Toda 3-point functions from topological strings

Journal of High Energy Physics, Jun 2015

Abstract We consider the long-standing problem of obtaining the 3-point functions of Toda CFT. Our main tools are topological strings and the AGT-W relation between gauge theories and 2D CFTs. In [1] we computed the partition function of 5D T N theories on S 4 × S 1 and suggested that they should be interpreted as the three-point structure constants of q-deformed Toda. In this paper, we provide the exact AGT-W dictionary for this relation and rewrite the 5D T N partition function in a form that makes taking the 4D limit possible. Thus, we obtain a prescription for the computation of the partition function of the 4D T N theories on S 4, or equivalently the undeformed 3-point Toda structure constants. Our formula, has the correct symmetry properties, the zeros that it should and, for N = 2, gives the known answer for Liouville CFT.

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Toda 3-point functions from topological strings

JHE Toda 3-point functions from topological strings 0 Notkestrasse 85 , D-22607 Hamburg , Germany 1 IRIS Haus , Zum Großen Windkanal 6, 12489 Berlin , Germany 2 Vladimir Mitev 3 DESY Hamburg, Theory Group 4 Physics Division, National Technical University of Athens 5 and Elli Pomoni 6 Open Access , c The Authors 7 15780 Zografou Campus , Athens , Greece We consider the long-standing problem of obtaining the 3-point functions of Toda CFT. Our main tools are topological strings and the AGT-W relation between gauge theories and 2D CFTs. In [1] we computed the partition function of 5D TN theories on S4×S1 and suggested that they should be interpreted as the three-point structure constants of q-deformed Toda. In this paper, we provide the exact AGT-W dictionary for this relation and rewrite the 5D TN partition function in a form that makes taking the 4D limit possible. Thus, we obtain a prescription for the computation of the partition function of the 4D TN theories on S4, or equivalently the undeformed 3-point Toda structure constants. Our formula, has the correct symmetry properties, the zeros that it should and, for N = 2, gives the known answer for Liouville CFT. topological; strings; Supersymmetry and Duality; Conformal and W Symmetry; Topological Strings 1 Introduction 2 The AGT dictionary 3 Toda 3-point functions 3.1 3.2 3.3 Enhanced symmetry of the Weyl invariant part Pole structure of the Weyl invariant part 3.4 The q-deformed Toda field theory 4 The TN partition function from topological strings The 5-brane webs The topological vertex computation 4.3 The 4D limit 5 Liouville from topological strings 6 W3 from topological strings 7 Conclusions and outlook A Parametrization of the TN junction B Conventions and notations for SU(N ) C Special functions C.3 The finite product functions D Computation of the TN partition function Introduction The AGT(-W) correspondence [2–4] is a relationship between, on one side, the 2D Liouville (Toda) CFT on a Riemann surface of genus g with n punctures and, on the other side, the SCFT on that same surface. The correlation functions of the 2D Toda WN conformal field S4 = [da] ZN4Dek(a, m, ǫ1,2) The conformal blocks of the 2D CFTs are given by the appropriate instanton partition functions, while the three point structure constants should be obtained by the S4 partition functions of the TN superconformal theories. These partition functions were until recently [1, 5] unknown, with the sole exception of the W2 case, i.e. the Liouville case, whose three point structure constants are given by the famous DOZZ formula [6, 7]. The AGT(-W) relation (1.1) holds after the mass parameters m of the gauge theory, the UV coupling constants and the vacuum expectation values a of the scalars in the vector multiplet (the Coulomb moduli) are appropriately identified with, respectively, the external ters) and the internal momenta over which we integrate. Finally, the IR regulators of the gauge theory, which are given by the Omega deformation parameters ǫ1,2, are identified specific cases exist) [14–16] in support of the AGT-W correspondence for N > 2. Similarly, there exists a 5D version of the AGT(-W) relation1 [18, 19] (see also [1, 20– 30]) which relates the 5D Nekrasov partition functions on S4 × S1 to correlation functions of q-deformed Liouville (Toda) field theory: S4×S1 = ∝ hVα1 (z1) · · · Vαn (zn)iq-Toda , squared of 5D Nekrasov partition function is the 5D superconformal index Z as discussed recently in [31] can be computed using the topological string partition function S4×S1 = [da] |ZN5Dek(a)|2 ∝ From both the 4D and the 5D AGT-W relations a very important element is missing: the three point functions of the WN Toda CFT. Computing the three point functions of the WN Toda CFT has been a long standing unsolved problem. From the the CFT side, the state of the art is due to Fateev and Litvinov, who in [32–34], were able to compute the 3-point functions of Toda primaries for the special case in which one of the fields is semi-degenerate, using [35]. On the gauge theory side, the 3-point functions correspond to the partition functions of the TN theories, but since these theories lack2 any known 1Originally suggested in [17]. 2After our paper, [36] discovered that the topological string partition functions of 5D mass-deformed TN theories, before the removal of the decoupled content (4.12), can be rewritten as the partition function removal of the decoupled content. One may interpret this proposal as providing a Lagrangian description of the 5D TN theories through these quivers. Lagrangian description, the usual methods of computing the partition functions are not In [1] we computed the partition functions of the 5D TN theories on S4×S1 by using the web diagram provided by [37] and by employing the refined topological vertex formalism of [38, 39]. We further argued that these partition functions should give the three point functions of q-deformed Toda, which was also proposed earlier in [40]. Our results were checked by computing the 5D superconformal index, i.e. the partition function on S4 × S1, using the prescription in [31] and comparing it to the result obtained via localization in [41]. The same partition functions were also obtained in [5] and the two computations agree. More comparisons with the superconformal index were given in the recent work [42]. In this paper we show how to, in principle,3 take the 4D limit, thus obtaining the 4D TN partition functions. Through the AGT(-W) relation, they are identified with the usual, undeformed Toda three point functions. Our formula has the correct symmetry