Abstract
We study the discrete flows generated by the symmetry group of the BPS quivers for Calabi–Yau geometries describing fivedimensional superconformal quantum field theories on a circle. These flows naturally describe the BPS particle spectrum of such theories and at the same time generate bilinear equations of qdifference type which, in the rank one case, are qPainlevé equations. The solutions of these equations are shown to be given by grand canonical topological string partition functions which we identify with \(\tau \)functions of the cluster algebra associated to the quiver. We exemplify our construction in the case corresponding to fivedimensional SU(2) pure super Yang–Mills and \(N_f=2\) on a circle.
Introduction
A crucial problem in quantum field theory (QFT) is understanding its nonperturbative aspects in the concrete terms of exact computations. QFTs can be embedded in string theory/Mtheory via geometric engineering [1]. Specifically, it can be obtained as the low energy limit of a compactified string theory in a large volume limit, which is needed to decouple its gravitational sector. When QFT is obtained in this way, the nature of exact dualities gets unveiled through the geometric properties of the string theory background behind it: the string theory on the noncompact Calabi–Yau (CY) background geometry encodes the spectral geometry of integrable systems whose solutions allow to obtain exact results. This is possible because the nonperturbative sector of string theory, described by Dbranes, gets transferred through this procedure to the geometrically engineered QFT. The setup engineered by Mtheory compactification on \(CY_3\times S^1\), in the limit of large \(CY_3\) volume and finite \(S^1\) radius, is that of a fivedimensional supersymmetric QFT on a circle, whose particles arise from membranes wrapped on the 2cycles of a suitable noncompact CY manifold. As such, the counting of the BPS protected sectors of the theory can be obtained by considering a dual picture given in terms of a topological string on \(CY_3\). The precise dictionary between the two descriptions is obtained by identifying the topological string partition function on the \(CY_3\) with the supersymmetric index of the gauge theory, which is conjectured to capture the exact BPS content of its 5d SCFT completion [2]. More generally, the supersymmetric index of the gauge theory with surface defects is matched with the corresponding Dbrane open topological string wave function [3, 4]. The coupling constants and the moduli of the QFT arise from the geometric engineering as CY moduli parameters (Kähler and complex in the A and Bmodel picture, respectively). Therefore, the QFT generated in this way is naturally in a generic phase in which all the coupling constants can be finite. To identify the weakly coupled regimes, one has to consider particular corners in the CY moduli space. In such corners, the topological string theory amplitudes allow a power expansion in at least one small parameter which is identified with the gauge coupling, while the others are fugacities of global symmetries of the QFT (masses and Coulomb parameters).
The problem we would like to face, in the general setup described so far, is that of understanding how to predict the properties of such a supersymmetric index, given the noncompact CY manifold which realizes the fivedimensional theory via geometric engineering. We will show that this index satisfies suitable qdifference equations which in the rank one case, namely for Calabi–Yau whose mirror is a local genus one curve, are well known in the mathematical physics literature as qPainlevé equations [5]. These are classified in terms of their symmetry groups as in Fig. 1. Remarkably, this classification coincides with the one obtained from string theory considerations in [6]. This allows to describe the grand canonical partition function of topological strings as \(\tau \)functions of a discrete dynamical system, whose solutions encode the BPS spectrum of the theory. From this viewpoint, the grand canonical partition function is actually vectorvalued in the symmetry lattice of the discrete dynamical system at hand. The exact spectrum of the relevant integrable system can be computed from the zeroes of the grand partition function.
The solutions of the discrete dynamical system are naturally parameterized in different ways according to the different BPS chambers of the theory. We will show that the Nekrasov–Okounkov [7] presentation of the supersymmetric index can be recovered in the large volume regions of the Calabi–Yau moduli space which allow the geometric engineering of fivedimensional gauge theories. The expansion parameter is schematically \(e^{V}\), V being the volume of the relevant cycle corresponding to the instanton counting parameter. Around the conifold point, the solution is instead naturally parameterized in terms of a matrix model providing the nonperturbative completion of topological string via topological string/spectral theory correspondence [8]. The case of local \({\mathbb {F}}_0\) geometry, which engineers pure SU(2) Yang–Mills in five dimensions at zero Chern–Simons level, was discussed in detail in [9]. In this case, the matrix model is a qdeformation of the O(2) matrix model describing 2d Ising correlators [10, 11]. The quantum integrable system arising from the quantum Calabi–Yau geometry is twoparticle relativistic Toda chain.
