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

Contents

1. Seiberg-Witten curve

2. SU(2) generalized quivers

3. SU(3) generalized quivers

4. SU(N) generalized quivers

5. Towards AGT relation

Seiberg-Witten curve

Low energy effective action (by Wilson’s renormalization : integration out of

massive fields)

prepotential

potential for scalar field

4-dim N=2 SU(2) supersymmetric gauge theory [Seiberg-Witten ’94]

classical 1-loop instanton

: energy scale

: Higgs potential (which breaks gauge symmetry)

This breakdown is parametrized by

u (VEV) : shift of color brane

mass : shift of flavor

brane

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. [Witten ’97]

: Seiberg-Witten curve in

coupling

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

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).

SU(2) gauge theory with massive fundamental hypermultiplets

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.

SU(2)1×SU(2)2×SU(2)3 gauge theory with fund. and bifund. flavors

(The similar discussion goes.)

■, ■ : weak : interchange

turn on/off a gauge coupling

For more generalized SU(2) quivers : more gauge groups, loops…

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 .

where are the solutions of

VEV coupling

polynomial of z of (n-1)-th order

divergent at punctures

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

Argyres-Seiberg duality for SU(3) gauge theory

infinite coupling

SU(3)1×SU(3)2 gauge theory with fundamental and bifundamental flavors

flavor symmetry of bifundamental

Argyres-Seiberg duality

For more generalized SU(3) quivers : more gauge groups, loops…

turn on/off a gauge coupling

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.

SU(N) generalized quivers

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…

Seiberg-Witten curve for massless SU(N) quiver gauge theories

SU(2) quiver case

• order of divergence :

• mass parameters :

• flavor symmetry : SU(2)

SU(3) quiver case

• order of divergence :

• mass parameters :

• flavor symmetry : U(1) SU(3)

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)

Classification of punctures : divergence of massless SW curve at punctures

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=∞.

Seiberg-Witten curve for linear SU(N) quiver gauge theories

Sorry, I write this on whiteboard…

Towards AGT relation

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…

AGT relation reveals the relation of 4-dim theory and SW curve concretely…