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AGT 関係式とその一般化に向けて (String Advanced Lectures No.22)

AGT 関係式とその一般化に向けて (String Advanced Lectures No.22). 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 7 月 5 日（月） 14:00-15:40. Contents. 1. Gaiotto’s discussion 2. AGT relation for SU(2) quiver theories 3. AGT-W relation for SU(N) quiver theories 4. AdS/CFT correspondence of AGT’s system.

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AGT 関係式とその一般化に向けて (String Advanced Lectures No.22)

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  1. AGT関係式とその一般化に向けて(String Advanced Lectures No.22) 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7月5日（月） 14:00-15:40

  2. Contents 1. Gaiotto’s discussion 2. AGT relation for SU(2) quiver theories 3. AGT-W relation for SU(N) quiver theories 4. AdS/CFT correspondence of AGT’s system

  3. Gaiotto’s discussion

  4. Seiberg-Witten curve 4-dim N=2 SU(2) supersymmetric gauge theory [Seiberg-Witten ’94] • Low energy effective action (by Wilson’s renormalization : integration out of massive fields) • prepotential • potential for scalar field : energy scale classical1-loopinstanton : Higgs potential (which breaks gauge symmetry) This breakdown is parametrized by

  5. Singular points of prepotential, Seiberg-Witten curve and S-duality • The singular points of prepotential on u-plane By studying the monodromy of and , we can find that the prepotential has singular points. This can be described as • These singular points means the emergence of new massless fields. • This means that the prepotential must become a different form near a different singular point. (S-duality) • M-theory interpretation : singular points are intersection points of M5-branes. • (or D4/NS5-branes) : Seiberg-Witten curve in u (VEV) : shift of colorbrane mass: shift of flavorbrane [Witten ’97] coupling

  6. SU(2) generalized quivers [Gaiotto ’09] SU(2) gauge theory with 4 fundamental flavors (hypermultiplets) • S-duality group SL(2,Z) • coupling const. : • flavor sym. : SO(8) ⊃ SO(4)×SO(4) ～ [SU(2)a×SU(2)b]×[SU(2)c×SU(2)d] • : (elementary) quark • : monopole • : dyon NS5 D4

  7. SU(2) gauge theory with massive fundamental hypermultiplets • Subgroup of S-duality without permutation of masses • In massive case, we especially consider this subgroup. • mass : mass parameters can be associated to each SU(2) flavor. • Then the mass eigenvalues of four hypermultiplets in 8v is , . • coupling : cross ratio (moduli) of the four punctures, i.e. z= • Actually, this is equal to the exponential of the UV coupling • → This is an aspect of correspondence between the 4-dim N=2 SU(2) gauge theory and the 2-dim Riemann surface with punctures.

  8. SU(2)1×SU(2)2 gauge theory with fundamental and bifundamental flavors • When each gauge group is coupled to 4 flavors, this theory is conformal. • flavor symmetry ⊃ [SU(2)a×SU(2)b]×SU(2)e×[SU(2)c×SU(2)d] • flavor sym. of bifundamental hyper. : Sp(1) ～ SU(2) i.e. real representation • S-duality subgroup without permutation of masses • When the gauge coupling of SU(2)2vanishes or is very weak, we can discuss it in the same way as before for SU(2)1. The similar discussion goes for (1 2). That is, this subgroup consists of the permutation of five SU(2)’s. • cf. Note that two SL(2,Z) full S-duality groups do not commute! Here, we analyze only the boundary of the gauge coupling moduli space.

  9. SU(2)1×SU(2)2×SU(2)3 gauge theory with fund. and bifund. flavors (The similar discussion goes.) ■, ■: weak : interchange

  10. For more generalized SU(2) quivers : more gauge groups, loops… turn on/off a gauge coupling

  11. where are the solutions of coupling VEV Seiberg-Witten curve for quiver SU(2) gauge theories • massless SU(2) case In this case, the Seiberg-Witten curve is of the form If we change the variable as , this becomes • massless SU(2)n case • or • mass deformation The number of mass parameters is n+3, because of the freedom . polynomial of z of (n-1)-th order divergent at punctures

  12. SU(3) generalized quivers SU(3) gauge theory with 6 fundamental flavors (hypermultiplets) • This theory is also conformal. • flavor symmetry U(6) : complex rep. of SU(3) gauge group • kind of S-duality group : Argyres-Seiberg duality [Argyres-Seiberg ’07] • coupling const. : • flavor : U(6) ⊃ [SU(3)×U(1)]×[SU(3)×U(1)] : weak coupling • U(6) ⊃ SU(6)×U(1) ～ [SU(3)×SU(3)×U(1)]×U(1) • SU(6)×SU(2) ⊂ E6 : infinite coupling of SU(3) theory • Moreover,weakly coupled gauge group becomes SU(2) instead of SU(3) ! breakdown by VEV

  13. Argyres-Seiberg duality for SU(3) gauge theory NS5 D4 infinite coupling

  14. SU(3)1×SU(3)2 gauge theory with fundamental and bifundamental flavors flavor symmetry of bifundamental Argyres-Seiberg duality

  15. For more generalized SU(3) quivers : more gauge groups, loops… turn on/off a gauge coupling

  16. Seiberg-Witten curve for SU(3) quiver gauge theories • massless SU(3)n case • massless SU(2)×SU(3)n-2×SU(2) case • mass deformation • massless : • massive : The number of mass parameters is n+3, because of the freedom . In both cases, SW curve can be rewritten as ( ), but the order of divergence of is different from each other.

