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AGT 関係式 (3) 一般化に向けて (String Advanced Lectures No.20)

AGT 関係式 (3) 一般化に向けて (String Advanced Lectures No.20). 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 23 日（水） 12:30-14:30. Contents. 1. AGT relation for SU(2) quiver theory 2. Partition function of SU(N) quiver theory 3. Toda theory and W-algebra

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AGT 関係式 (3) 一般化に向けて (String Advanced Lectures No.20)

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  1. AGT関係式(3) 一般化に向けて(String Advanced Lectures No.20) 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23日（水） 12:30-14:30

  2. Contents 1. AGT relation for SU(2) quiver theory 2. Partition function of SU(N) quiver theory 3. Toda theory and W-algebra 4. Generalized AGT relation for SU(N) case 5. Towards AdS/CFT duality of AGT relation

  3. AGT relation for SU(2) quiver We now consider only the linear quiver gauge theories in AGT relation. Gaiotto’s discussion

  4. An example : SW curve is a sphere with multiple punctures. • The Seiberg-Witten curve in this case corresponds to • 4-dim N=2 linear quiver SU(2) gauge theory. • Nekrasov instanton partition function • where equals to the conformal block of Virasoro algebra with for the vertex operators which are inserted at z= • Liouville correlation function (corresponding n+3-point function) • where is Nekrasov’s full partition function. • (↑including 1-loop part) U(1) part

  5. AGT relation : SU(2) gauge theory  Liouville theory! [Alday-Gaiotto-Tachikawa ’09] • 4-dim theory : SU(2) quiver gauge theory • 2-dim theory : Liouville (A1Toda) field theory In this case, the 4-dim theory’s partition function Zand the 2-dim theory’s correlation function correspond to each other : central charge :

  6. SU(N) partition function Nekrasov’s partition function of 4-dim gauge theory • Now we calculate Nekrasov’s partition function of 4-dim SU(N) quiver gauge theory as the quantity of interest. • SU(2) case : We consider only SU(2)×…×SU(2) quiver gauge theories. • SU(N) case : According to Gaiotto’s discussion, we consider, in general, the • cases of SU(d1) x SU(d2) x … x SU(N) x … x SU(N) x … x SU(d’2) x SU(d’1) group, • whereis non-negative. … … … x x x x x x * * … … d3–d2 d2–d1 d1 … … … … … d’3–d’2 d’2–d’1 d’1 … … … …

  7. gauge bifund. fund. antifund. 1-loop part of partition function of 4-dim quiver gauge theory We can obtain it of the analytic form : where each factor is defined as VEV # of d.o.f. depends on dk mass flavor symm. of bifund. is U(1) mass mass deformation parameters : each factor is a product of double Gamma function! ,

  8. Instanton part of partition function of 4-dim quiver gauge theory We obtain it of the expansion form of instanton number : where : coupling const.and and Young tableau arm leg < Young tableau> instanton # = # of boxes

  9. … … … … What kind of 2-dim CFT corresponds to 4-dim SU(N) quiver gauge theory? • Naive assumption is 2-dim AN-1Toda theory, since Liouville theory is nothing but A1Toda theory. This means that the generalized AGT relation seems • Difference from SU(2) case… • VEV’s in 4-dim theory and momentain 2-dim theory have more than one d.o.f. • In fact, the latter corresponds to the fact that the punctures are classified with more than one kindsof N-box Young tableaux : < full-type > < simple-type > < other types > (cf. In SU(2) case, all these Young tableaux become ones of the same type .) • In general, we don’t know how to calculate the conformal blocks of Toda theory. … … …

  10. Toda theory and W-algebra What is AN-1 Toda theory? : some extension of Liouville 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 :

  11. What is AN-1 Toda field theory? (continued) • In this theory, there are energy-momentum tensor and higher spin fields • as Noether currents. • The symmetry algebra of this theory is called W-algebra. • For the simplest example, in the case of N=3, the generators are defined as • And, their commutation relation is as follows: • which can be regarded as the extension of Virasoro algebra, and where • , We ignore Toda potential (interaction) at this stage.

  12. As usual, we compose the primary, descendant, and null fields. • The primary fields are defined as ( is called ‘momentum’) . • The descendant fields are composed by acting / on the primary fields as uppering / lowering operators. • First, we define the highest weight state as usual : • Then we act lowering operators on this state, and obtain various descendant fields as . • However, some linear combinations of descendant fields accidentally satisfy the highest weight condition. They are called null states. For example, the null states in level-1 descendants are • As we will see next, we found the fact that these null states in W-algebra are closely related to the singular behavior of Seiberg-Witten curve near the punctures. That is, Toda fields whose existence is predicted by AGT relation may in fact describe the form (or behavior) of Seiberg-Witten curve.

