agt 関係式 (4) ads/cft 対応 (string advanced lectures no.21)...
TRANSCRIPT
AGT 関係式 (4) AdS/CFT 対
応(String Advanced Lectures No.21)
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 30 日(水) 12:30-14:30
Contents
1. Generalized AGT relation for SU(N)
quiver
2. AdS/CFT correspondence for AGT
relation
3. Discussion on our ansatz
In AGT context, we concentrate on the linear (or necklace) quiver
gauge theory with SU(d1) x SU(d2) x … x SU(N) x … x SU(N) x … x
SU(d’2) x SU(d’1) group.
The various S-duality transformation can be realized as the shift or
interchange of various kinds of punctures on 2-dim Riemann surface
(Seiberg-Witten curve).
Here, is non-negative.
Generalized AGT relation
Gaiotto’s discussion on 4-dim N=2 SU(N) quiver gauge theoriesGaiotto’s discussion on 4-dim N=2 SU(N) quiver gauge theories
xx xxx
*
… …x
*
…
…
d’3 – d’2d’2 – d’1d’1
… ………
…
d3 – d2
d2 – d1
d1… ………
Now we are interested in the Nekrasov’s partition function of 4-dim
SU(N) quiver gauge theory.
It seems natural that generalized AGT relation (or AGT-W relation)
clarifies the correspondence between Nekrasov’s function and some
correlation function of 2-dim AN-1 Toda theory:
Main difference from SU(2) case:
Not all flavor symmetries are SU(N), e.g. bifundamental flavor
symmetry.
Therefore, we need the condition which restricts the d.o.f. of
momentum β in Toda vertex which corresponds to
each (kind of) puncture.
→ level-1 null state condition
[Wyllard ’09][Kanno-Matsuo-SS-Tachikawa ’09]
N-1 Cartans
SU(N)
SU(N)
SU(N)
U(1)
SU(N)
U(1) U(1)
SU(N)U(1)
SU(N)…
N-1 d.o.f.
AGT relation : 4-dim SU(N) quiver gauge and 2-dim AAGT relation : 4-dim SU(N) quiver gauge and 2-dim AN-1N-1 Toda theoryToda theory
Correspondence between each kind of punctures and vertices
:
we conjectured it, using level-1 null state condition for non-full-type
punctures.
• full-type : correponds to SU(N) flavor symmetry (N-1
d.o.f.)
• simple-type : corresponds to U(1) flavor symmetry (1 d.o.f.)
• other types : corresponds to other flavor symmetry
The corresponding momentum is of the form
which naturally corresponds to Young tableaux .
More precisely, the momentum is , where
[Kanno-Matsuo-SS-Tachikawa ’09]
…
…
…
…
………
Level-1 null state condition resolves the problems of AGT-W relation.Level-1 null state condition resolves the problems of AGT-W relation.
Difficulty for calculation of conformal blocks :
Here we consider the case of A2 Toda theory and W3-algebra. In usual,
the conformal blocks are written as the linear combination of
which cannot be determined by recursion formula.
However, in this case, thanks to the level-1 null state condition
we can completely determine all the conformal blocks.
Also, thanks to the level-1 null state condition, the 3-point function of
primary vertex fields can be determined completely:
Level-1 null state condition resolves the problems of AGT-W relation.Level-1 null state condition resolves the problems of AGT-W relation.
AdS/CFT for AGT relation
CFT side : 4-dim SU(N≫1) quiver gauge theory and 2-dim AN-1Toda
theory
• 4-dim theory is conformal.
• The system preserves eight (1/2×1/2) supersymmetries.
AdS side : the system with AdS5 and S2 factor and 1/2 BPS state of
AdS7×S4
• This is nothing but the analytic continuation of LLM’s system in M-
theory.
• Moreover, when we concentrate on the neighborhood of punctures
on Seiberg-Witten curve, the system gets the
additional S1 ~ U(1) symmetry.
• According to LLM’s discussion, such system can
be analyzed using 3-dim electricity system:
[Gaiotto-Maldacena ’09]
[Lin-Lunin-Maldacena ’04]
On the near horizon (dual) spacetime and its symmetryOn the near horizon (dual) spacetime and its symmetry
The near horizon region of M5-branes is AdS7×S4 spacetime.
Then, what is the near horizon of intersecting M5-branes like?
0,1,2,3-direction : 4-dim quiver gauge theory lives here.
All M5-branes must be extended.
7-direction : compactification direction of M → IIA Only M5(D4)-branes must be extended.