properties, zeros and reproduces the known answer for the Liouville CFT. Furthermore, we carefully study the 5D AGT-W dictionary. For that, it was very important to examine the known q-Liouville case [23, 40] for which for the first time we were able to write the formula with Our method of attacking the problem of solving Toda, even though indirect, is very powerful for the following reasons. For 2D CFTs with only Virasoro symmetry the multipoint correlation functions of Virasoro descendants can be obtained from the ones containing only Virasoro primary fields [44]. On the other hand, for the WN Toda CFTs with N > 2 complete knowledge of the correlation functions of WN primary fields is not enough to obtain the correlation functions of descendents. Fully solving Toda means being able to construct the complete set of correlation functions both of primaries and descendants. Obtaining the three point functions with descendants is very naturally done using topological strings and is work in progress [45]. Since this article relates two somewhat disjointed fields, each used to its own notations, we wish to include a reader’s guide to the other sections. We begin in section 2 with a presentation of the parametrizations and the precise relations between the partition functions of section 4 and the correlators of section 3. In the following section 3, we review shortly the Toda CFT, introduce the associated notation and make some observations regarding the symmetries of the correlation functions that to our knowledge are not available in the literature. We finish section 3 by a discussion of the pole structures and the q-deformations of the correlation functions. In section 4, we give a short review of the derivation of the opinion are the appropriate tools to use in this context. We then discuss their 4D limit. In The reader can find a collection of useful formulas, notations and parametrizations in the together with a discussion of their properties. 3The specification “in principle” refers to the fact that there is still a missing ingredient which is to perform the sums in (4.10). This work can be found in a separate [43] publication, where we compute some of the sums. The AGT dictionary The main goal of this section is to provide the dictionary needed to relate the topological string amplitudes of section 4 to the Toda CFT correlation functions of section 3. First, we review the parameters of the Omega deformation. The circumference of the 5D circle Furthermore, we need to also define4 x := = q 2 , the natural variables, fugacities of the 5D superconformal index. When we need to relate parameters need to be specialized as ǫ1 = b, ǫ2 = b−1, which implies in particular that |q| < 1, |q| < 1, |t| > 1 and |x| < 1 since we take b to be label the primary fields, while on the TN theory side, one uses the positions of the exterior branes, see section 4 and appendix A, as parameters. The rough relationship is illustrated in figure 1 and the precise identifications are are separately dimensionless. j mi = (α1 − Q, hi) = N X α1 − ni = − (α2 − Q, hi) = −N X α2j + X jα2j + li = − (α3 − Q, hN+1−i) = −N N + 1 − 2i N + 1 − 2i where hi are the weights of the fundamental representation of SU(N ). In appendix B the m1 = −m2 = α11 − 2 while for N = 3 we have with similar expressions for the ni and li. Having set up the parametrization, we are ready to present our full claim. For that it is important to stress that from the Toda CFT 3-point structure constants C, see (3.10), we can extract the Weyl-invariant structure constants C as The partition function ZN b = ǫ1 = ǫ2−1 and obtain Cq(α1, α2, α3) = const × ZN 1 − qb−1 2bi− P3k=1(αk,ρ) partition function on S4 (ZNS4) as transformation. All the details needed are introduced in section 3. We claim that the exact AGT-W dictionary relates the Weyl-invariant structure constants C to the 4D TN where the constant part can be a function of N and of the Omega deformation parameters but cannot be a function of the masses. The partition function on S4 itself is obtained from the partition function on S4 × S1, also called the 5D superconformal index, by taking ZNS4 = const × βli→m0 β− ǫχ1Nǫ2 ZNS4×S1. in (4.24). Moreover, as far as the 5D AGT-W dictionary is concerned, we need (2.3) to set where again the constant parts can only depend on N and of the Omega deformation parameters but cannot be functions of the parameters that define the theory, i.e. the masses. The Cq are the q-deformed Weyl-invariant structure constants (3.42), Jq the q-deformation of the Weyl-covariant part of the structure constants (3.41) and ZN tion of some extra degrees of freedom (4.12) that are included in the topological string calculation but then decouple from the 5D theory. In [1] we refer to them as non-full spin content. Note that the second line of (2.10) is the same as equation (4.13), where the dec the partition funcconstant factor is explicitly written. Finally, putting (2.9) and (2.10) together, we obtain the final identification ZN (2.11) b. The above equation gives the complete relationship between the Toda 3-point structure constants and the partition functions of the TN theories. Toda 3-point functions We begin this section with a review of known facts about Toda 3-point functions of three primaries that we will need in later sections. We follow [32–34] whenever possible. We then discuss the symmetry enhancement of the Weyl invariant part of the 3-point functions as well as it’s pole structure. We conclude the section with a generalization of these facts for the q-deformed Toda. The Lagrangian of the Toda CFT theory is given by L = mental weights of SU(N ). We have collected all useful definitions and notations in in appendix B for the convenience of the reader. The parameter µ is called the cosmological the cosmological constant to its dual µ˜, defined in such a way that = πµγ (b2) b =⇒ µ˜ = The Toda CFT has a WN higher spin chiral symmetry generated by the spin k fields The central charge of the Toda CFT and the conformal dimension of the primary fields are dimension, as well as the eigenvalues of all the other higher spin currents Wk are invariant with the reflection amplitude R given by the expression Here, as in [34], we define the function Y Γ (1 − b (α − Q, e)) Γ −b−1 (α − Q, e) . The 2-point correlation functions of primary fields are fixed by conformal invariance and by the normalization (3.3). They read (2π)N−1δ(α1 + α2 − 2Q) + Weyl-reflections identifications (3.6). In this article, we shall be mostly interested in the three point functions of primary Their coordinate dependence is fixed by conformal symmetry up to an overall |z12|2(Δ1+Δ2−Δ3)|z13|2(Δ1+Δ3−Δ2)|z23|2(Δ2+Δ3−Δ1) where we have used zij := zi − zj . Due to the property (3.6), the 3-point structure constants are not invariant under 5One should not confuse the affine Weyl tranformation, i.e. Weyl reflections accompanied by two translations, with Weyl reflections belonging to the Weyl group of the affine Lie algebra. the primaries themselves. As [34], we will find it advantageous to talk about the Weyl invariant part of the 3-point structure constants. For that purpose, it is useful to define the functions6 Y as Y (α) := hπµγ (b2)b2−2b2 i− (αb,ρ) = hπµγ (b2)b2−2b2 i− 2Nb PjN=−11 αjj(N−j) N−1 N−k Y Y Υ kQ − N (αi + · · · + αi+k−1) , k=1 i=1 e6=ej functions obeys the same reflection property as the primary fields, i.e. Y (w ◦ α) = Rw(α)Y (α) . The transformation property (3.12) under affine Weyl transformation can be easily derived for reflections on the simple roots ei by noting that for any function f Y f ((Q − α, e − ej (ej , e))) = Y f ((Q − α, e)) × f (− (Q − α, ej )), a zero of order (N−1)(N−2) if we set α = κω1 or α = κωN−1. 2 Now, we can introduce the Weyl invariant part of the structure constants by dividing out the piece that transforms non-trivially under Weyl transformations. The that the Weyl invariant part of the 3-point structure constants has a higher symmetry than While the general formula for the 3-point structure constants of Toda CFT is not known, they have been computed in special cases. The formula for the structure constants of WN for the degenerate case in which one of the three weights becomes proportional to and reads7 C(α1, α2, κωN−1) = πµγ (b2)b2−2b2 Υ′(0)N−1Υ(κ) Qe>0 Υ((Q − α1, e))Υ((Q − α2, e)) QiN,j=1 Υ( Nκ + (α1 − Q, hi) + (α2 − Q, hj)) take the form Setting then one can show that the 3-point structure constants (3.15) converge to (3.9). C(α1, α2, α3) = πµγ (b2)b2−2b2 Υ(Pi3=1 αi − Q) Qj3=1 Υ(Pi3=1 αi − 2αj) which was conjecture by [6, 7] and derived by [46, 47]. Enhanced symmetry of the Weyl invariant part In this subsection, we shall make a couple of observations on the symmetries of the Weyl invariant part of the structure constants that to our knowledge are not found in the literthe Weyl invariant structure constants of the Liouville CFT X ui = 0. We observe that the above is invariant under the SU(4) Weyl group that acts as the permutation group S4 on the variables ui. We have thus uncovered the presence of a κω2 − mbω1 with m integer. We will not need it here. m1 = , n1 = , l1 = “hidden” symmetry group. In fact, in the next section, we shall find that the enhanced symmetry becomes an SO(8), see (3.29). Using [34], we know that C is invariant not only under SU(3) affine Weyl reflections of the where i, j and k are fixed and we have defined We can now make the following set of observations. First, the affine SU(3) Weyl transformi, ni and li defined via (2.4), i.e. they act as the S3 permutations. Using the parametrization (2.4), we then observe that where we refer to figure 2 for the numbering of the E6 simple roots. We observe that is the Cartan matrix of E6, if we require = 0 if k 6= l. Therefore, we have constructed the E6 root system within the space spanned by the hi(j). Furthermore, we can obtain all the variables mi, ni and li by taking the scalar products X mi = X ni = X li = 0. Therefore, the transformation (3.21) for a given choice of i, j and k acts of the variables ma, nb and lc as ma → ma − (mi − nj − l4−k) δai − 3 nb → nb + (mi − nj − l4−k) δbj − 3 lc → lc + (mi − nj − l4−k) δc,4−k − 3 where no sum over i, j, k is to be taken. We now want to interpret the new transformations (3.21) as being the result of the (non-affine) action of the Weyl group of E6. Since the Weyl group is generated by the Weyl reflections associated to the simple roots, we only need and we can build the E6 root system from them as eE6 = h(1) − h(21), eE6 = h(1) − h(31), eE6 = h(2) 2 − h(32), the m’s, n’s and l’s among themselves. However, the Weyl reflection corresponding to e3E6 transforms the weights as hi(j) 7→ e3E6 · hi(j) = hi(j) − e3E6, hi(j) eE6 3 which implies that the variables change as li 7→ −  (3) X(αj − Q), e3E6 · h4−i . X(αj − Q), e3E6 · hi(1) ni 7→ −  X(αj − Q), e3E6 · hi(2) Going through the computations, we find explicitly k can be obtained by acting with some other elE6 first. Hence, the Weyl transformations of the three SU(3) can be combined with (3.21) to generate the Weyl group of the entire E6. constants is not completely known. We shall argue in the conclusions that the enhanced Pole structure of the Weyl invariant part Q #−1 =  !!−1 1 where we used the function G(x) = Γb(x) with Υ(x) = G(x)G(Q − x) introduced in [34], see (C.11). Using the functions G allows us to see that the enhanced symmetry group is bigger than SU(4); specifically it is SO(8). The representation 8v appearing on the right SU(4). The weights h are defined as SU(4) weights and in the fundamental representation 4 they are9 hSU(4) = h(1) − h(12) + h(13), contained in C(α1, α2, α3) ∼ Z(0)3 Z(mi − nj − lk) Y Z(mi − mj )Z(ni − nj )Z(li − lj ) with the weights of 4 being the negatives of the above. i1,i2,i3=1 = F Z(mi − nj − lk) −1 −1 where Z(0)3 is just convenient normalization. We recognize in this expression the weights of Qh∈78 G The additional poles introduced in (3.32) are completely canceled by the Weyl covariant part in the formula (3.14) relating them to the 3-point structure constants, because Y Y (αk) ∝ Q3k=1 Qe>0 (Q − αk, e) Γ b (Q − αk, e) Γ b−1 (Q − αk, e) k=1 Qi3<j=1 Z(mi − mj )Z(ni − nj )Z(li − lj ) where we have used (3.11) and (C.12). The proportionality factor in (3.34) depends only covariant part, see (3.14), in order to get the full 3-point structure constants will cancel the extra poles that we introduced. 