1. AGT 関係式(1) Gaiotto の議論 (String Advanced Lectures No.18)
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2日（水） 12:30-14:30

2. Contents
1. Seiberg-Witten curve 2. SU(2) generalized quivers 3. SU(3) generalized quivers 4. SU(N) generalized quivers 5. Towards AGT relation

3. Seiberg-Witten curve 4-dim N=2 SU(2) supersymmetric gauge theory
Low energy effective action (by Wilson’s renormalization : integration out of massive fields)
prepotential
potential for scalar field
: energy scale
classical loop instanton
: Higgs potential (which breaks gauge symmetry)
This breakdown is parametrized by

4. Singular points of prepotential, Seiberg-Witten curve and S-duality
The singular points of prepotential on u-plane
By studying the monodromy of and , we can find that the prepotential has singular points. This can be described as
These singular points means the emergence of new massless fields.
This means that the prepotential must become a different form near a different singular point. ( S-duality)
M-theory interpretation : singular points are intersection points of M5-branes.
: Seiberg-Witten curve in
[Witten ’97]
u (VEV) : shift of color brane
mass : shift of flavor brane
coupling

5. SU(2) generalized quivers
SU(2) gauge theory with 4 fundamental flavors (hypermultiplets)
This theory is conformal.
flavor symmetry SO(8) : pseudoreal representation of SU(2) gauge group
S-duality group SL(2,Z)
coupling const. :
flavor : SO(8) ⊃ SO(4)×SO(4) ～ [SU(2)a×SU(2)b]×[SU(2)c×SU(2)d]
: (elementary) quark
: monopole
: dyon

6. SU(2) gauge theory with massive fundamental hypermultiplets
In the following, we consider, in particular,
subgroup of S-duality without permutation of masses
mass : VEV of vector multiplet (adjoint) scalar
Then, there are three possible degeneration (i.e. weak coupling) limits of a sphere with four punctures (i.e. fundamentals).

7. SU(2)1×SU(2)2 gauge theory with fundamental and bifundamental flavors
When each gauge group is coupled to 4 flavors, this theory is conformal.
flavor symmetry ⊃ [SU(2)a×SU(2)b]×SU(2)e×[SU(2)c×SU(2)d]
flavor sym. of bifundamental hyper. : Sp(1) ～ SU(2) i.e. real representation
S-duality subgroup without permutation of masses
When the gauge coupling of SU(2)2 vanishes or is very weak, we can discuss it in the same way as before for SU(2)1. The similar discussion goes for (1 2). That is, this subgroup consists of the permutation of five SU(2)’s.
cf. Note that two SL(2,Z) full S-duality groups do not commute! Here, we analyze only the boundary of the gauge coupling moduli space.

8. SU(2)1×SU(2)2×SU(2)3 gauge theory with fund. and bifund. flavors
(The similar discussion goes.)
■, ■ : weak
: interchange

9. For more generalized SU(2) quivers : more gauge groups, loops…
turn on/off a gauge coupling

10. Seiberg-Witten curve for quiver SU(2) gauge theories
massless SU(2) case
In this case, the Seiberg-Witten curve is of the form
If we change the variable as , this becomes
massless SU(2) n case or
mass deformation
The number of mass parameters is n+3, because of the freedom
VEV
coupling
where are
the solutions of
polynomial of z of (n-1)-th order
divergent at punctures

11. SU(3) generalized quivers
SU(3) gauge theory with 6 fundamental flavors (hypermultiplets)
This theory is also conformal.
flavor symmetry U(6) : complex rep. of SU(3) gauge group
kind of S-duality group : Argyres-Seiberg duality [Argyres-Seiberg ’07]
coupling const. :
flavor : U(6) ⊃ [SU(3)×U(1)]×[SU(3)×U(1)] : weak coupling
U(6) ⊃ SU(6)×U(1) ～ [SU(3)×SU(3)×U(1)]×U(1)
SU(6)×SU(2) ⊂ E6 : infinite coupling of SU(3) theory
Moreover, weakly coupled gauge group becomes SU(2) instead of SU(3) !
breakdown by VEV

12. Argyres-Seiberg duality for SU(3) gauge theory
infinite coupling

13. SU(3)1×SU(3)2 gauge theory with fundamental and bifundamental flavors
flavor symmetry of bifundamental
Argyres-Seiberg duality

14. For more generalized SU(3) quivers : more gauge groups, loops…
turn on/off a gauge coupling

15. Seiberg-Witten curve for SU(3) quiver gauge theories
massless SU(3) n case
massless SU(2)×SU(3) n-2×SU(2) case
mass deformation
massless :
massive :
The number of mass parameters is n+3, because of the freedom
In both cases, SW curve can be rewritten as ( ),
but the order of divergence of is different from each other.

16. SU(N) generalized quivers
Seiberg-Witten curve for massless SU(N) quiver gauge theories
Seiberg-Witten curve in this case is of the form
The variety of quiver gauge group
where
is reflected in the various order of divergence of　　 at punctures.
For example…

17. Classification of punctures : divergence of massless SW curve at punctures
SU(2) quiver case
order of divergence :
mass parameters :
flavor symmetry : SU(2)
SU(3) quiver case
flavor symmetry : U(1) SU(3)

18. Classification of punctures : divergence of massless SW curve at punctures
SU(3) quiver case
corresponding puncture :
SU(4) quiver case (and the natural analogy is valid for general SU(N) case)

19. Sorry, I write this on whiteboard…
Seiberg-Witten curve for linear SU(N) quiver gauge theories
quiver gauge group (as a quite general case)
Seiberg-Witten curve (type of each puncture)
Seiberg-Witten curve in a massive case (concrete form of equation)
where , which corresponds the Young tableau at z=∞.
Sorry, I write this on whiteboard…

20. Towards AGT relation
AGT relation reveals the relation of 4-dim theory and SW curve concretely…
4-dim linear SU(2) quiver gauge theory :
We can calculate the partition functions by Nekrasov’s formula.
2-dim conformal field theory on Seiberg-Witten curve :
We calculate the correlation functions with vertex operators at punctures.
AGT relation :
Both functions correspond to each other.
to be continued…