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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…