Recurrence relations for the \( {\mathcal{W}}_3 \) conformal blocks and \( \mathcal{N}=2 \) SYM partition functions

Journal of High Energy Physics, Nov 2017

Recursion relations for the sphere 4-point and torus 1-point \( {\mathcal{W}}_3 \) conformal blocks, generalizing Alexei Zamolodchikov’s famous relation for the Virasoro conformal blocks are proposed. One of these relations is valid for any 4-point conformal block with two arbitrary and two special primaries with charge parameters proportional to the highest weight of the fundamental irrep of SU(3). The other relation is designed for the torus conformal block with a special (in above mentioned sense) primary field insertion. AGT relation maps the sphere conformal block and the torus block to the instanton partition functions of the \( \mathcal{N}=2 \) SU(3) SYM theory with 6 fundamental or an adjoint hypermul-tiplets respectively. AGT duality played a central role in establishing these recurrence relations, whose gauge theory counterparts are novel relations for the SU(3) partition functions with N f = 6 fundamental or an adjoint hypermultiplets. By decoupling some (or all) hypermultiplets, recurrence relations for the asymptotically free theories with 0 ≤ N f < 6 are found.

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Recurrence relations for the \( {\mathcal{W}}_3 \) conformal blocks and \( \mathcal{N}=2 \) SYM partition functions

Received: June W3 conformal blocks and Rubik Poghossian 0 Yerevan Physics Institute 0 Supersymmetric Gauge Theory 0 Alikhanian Br. 2, AM-0036 Yerevan , Armenia Recursion relations for the sphere 4-point and torus 1-point W3 conformal blocks, generalizing Alexei Zamolodchikov's famous relation for the Virasoro conformal blocks are proposed. One of these relations is valid for any 4-point conformal block with two arbitrary and two special primaries with charge parameters proportional to the highest weight of the fundamental irrep of SU(3). The other relation is designed for the torus conformal block with a special (in above mentioned sense) primary relation maps the sphere conformal block and the torus block to the instanton partition functions of the N = 2 SU(3) SYM theory with 6 fundamental or an adjoint hypermultiplets respectively. AGT duality played a central role in establishing these recurrence relations, whose gauge theory counterparts are novel relations for the SU(3) partition functions with Nf = 6 fundamental or an adjoint hypermultiplets. By decoupling some (or all) hypermultiplets, recurrence relations for the asymptotically free theories with 0 Conformal and W Symmetry; Extended Supersymmetry; Gauge Symmetry - N Nf < 6 are found. To the memory of Alexei Zamolodchikov 1 Introduction 2 Instanton partition function in background 2.1 SU(3) theory with Nf = 6 fundamental hypermultiplets 2.1.1 2.1.2 2.1.3 The residues Large v limit The recurrence relation 2.2 2.3 Recurrence relation for W3 conformal blocks 3.1 Preliminaries on A2 Toda CFT 3.1.1 3.1.2 Sphere 4-point block Torus 1-point block 4 Summary and discussion theory [4], 4d N = 2 SYM [5], topological strings [6, 7], partition function and Donaldson polynomials on CP2 [8] et al. Analogous recurrence relations has been found much later also for torus 1-point Virasoro block [5] (see also [9]) and for N = 1 Super-conformal blocks [10, 11]. The case when the theory admits higher spin W-algebra symmetry [12{14] is much more complicated. Holomorphic blocks of correlation functions of generic W-primary elds can not be found on the basis of the W-algebra Ward identities solely. Still, it is known { 1 { that if an n-point (n 4) contains n 2 partially degenerate primaries,1 the W-algebra is restrictive enough to determine (in principle) such blocks. It appears that exactly at this situation an alternative way to obtain W-conformal blocks based on AGT relation [15{17] is available. Note that though AGT relations provide combinatorial formulae for computing such conformal blocks, a recursion formulae like the one originally proposed by Zamolodchikov have an obvious advantage. Besides being very e cient for numerical calculations [4], such recursive formulae are very well suited for the investigation of analyticity properties and asymptotic behavior of the conformal blocks (or their AGT dual instanton partition functions [5]). Instead the individual terms of the instanton sum have many spurious poles HJEP1(207)53 that cancel out only after summing over all, rapidly growing number of terms of given order which leaves the nal analytic structure more obscure. In this paper recursion formulae are proposed for N = 2 SU(3) gauge theory instanton partition function in -background (Nekrasov's partition function) with 0 Nf 6 fundamental hypermultiplets as well