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AGT 関係式 (2) AGT 関係式 (String Advanced Lectures No.19)

AGT 関係式 (2) AGT 関係式 (String Advanced Lectures No.19). 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 9 日（水） 12:30-14:30. Contents. 1. Gaiotto’s discussion for SU(2) 2. SU(2) partition function 3. Liouville correlation function 4. Seiberg-Witten curve and AGT relation

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AGT 関係式 (2) AGT 関係式 (String Advanced Lectures No.19)

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  1. AGT関係式(2) AGT関係式(String Advanced Lectures No.19) 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9日（水） 12:30-14:30

  2. Contents 1. Gaiotto’s discussion for SU(2) 2. SU(2) partition function 3. Liouville correlation function 4. Seiberg-Witten curve and AGT relation 5. Towards generalized AGT relation

  3. Gaiotto’s discussion for SU(2) [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

  4. 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.

  5. SU(2) partition function 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. • Action • classical part • 1-loop correction : more than 1-loop is cancelled, because of N=2 supersymmetry. • 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 of extra dimensions (Note that they are different from Gaiotto’s ones!)

  6. 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! ,

  7. 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

  8. 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)].

  9. 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)

  10. 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

  11. 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 :

  12. 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.

  13. 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.

  14. 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)

  15. 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 .

  16. … … … … Towards generalized AGT • Natural generalization of AGT relation seems the correspondence between partition function of 4-dim SU(N) quiver gauge theory and correlation function of 2-dim AN-1 Toda theory : • This discussion is somewhat complicated, since in SU(N>2) case, the punctures are classified with more than one kinds of 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 .) [Wyllard’09] [Kanno-Matsuo-SS-Tachikawa’09] … … …

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