In this paper, we show that the discrete dynamics is determined from the analysis of the extended automorphism group of the BPS quiver associated to the Calabi–Yau geometry. In this respect let us recall the results [12,13,14,15] where the BPS state spectrum of a class of fourdimensional supersymmetric theories is generated through quiver mutations. The quiver describes the BPS vacua of the supersymmetric theory and encodes the Dirac pairing among the stable BPS particles. The consistency of the Kontsevich–Soibelman formula [16] for the wallcrossing among the different stability chambers is encoded in Y and Q systems of Zamolodchikov type. While this program has been mostly studied for fourdimensional theories, recently a proposal for BPS quivers for the fivedimensional theory on a circle has been advanced in [17]. The fivedimensional BPS quivers are conjectured to describe the BPS spectrum of the fivedimensional theory on \({\mathbb {R}}^4\times S^1\) and have two extra nodes with respect to the corresponding fourdimensional ones, representing, in properly chosen regimes, the KK tower of states and the fivedimensional instanton monopole which characterize the theory on a circle.
The proposal we make in this paper is that these very same quivers also encode the qdifference equations satisfied by the SUSY index. These are generated by studying the application of extended quiver symmetries on the relevant cluster algebra variables \(\tau \), the latter being identified with a vectorvalued topological string grand partition function. The action of the symmetry generators on the cluster algebra coefficients y keeps track of the discrete flows for the \(\tau \)functions. As such, once the gauge theory is considered on a selfdual \(\Omega \)background, we obtain that its supersymmetric index satisfies a proper set of qPainlevé equations generated by the extended automorphisms of the quiver. More precisely, we identify different dynamics corresponding to different generators of the extended automorphism group. In a given patch, in which the topological string theory engineers a weakly coupled fivedimensional theory, the generator shifting the chosen gauge coupling induces the qPainlevé dynamics, while the other independent ones act as Bäcklund transformations of the former.
In this paper, we make a first step towards realizing the above proposal by showing that the discrete flows induced by the extended automorphism group on the BPS quiver generate in a simple way the full BPS spectrum of the 5d SCFT for some examples in the rank one case. At the same time we show that the Nekrasov–Okounkov dual partition function of the 5d gauge theory obtained by relevant deformation of those theories solves the qPainlevé equations associated to the same discrete flows. This will be accompanied also by the study of the degeneration of the fivedimensional cluster algebra into the fourdimensional one by appropriate decoupling limits. More specifically, we explore the above connection by considering in detail the case of pure SU(2) gauge theory, engineered by local \({\mathbb {F}}_0\) and local \({\mathbb {F}}_1\) depending on the value of the Chern–Simons level, as well as the SU(2) gauge theory with two fundamental flavors, or equivalently the one engineered by the local Calabi–Yau threefold over \(dP_3\). This case gives a much richer lattice of bilinear equations than the case of pure gauge, with four independent discrete time evolutions.
It was noticed in [18,19,20] that cluster algebras provide a natural framework to describe qdeformed Painlevé equations, together with their higher rank generalizations and quantization (crucial to describe the refined topological string set up). Further, following the results of [9, 21] evidence was provided for the identification of the qPainlevé tau function with Topological String partition function on toric Calabi–Yau threefold, or qdeformed conformal blocks. However, while the connection with qPainlevé equations was derived in many cases, only in the case of pure SU(N) gauge theory (corresponding in the SU(2) case to qPainlevé \(\text {III}_3\)) bilinear equations were derived from the cluster algebra. In this paper, we derive from the cluster algebra bilinear equations for the SU(2) theory with two flavors, as well as a bilinear form the qPainlevé IV equation from the cluster algebra of the local \(dP_3\) geometry which to our knowledge did not appear in the literature, and we discuss its physical interpretation in terms of the \((A_1,D_4)\) Argyres–Douglas theory.
