 ψ(z) +
+
ψ(z) = 0
(z − zk )2 z − zk
k=1
with ψ1 ψ20 − ψ10 ψ2 = 1 and SL(2, R) monodromy w.r.t. all punctures.
5 / 22
Solution of the Liouville equation on C (0, n), n ≥ 3
THEOREM (Picard 1893, 1905):
The Liouville equation ∂z ∂z̄ ϕ(z, z̄) =
asymptotics:
1 ϕ(z,z̄)
2e
on C (0, n), n ≥ 3 with
elliptic singularities
(
−2 (1 − ξj ) log |z − zj | + O(1) as z → zj ,
ϕ(z, z̄) =
−2 (1 + ξn ) log |z| + O(1)
as z → ∞,
parabolic singularities (ξi → 0)
(
−2 log |z − zj | − 2 log |log |z − zj || + O(1) as z → zj ,
ϕ(z, z̄) =
−2 log |z| − 2 log |log |z|| + O(1)
as z → ∞
has a unique (real) solution.
6 / 22
Solution of the Liouville equation on C (0, n), n ≥ 3
PROPOSITION:
Locally, general solution of the Liouville equation on C (0, n) with elliptic
or parabolic asymptotics is given by
2
−1 0
(z)
λ
ϕ(z,z̄)
e
=
,
(Im λ−1 (z))2
where z is a local complex coordinate on C (0, n).
7 / 22
Polyakov’s solution
THEOREM (Polyakov 800 s, Takhtajan, Zograf 1993-96):
The properly defined and normalized Liouville action functional evaluated
on the solution ϕ(z, z̄) of the Liouville equation on C (0, n), n ≥ 3 is the
generating function for the accessory parameters:
cj = −
∂SLcl [ϕ]
.
∂zj
How to compute the classical action SLcl [ϕ] ?
8 / 22
Four-point case: n = 4
One can get an analytical expression of (at least) the 4-point action
cl,(4)
SL
computing the classical limit of the 4-point Liouville correlator
(Zamolodchikov’s conjecture, 1995) and using classical version of the
AGT conjecture.
The Fuchs equation on C (0, 4) (z4 = ∞, z3 = 1, z2 = q, z1 = 0):
∂z2 ψ(z) +
+
δ2
δ3
δ1 + δ2 + δ3 − δ4
+
+
2
2
(z − q)
(1 − z)
z(1 − z)
q(1 − q)c2 (q) i
ψ(z) = 0.
z(z − q)(1 − z)
hδ
1
z2
+
9 / 22
Zamolodchikov’s conjecture
Liouville correlators on the sphere in the geometric path integral formulation of
QLFT (Polyakov, Takhtajan):
Z
2
1
Dϕ e−Q SL [ϕ] ,
Q = b+ .
hC (0, n)i :=
b
M[C (0,n)]
M[C (0, n)] is the space of conformal factors of the metrics eϕ(z,z̄) |dz|2 on C (0, n)
with elliptic or parabolic singularities at punctures.
In the limit Q → 0 ⇐⇒ b → 0 the correlation function hC (0, n)i is dominated by
the saddle point of the functional integral:
b→0
− 12 SLcl [ϕ]
hC (0, n)i ∼ e
b
.
Correlators hC (0, n)i correspond to n-point functions of primary Liouville vertex
operators of the operator formulation of QLFT:
D
E
hC (0, n)i = Vαn (∞, ∞) . . . Vα1 (z1 , z̄1 ) ,
where ∆j = αj (Q − αj ), αj =
Q
2
(1 + ξj ) .
10 / 22
Zamolodchikov’s conjecture
DOZZ 4-point function (Dorn, Otto, Zamolodchikov’s 1995):
Gc∆4 ,...,∆1 (q, q̄)
:=
=
h Vα4 (∞, ∞)Vα3 (1, 1)1Vα2 (q, q̄)Vα1 (0, 0)i
Z
2
h
i
3 ∆2
dP C (α4 , α3 , αP )C (ᾱP , α2 , α1 ) Fc,∆(αP ) ∆
∆4 ∆1 (q) ,
R+
2
c = 1 + 6Q ,
¯ i = ∆i = ∆(αi ) = αi (Q − αi ).
∆
αP = Q/2 + iP,
Assuming a path integral representation of the l.h.s. one should expect that
1
b→0 − 1 S cl (δ ,...,δ1 ,q)
Gc∆4 ,...,∆1 (q, q̄) ∼ e b2 L 4
,
∆i = 2 δi .
b
In the limit b → 0 the r.h.s. behaves as follows (P = p/b, δ =
Z∞
dp e
− 12 Ŝ(δi ,q;δ)
b
,
1
4
+ p2 )
Ŝ(δi , q; δ) = S (3) (δ4 , δ3 , δ) + S (3) (δ, δ2 , δ1 ) − 2Re fδ
0
C ∼e
− 12
b
S
(3)
,
F ∼e
1
b2
h
δ3 δ2
δ4 δ1
i
(q),
f
