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(d 1 ) x SU(d 2 ) 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 theories x xx x x * … … x * … … … d’ 3 – d’ 2 d’ 2 – d’ 1 d’ 1 …… … … … d3 – d2d2 – d1d1d3 – d2d2 – d1d1 …… … …

 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 A N-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) U(1) SU(N) U(1) SU(N) U(1) SU(N) … N-1 d.o.f. AGT relation : 4-dim SU(N) quiver gauge and 2-dim A N-1 Toda 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.

 Difficulty for calculation of conformal blocks : Here we consider the case of A 2 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.

AdS/CFT for AGT relation  CFT side : 4-dim SU(N ≫ 1) quiver gauge theory and 2-dim A N-1 Toda theory 4-dim theory is conformal. The system preserves eight (1/2×1/2) supersymmetries.  AdS side : the system with AdS 5 and S 2 factor and 1/2 BPS state of AdS 7 ×S 4 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 S 1 ~ 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 symmetry The near horizon region of M5-branes is AdS 7 ×S 4 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 AdS 7 × S 4 ) 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 AdS 5 x S 2 factor and 8 SUSY [Lin-Lunin-Maldacena ’04]

The 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 AdS 5 ×S 1, which is conformal to R 1,5. So, including the effect of M5(D4)-branes, the near horizon geometry is also AdS 7 ×S 4 : 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: S1S1 S1S1

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. 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.

The 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 A k-1 singularity at and, since the period of β becomes. In general, if the slope changes by k units, we get A k-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 ! integer

 Action : Toda field with : It parametrizes the Cartan subspace of A N-1 algebra. simple root of A N-1 algebra : Weyl vector of A N-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 A N-1 Toda theory

3-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 elementcoordinate

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 equation source??

Towards the correspondence of “source” in AdS/CFT context…? For full [1,…,1]-type puncture: For simple [N-1,1]-type puncture : For [l 1,l 2,…]-type puncture :