as for the case with an adjoint hypermultiplet (N = 2 theory). As a byproduct all instanton exact formula is conjectured for the partition function in an one-parameter family of vacua, which is a natural generalization of the special vacuum introduced in [18] and recently investigated in [19]. The IR-UV relation discovered in [19, 20] was very helpful in nding these results. Using AGT relation the analogs of Zamolodchikov's recurrence relations are proposed for the (special) W3 4-point blocks on sphere and for the torus 1-point block. Though CFT point of view makes many of the features of the recurrence relations natural, unfortunately rigorous derivations are still lacking. The organization of the paper is as follows. In section 2. After a short review of instanton counting in the theory with 6 fundamentals, it is shown how investigation of the poles and residues of the partition function incorporated with the known UV - IR relation and the insight coming from the 2d CFT experience leads to the recurrence relation. Then, subsequently decoupling the hypermultiplets by sending their masses to in nity corresponding recurrence relations for smaller number of avours are found. The simplest case of pure theory (Nf = 0) is presented in more details. Then a similar analysis is carried out and as a result, corresponding recurrence relation is found for the SU(3), N = 2 theory. In section 3. Using AGT relation, the recurrence relations are constructed for the 4point W3 sphere blocks with two arbitrary and two partially degenerate insertions and for the torus block with a partially degenerate insertion. In both cases exact formulae for the large W3 current zero mode limit are presented. It is argued that the location of the poles as well as the structure of the residues which were instrumental in nding the recurrence relations of section 2., are related to the degeneracy condition and the structure of OPE of W3 CFT. on level 1. 1In this paper the term partially degenerate refers to the primary elds which admit a single null-vector { 2 { where hypermultiplets. On the right: the dual W3 conformal block. HJEP1(207)53 Instanton partition function in background SU(3) theory with Nf = 6 fundamental hypermultiplets Graphically this theory can be depicted as a quiver diagram on the left side of gure 1. The parameters a0;i, a2;i are related to the hypermultiplet masses while ai (i runs over 1; 2; 3) are the expectation values of the vector multiplet. The instanton part of the partition function is given as a sum over triple of Young diagrams Y~ = (Y1; Y2; Y3) (see [21{23]) Z = X ZY~ xjY j; ~ ~ Y where x is the exponentiated coupling (the instanton counting parameter); jY~ j is the total number of boxes of Young diagrams. The coe cients ZY~ can be represented as ZY~ = 3 Y i;j=1 Zbf (;; ai;0jYj ; aj )Zbf (Yi; aij;; a2;j ) Zbf (Yi; aijYj ; aj ) x SU(3) ai () a2;i 1 (3)!1 (4) (2)!1 (1) 0 (2.1) (2.2) (2.3) (2.4) (2.5) Y(a s2 Y(a s2 Zbf ( ; aj ; b) = b L (s) 1 + (1 + A (s)) 2) b + (1 + L (s)) 1 A (s)) 2) : Here A (s) (L (s)) is the distance in vertical (horizontal) direction from the upper (right) border of the box s to the outer boundary of the diagram as demonstrated in gure 2. As usual 1 and 2 denote the parameters of the background. Without loss of generality one may assume that a1 + a2 + a3 = 0. Then this parameters can be reexpressed in terms of the independent di erences a12 a1 a2 and a23 a2 a3 (a1; a2; a3) = 2a12 + a23 , 3 a12 a23 , a12 + 2a23 3 : The masses of 6 fundamental hypermultiplets can be identi ed as mi = a0;i mi = 1 + 2 i = 1; 2; 3 ; i = 4; 5; 6 : 3 for a0;i 3 for { 3 { Arm and leg length with respect to the Young diagram with column lengths f4; 3; 3; 1; 1; 1g. The thick solid line outlines its outer border. A(s1) = 2, L(s1) = 2, A(s2) = 2, L(s2) = 3, A(s3) = 3, L(s3) = 4. The advantage of the de nition above is that the partition function is symmetric with respect to permutations of Nf = 6 masses m1; : : : ; m6. For later convenience let us introduce also notations (elementary symmetric functions of masses) Tn = X 1 i1<i2< <in Nf mi1 min : Let us x an instanton number k and perform partial summation in (2.1) over all diagrams with total number of boxes equal to k. Many spurious poles present in individual terms cancel and one gets a rational expression whose denominator is where the product is over the positive integers r It is not di cult to check this statement explicitly for small k. Under AGT map this is equivalent to the well known fact that the 2d CFT blocks as a function of the parameters of the intermediate state acquire poles exactly at the degeneration points. Anticipating this relation let us introduce parameters u = a122 + a12a23 + a223; v = (a12 a23)(2a12 + a23)(a12 + 2a23): We'll see