AGT 関係式 (4) AdS/CFT 対応 (String Advanced Lectures No.21)

1. 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

2. Contents 1. Generalized AGT relation for SU(N) quiver 2. AdS/CFT correspondence for AGT relation 3. Discussion on our ansatz

3. Generalized AGT relation Gaiotto’s discussion on 4-dim N=2 SU(N) quiver gauge theories 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. … … … x x x x x x * * … … d3–d2 d2–d1 d1 … … … … … d’3–d’2 d’2–d’1 d’1 … … … …

4. AGT relation : 4-dim SU(N) quiver gauge and 2-dim AN-1Toda theory • 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] SU(N) SU(N) … SU(N) SU(N) SU(N) SU(N) U(1) U(1) N-1 d.o.f. U(1) U(1) N-1 Cartans

5. … … … Level-1 null state condition resolves the problems of AGT-W relation. • 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] … …

6. 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:

7. 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]

8. On 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 … r (original AdS7 × S4)

9. LLM ansatz : 11-dim SUGRA solution with AdS5 x S2 factor and 8 SUSY 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: [Lin-Lunin-Maldacena ’04]

10. 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 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

11. For simplicity, we concentrate on the neighborhood of the punctures. 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. η ρ

12. For more simplicity, we concentrate on the neighborhood of the punctures. • 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 integerslopes correspond to the size of quiver gauge groups. • (→ the next page…)

13. 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 Ak-1 singularity at and , • since the period of βbecomes . • 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

14. On the source term : AdS/CFT correspondence for AGT relation ! Near , the potential can be written as (since , ) Then we obtain , So the boundary condition (~ source at r=0) is integer

15. Discussion on our ansatz CFT side : 2-dim AN-1 Toda theory • 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 :

16. 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: element coordinate differential of differential

17. Source term from 2-dim Toda equation 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/CFTcorrespondence). According to our ansatz, this is of the form where : N elements (Weyl vector) : k elements source??

18. 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 :