and the saddle point approximation gives
SLcl (δ4 , . . . , δ1 , q) = Ŝ(δi , q; δs ),
δs =
1
4
+ ps (q)2 ,
∂
Ŝ(δi , q;
∂p
1
4
+ p 2 )
= 0.
p=ps
11 / 22
Classical block and accessory parameter
The accessory parameter for the Fuchs equation on C (0, 4):
∂ cl,(4)
S
(δi , q)
∂q L
∂ps (q)
∂
∂ cl
= − SLcl (δi , q, 14 + p 2 )
−
SL (δi , q, 14 + p 2 )
∂p
∂q
∂q
p=ps (q)
p=ps (q)
h
i
∂
∂
.
=
f 1 2 δ3 δ2 ( q)
= − SLcl (δi , q, 14 + p 2 )
∂q
∂q 4 +p δ4 δ1
p=ps (q)
p=ps (q)
c2 (q)
=
−
The classical 4-point conformal block:
h
fδ δδ34 δδ21
i
2
( q) = (δ − δ1 − δ2 ) log q + lim b log 1 +
b→0
∞
X
h
n
∆3 ∆2
Fc,∆
∆4 ∆1
i
!
q
n
n=1
(δ + δ3 − δ4 )(δ + δ2 − δ1 )
q + ... .
2δ
h
i
The task: to sum up the 4-point classical block fδ δδ43 δδ12 ( q).
= (δ − δ1 − δ2 ) log q +
12 / 22
The Alday-Gaiotto-Tachikawa (AGT) conjecture
Liouville correlation function
LFT
n
Q
Vβa
a=1
C (g ,n)
on C (g , n) − Riemann surface
with genus g and n punctures
Virasoro conformal block
σ
F1+6Q
2 ,α [β](Z) on C (g , n);
α ≡ (α1 , . . . , α3g −3+n ),
β ≡ (β1 , . . . , βn ),
Z = (z1 , . . . , z3g −3+n ),
∆α = α(Q − α), Q = b + b −1
Partition function ZTσ (g ,n)
=
of a class T (g , n) of 4d
(Ω − deformed) N = 2 SUSY
SU(2) quiver gauge theories
Nekrasov instanton partition function
Zinst (Z, a, m; 1 , 2 );
a = (a1 , . . . , a3g −3+n ),
=
m = (m1 , . . . , mn ),
Z = (zi = exp 2πτi )i=1,...,3g −3+n ,
p
(1 +2 )2
= Q ⇐⇒
2 /1 = b
1 2
Classical limit b → 0 ⇐⇒ Nekrasov-Shatashvili limit 2 → 0, 1 = const. .
13 / 22
Nekrasov instanton partition function
U(2),Nf =4
Zinst
Ωk
=
= 1+
k
Y
I =1
I
I
∞
X
dφ1
dφk
q k 1 + 2 k
...
Ωk ,
k!
1 2
2πi
2πi
k=1
4
Q
(φI + mα )
α=1
2
Q
(φI − au − i0) (φI − au + 1 + 2 + i0)
u=1
×
k
Y
(φI − φJ ) (φI − φJ + 1 + 2 )
,
(φI − φJ + 1 + i0) (φI − φJ + 2 + i0)
au , 1 , 2 ∈ R.
I ,J=1
I 6=J
Poles which contribute to the integral are in correspondence with pairs of Young
diagrams Y = {Y1 , Y2 }:
Y −→ φI = φu,r ,s = au + (r − 1)1 + (s − 1)2 ,
u = 1, 2.
14 / 22
Result
∆i ∼
fδ
h
1
δ,
b2 i
δ3 δ2
δ4 δ1
i
∆∼
1
δ,
b2
( q) =
b2 =
2
1
JHEP 05 (2012) 025
lim b 2 log F1+6Q 2 ,∆
b→0
h
∆3 ∆2
∆4 ∆1
i
(q)
1
SU(2),Nf =4
lim 2 log Zinst
(q, a, m1 , . . . , m4 ; 1 , 2 )
1 2 →0
(m1 + m2 )(m3 + m4 )
= (δ − δ1 − δ2 ) log q −
log(1 − q)
221
1 U(2),Nf =4 ∗
+
H
(xu,r ( q)),
1 inst
=
1
1
m 1 = 1
, m2 = 1 η2 − η1 +
, a = 1 η −
,
2
2
1
1
m3 = 1 η3 + η4 −
, m4 = 1 η3 − η4 +
, δ = η(1 − η), δi = ηi (1 − ηi ),
2
2
u = 1, 2,
1
η1 + η2 −
2
r = 1, . . . , ∞.
15 / 22
Instanton free energy
U(2),Nf =4
Hinst
(xu,r ) =
2
∞ h
X
X
− F (xu,r − xv ,l + 1 ) + F xu,r − xv0,l + 1
u,v =1 r ,l=1
0
0
+ F xu,r
− xv ,l + 1 − F xu,r
− xv0,l + 1 + F (xu,r − xv ,l )
i
0
0
− F xu,r − xv0,l − F xu,r
− xv ,l + F xu,r
− xv0,l
+
2 X
∞ h
X
0
− F (xu,r − av ) + F xu,r
− av
u,v =1 r =1
i
0
− av + 1
−F (xu,r − av + 1 ) + F xu,r
+
2 X
∞ X
4 h
i
X
0
F (xu,r + mα ) − F xu,r
+ mα
u=1 r =1 α=1
+
2 X
∞
X
(xu,r − (r − 1)1 − au ) log | q| ,
F (t) = t(log | t| − 1).
u=1 r =1
16 / 22
Critical Young diagrams
∗
xu,r
≡ xu,r is a solution of the saddle point equation ∂Hinst (xu,r ) /∂xu,r = 0 ⇒
−q
2 Y
∞
Y
(xu,r − xv ,l − 1 )(xu,r − xv0,l + 1 )
v =1 l=1
!
(xu,r − xv ,l + 1 )(xu,r − xv0,l − 1 )