8,9,10-direction and 5-direction : corresponding to SU(2)×U(1) R-
symmetry
No M5-branes are extended to the former, and only M5(NS5)-branes
are to the latter.
Then the result is …
(original AdS7 × S4)
r
The most general gravity solution with such symmetry is
Note that the spacetime solution is constructed from a single
function
which obeys 3-dim Toda equation
(In the following, we consider the cases where the source term is
non-zero.)
cf. coordinates of 11-dim spacetime:
LLM ansatz : 11-dim SUGRA solution with AdSLLM ansatz : 11-dim SUGRA solution with AdS55 x S x S22 factor and 8 SUSY factor and 8 SUSY
[Lin-Lunin-Maldacena ’04]
The neighborhood of punctures : Toda equation with source termThe neighborhood of punctures : Toda equation with source term
We consider the system of N M5(D4)-branes and K M5(NS5)-branes (N
≫K≫1), and locally analyze the neighborhood of punctures
(intersecting points).
• M5(NS5)-branes wrap AdS5×S1, which is conformal to R1,5.
• So, including the effect of M5(D4)-branes, the near horizon geometry
is also AdS7×S4 :
When we set the angles and (i.e. U(1) symm. for β-
direction), we can determine the correspondence to LLM ansatz
coordinates as
where .
Note that D→∞ along the segment r=0 and 0≦y ≦1. This means that
Toda equation must have the source term, whose charge density is
constant along the segment:
S1 S1
In this simplified situation, 11-dim spacetime has an additional U(1) symmetry.
Moreover, the analysis become much easier, if we change the variables:
Note that this transformation mixes the free and bound variables: (r, y, D) → (ρ, η, V)…
Then LLM ansatz and Toda equation becomes ( )
and
i.e.
This is nothing but the 3-dim cylindrically symmetric Laplace equation.
For simplicity, we concentrate on the neighborhood of the punctures.For simplicity, we concentrate on the neighborhood of the punctures.
ρ
η
From the U(1) symmetry of β-direction, the source must exist at
ρ=0.
Near , LLM ansatz becomes more simple form (using
)
Note that at (i.e. at the puncture),
• The circle is shrinking
• The circle is not shrinking.
This makes sense, only when the constant slope is integer.
In fact, this integer slopes correspond to the size of quiver gauge
groups.(→ the next page…)
For more simplicity, we concentrate on the neighborhood of the punctures.For more simplicity, we concentrate on the neighborhood of the punctures.
The neighborhood of punctures : Laplace equation with source termThe neighborhood of punctures : Laplace equation with source term
We consider the such distribution of source charge:
When the slope is 1, we get smooth geometry.
When the slope is k, which corresponds to the
rescale and ,
we get Ak-1 singularity at and ,
since the period of β becomes .
In general, if the slope changes by k units, we get Ak-1 singularity there.
This can be regard the flavor symmetry of
additional k fundamental hypermultiplets.
This means the source charge corresponds to
nothing but the size of quiver gauge group.
N
Near , the potential can be written as (since ,
)
Then we obtain
,
So the boundary condition (~ source at r=0) is
On the source term : AdS/CFT correspondence for AGT relation !On the source term : AdS/CFT correspondence for AGT relation !
integer
Action :
Toda field with :
It parametrizes the Cartan subspace of AN-1 algebra.
simple root of AN-1 algebra :
Weyl vector of AN-1 algebra :
metric and Ricci scalar of 2-dim surface
interaction parameters : b (real) and
central charge :
Discussion on our ansatz
CFT side : 2-dim CFT side : 2-dim AAN-1N-1 Toda theory Toda theory
3-dim Toda equation, 2-dim Toda equation and their correspondence3-dim Toda equation, 2-dim Toda equation and their correspondence
3-dim Toda equation :
2-dim Toda equation (after rescaling of μ) :
Correspondence : or
[proof] The 2-dim equation (without curvature term, for simplicity) says
Therefore, under the correspondence, this 2-dim equation exactly
becomes the 3-dim equation:
differential of differential
element coordinate
To obtain the source term, we consider OPE of kinetic term of 2-dim
equation and the vertex operator :
( )
Then using the correspondence , we obtain
In massless case, (since we consider AdS/CFT
correspondence). According to our ansatz, this is of the form
where
: N elements (Weyl
vector)
: k elements
Source term from 2-dim Toda equationSource term from 2-dim Toda equation
source??
Towards the correspondence of “source” in AdS/CFT context…?Towards the correspondence of “source” in AdS/CFT context…?
• For full [1,…,1]-type puncture:
• For simple [N-1,1]-type puncture :
• For [l1,l2,…]-type puncture :