On non-conformal limit of the AGT relations
aa r X i v : . [ h e p - t h ] S e p On non-conformal limit of the AGT relations
A. Marshakov § , A. Mironov ¶ Theory Department, Lebedev Physics Institute and
ITEP, Moscow, Russia
A.Morozov k ITEP, Moscow, Russia
FIAN/TD-19/09ITEP/TH-41/09
The Seiberg-Witten prepotentials for N = 2 SUSY gauge theories with N f < N c fundamentalmultiplets are obtained from conformal N f = 2 N c theory by decoupling 2 N c − N f multiplets of heavymatter. This procedure can be lifted to the level of Nekrasov functions with arbitrary backgroundparameters ǫ and ǫ . The AGT relations imply that similar limit exists for conformal blocks (or,for generic N c >
2, for the blocks in conformal theories with W N c chiral algebra). We consider thelimit of the four-point function explicitly in the Virasoro case of N c = 2, by bringing the dimensionsof external states to infinity. The calculation is performed entirely in terms of representation theoryfor the Virasoro algebra and reproduces the answers conjectured in arXiv:0908.0307 with the helpof the brane-compactification analysis and computer simulations. In this limit, the conformal blockinvolving four external primaries, corresponding to the theory with vanishing beta-function, turnseither into a 2-point or 3-point function, with certain coherent rather than primary external states. The AGT relations [1]-[11] express generic 2 d conformal blocks through the Nekrasov functions[12]-[20] Z ( Y ), associated with N = 2 SUSY quiver 4 d gauge theories with extra fundamental mul-tiplets, generalizing the earlier predictions of [14, 15]. Most commonly these theories have vanishingbeta-functions and possess conformal invariance in four dimensions. In the simplest case of the 4-pointVirasoro conformal block, this is the conformal SU (2) model with N f = 2 N c = 4 flavors. The masses µ , . . . , µ N f of the four fundamentals are related to the dimensions of four external states operators: µ = α − α + ǫ , µ = α + α − ǫ , µ = α − α + ǫ , µ = α + α − ǫ , ∆ k = α k ( ǫ − α k ) ǫ ǫ , c = 1 + 6 ǫ ǫ ǫ , ǫ = ǫ + ǫ (1)and the gauge theory condensate (modulus) a = a = − a is related to that of the intermediate state: a = α − ǫ µ k → ∞ the fundamental fields in 4 d theory decouple, and one gets an asymptoticallyfree pure gauge N = 2 SUSY theory, with prepotential expressed through (the ǫ = − ǫ → pure gauge Nekrasov functions Z ( Y ): Z ( Y ) ∼ lim µ k →∞ Z ( Y ) (3)The AGT relation implies that the associated limit of conformal block corresponds to this Z ( Y ). Anatural question is how does this limit look like from the point of view of 2 d conformal theory itself. § E-mail: [email protected]; [email protected] ¶ E-mail: [email protected]; [email protected] k E-mail: [email protected] direct way, by taking a particular limit of the 4-pointconformal block with generic µ ’s. Instead, the conclusion was based on analysis of the underlying5-brane configurations [21], which was also the original source of the AGT relations in [1]. In thisletter, we fill the gap and derive the result of [4] straightforwardly, making use of explicit knowledge ofthe Virasoro conformal blocks from [3]. A similar analysis is possible for conformal blocks with moreexternal states and for some W -algebra blocks N c >