4
Q
(xu,r + mα )




α=1
× 2
 = 1.

Q
(xu,r − av )(xu,r − av + 1 )
v =1
One can assume that ∃ L ∈ N such that ωu,r = 0 for r > L then l = 1, . . . , L and
0
xu,r
=
au + (r − 1)1 ,
xu,r
=
au + (r − 1)1 + ωu,r ( q)
=
au + (r − 1)1 +
L
X
ωu,r ,n q n ,
ωu,r ∼ O(q r ).
n=r
17 / 22
Column length function coefficients of critical diagrams
For L = 1
q(xu,r − x1,1 − 1 )(xu,r − x2,1 − 1 )(xu,r + m1 )(xu,r + m2 )(xu,r + m3 )(xu,r + m4 )
+
(xu,r − a1 − 1 )(xu,r − a2 − 1 )(xu,r − x1,1 + 1 )(xu,r − x2,1 + 1 )
×
(xu,r − a1 )(xu,r − a2 ) = 0.
Setting xu,r = au + (r − 1)1 + ωu,1,1 q where u = 1, 2 and (a1 , a2 ) = (a, −a) one
finds
4
Q
ω1,1,1 = −
4
Q
(a + mα )
α=1
1 2a(2a + 1 )
,
ω2,1,1 = −
(a − mα )
α=1
1 2a(2a − 1 )
.
For L = 2 the system of equations yields
the second order corrections to the length of the first column,
the length of the second column at the leading order in q 2 .
. . . etc.
18 / 22
Saddle point equation
(Fucito,. . . , Poghossian 2011)
Let us define
Y
∞ z
z
Y (z) =
exp
1−
exp
,
0
xu,r
xu,r
u=1
r =1
Y
∞ 2
Y
au
z
z
z
ψ
1− 0
Y0 (z) =
exp
exp
,
0
1
1
xu,r
xu,r
2
Y
z
ψ
1
au
1
r =1
u=1
ψ(z) = ∂z log Γ(z).
The saddle point equation can be rewritten to the form:
q
M(x) =
4
Q
M(xu,r ) Y (xu,r − 1 )
= −1,
41 Y (xu,r + 1 )
(x + mα ), u = 1, 2, r = 1, . . . , ∞.
α=1
19 / 22
Accessory parameter from twisted superpotential
fδ
h
δ3 δ2
δ4 δ1
i
( q)
(δ − δ1 − δ2 ) log q −
+
1 U(2),Nf =4 ∗
H
(xu,r ( q)).
1 inst
∂ U(2),Nf =4 ∗
H
(xu,r ( q)) =
∂q inst
c2 (q)
=
+
X
u,r
(m1 + m2 )(m3 + m4 )
log(1 − q)
221
=
ωu,r
∂Hinst
∂Hinst ∂xu,r
+
∂xu,r ∂q
∂q
∗
xu,r =xu,r
=
1X ∗
ωu,r
q u,r
⇒
h
i
1
∂
f 1 +p2 δδ43 δδ12 ( q)
= (δs − δ1 − δ2 )
∂q 4
q
p=ps (q)
!
1 X ∗
1
ωu,r (q, a = −i1 ps , mi ; 1 ) + 2η2 η3
.
1 u,r
1−q
=
I dz
X
Y (z)
0
xu,r − xu,r
=
z∂z log
2πi
Y
0 (z)
u,r
γ
0
γ encloses all the points xu,r , xu,r
, u = 1, 2, r = 1, . . . , ∞.
20 / 22
Concluding remarks
The main results:
an expression of the accessory parameter c2 (q) in terms of the solution
of the saddle point equation (TBA-like equation);
the saddle point equation has been solved by a power expansion in the
modular parameter q (other solutions?!);
a contour integral representation for c2 (q).
21 / 22
Open problems
1
Accessory parameters for the Fuchs equation on C (0, n), n > 4
n-point classical spherical block from NS limit of the instanton
partition function of the U(2)1 × . . . × U(2)n−3 linear quiver
2
Accessory parameters for Lamé and Mathieu equations
Lamé accessory parameter from torus classical block/N = 2∗ U(2)
twisted superpotential
3
Relation to quantum integrable systems (Gaudin, eCM, pToda chain)
4
Classical conformal blocks from N large limit of β-ensembles
22 / 22

Classical conformal blocks, twisted superpotentials

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