  17. SU(N) generalized quivers Seiberg-Witten curve for massless SU(N) quiver gauge theories Seiberg-Witten curve in this case is of the form The variety of quiver gauge group where is reflected in the various order of divergence of at punctures. For example…

  18. Classification of punctures : divergence of massless SW curve at punctures • SU(2) quiver case • order of divergence : • mass parameters : • flavor symmetry : SU(2) • SU(3) quiver case • order of divergence : • mass parameters : • flavor symmetry : U(1) SU(3)

  19. Classification of punctures : divergence of massless SW curve at punctures • SU(3) quiver case • corresponding puncture : • SU(4) quiver case (and the natural analogy is valid for general SU(N) case)

  20. AGT relation for SU(2) quivers

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

  22. Action • classical part • 1-loop correction : more than 1-loop is cancelled, because of N=2 SUSY. • instanton correction : Nekrasov’s calculation with Young tableaux • Parameters • coupling constants • masses of fundamental/antifund./bifund. fields and VEV’s of gauge fields • deformation parameters : background of graviphoton or deformation (rotation) of extra dimensions (Note that they are different from Gaiotto’s ones!) Nekrasov’s partition function of 4-dim gauge theory Now we calculate Nekrasov’s partition function of 4-dim SU(2) quiver gauge theory as the quantity of interest. NS5 D4

  23. 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 mass mass mass deformation parameters : each factor is a product of double Gamma function! ,

  24. 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

  25. SU(2) with four flavors : Calculation of Nekrasov function for U(2) • The Nekrasov partition function for the simple case of SU(2) with four flavors is • Since the mass dimension of is 1, so we fix the scale as , . • (by definition) • Mass parameters : mass eigenvalues of four hypermultiplets • : mass parameters of • : mass parameters of • VEV’s : we set --- decoupling of U(1) (i.e. trace) part. • We must also eliminate the contribution from U(1) gaugemultiplet. • This makes the flavor symmetry SU(2)i×U(1)i enhanced to SU(2)i×SU(2)i. • (next page…) U(2), actually Manifest flavor symmetry is now U(2)0×U(2)1 , while actual symmetry is SO(8)⊃[SU(2)×SU(2)]×[SU(2)×SU(2)].

  26. SU(2) with four flavors : Identification of SU(2) part and U(1) part • In this case, Nekrasov partition function can be written as • where and • is invariant under the flip(complex conjugate representation) : • which can be regarded as the action of Weyl group of SU(2) gauge symmetry. • is not invariant. This part can be regarded as U(1) contribution. • Surprising discovery by Alday-Gaiotto-Tachikawa • In fact, is nothing but the conformal block of Virasoro algebra with • for four operators of dimensions inserted at : (intermediate state)

  27. Liouville correlation function Correlation function of Liouville theory with. • Thus, we naturally choose the primary vertex operator as the examples of such operators. Then the 4-point function on a sphere is • 3-point function conformal block • where • The point is that we can make it of the form of square of absolute value! • … only if • … using the properties : and

  28. Example 1 : SU(2) with four flavors (Sphere with four punctures) • As a result, the 4-point correlation function can be rewritten as • where and • It says that the 3-point function (DOZZ factor) part also can be written as the product of 1-loop part of 4-dim SU(2) partition function : • under the natural identification of mass parameters :

  29. Example 2 : Torus with one puncture • The SW curve in this case corresponds to 4-dim N=2* theory : • N=4 SU(2) theory deformed by a mass for the adjoint hypermultiplet • Nekrasov instanton partition function • This can be written as • where equals to the conformal block of Virasoro algebra with • Liouville correlation function (corresponding 1-point function) • where is Nekrasov’s partition function.

  30. Example 3 : 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

  31. 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 :

  32. SW curve and AGT relation Seiberg-Witten curve and its moduli • According to Gaiotto’s discussion, SW curve for SU(2) case is . • In massive cases, has double poles. • Then the mass parameters can be obtained as , • where is a small circle around the a-th puncture. • The other moduli can be fixed by the special coordinates , • where is the i-th cycle (i.e. long tube at weak coupling). • Note that the number of these moduli is 3g-3+n. • (g : # of genus, n : # of punctures)

  33. 2-dim CFT in AGT relation : ‘quantization’ of Seiberg-Witten curve?? • The Seiberg-Witten curve is supposed to emerge from Nekrasov partition function in the “semiclassical limit” , so in this limit, we expect that . • In fact, is satisfied on a sphere, • then has double poles at zi . • For mass parameters, we have , • where we use and . • For special coordinate moduli, we have , • which can be checked by order by order calculation in concrete examples. • Therefore, it is natural to speculate that Seiberg-Witten curve is ‘quantized’ to at finite .

  34. AGT-W relation for SU(N)

  35. 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 … … … …

  36. 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! ,

  37. 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

  38. … … … … 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. … … …

  39. 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 :

  40. 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.

  41. 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.

  42. 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

  43. 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]

  44. 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 .

  45. 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

  46. 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.

  47. 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

  48. … … … 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] … …

  49. 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:

  50. 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 section…

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