2, in the last case the results of [2, 5, 7] shouldbe used. These generalizations are, however, beyond the scope of this paper. We use notations from [3] and refer for details and explanations to that paper. The 4-pointconformal block is given by the sum over Young diagrams B ∆ ∆ ;∆ ∆ ;∆ ( x ) = X | Y | = | Y ′ | x | Y | γ ∆∆ ∆ ( Y ) Q − ( Y, Y ′ ) γ ∆∆ ∆ ( Y ′ ) (4)with the inverse Shapovalov form Q ∆ ( Y, Y ′ ) = h ∆ | L Y ′ L − Y | ∆ i , where L − Y = L − k ℓ . . . L − k L − k forthe Young diagram Y = { k ≥ k ≥ . . . ≥ k ℓ > } are made from the Virasoro operators L k , k ∈ Z ,satisfying [ L m , L n ] = c n ( n − δ m + n, + ( m − n ) L m + n (5)and the three-point functions [22, 3] are γ ∆∆ ∆ ( Y ) = ℓ ( Y ) Y i =1 ∆ + k i ∆ − ∆ + X j
1, the γ -factor reduces to γ ( Y ) ∼ ℓ ( Y ) Y i =1 ( k i ∆ − ∆ ) (9)2nd of all diagrams of a given size | Y | , the sum in (4) is saturated by the terms, where γ ( Y )’s (9)contain maximal possible number of factors, i.e. when ℓ ( Y ) = | Y | , or Y is a single-column diagram[1 | Y | ] = [1 , . . . , | {z } | Y | times ]: x | Y | / γ ∆∆ ∆ ( Y ) → (cid:16) √ x (∆ − ∆ ) (cid:17) | Y | δ ( Y, [1 | Y | ]) == (cid:18) √ xµ µ − ǫ ǫ (cid:19) | Y | δ ( Y, [1 | Y | ]) → (cid:18) Λ − ǫ ǫ (cid:19) | Y | δ ( Y, [1 | Y | ]) (10)In what follows we often omit the powers of − ǫ ǫ , which can be easily restored from dimensionalconsideration. Since the Shapovalov form does not depend on ∆ , . . . , ∆ , this means that the limitof conformal block B ∆ (Λ) = lim ∆ i →∞ B ∆ ∆ ;∆ ∆ ;∆ ( x ) = X | Y | = | Y ′ | Λ | Y | Q − ( Y, Y ′ ) δ (cid:16) Y, [1 | Y | ] (cid:17) δ (cid:16) Y ′ , [1 | Y | ′ ] (cid:17) = X n Λ n Q − ([1 n ] , [1 n ])(11)The r.h.s. of this expression can be treated as a norm (scalar square) of a peculiar vector in the VirasoroVerma module H ∆ with the highest weight ∆. Following [4], we denote it | Λ , ∆ i = P Y C Y L − Y | ∆ i ∈H ∆ . Then k| ∆ , Λ ik = h ∆ , Λ | ∆ , Λ i = X Y,Y ′ C Y Q ( Y, Y ′ ) C Y ′ (12)and, in order to reproduce the r.h.s. of (11), one should take C Y = Λ | Y | Q − ([1 | Y | ] , Y ), so that | ∆ , Λ i = X Y Λ | Y | Q − (cid:16) [1 | Y | ] , Y (cid:17) L − Y | ∆ i (13)This vector can be characterized as being orthogonal to all non single-column states | ∆ , Y i = L − Y | ∆ i ∈ H ∆ with Y = [1 | Y | ], since h ∆ | L Y | ∆ , Λ i = X Y ′ Λ | Y ′ | Q − (cid:16) [1 | Y ′ | ] , Y ′ (cid:17) h ∆ | L Y L − Y ′ | ∆ i == X Y ′ Λ | Y ′ | Q − (cid:16) [1 | Y ′ | ] , Y ′ (cid:17) Q ∆ ( Y ′ , Y ) = Λ | Y | δ (cid:16) Y, [1 | Y | ] (cid:17) (14)This means, in particular, that it is a kind of a “coherent” state, satisfying L | ∆ , Λ i = Λ | ∆ , Λ i ,L k | ∆ , Λ i = 0 , ∀ k ≥ ⇒ (15) deserves more detailed explanation. Consider the vector L k | ∆ , Λ i ∈ H ∆ for k >