9Note that in order to get the suitably normalized scalar product for SU(4), we need to define Finally, it is compelling to conjecture that for any N the poles of the Weyl invariant structure constants should behave as Qh∈R G N2−1 Q + for an appropriate representation R of SU(N )3. The q-deformed Toda field theory One of our goals in this paper is to show how to use the topological string formalism to solve the Toda field theory. This will require a careful study of the q-deformed Toda correlation functions which topological strings naturally provide and to then learn how to take the q → 1 limit. For this purpose here we generalize some of the formulas that we discussed in the previous sections. An incomplete list of references includes [1, 18–27]. This section goes hand in hand with appendix C.2, where we define the q-deformed version of knowledge these formulas do not exist in the literature. We begin by stressing some defining properties that all the q-deformed formulas • They must reproduce the exact undeformed formula in the q → 1 limit. With no further prefactor, unless stated otherwise. That will be the case of the Cq (3.36). • They must have exactly the same symmetries and transformation properties as the undeformed ones under the (affine) Weyl, as well as the enhanced symmetry group. • They must have their poles and zeros in the same place with the undeformed ones. To be more precise, the q-deformed functions have more zeroes/poles, specifically a whole tower of zeroes/poles for each zero/pole of the undeformed function as discussed in (C.31). The tower is generated by beginning with the undeformed zero/pole and We moreover want to stress that the q-deformed version of Toda field theory does not have a known Lagrangian description. Everything is defined algebraically in analogy to the usual case via a deformation of the WN algebra, see [48] and references therein. Since no Lagrangian description is known for the q-deformed Toda field theory, we can compute everything up to overall factors that in the q → 1 limit give the cosmological constant. structure constants Obviously, after the q → 1 limit is taken and the undeformed answer is obtained, it is clear obtaining the full result with all the factors. As we already said in section 3.1 the Weyl invariant part C is independent of the cosmological constant and thus it’s q-deformed version should be straightforward. However, the Weyl covariant part with which we need to multiply in order to obtain the full Cq will version of the functions Y defined in (3.11) is " 1 − q b 2b−1 1 − q (1 − q)2Q behaves under affine Weyl transformations as with the q-deformed version of the reflection amplitude Yq(w ◦ α) = Rqw(α)Yq(α) being composed out of so that like in (3.14) Note that also the q-deformed version of the reflection amplitude in the q → 1 limit gives order to get the Weyl invariant structure constants is " 1 − q (1 − q)2(1+b2) Y Yq(αi) = const × Y Yq(αi), (3.41) The q-deformation version of the Fateev and Litvinov formula (3.15) for the 3-point correlation functions with one degenerate insertion reads 1 − q b−1 2b2 ! (1 − q)2(1+b2) QiN,j=1 Υq( Nκ + (α1 − Q, hi) + (α2 − Q, hj )) This formula to our knowledge does not appear anywhere else in the literature. We write it down as the unique formula that has the properties mentioned at the beginning of the section. First, it has its poles and zeros in the correct positions, see (C.5). Second, it has the correct covariance properties under the affine Weyl symmetries of the non-degenerate formula (up to the µ dependence) 1 − q b−1 2b2 ! (1 − q)2(1+b2) × Υq(Pi3=1 αi − Q) Qj3=1 Υq(Pi3=1 αi − 2αj ) , (3.44) derived in [40]. From it we can extract the q-deformed version of the Weyl invariant part using equation (3.42), Υq(Pi3=1 αi − Q) Qj3=1 Υq(Pi3=1 αi − 2αj ) Qi4=1 Υq(ui + Q2 ) , (3.45) which immediately gives the he correct undeformed C as it is in equations (3.17) and (3.20) with no further factors. N partition function from topological strings In this section we introduce the formula for the 5D TN partition functions that we computed in [1] and we discuss how they can be brought to a form that allows us to take the 4D limit crucial, we begin by carefully discussing it and the way it is read off from the web diagrams. Some details of the computations are presented in appendix D. The TN theories are isolated strongly coupled fixed points that one can discover by linear quivers. The calculation of the TN partition function is not possible using any purely field theoretic method currently known, because the TN theories have no known Lagrangian description. The only applicable method is string theory and in particular 5-brane webs [49, 50] from which the answer is derived using topological strings. The 5-brane webs bedded in string theory by using type IIB (p, q) 5-brane webs [49, 50]. All the information needed to describe the low energy effective theory on the Coulomb branch is encoded in the web diagrams, through which the 5D SW curves can be easily derived [22, 49–51]. Furtherin particular M-theory compactified on Calabi-Yau threefolds. This alternative description provides an efficient way of computing the Nekrasov partition functions of the gauge theories by computing the partition functions of topological strings living on these backgrounds. Recent reviews on the subject can be found in [54, 55]. In particular, the dual to the Calabi-Yau toric diagram is exactly equal to the web diagram of the type IIB (p, q) 5-brane systems [56]. The SW curves and the Nekrasov partition functions are parametrized by the Coulomb parameters are encoded in the web diagrams as follows. On the one hand, deformations of the webs that do not change the asymptotic form of the 5-branes correspond to the Coulomb moduli a and their number is the number of faces of the web diagram. On the other hand, deformations of the webs that do change the asymptotic form of the 5-branes correspond to parameters that define the theory, namely masses and coupling constants and they are equal to the number of external branes minus three. Note that at each vertex there is a no-force condition (D5/NS5 (p, q) charge conservation) that serves to preserve 8 supersymmetries. Having said all the above, we can now return to the TN theories. The first step towards being able to calculate the TN partition functions was taken by Benini, Benvenuti and Tachikawa, who gave in [37] the web diagrams of the 5D TN theories. Subsequently, in [1] we tested their proposal by deriving the corresponding SW curves and Nekrasov partition Most importantly, we were able to cross-check our results for the partition functions against the 5D superconformal index that was recently calculated in [41]. For similar work see also [5, 42]. We now turn to the parametrization of the TN web diagrams. The general parametrization in contained in the appendix, see figure 6 and here we just give a short introduction. We have one parameter ai(j) for each face, or hexagon, of the diagram, that will also appear as A˜(j) = e−βai(j). They can be thought of as Coulomb moduli that will be integrated over i and are called breathing modes. The number of faces in the web diagram of the TN theory exterior flavor branes for the branes on the, respectively, left, lower and upper right side of the diagram. From them, we define the fugacities that are subject to the relation Y M˜ k = Y N˜k = Y L˜k = 1 ⇐⇒ X mk = X nk = X lk = 0. From the mass parameters, we also define the “boundary” Coulomb parameters. They are of the flavor branes