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AGT関係式(2) AGT関係式
(String Advanced Lectures No.19)
高エネルギー加速器研究機構(KEK)
素粒子原子核研究所(IPNS)
柴 正太郎
2010年6月9日(水) 12:30-14:30
Contents
1. Gaiotto‟s discussion for SU(2)
2. SU(2) partition function
3. Liouville correlation function
4. Seiberg-Witten curve and AGT relation
5. Towards generalized AGT relation
Gaiotto‟s discussion for SU(2)[Gaiotto ‟09]
SU(2) gauge theory with 4 fundamental flavors (hypermultiplets)
S-duality group SL(2,Z)
coupling const. :
flavor sym. : SO(8) ⊃ SO(4)×SO(4) ~ [SU(2)a×SU(2)b]×[SU(2)c×SU(2)d]
: (elementary) quark
: monopole
: dyon
Subgroup of S-duality without permutation of masses
In massive case, we especially consider this subgroup.
• mass : mass parameters can be associated to each SU(2) flavor.
Then the mass eigenvalues of four hypermultiplets in 8v is , .
• coupling : cross ratio (moduli) of the four punctures, i.e. z=
Actually, this is equal to the exponential of the UV coupling
→ This is an aspect of correspondence between the 4-dim N=2 SU(2) gauge
theory and the 2-dim Riemann surface with punctures.
SU(2) gauge theory with massive fundamental hypermultiplets
SU(2) partition function
Action
classical part
1-loop correction : more than 1-loop is cancelled, because of N=2 supersymmetry.
instanton correction : Nekrasov‟s calculation with Young tableaux
Parameters
coupling constants
masses of fundamental / antifund. / bifund. fields and VEV‟s of gauge fields
deformation parameters : background of graviphoton or deformation of extra dimensions
Nekrasov’s partition function of 4-dim gauge theory
(Note that they are different from Gaiotto‟s ones!)
Now we calculate Nekrasov‟s partition function of 4-dim SU(2) quiver gauge
theory as the quantity of interest.
1-loop part of partition function of 4-dim quiver gauge theory
We can obtain it of the analytic form :
where each factor is defined as
: each factor is a product of double Gamma function!
,
gauge antifund. bifund. fund.
mass massmassVEV
deformation parameters
We obtain it of the expansion form of instanton number :
where : coupling const. and
and
Instanton part of partition function of 4-dim quiver gauge theory
Young tableau
< Young tableau >
instanton # = # of boxes
leg
arm
The Nekrasov partition function for the simple case of SU(2) with four flavors is
Since the mass dimension of is 1, so we fix the scale as , .(by definition)
Mass parameters : mass eigenvalues of four hypermultiplets
• : mass parameters of
• : mass parameters of
VEV‟s : we set --- decoupling of U(1) (i.e. trace) part.
We must also eliminate the contribution from U(1) gauge multiplet.
This makes the flavor symmetry SU(2)i×U(1)i enhanced to SU(2)i×SU(2)i .(next page…)
SU(2) with four flavors : Calculation of Nekrasov function for U(2)
U(2), actually
Manifest flavor symmetry is now
U(2)0×U(2)1 , while actual symmetry is
SO(8)⊃[SU(2)×SU(2)]×[SU(2)×SU(2)].
In this case, Nekrasov partition function can be written as
where and
is invariant under the flip (complex conjugate representation) :
which can be regarded as the action of Weyl group of SU(2) gauge symmetry.
is not invariant. This part can be regarded as U(1) contribution.
Surprising discovery by Alday-Gaiotto-Tachikawa
In fact, is nothing but the conformal block of Virasoro algebra with
for four operators of dimensions inserted at :
SU(2) with four flavors : Identification of SU(2) part and U(1) part
(intermediate state)
Correlation function of Liouville theory with .
Thus, we naturally choose the primary vertex operator as the
examples of such operators. Then the 4-point function on a sphere is
3-point function conformal block
where
The point is that we can make it of the form of square of absolute value!
… only if
… using the properties : and
Liouville correlation function
As a result, the 4-point correlation function can be rewritten as
where and
It says that the 3-point function (DOZZ factor) part also can be written as
the product of 1-loop part of 4-dim SU(2) partition function :
under the natural identification of mass parameters :
Example 1 : SU(2) with four flavors (Sphere with four punctures)
Example 2 : Torus with one puncture
The SW curve in this case corresponds to 4-dim N=2* theory :
N=4 SU(2) theory deformed by a mass for the adjoint hypermultiplet
Nekrasov instanton partition function
This can be written as
where equals to the conformal block of Virasoro algebra with
Liouville correlation function (corresponding 1-point function)
where is Nekrasov‟s partition function.
Example 3 : Sphere with multiple punctures
The Seiberg-Witten curve in this case corresponds to 4-dim N=2 linear quiver
SU(2) gauge theory.
Nekrasov instanton partition function
where equals to the conformal block of
Virasoro algebra with for the vertex operators which are inserted
at z=
Liouville correlation function (corresponding n+3-point function)
where is Nekrasov‟s full partition function.
• According to Gaiotto‟s discussion, SW curve for SU(2) case is .
• In massive cases, has double poles.
• Then the mass parameters can be obtained as ,
where is a small circle around the a-th puncture.
• The other moduli can be fixed by the special coordinates ,
where is the i-th cycle (i.e. long tube at weak coupling).
Note that the number of these moduli is 3g-3+n.
(g : # of genus, n : # of punctures)
SW curve and AGT relation
Seiberg-Witten curve and its moduli
• The Seiberg-Witten curve is supposed to emerge from Nekrasov partition
function in the “semiclassical limit” , so in this limit, we expect
that .
• In fact, is satisfied on a sphere,
then has double poles at zi .
• For mass parameters, we have ,
where we use and .
• For special coordinate moduli, we have ,
which can be checked by order by order calculation in concrete examples.
• Therefore, it is natural to speculate that Seiberg-Witten curve is „quantized‟
to at finite .
2-dim CFT in AGT relation : „quantization‟ of Seiberg-Witten curve??
Towards generalized AGT
Natural generalization of AGT relation seems the correspondence between
partition function of 4-dim SU(N) quiver gauge theory and correlation function
of 2-dim AN-1 Toda theory :
• This discussion is somewhat complicated, since in SU(N>2) case, the
punctures are classified with more than one kinds of N-box Young tableaux :
< full-type > < simple-type > < other types >
(cf. In SU(2) case, all these Young tableaux become ones of the same type .)
[Wyllard ’09]
[Kanno-Matsuo-SS-Tachikawa ’09]
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