0. The coefficients of its expansion over the basis | ∆ , Y i = L − Y | ∆ i in H ∆ are characterizedtotally by the scalar products h ∆ , Y | L k | ∆ , Λ i = h ∆ | L Y L k | ∆ , Λ i = X Y ′ b ( k ) Y Y ′ h ∆ | L Y ′ | ∆ , Λ i = ( ) = X Y ′ b ( k ) Y Y ′ Λ | Y ′ | δ (cid:16) Y ′ , [1 | Y ′ | ] (cid:17) = X ℓ ′ b ( k ) Y [1 ℓ ′ ] Λ ℓ ′ (16)where ℓ ′ = ℓ ( Y ′ ) = | Y ′ | , i.e. only the Young diagrams Y ′ = [1 | Y ′ | ] = [1 ℓ ( Y ′ ) ] can contribute. It isimportant, however, that due to the Virasoro commutation relations, (5) the sum in (16) is restrictedby | Y ′ | ≤ | Y | + k and ℓ ( Y ′ ) ≤ ℓ ( Y ) + 1, meaning that both the number of boxes in Y ′ and the number3f elementary Virasoro generators in L Y ′ is less or equal to those in L Y L k ; moreover, the structure ofVirasoro algebra (5) requires necessarily k i ( Y ′ ) ≥ k j ( Y ), i, j = 1 , . . . , ℓ ( Y ′ ) , ℓ ( Y ). Hence, one gets for(16) h ∆ , Y | L k | ∆ , Λ i = X ℓ ′ b ( k ) Y [1 ℓ ′ ] Λ ℓ ′ = δ (cid:16) Y, [1 ℓ ] (cid:17) X ℓ ′ ≤ ℓ +1 b ( k )[1 ℓ ][1 ℓ ′ ] Λ ℓ ′ = δ (cid:16) Y, [1 ℓ ] (cid:17) δ k, Λ ℓ +2 (17)and this immediately leads to (15), since for k > L k | ∆ , Λ i is orthogonal to all vectors in H ∆ , while for k = 1 it coincides with the vector | ∆ , Λ i up to a numerical factor Λ .Differently, expanding | ∆ , Λ i = P n ≥ Λ n | ∆ , n i , one gets for | ∆ , n i = X | Y | = n Q − (cid:16) [1 n ] , Y (cid:17) L − Y | ∆ i (18)that L | ∆ , n i = | ∆ , n − i , n ≥ L k | ∆ , n i = 0 , ∀ k ≥ , n ≥ proved it directly, taking the limit of the4-point conformal block with arbitrary dimensions. Note that the whole reasoning is valid for any ǫ , ǫ and ǫ , i.e. for conformal theory with arbitrary central charge c (the central charge dependencearises in | ∆ , Λ i through the inverse matrix of the Shapovalov form). In a similar way, one can consider partial decoupling of the fundamental matter, correspondingto the models with N f = 1 , ,
3, when remaining masses (and related combinations of conformaldimensions) are preserved as free parameters. Let us start with the case of N f = 1. In such a limit, µ , , → ∞ with finite x Q I =2 , , µ I = Λ , but µ remains finite itself. According to (1), this meansthat α and α go to infinity, but not independently: their difference remains finite. In terms ofconformal dimensions it means that∆ − ∆ = ( α − α )( α + α − ǫ ) − ǫ ǫ ∼ (2 µ − ǫ ) √ ∆ √− ǫ ǫ (20)i.e. all dimensions are infinite, but ∆ − ∆ √ ∆ remains finite. Hence, in this limit the single-columndiagrams still dominate to contribute into γ ∆∆ ∆ ( Y ), like it has been considered above in the case offlow into pure gauge theory, but the factor γ ∆∆ ∆ ( Y ) is now dominated by a different sort of Youngdiagrams. The reason is that for k i = 1 the factor k i ∆ − ∆ turns into ∆ − ∆ and grows not as fastas ∆ and ∆ themselves. Instead, the dominant contribution comes now from the Young diagramsof the form Y = [2 