in (A.1). In the dual, geometric engineering description, the parameters above correspond to the K¨ahler parameters of the Calabi-Yau threefold. On the web diagram, the K¨ahler brane positions. A straightforward computation gives the nine K¨ahler parameters of the web diagram as “mass” parameters are shown in red, the “face” moduli in blue and the “edge” ones in black. parameters correspond to the horizontal, the diagonal and the vertical lines and are labeled by Q(nj;)i, Q(mj);i and Ql(;ji) respectively. They are derived quantities through the equations (A.4) and are useful because they are the ones that enter in the computation of the partition function via the topological vertex. In order to familiarize the reader with the parametrization, we shall illustrate the K¨ahler parameters obey the relation Q(m1;)1 = A−1M˜ 1N˜1, Q(n1;1) = AM˜ 1−1N˜2−1, Ql(;11) = AM˜ 2−1N˜1−1, Q(m1;)2 = AM˜ 2L˜3, Q(n1;2) = AM˜ 3L˜2, Ql(;12) = A−1M˜ 3−1L˜3−1, Q(m2;)1 = AN˜2L˜1−1, Ql(;21) = AN˜3L˜2−1. It is easy to check that the above solutions obey the set of equations (A.6) relating them to the brane position parameters and that they furthermore satisfy the two constraints coming from matching the height and widths of the hexagon of figure 3 Q(m1;)2Ql(;11) = Q(m2;)1Ql(;21). The topological vertex computation Now that we have gained some understanding of the parametrization of the TN web diagram, we would like to compute its refined topological string amplitude. For this, we dissection of the TN diagram into N strips. The partitions associated with the horizontal, diagonal Ka¨hler parameters of the horizontal, diagonal and vertical lines are Q(nj;)i , Q(mj);i, Ql(;ji) respectively with the same range of indices. use the refined topological vertex, choose the preferred direction to be the horizontal one and cut the toric diagram diagonally into sub-diagrams called strips. The calculation was carried out in [1], here we just reproduce the results for the reader’s convefigure 4. The corresponding partition function depends on the external horizontal partiand Ql = (Ql;1, . . . , Ql;L). It takes the form top = ZN full topological string partition function is then given by ZN top = X Zν(r−1)ν(r) (Q(mr), Ql(r); t, q). The strip partition function (4.6) was computed in [1]. In appendix D, we show that it is useful to redefine the strip slightly, i.e. to “cut” the TN junction in a different way by moving some factors from one strip to its neighbors. These redefinitions do not change the full topological string partition function of the TN junction. The technical details are left to appendix D. Combining everything, we obtain where we have defined the “perturbative” partition function pert := N−1 N−r r=1 i≤j=1 M and the “instanton” one q t A˜i(r−1)A˜(r) ˜(r−1) q A˜(r) A˜(r) ZN inst := X N N−r N˜rL˜N−r − a(jr) − ǫ+/2 − ai−1 − aj+1 (r) (r−1) − ai−1 − aj+1 − ǫ+/2  where the ai(j) are defined via A˜(j) = e−βai(j) . i we have introduced the notation10 “instanton” inside quotation marks because for the TN there is not really a notion of instanton expansion. There is no coupling constant, since there is no gauge group. We recall that the boundary ai(j) are related to the masses via (A.1). In writing (4.9) and (4.10) We put the words “perturbative” and M(u; t, q) ≡M(u) = Y (1 − ut−iqj ), m + ǫ1(λi − j + 1) + ǫ2(i − µ tj ) m + ǫ1(j − µ i) + ǫ2(λtj − i + 1) . We refer to appendix C.2, respectively C.3 for more details concerning M, respectively As in [1], we define the non-full spin content (also called U(1) factor in [5]) ZNdec := M(M˜ iM˜ j−1)M(t/qN˜iN˜j−1)M(L˜iL˜j−1). (1 − q) N(N−1)(2N2−2N−1) Q2 Y (1 − q)N(αk,αk−2Q) 1 − q 1 − q 10We often drop the explicit dependence of these functions on the parameters t and q. b 2b−1 top = Each one is labeled by the Ka¨hler moduli of the non-full spin content that is factorized. We also indicate the names of the partitions entering the instanton sums and to avoid clutter, we only do it for the middle one. From equations (4.9) and (4.10), we can read off the partition function for the M˜ choice. After some rearrangements, we find the cumbersome expression for Z3 ≡ Z3,M˜ A2N˜1−1L˜3 qt A2N˜1−1L˜3 q qt A−1M˜ 1N˜1 q qt AN˜3L˜2−1 q qt AM˜ 1L˜3 q qt AM˜ 2L˜3 q qt AN˜2L˜1−1 M˜ 1M˜ 2−1 M˜ 1M˜ 3−1 M˜ 2M˜ 3−1 Q3k=1 ν1(1)∅ (a − n1 − mk − ǫ+/2) Nβ Nβ ∅ν2(1) (a + l3 + mk − ǫ+/2) We can also compute the topological string partition function for the choice N˜ or L˜ of the preferred direction. A straightforward computation as in [1], which we skip here, shows that the topological amplitude Z3to,Np˜ for the choice N˜ of preferred direction can be obtained from equation (6.1) by simply making the substitutions mk → n4−k, lk → −l4−k, and exchanging t ↔ q. Furthermore, the amplitude Z3to,L˜p for the last remaining possible It is thought and in some cases shown [39, 57] that the choice of preferred direction is irrelevant. Thus, we may assume that the transformations (6.2) and (6.3) are symmetries of the topological amplitude and conjecture that (6.1) is invariant under them: Checking this conjecture turns out to be quite involved since it requires computing the sums relating the topological string amplitude to the Toda structure constants, we only need a much weaker statement that is easier to check, namely that Z3top(n4−k, mk, −l4−k) Z3top(−l4−k, −m4−k, nk) . (6.6) choice of preferred direction is obtained by setting in (6.1) mk → −l4−k, nk → −m4−k, without exchanging t ↔ q. We can express this succinctly as Z3to,Np˜ (mk, nk, lk) = Z3 top(n4−k, mk, −l4−k) , Z3to,L˜p(mk, nk, lk) = Z3top(−l4−k, −m4−k, nk) . We now shall perform a direct check of the above relations. We first observe (4.22) that 2πiA |Z3top|2 = |M (t, q)|−2 conformal index Z3 computed in [41], that reads The decoupled part Z3 dec was defined in (4.12) and we easily find that |Z3 under the transformations (6.2) and (6.3) if we use the symmetries of the functions M, see (C.24), as well as the definition (4.16) of the norm squared. Since M (t, q) is independent of the fugacities, we only need to check that the super S4×S1 is invariant under (6.2) and (6.3). For this, we expand in powers of x = e The fugacities M˜ , N˜ and L˜ enter the E6 characters as follows. We embed SU(3)3 into E6 so that the character of the 78-dimensional adjoint representation of E6 decomposes as the character χSU(3) = M˜ 1 + M˜ 2 + M˜ 3 with similar expressions for the other SU(3) factors. 3 The other characters can be decomposed in a similar fashion, see appendix C of [1] for more details. One easily checks that (6.2) acts on the SU(3) characters as χSU(3)(M˜ ) 7→ χSU(3)(N˜ ) , 3 3 χSU(3)(N˜ ) 7→ χSU(3)(M˜ ) 3 3 χSU(3)(L˜) 7→ χS¯U(3)(L˜) . 