in section 3.1 that u is closely related to the dimension and v to the W zero mode eigenvalue of the intermediate state. For what follows it will be crucial to note that the factors of (2.7) in terms of newly introduced parameters can be rewritten as 2 27 a12 2 r;s Using (2.9) also in the numerator we can expel the parameters a12, a23 in favor of v and u. Moreover for xed u one gets a polynomial dependence on v. Thus, to recover the partition function one needs (2.6) (2.7) (2.8) (2.9) (2.10) (2.11) the residues at v = vr;s(u); the asymptotic behaviour of the partition function for a xed value of u and large v. 2.1.1 The residues It follows from the remarkable identity (2.10) that the residues at v = the residue with respect to the variable a12 at a12 = r;s in a simple way:2 vr;s is related to To restore the u-dependence in right hand side of (2.12) due to (2.10) one should substitute Resjv= vr;s = Resja12= r;s : 27 r;s r;s + 2a23 a23 = r;s q 4u 2 A careful examination shows that the residue of k = rs instanton term at a12 = r;s receives a nonzero contribution only from the triple (Y1; ;; ;), where Y1 is a rectangular diagram of size r s. Using eqs. (2.1), (2.2), (2.3) it is straightforward to evaluate this contribution. The result has a nice factorized form where the prime over the product means that the term with i = j = 0 should be omitted.3 2.1.2 Large v limit Now let us consider the limit v ! 1 for xed u. This is equivalent to choosing and taking large a12 limit. Here are the rst few terms of this expansion 2This is a choice of branch of the inverse map (v; u) ! (a12; a23). We could consider the poles at a23 = r;s or a12 + a23 = r;s instead. 3In generic SU(n) case with no hypers a nice formula has been found earlier [24] for the multiple residues are not su cient to derive a recurrence relation for the partition function. (2.12) I performed instanton calculation in this limit up to the order x5. The result up to the order x4 reads: 1 2 log Z x m1 3 m2 3 the choice of VEV (2.15), (2.16) coincides with the special vacuum investigated in [18, 19]. In [19, 20] an exact relation between the UV coupling and e ective IR coupling has been established. It was shown that a central role is played by the congruence subgroup 1(3) of the duality group SL(2; Z) [ 25, 26 ] and that the relation (2.18) (2.19) (2.20) (2.21) between x = exp 2 i uv and q = exp 2 i ir, where (q) is Dedekind's eta function x = 27 is valid. It should not come as a surprise also that the unique degree 1 modular form of 1(3) and its \ingredients" have a role to play. Indeed the expression f1(q) = 3(q) 3 (q3) + 27 3(q3) 3!1=3 (q) 1 2 log B 0 27q u (q3) 3(q) nicely matches the expansion (2.17) up to quite high orders in q and there is little doubt that the argument of logarithm in (2.21) indeed gives the large v limit of the partition function exactly. 2.1.3 The recurrence relation Using AGT relation it is not di cult to establish that the residue of the partition function at v = vr;s(u) is proportional to the partition function with expectation values speci ed as v ! given degenerate intermediate state related to the choice v = vr;s(u). Let us represent the partition function as Z(v; u; q) = x 27q u Incorporating information about residues establish above we nally arrive at the recurrent Using the recurrence relation I have computed the partition function up to the order x8 and compared it with the result of the direct instanton calculation. The agreement was perfect. 2.2 , renormalize the coupling constant as x ! limit.4 The net e ect is that instead of the recursion relaH(v; ujx) = 1 + where for the residues the same formula (2.26) with appropriate number of hypermultiplets Nf is valid. The relation between Z and H becomes much simpler. Using eq. (2.17) we immediately see that for Nf = 5 the appropriate relation is ZNf =5 = exp x (18(T1 ) x) 4The minus sign is due to a subtle di erence between fundamental and anti-fundamental hypermultiplets. With this sign included we get Nf anti-fundamentals in conventions of [15]. { 7 { and, for Nf = 4: ZNf =4 = exp H(v; ujx): x respect to the parameter v, so that the expansion (2.25) can be organized according to the poles in the variable v2: Z(v2; ujx) = 1 + where r;s=1 X1 ( x)rsRr;s(u) v2 vr2;s(u) Rr;s = 54 r;s u 2 r;s u similar manner. The coe cients ZY~ of the instanton partition function (2.1) in this case is given by ZY~ = 3 Y i;j=1 Zbf (Yi; ai mjYj ; aj ) ; where m is the mass of the adjoint hypermultiplet. The structure of poles is the same as in the previous cases. Due to symmetry under permutation a12 $ a23 the partition function, as in the case of pure theory, is a function of v2. The residue of the k = rs instanton charge sector of the partition function at v2 = vr2;s and xed u is