p , q ] with | Y | = 2 p + q , ℓ ( Y ) = p + q , since for all of them γ ∆∆ ∆ ( Y ) ∼ (2∆ − ∆ ) p (∆ − ∆ ) q ∼ (cid:18) µ − ǫ √− ǫ ǫ (cid:19) q ∆ p + q/ ∼ (2 µ − ǫ ) q ( µ / | Y | ( − ǫ ǫ ) p + q (21)Instead of (11), the limit of conformal block is now given by (we again omit the powers of − ǫ ǫ ) B N f =1∆ (Λ , m ) = lim ∆ I →∞ ∆ − ∆ ∼ m √ ∆ B ∆ ∆ ;∆ ∆ ;∆ ( x ) == X | Y | = | Y ′ | X p (2 m ) | Y |− p (cid:16) xµ µ µ (cid:17) | Y | Q − ( Y, Y ′ ) δ (cid:16) Y, [2 p , | Y |− p ] (cid:17) δ (cid:16) Y ′ , [1 | Y | ′ ] (cid:17) == X n,p (2 m ) n − p (cid:18) Λ (cid:19) n Q − (cid:16) [2 p , n − p ] , [1 n ] (cid:17) = h ∆ , Λ / , m | ∆ , Λ i , (22)4here m = µ − ǫ , Λ = xµ µ µ to be fixed when taking the limit of x → µ I → ∞ , I = 2 , , | ∆ , Λ , m i = X Y X p m | Y |− p Λ | Y | Q − (cid:16) [2 p , | Y |− p ] , Y (cid:17) L − Y | ∆ i (23)while the vector | ∆ , Λ i has been already defined in (13). Considering the matrix elements h ∆ | L Y | ∆ , Λ , m i = X p m | Y |− p Λ | Y | δ (cid:16) Y, [2 p , | Y |− p ] (cid:17) (24)and h ∆ | L Y L k | ∆ , Λ , m i = X Y ′ b ( k ) Y Y ′ h ∆ | L Y ′ | ∆ , Λ , m i = ( ) X Y ′ b ( k ) Y Y ′ X p m | Y ′ |− p Λ | Y ′ | δ (cid:16) Y ′ , [2 p , | Y ′ |− p ] (cid:17) == δ k, b (1)[2 p , | Y |− p ][2 p , | Y | +1 − p ] m | Y | +1 − p Λ | Y | +1 + δ k, b (2)[2 p , | Y |− p ][2 p +1 , | Y | +2 − p +1) ] m | Y | +2 − p +1) Λ | Y | +2 (25)one proves exactly in the same way as before that L | ∆ , Λ , m i = m Λ | ∆ , Λ , m i ,L | ∆ , Λ , m i = Λ | ∆ , Λ , m i ,L k | ∆ , Λ , m i = 0 for k ≥ m → ∞ together with Λ → m Λ = Λ N f =0 , only theterm with p = 0 survives in the sum (23) and this state turns into (13): | ∆ , m, Λ i → | ∆ , Λ N f =0 i , whileconstraints (26) turn into (15). It deserves mentioning that, due to separation of powers of Λ in (22)between two vectors in the scalar product (which is, of course, ambiguous), this limit is a little bitdifferent from the conventional “physical” limit in Seiberg-Witten theory, µ Λ → Λ N f =0 .The calculation is very similar in the case of N f = 2, if keeping finite the masses µ and µ . Then,both the factors γ ∆∆ ∆ and γ ∆∆ ∆ behave according to (21), when taking µ → ∞ and µ → ∞ ,and one gets that the conformal block (4) B N f =2∆ (Λ , m , m ) = lim ∆ I →∞ ∆1 − ∆2 ∼ µ √ ∆1 ∆ − ∆ ∼ µ √ ∆ B ∆ ∆ ;∆ ∆ ;∆ ( x ) == X | Y | = | Y ′ | X p,p ′ (2 m ) | Y |− p (2 m ) | Y |− p ′ (cid:18) Λ (cid:19) | Y | Q − ( Y, Y ′ ) δ (cid:16) Y, [2 p , | Y |− p ] (cid:17) δ (cid:16) Y ′ , [2 p ′ , | Y ′ |− p ′ ] (cid:17) == X n,p,p ′ (2 µ ) n − p (2 µ ) n − p ′ (cid:18) Λ (cid:19) n Q − (cid:16) [2 p , n − p ] , [2 p ′ , | Y ′ |− p ′ ] (cid:17) == h ∆ , Λ / , m | ∆ , Λ / , m i (27)in this limit is a scalar product of two states (23), where m , = µ , − ǫ , Λ = xµ µ , are again to befixed finite in the limit of x → µ , → ∞ . In the case of “asymmetric limit”, i.e. if instead of taking µ , → ∞ , one decouples, say, µ , → ∞ , no simplification occurs in the factor γ ∆∆ ∆ ( Y ) in (4), while