3 3 E6 characters appearing in (6.8) are invariant as well and the same holds for the transformation (6.3). Hence, we are confident that for the Toda structure constants (6.2) and (6.3) hold, making our proposal (2.11) independent of the choice of slicing. The weaker form (6.6) of the slicing invariance conjecture for the partition function can help us prove the Weyl covariance of the structure constants. Using (6.1) and the S4×S1 = S4×S1 = dA (1 − q) ǫ1ǫ2 Υq(ǫ1) hQi3<j=1 Υq(ni − nj )Υq(lj − li)i−1 Q3k=1 Υq(a − mk − n1 + ǫ+/2)Υq(a + mk + l3 + ǫ+/2) × Υq(a+n3 −l2 +ǫ+/2)Υq(a+n2 −l1 +ǫ+/2) Q3k=1 ν1(1)∅ (a − n1 − mk − ǫ+/2) Nβ Nβ ∅ν2(1) (a + l3 + mk − ǫ+/2) where we the exponent of (1 − q) is − N (nj − ni)2 + (li − lj )2 − 2 X nil4−i + nj − ni + + li − lj + |M (t, q)|2. In deriving expression (6.12), we have used (4.15) and (4.18). Now the invariance of Z3 the mi, ni and li, i.e. they simply permute them. For the choice M˜ of preferred direction shown in (6.12), we can easily see that the expression is invariant. However, while the invariance of (6.12) under the Weyl group of the first SU(3) is easy, the Weyl reflections of the remaining Weyl groups act non-trivially. At this point we need to use the fact that slicing invariance is a symmetry of the problem and by applying first (6.2) or (6.3) on (6.12) before acting with the Weyl reflection we can prove the complete invariance under Weyl reflections. stants and since we showed in section 3.2 that the Weyl invariant structure constants have an E6 symmetry, we have an additional piece of evidence in our favor. Furthermore, we can use the fact that (3.32) captures all the poles of the Weyl invariant structure constants and that the position of the poles does not change under q-deformation to write another formula for the index.15 Specifically, we make a guess for the q-deformation of (3.32) and write S4×S1 = Since we wish to identify the index as the q-deformed Weyl invariant structure conZq(u; t, q) := Y (1 − ut−i−1qj+1)(1 − u−1t−i−1qj+1) entire16 function with the expansion in x given by is up to a constant the q-deformation of Z and the compensating factor F3 is an unknown F3 =1 + χ2(y)x3 + hχ3(y) − χ6E560i x4 + hχ4(y) − So far, we have no closed expression for the function F3. We end this section with one last remark. Our claim (2.8) states that C(α1, α2, α3) = const × βli→m0 β−χ3 Z3S4×S1 . We see in (6.8) that Z3 S4×S1 is invariant under an E6 symmetry and we saw in section 3.2 (mi − mj )2 + (nj − ni)2 + (li − lj )2 is invariant under the E6 Weyl tranformations (3.24) as well. 15To be more precise, as we discussed in section 3.4 and can be seen in equation (C.5) of the appendix C.2, after the q-deformed versions of the functions have more poles. For for each single pole of the undeformed function, they have a whole tower of poles. 16That the function F3 is entire follows from the facts that 1) the function F in (5.5) of [34] is entire and 2) the Weyl covariant part has no poles. Conclusions and outlook In [1] we calculated the 5D partition function of the non-Lagrangian TN theories on S4 × S1 using topological strings. In this paper we take the next very important step and argue → 1), thus obtaining the partition function of the 4D non-Lagrangian TN theories on S4. Taking the 4D limit is not as simple as one might naively think and it is definitely not as easy as for theories with a Lagrangian description. The first step in overcoming this difficulty was realizing that one can bring formula (4.7) and the “Nekrasov functions” Nβ for which the 4D limit is well defined as individual µν functions (C.29) (C.42). Our formula for the partition function (4.22) is then written as a product of the factors17 ZNpert/ZNdec 2 inst 2. The first factor ZNpert/ZNdec 2 (1 − q)χ′N/ǫ1ǫ2 . However, for the ZN inst 2 piece we have a further obstacle to overcome. The in contrast with the usual sums in theories with Lagrangian description. Schematically, description, where one can commute the limit with the sum, as for example in for the case of the TN theories (that are isolated non trivial fixed points) there is no of qUV e−βx |µ | Nµνβ 1 (a1) · · · Nµνβ L (aL) β−→→0 βpower in (4.13) the decoupled part ZN where the sum is given by (C.40). However, by carefully studying the symmetry properties against the q-deformed version of the Fateev-Litvinov formula with one semi-degenerate insertion (3.43). Combining everything, we propose that the 4D limit of the superconforequal to the partition function of the TN theory on S4. Moreover, we explicitly computed and it is finite after extracting a divergent factor of 17As we already stress in the main text, using the worlds “perturbative” and “instanton” is an abuse of Via the AGT-W correspondence, we translate our formula for the TN partition function to the 3-point structure constants of three generic primaries of the Toda field theories, both for the undeformed (4D AGT-W) as well as for the deformed (5D AGT-W) WN algebra. We give explicitly the parameter identification from the topological string parameters to the gauge theory ones in appendix A and then to the 2D Toda parameters in equations (2.4). We identify the 3-point structure constants of the Toda CFT with the topological string partition function in (2.11). A very nice byproduct of our work is our ability to give the knowledge do not appear in the literature. This discussion appears in appendix C.2. Moreover, we identified in (2.10) Cq, the Weyl invariant part of the q-deformed 3-point structure constants, with the 5D superconformal index Z object.18 This identification allows us to predict that the Weyl invariant part of the qdeformed 3-point structure constants should have not just SU(N)3 symmetry but also an S4×S1 , a powerful gauge theory extended symmetry as predicted by [67–69] due to the existence of non-trivial UV fixed points for the 5D gauge theories. We have explicitly checked in the Liouville case that the Weyl invariant part of the DOZZ formula (3.20) enjoys SU(4) enhanced symmetry, and Checking that the Weyl invariant 3-point structure constants for higher N enjoy some, other than just the Weyl group of SU(N)3, enhanced symmetry is an important future direction.19 The formula we give for the 3-point functions at this point is very implicit, since there are still integrals and sums that need to be performed. In a separate publication [43], we show how at least some of the sums and integrals can be performed. In so doing, we are able to explicitly reproduce the formula (3.15) of Fateev and Litvinov [32–34] for degenerate fields. Furthermore, our formulas predict highly nontrivial relations for the