related to the residue in variable a12 at a12 = r;s (with a23 xed) Resjv2=vr2;s = 54 r;s(a223 r2;s)(2 r;sa23 + a223) Resja12= r;s : (2.33) As in the case of fundamental hypermultiplets the residue of k = rs instanton term at a12 = r;s receives a nonzero contribution only from the triple of Young diagrams (Y1; ;; ;) with Y1 being a rectangular diagram of size r s. A direct calculation, using eqs. (2.3), (2.32) shows that { 8 { (2.29) (2.31) (2.32) HJEP1(207)53 Investigation of the large v2 behavior in this case is simpler compared to the theory with 6 fundamentals. Computations in rst few instanton orders shows that (in this section a more conventional notation q instead of x for the instanton counting parameter is restored) 1 2 log ZN =2 = 3(m 1)(m 2) log q 214 (q) + O(v 2): (2.35) This is a suggestive result. Recall that in the case of SU(2) gauge group one gets the same answer with the only di erence that the overall factor 3 is replaced by 2 [5]. Further steps are straightforward. Introducing the function H via we get the recurrence relation This recurrence relation has been checked by instanton calculation up to the order q10. 3 Recurrence relation for W3 conformal blocks In this section using AGT relations [15{17] the recurrence relations for N = 2 SYM partition functions will be translated into recurrence relations for certain W3-algebra four-point conformal blocks on sphere (AGT counterpart of Nf = 6 theory) and one-point torus blocks (AGT dual of N = 2 ). This recurrence relations generalize Alexei Zamolodchikov's famous relation established for the four point Virasoro conformal blocks [2, 3]. The recurrent relation for Virasoro 1-point torus block was proposed in [5] (see also [9]). It should be emphasised nevertheless, that the W3 blocks considered here are not quite general, two of four primary elds of the sphere block as well as that of the 1-point torus block are speci c. The charge vectors de ning their dimensions and W3 zero-mode eigenvalues are taken to be multiples of the highest weight of the fundamental (or anti-fundamental) representation of SU(3). Unfortunately e ective methods to understand generic W-blocks (to my knowledge) are still lacking. 3.1 These are 2d CFT theories which, besides the spin 2 holomorphic energy momentum current W(2)(z) T (z) are endowed with additional higher spin s = 3 current W(3) [ 12, 13, 27 ]. The Virasoro central charge is conventionally parameterised as c = 2 + 24Q2 ; { 9 { where the \background charge" Q is given by Q = b + 1 b ; and b is the dimensionless coupling constant of Toda theory. In what follows it would be convenient to represent roots, weights and Cartan elements of the Lie algebra A2 as 3-component vectors satisfying the condition that the sum of the components is zero. It is assumed also that the scalar product is the usual Kronecker one. Obviously this is equivalent to a more conventional representation of these quantities as diagonal traceless 3 3 matrices with pairing given by trace. In this representation the Weyl vector is given by HJEP1(207)53 For further reference let us quote here explicit expressions for the highest weight !1 of the rst fundamental representation and for its complete set of weights h1; h2; h3 where v is de ned in terms of the momentum vector p as v = 27p1p2p3 = (p12 p23)(p12 + 2p23)(2p12 + p23) and p12 = p1 p2, p23 = p2 p3. It is convenient to introduce also the parameter so that the conformal dimension (3.4) can be rewritten as The pair v; u characterizes primary elds more faithfully, than the charge vector, since they are invariant under the Weyl group action. The primary elds V (in this paper we concentrate only on the left moving holomorphic parts) are parameterized by vectors with vanishing center of mass. Their conformal wights are given by Sometimes it is convenient to parameterize primary elds (or states) in terms of the Toda momentum vector p = Q the elds V !1 with dimensions instead of . In what follows a special role is played by For generic these elds admit a single null vector at the rst level. Besides the dimension, the elds are characterized also by the zero mode eigenvalue of the W3 current , 1 3 , (3.1) (3.2) (3.3) (3.4) (3.5) (3.6) (3.7) (3.8) The object of our interest in this section will be the conformal block h V 4 (1)V 3!1 (1)V 2!1 (x)V 1 (0)ip x h h1 h2 G(v; ujx) ; (3.9) where h ip denotes the holomorphic part of the correlation function with a speci ed intermediate state of momentum p = Q . It is assumed that the function G(v; ujx) is normalized so that G(v; ujx) = 1 + O(x) (we explicitly display only dependence on the parameters v; u, which specify the intermediate state). Due to AGT relation, the function G(v; ujx) is directly connected to the instanton partition function of