Notice that, given the geometrical datum of the toric Calabi–Yau, it is possible to obtain its associated quiver from the corresponding dimer model [22, 23], and the Acluster variables defined from this quiver lead to bilinear equations. In many cases these have been shown to be satisfied by dual partition functions of Topological String theory on this same Calabi–Yau [24], or by qdeformed Virasoro conformal blocks [21, 25,26,27]. These can also be rephrased in terms of Ktheoretic blowup equations [24, 28, 29].
Here is the plan of the paper. In Sect. 2 we discuss BPS quiver spectroscopy, by first checking our perspective with the known case of local \({\mathbb {F}}_0\) which correctly reproduces the results of [17], before turning to discuss the new cases of local \({\mathbb {F}}_1\) and \(dP_3\) fivedimensional gauge theories. In Sect. 3 we study the discrete BPS quiver dynamics for the above examples and the related cluster algebras and obtain the explicit qPainlevé equations in bilinear form. In Sect. 4 we show that indeed the \(\tau \)functions of the specific qPainlevé flows can be matched with shifted Nekrasov–Okounkov partition function of the corresponding gauge theory. In Sect. 5 we make more precise the observation that the fourdimensional quiver can be obtained by removing two nodes from the fivedimensional one, and show, for all the examples previously considered that appropriate scaling limits of the cluster variables correctly reproduce the cluster algebra of the fourdimensional subquiver, which is known to describe the 4d BPS spectrum. In “Appendix A” we collect all the necessary formulas for fivedimensional Nekrasov functions, while in “Appendix B” we show how the bilinear equations of [30] can be also recovered from the cluster algebra approach. These are seemingly different from the ones of [26], but we show that they correspond to a different choice of initial coefficients for the cluster algebra. “Appendix C” collects few technical points related to improved saddle point expansion of NO partition functions used in the paper.
BPS Spectrum of 5d SCFT on \(S^1\) and Quiver Mutations
The construction that generates the BPS spectrum of a supersymmetric theory through mutations of its BPS quiver is known as mutation algorithm, and was widely employed in the case of fourdimensional \({\mathcal {N}}=2\) theories [13, 14, 31]. We recall that, given a quiver with adjacency matrix \(B_{ij}\), the mutation at its kth node^{Footnote 1} is defined by
The mutations of the BPS charges \(\gamma _i\) are given by
where we defined \([x]_+=\max (x,0) \). In this context, each node of the quiver represents a BPS charge in the upperhalf plane, and a mutation \(\mu _k\) encodes the rotation of a BPS ray vector out of the upper half central charge Zplane (see [13, 31] for a detailed description) in counterclockwise sense. If the charge is rotated out of the upperhalf plane clockwise instead, one has to use a slightly different mutation rule
This construction is most effective when the BPS states lie in a “finite chamber”, i.e. when the BPS spectrum consists entirely of hypermultiplets. This is not the case for the 5d theories we are considering: due to the intrinsically stringy origin of the UV completion of these theories, in general the BPS spectrum is organized in Regge trajectories of particles with arbitrary higher spin [32, 33]; such chambers of the moduli space are known as “wild chambers”. In [17] an argument was put forward for the existence of a “tame chamber” of the moduli space. Such a region is characterized by the fact that the higherspin particles are unstable and decay, and one is left with hypermultiplet and vector multiplets only, giving a situation much similar to the fourdimensional weakly coupled chambers.