sums in the semi-degenerate case. In this paper we give the Toda 3-point functions with three primaries, which however is not enough to solve Toda. To achieve that, we need to also compute the correlation functions of descendants, which as we discussed in the introduction is not as immediate as in the Liouville case. However, it is straightforward to see from the point of view of the topological strings what needs to be done in order to compute them. Specifically, we need 18The superconformal index in any dimension is the partition function of protected operators and is independent of the coupling constants of the theory, implying that it remains invariant under S-duality. In 4D the superconformal index S 1 was proven to be equivalent to a 2D TQFT [61–65]. It is very possible that something very similar will also be proven for the 5D superconformal index S progress in this direction) and thus the Weyl invariant part of the q-deformed 3-point function could be discovered to obey special properties not visible from the CFT point of view, but realized only once one is using the superconformal index interpretation. 19The authors of [5] were able to discover that some specialization (called “Higgsing” in the gauge theory jargon) of the parameters in the T4 theory leads to an E7 symmetry, while in [42] a similar specialization of the parameters in T6 leads to an E8 symmetry. These specializations change the TN geometry significantly and in particular reduce the number of Coulomb moduli to one. It would be very interesting to see the meaning of this on the CFT side. to take the TN web diagrams from figure 4 and evaluate them with the refined topological vertex without putting empty Young diagram to the external legs. This will provide the general Ding-Iohara algebra interwiners. The Ding-Iohara algebra [70] in the free boson representation (with N free bosons) is known to become A = WN ⊗ H where H is the Heisenberg algebra which is exactly the algebra that is needed to describe what is obtained from AGT-W [11, 14]. In particular, it is quite easy to obtain the 3point function of two primaries and one descendant and in fact the answer is just (4.8) to give us, via bootstrapping many higher point functions. Solving this problem is work in progress [45]. We would like to finish by remarking that for many reasons it seems to be much more advantageous to study the q-deformed version of the Toda field theories instead of the undesince for example the product formula (C.26) is much simpler that (C.17) and (C.10). Furthermore, in the q-deformed case, we can use the topological string formalism to compute the partition functions, tools that are not directly available in the undeformed theory. Acknowledgments We would like to thank first our collaborators on closely related projects Mikhail Isachenkov, Masato Taki and Futoshi Yagi. In addition, we are thankful to Sara Pasquetti, Paulina Suchanek and J¨org Teschner for insightful comments and discussions. Finally, we are very grateful to Sylvain Ribault for making several useful comments on our draft. We thank CERN, the CERN-Korea Theory Collaboration funded by the National Research Foundation (Korea) and the C.N.Yang Institute for Theoretical Physics (Stony Brook) for their hospitality during the finishing stage of this work. EP is partially supported by the Humboldt Foundation, the Marie Curie grant FP7-PEOPLE-2010-RG. The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme FP7/2007-2013/ under REA Grant Agreement No 317089 (GATIS). Parametrization of the TN junction We gather in this appendix all necessary formulas for the parametrizations of the K¨ahler are independent, while the “border” ones are given by horizontal lines as Q(nj;)i, to the vertical lines as Ql(;ji), and to tilted lines as Q(mj);i. We denote the breathing modes as A˜i(j). The index j labels the strips in which the diagram can be decomposed. The parameters labeling the positions of the flavors branes obey the relations Y M˜ k = Y N˜k = Y L˜k = 1 ⇐⇒ X mk = X nk = X lk = 0. Therefore, A˜(0) = A˜(N0) = A˜(N) = 1 and we can invert relation (A.1) as 0 0 M˜ i = N˜i = A˜(i−1) L˜i = ˜(N−i) Ai−1 ˜(N−i+1) All placements are illustrated in figure 6. Q(nj;)i = The K¨ahler parameters associated to the edges of the TN junction are related to the A˜(j−1)A˜(j+1) Ql(;ji) = A˜(j)A˜(j−1) Q(mj);i = A˜(j−1)A˜(j) A˜(j)A˜(j−1) i−1 . For each inner hexagon of (6), the following two constraints are satisfied Ql(;ji)Q(mj);i+1 = Q(mj+;i 1)Ql(;ji+1) Q(nj;)iQ(mj+;i 1) = Q(mj);i+1Q(nj;)i+1. Furthermore, we find the following identities relating them to the masses: Q(m1;)iQl(;1i) = Q(mi);1Q(ni;)1 = (N−i)Ql(;Ni−i) = Using the above, we find the following expressions for the products appearing in the TN partition function: Y Ql(;rk)Q(mr);k = Ql(;rk)Q(mr);k = Y Ql(;rk)Q(mr);k+1 = Y Y Ql(;rk)Q(mr);k = Y k=i A˜(kr−)1 A˜(r−1) 2 k=i A˜(r−1)A˜(r−1) k−1 k+1 j−1 A˜(r−1)A˜(r−1) i j−1 A˜(r)A˜(r−1) A˜(r−1)A˜(r−1) i i i+1 j A˜i(−r)1A˜(r−1) A˜(r−1)A˜(r−1) i+1 i j+1 A˜i(−r)1A˜(r) Furthermore, the following two follow directly from (A.6) and are used in the derivation of the “perturbative” part of the topological string partition function (4.9) Y Q(mr);j = Y Ql(;rk) = A˜i(−r)1A˜(Nr−−1r)+1 Conventions and notations for SU(N ) product is defined via (hi, hj) = δij − N 1 . The simple roots are The purpose of this appendix is to summarize our SU(N ) conventions. The weights of the ek := hk − hk+1, k = 1, . . . , N − 1, and the positive roots e > 0 are contained in the set N N−1 N−2 Δ+ := {hi − hj}i<j=1 = {ei}i=1 ∪ {ei + ei+1}i=1 ∪ · · · ∪ {e1 + · · · + eN−1}. X e = X (hi − hj) = X N + 1 − 2i as well as − j, After some work, one can prove using the scalar products (B.5) and (B.6) that i(N − i) min(i, j) (N − max(i, j)) 1 X (α1, e) (α2, e) = (α1, α2) , ( 1 − Ni j ≤ i − N N (N 2 − 1) i = 1, . . . , N − 1 and the corresponding finite dimensional representations are the i-fold antisymmetric tensor i.e. they are a dual basis. Furthermore, we find the following scalar products useful with respect to the simple root ei wi · α := α − 2 (ei, ei) ei = α − (ei, α) ei. Furthermore, we define the affine Weyl reflections with respect to ei as follows where Q := Qρ = (b + b−1)ρ. Special functions functions used in the main text. For the reader’s convenience, we gather here the definitions and properties of all special The purpose of this part of the appendix is to summarize the known identities for the functions used in the undeformed Liouville and Toda CFT. We begin with the function Z ∞ dt  e−t 2 − x 2t i  It is clear from the definition that = 1. One can show from the alternative definition below that