SU(3) gauge theory with Nf = 6 hypermultiplets discussed earlier (see gure 1). Here is the map between parameters of the CFT and Gauge Theory (GT) sides: Under this identi cation of parameters the relation between the gauge theory (with Nf = 6 fundamentals) partition function and the CFT conformal block is very simple: Z = (1 x) (3)(Q 13 (2)) G : Now it is quite easy to rephrase the recurrence relation for the partition function in terms of CFT language. De ne a function H(v; ujq) through G(v; ujx) = x 27q 3(q) u 3 f1(q) 3(h2+h23)+2Q2 H(v; ujq); (3.15) where q and x are related as in (2.18). Then, due to (2.23), (2.25), (3.14) and (3.15) for H(v; ujq) we get essentially the same recurrence relation (2.25) where similar to (2.11) vr;s(u) = (3Qr2;s q u) 4u 3Qr2;s (3.17) (3.10) (3.11) (3.12) (3.13) (3.14) with (cf. (2.8) ) and the residues are given by Qr;s = br + s b where CFT counterparts of gauge theory masses l = ml=p 1 2 are related to the parameters of the inserted elds via (3.11){(3.13). It follows from the analog of the Kac determinant for W3-algebra [ 28 ], that the conformal block truncated up to the order xk should have simple poles in the variable v (for u xed) located at v = vr;s(u) with r among parameters v, u is the condition of existence of a null vector at the level rs. This null vector originates a W3-algebra representation with parameters u ! u 3rs ; v ! Though we arrived to the recurrence relation starting from the gauge theory side, in fact many features of this relation are transparent from the CFT side and it is reasonable to expect that a rigorous proof may be found generalizing arguments of Alexei Zamolodchikov from Virasoro to the W-algebra case. Indeed (3.16) states that the residues at the poles v = vr;s(u) (3.17), are proportional to the conformal block with internal channel parameters (3.21) corresponding to the null vector at the level rs. The factor Rr(;s) (3.19) also has many expected features. Its denominator vanishes exactly when the parameter u is speci ed so that a second independent degenerate state arises. The factors in the numerator re ect the structure of OPE with degenerate eld (see [14]) exactly as it was in the case of Virasoro block considered by Alexei Zamolodchikov. It seems more subtle to justify presence of the u independent factors Qi;j1. Our result predicts the following large v behavior of the W3 block G(v; ujx) x 27q u 3 3(q) f1(q) 3(h2+h23)+2Q2 + O(v 1): (3.22) A good starting point to prove this relation might be the deformed Seiberg-Witten curve DSFT [29{31] or, equivalently, the quasiclassical null vector decoupling equation for Wblocks derived in [32]. (3.18) (3.20) (3.21) Since the torus 1-point block (below is the charge parameter of the intermediate states) is related to the partition function of the gauge theory with adjoint hypermultiplet via [33] is related to the adjoint hypermultiplet mass m: and as earlier the intermediate momentum parameter p = Q is related to the VEV of the vector multiplet a as Thus, comparing with (2.36), (2.37), (2.38), we see that the function H(v2; u; q) de ned by (3.23) (3.24) (3.25) (3.26) (3.27) (3.29) the equality relation where F (q) = q 24 (q) 1 2 H(v2; u; q) ; (v and u in terms of the momentum p were de ned in (3.6), (3.7)) satis es the recurrence Summary and discussion To summarize let me quote the main results of this paper: the recurrence relation (see (2.21), (2.25), (2.26)) for the instanton partition function of N = 2 SU(3) gauge theory with 6 fundamental hypermultiplets. This recurrence relation suggests an exact in all instanton orders formula for the partition function and prepotential for the theory in a generalized version of the special vacuum considered in [18, 19]; recurrence relations for smaller number of hypermultiplets (see section 2.2) and for pure Nf = 0 theory (section 2.2.1); recurrence relations for the theory with an adjoint hypermultiplet, commonly referred as N = 2 theory (see section 2.3); the analogs of Zamolodchikov's recurrence relations are constructed for 4-point sphere W3-blocks with two arbitrary and two partially degenerate insertions (see (3.15), (3.16), (3.19)) and for the torus W3-block with a partially degenerate insertion (see (3.27), (3.28), (3.29)). For both cases recursion formulae provide explicit expressions for the large W3 zero mode limit. Though many details of the recurrence relations are transparent either from the 4d gauge theory or from the 2d CFT points of view, still full derivation is lacking. 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Rubik Poghossian. Recurrence relations for the \( {\mathcal{W}}_3 \) conformal blocks and \( \mathcal{N}=2 \) SYM partition functions, Journal of High Energy Physics, 2017, 53, DOI: 10.1007/JHEP11(2017)053