the second factor degeneratesaccording to (10), i.e. x | Y ′ | γ ∆∆ ∆ ( Y ′ ) → Λ | Y ′ | δ (cid:16) Y ′ , [1 | Y ′ | ] (cid:17) . This means that the conformal block5implifies, though not as drastically as in the symmetric limit:˜ B N f =2∆ (Λ , µ , µ ) = lim ∆ , →∞ B ∆ ∆ ;∆ ∆ ;∆ ( x ) = X Y Λ | Y | γ ∆∆ ∆ ( Y ) Q − (cid:16) Y, [1 | Y | ] (cid:17) == h ∆ , Λ | V ∆ (1) V ∆ (0) i (28)since [3] γ ∆∆ ∆ ( Y ) = h L − Y V ∆ | V ∆ (1) V ∆ (0) i (29)Thus, the 4-point conformal block in this limit reduces to a triple vertex, as was conjectured in [4]. Itdepends on µ and µ through ∆ and ∆ .Similarly, if only one mass, say, µ → ∞ , one obtains B N f =3∆ (Λ , µ , µ , µ ) = lim ∆3 , →∞ ∆ − ∆ ∼ µ √ ∆ B ∆ ∆ ;∆ ∆ ;∆ ( x ) == X Y X p (2 µ − ǫ ) | Y |− p (cid:18) Λ2 (cid:19) | Y | γ ∆∆ ∆ ( Y ) Q − (cid:16) Y, [2 p , | Y |− p ] (cid:17) == h ∆ , Λ / , µ − ǫ | V ∆ (1) V ∆ (0) i (30)which is again a reduction from the 4-point function to a 3-point one. To conclude, in this paper we have studied the non-conformal limits (in the sense of 4 d su-persymmetric gauge theory) of conformal blocks related to Nekrasov partition functions by the AGTcorrespondence. We have derived directly from 2 d CFT analysis the results, conjectured in [4] frombrane considerations and confirmed by computer simulations, for the asymptotically free limit of con-formal blocks. The proof holds at the level of Nekrasov functions for arbitrary values of ǫ , ǫ and ǫ = ǫ + ǫ , the result for the Seiberg-Witten prepotentials [23] follows [15, 16] after taking the limitof ǫ , ǫ →
0. The proof is self-consistent within 2 d CFT, and, in application to Nekrasov functions,it assumes that the original AGT relation is correct. After numerous checks in [1]-[11] this looksindisputably true, though so far has been proven exactly [6, 7] only in the hypergeometric case forthe W -algebra blocks with one special, one fully-degenerate external state and a free field theory likeselection rule imposed on the intermediate state.There is a number of other interesting limits, which are natural and well understood from thepoint of view of 2 d CFT (e.g. large intermediate dimension ∆ or the central charge c ). It can beinteresting to find their interpretation in terms of the Nekrasov functions and/or instanton expansionsin 4 d SUSY models.Our work was partly supported by Russian Federal Nuclear Energy Agency and by the joint grants09-02-90493-Ukr, 09-02-93105-CNRSL, 09-01-92440-CE. The work of A.Mar. was also supported byRussian President’s Grants of Support for the Scientific Schools NSh-1615.2008.2, by the RFBR grant08-01-00667, and by the Dynasty Foundation. The work of A.Mir. was partly supported by the RFBRgrant 07-02-00878, while the work of A.Mor. by the RFBR grant 07-02-00645; the work of A.Mir.and A.Mor. was also supported by Russian President’s Grants of Support for the Scientific SchoolsNSh-3035.2008.2 and by the joint program 09-02-91005-ANF.
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