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 :