  13. The singular behavior of SW curve is related to the null fields of W-algebra. [Kanno-Matsuo-SS-Tachikawa ’09] • As we saw, Seiberg-Witten curve is generally represented as • and Laurent expansion near z=z0 of the coefficient function is generally • This form is similar to Laurent expansion of W-current (i.e. W-generators) • Also, the coefficients satisfy similar equations, except full-type puncture’s case • This correspondence becomes exact, in some kind of ‘classical’ limit: • (which is related to Dijkgraaf-Vafa’s discussion on free fermion’s system?) • This fact strongly suggests that vertex operators corresponding non-full-type punctures must be the primary fields which has null states in their descendants. ~ direction of D4~ direction of NS5 null condition

  14. The punctures on SW curve corresponds to the ‘degenerate’ fields! • If we believe this suggestion, we can conjecture the form of • momentum of Toda field in vertex operators , which corresponds to each kind of punctures. • To find the form of vertex operators which have the level-1 null state, it is useful to consider the screening operator (a special type of vertex operator) • We can show that the state satisfies the highest weight condition, since the screening operator commutes with all the W-generators. • (Note a strange form of a ket, since the screening operator itself has non-zero momentum.) • This state doesn’t vanish, if the momentum satisfies • for some j. In this case, the vertex operator has a null state at level . [Kanno-Matsuo-SS-Tachikawa ’09]

  15. The punctures on SW curve corresponds to the ‘degenerate’ fields! • Therefore, the condition of level-1 null state becomes for some j. • It means that the general form of mometum of Toda fields satisfying this null state condition is . • Note that this form naturally corresponds to Young tableaux . • More generally, the null state condition can be written as • (The factors are abbreviated, since they are only the images under Weyl transformation.) • Moreover, from physical state condition (i.e. energy-momentum is real), we need to choose , instead of naive generalization: • Here, is the same form of β, is Weyl vector, and .

  16. … … … … Generalized AGT relation Correspondence : 4-dim SU(N) quiver gauge and 2-dim AN-1Toda theory • Natural form : former’s partition function and latter’s correlation function • Problems and solutions for its proof • correspondence between each kind of punctures and vertices: • we can conjecture it, using level-1 null state condition. < full-type >< simple-type >< other types > • difficulty for calculation of conformal blocks: null statecondition resolves it again! [Wyllard ’09] [Kanno-Matsuo-SS-Tachikawa ’09] … … …

  17. On calculation of correlation functions of 2-dim AN-1 Toda theory • We put the (primary) vertex operators at punctures, and consider the correlation functions of them: • In general, the following expansion is valid: • where • and for level-1 descendants, • : Shapovalov matrix • It means that all correlation functions consist of 3-point functions and inverse Shapovalov matrices (propagator), where the intermediate states (descendants) can be classified by Young tableaux. descendants primaries

  18. On calculation of correlation functions of 2-dim AN-1 Toda theory • In fact, we can obtain it of the factorization form of 3-point functions and inverse Shapovalov matrices : • 3-point function : We can obtain it, if one entry has a null state in level-1! • where ’ highest weight ~ simple punc.

  19. Our plans of current and future research on generalized AGT relation • Case of SU(3) quiver gauge theory • SU(3) : already checked successfully.[Wyllard ’09] [Mironov-Morozov ’09] • SU(3) x … x SU(3) : We have checked successfully. [Kanno-Matsuo-SS ’10] • SU(3) x SU(2) : We could check it, but only for restricted cases. [Kanno-Matsuo-SS ’10] • Case of SU(4) quiver gauge theory • In this case, there are punctures which are not full-type nor simple-type. • So we must discuss in order to check our conjucture (of the simplest example). • The calculation is complicated because of W4 algebra, but is mostly streightforward. • Case of SU(∞) quiver gauge theory • In this case, we consider the system of infinitely many M5-branes, which may relate to AdS dual system of 11-dim supergravity. • AdS dual system is already discussed using LLM’s droplet ansatz, which is also governed by Toda equation. [Gaiotto-Maldacena ’09]→ subject of next talk…

  20. Towards AdS/CFT of AGT • CFT side : 4-dim SU(N) quiver gauge theory and 2-dim AN-1Toda theory • 4-dim theory is conformal. • The system preserves eight supersymmetries. • AdS side : the system with AdS5 and S2 factor and eight supersymmetries • 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]

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