the following shift identities are Υ(x + b−1) = γ(xb−1)b2xb−1−1Υ(x). where γ(x) := Γ(Γ1(−x)x) . An useful implication is Υ(x + Q) = b2(b−1−b)x Γ 1 + bx Γ b−1x which is used in the derivation of the reflection amplitude (3.7). It follows from (C.3) that x = −n1b − n2b−1, x = (n1 + 1)b + (n2 + 1)b−1, where ni ∈ N0. ℜ(t) > 2) of From this definition, one can prove (see A.54 of [71]) the difference property first define the normalized function Z ∞ dt e−xt − e− Q2t t  (1 − e−tb)(1 − e−tb−1) − 2 − x e−t interest is the function G(x) introduced in [34] with the properties b1/2−bx bb−1x−1/2 G(x + b) = G(x + b−1) = bb−1x−bx Z(x) = G(Q + x)G(Q − x) = working, one has to use (A.62) of [71]. Specifically, we set for ℜ(s) > 2 n1,n2≥0 (ω1n1 + ω2n2)s poles. We have the residues n=0 (q + n)s n1,n2≥0 and the finite parts s − 1 s − 2 = − i Z ∞ ψ(i ωω12 y + 1) − ψ(−i ωω21 y + 1) − 1 − 1 i Z ∞ ζH (2, i ωω12 y + 1) − ζH (2, −i ωω21 y + 1) First we begin by defining the shifted factorials20 (we require for convergence that |qi| < 1 for all i) (x; q1, . . . , qr)∞ := i1=0,...,ir=0 (1 − xq1i1 · · · qrir ). (x; q1, . . . , qi−1, . . . , qr)∞ = (xqi; q1, . . . , qr)∞ We can extend the definition of the shifted factorial for all values of qi by imposing the Furthermore, they obey the following shifting properties (qj x; q1, . . . , qr)∞ = (x; q1, . . . , qr)∞ (x; q1, . . . , qj−1, qj+1, . . . , qr)∞ We then define the function M(u; t, q) as Qi∞,j=1(1 − ut−iqj )  Qi∞,j=1(1 − uti−1qj )−1 for |t| < 1, |q| < 1 for |t| < 1, |q| > 1 for |t| > 1, |q| < 1 converging for all u. This function can be written as a plethystic exponential M(u; t, q) = exp " ∞ m=1 m (1 − tm)(1 − qm) Here and the definition M(u; q, t) = M(ut/q; t, q). From the analytic properties of the shifted factorials (C.19), we read the identities M(u; t−1, q) = M(ut; t, q) M(u; t, q−1) = M(uq−1; t, q) while from (C.20) we take the following shifting identities M(ut; t, q) = (uq; q)∞M(u; t, q), M(uq; t, q) = (uq; t)∞M(u; t, q). Υq(x|ǫ1, ǫ2) =(1 − q)− ǫ11ǫ2 (x− ǫ2+ )2 =(1 − q)− ǫ11ǫ2 (x− ǫ2+ )2 20A good source for the properties of the shifted factorials is [72]. n1,n2=0 (1 − qx+n1ǫ1+n2ǫ2 )(1 − qǫ+−x+n1ǫ1+n2ǫ2 ) (1 − qǫ+/2+n1ǫ1+n2ǫ2 )2 M(q−x; t, q) where we have used the definition (4.16) for the norm squared. If follows from the definition 1 − q 1 − qǫ2 1−2ǫ2−1x together with a similar equation for the shift with ǫ2. Here, we have used the definition of Γq(x) := (1 − q)1−x (q; q)∞ , (qx; q)∞ = (1 − q)1−2x (q1−x; q)∞ (qx; q)∞ of (C.27) have a well defined limit for q → 1, we find by comparing functional identities Γ2 x|ǫ1, ǫ2 Γ2 ǫ+ − x|ǫ1, ǫ2 for the derivation of the reflection amplitude (3.39) " 1 − q (1 − q)Q Γqb−1 (1 − bx)Γqb (−b−1x) Υq(x), which reduces to (C.4) in the limit q → 1. We finish this part of the appendix with two x = −n1ǫ1 − n2ǫ2 + x = (n1 + 1)ǫ1 + (n2 + 1)ǫ2 + where ni ∈ N 0 and m and m′ are integer. We thus see by comparing with (C.5) that are q-independent. The tower of zeros is obtained by beginning with the q-independent in the numerator of (C.26), we find that the only piece of the derivative that survives is 1 − q i=1 j=1 1 − Qti−1qj The finite product functions In this subsection ǫ1 and ǫ2 are general. In the definition of the topological string amplitudes, we often need to use the following two functions given by finite products Y (1 − Qtµ tj−iqλi−j+1) Y (1 − Qt−λtj+i−1q−µ i+j). From [1, 40] we take the following summation formula q qt Q1 N∅µ In the 4D limit, it is often useful to use the rescaled N functions that we refer to as ||µ t||2 −4||λt||2 q ||λ||2−4||µ ||2 Nβλµ (m; ǫ1, ǫ2), where, using the parametrization (2.1), the new functions are given by m + ǫ1(λi − j + 1) + ǫ2(i − µ tj) allowing us to express the product of two Z˜µ through t a delta function, for instance Nλ∅( q ) = N∅λ(1) = δλ∅. Furthermore, we find a relation Nµ (1; t, q) = − ||µ t||2 Z˜µ (t, q)Z˜µ t(q, t) Using the identities X i = X µ i = we find the exchange identities Nλµ (Q; q, t) = Nµ tλt(Q ; t, q), −Q−1r q |λ|+|µ | −||λt||2+||µ t||2 ||λ||2−||µ ||2 t 2 q 2 m + ǫ1(j − µ i) + ǫ2(λtj − i + 1) . The new function obeys the simpler exchange identities as well as the summation formula m2 − ǫ2+ Nβλµ (m; −ǫ2, −ǫ1) = Nµβtλt(m − ǫ1 − ǫ2; ǫ1, ǫ2), Nβλµ (−m; ǫ1, ǫ2) = (−1)|λ|+|µ |Nµβλ (m − ǫ1 − ǫ2; ǫ1, ǫ2), Nβλµ (m; ǫ2, ǫ1) = Nβλtµ t(m; ǫ1, ǫ2), M e−β(m1+m2+m3− ǫ2+ ) . (C.40) where we have defined m + ǫ1(λi − j + 1) + ǫ2(i − µ tj) m + ǫ1(j − µ i) + ǫ2(λtj − i + 1) . Computation of the TN partition function In this part of the appendix, we wish to put together the computations that bring us from equations (4.6) and (4.7) to (4.8), (4.9) and (4.10). We define the function 1 − Qti− 12 −λj qj− 21 −µ i = M(Q ; t, q), (D.1) and, after using some Cauchy identities, we rewrite (4.6) as equation (4.67) of [1]: Zνstτrip(Qm, Ql; t, q) = Y t 2 Rνitτj Qm;j Qjk−=1i Qm;kQl;k Rτitνj+1 Ql;i Qk=i+1 Qm;kQl;k j Y Ql;kQm;k+1 r t −1 Qk=i Ql;kQm;k t partition function, The complete TN diagram is made out of N such strips, as depicted in figure 4 and written can redefine the strip partition function without affecting the topological string partition function (4.7), by moving half of the Z˜ from one strip to the another. Specifically, we move for the strip on the left. This redefinition doesn’t change ZN , since the partitions to the extreme left of the TN diagram are all empty. Putting it all together, we get a new strip strip ′(Qm, Ql; t, q) = Y t 2 Rνitτj Qm;j Qjk−=1i Qm;kQl;k Rτitνj+1 Ql;i Qk=i+1 Qm;kQl;k j Y Ql;kQm;k+1 r t −1 Qk=i Ql;kQm;k We can get rid of the Z˜ functions using (C.34). Putting things together in the products and replacing the R functions by using (D.1), we get Qk=i Ql;kQm;k We can straightforwardly obtain (4.9) by taking only the M dependent terms of the strip partition functions, plugging them in (4.7) and replacing the K¨ahler parameters Qm, and Ql by the A˜’s using the formulas (A.8) of appendix A. Thus, we get the “perturbative part” of the topological string TN partition (4.9), i.e. the part that is independent of the partitions entering the sum. of the factors from one strip to the one standing on its left, which implies the following strip ′(Qm, Ql; t, q) = we can write the “instanton” part of the redefined strip as strip, inst ′′(Qm, Ql; t, q) = Y(−1)|τk| Y let us remark that i=1 j=1 = Y  2 L+1 i−1 i=1 j=1 Armed with (D.6), we can compute (4.10). We have inst = Y − Q(nr) |ν(r)|Zνst(rri−p1,)iνns(rt)′′(Q(mr), Ql(r); t, q). 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Vladimir Mitev, Elli Pomoni. Toda 3-point functions from topological strings, Journal of High Energy Physics, 2015, 49, DOI: 10.1007/JHEP06(2015)049