Super Yang–Mills, \(k=0\)
As an example, Closset and Del Zotto argued that the spectrum for local \({\mathbb {F}}_0\), engineering pure SU(2) SYM on \({\mathbb {R}}^4\times S^1\) with Chern–Simons level \(k=0\), in such a tame chamber is organized as two copies of the weakly coupled chamber of the fourdimensional pure SU(2) gauge theory. The relevant quiver is depicted in Fig. 2, and its adjacency matrix is
The spectrum of this theory was originally derived by using the mutation algorithm in [17] , by applying the sequence of mutations
to the BPS charges. Each mutation in this context describes the change of elementary BPS states under a change of duality frame, so that the transformation \(\mathbf{m }\) amounts to the action of a selfduality on the BPS spectrum. The nth iteration of this operator has the following effect on the charges \(\gamma _i\), \(i=1,\dots ,4\) of the BPS states:
with
The action of \(\mathbf{m }\) corresponds to rotating out of the upperhalf plane the BPS charges in the order 1342. The towers of states obtained in this way accumulate on the vector multiplets from one side only. Because of this, the operator \(\mathbf{m }\) is not sufficient: in order to construct the full spectrum in this chamber, it is necessary to use also the second operator
constructed from right mutations (2.3). The shifts obtained from this operator are
The resulting BPS spectrum consists of two vector multiplets \(\delta _u,\delta _d\), and two towers of hypermultiplets
These are two copies of the weakly coupled spectrum of fourdimensional \({\mathcal {N}}=2\) SU(2) pure SYM, which can be thought as being associated to the decomposition of the quiver in Fig. 2 into two fourdimensional Krönecker subquivers, as depicted in Fig. 3a.
Let us show an alternative derivation of the above result, making use of the group \({\mathcal {G}}_Q\) of quiver automorphisms. This contains the semidirect product \(Dih_4\ltimes W(A_1^{(1)})\), where \(Dih_4\) is the dihedral group of the square, which consists only of permutations. The automorphisms group is generated by
The operator \(T_{{\mathbb {F}}_0}\) is a Weyl translation on the \(A_1^{(1)}\) lattice.
This operator directly generates the whole BPS spectrum of the theory by acting on the charges. Indeed, by applying the mutation rules (2.2) we obtain
By computing the Dirac pairing of these states, which is given by the adjacency matrix of the quiver, in particular
we see that \(T_{{\mathbb {F}}_0}\) generates mutually local towers of states
and
As \(n\rightarrow \infty \), the towers of states accumulate to the vectors \(\delta _u,\delta _d\), which are vector multiplets for the fourdimensional quivers that decompose the fivedimensional quiver as in Fig. 3a. The mutation operators \(\mathbf{m }, \hat{\mathbf{m }}\) are related in a simple way to the time evolution operator:
where \(\iota \) is the inversion. From the perspective of the full automorphism group
it is natural to consider also another translation operator
whose action on the charges is
This generates different towers of hypermultiplets, which are still organized as two copies of the weakly coupled chamber of fourdimensional super Yang–Mills, with vector multiplets
In this way, we find a different infinite chamber, corresponding to the decomposition of the 5d BPS quiver as in Fig. 3b.
We see that considering the natural translation operators associated to the quiver automorphisms builds the correct spectrum for the tame chambers in a simpler way, without the need to consider both left and right mutations. This simplification occurs because we are allowing not just mutations, but also permutations, which are relabelings of the BPS charges. This operation of course has no effect on the resulting spectrum, which is the same as the one emerging from using just the mutation algorithm. However, by using quiver automorphisms, it is possible to construct more elementary dualities of the theory, and the spectrum can be constructed more simply. This plays a crucial rôle in more complicated cases. To illustrate this point, we discuss in the following the cases of local \({\mathbb {F}}_1\) and \(\text {dP}_3\).
Super Yang–Mills, \(k=1\)
The local \({\mathbb {F}}_1\) quiver is displayed in Fig. 4a. This engineers pure SU(2) SYM with 5d Chern–Simons level \(k=1\). The adjacency matrix is
The translation operator is given by
As far as the spectrum is concerned, this is the same as the local \({\mathbb {F}}_0\) one. The operators \(\mathbf{m },\tilde{\mathbf{m }}\) are easy to build
and the associated evolution on the vector of charges \(\varvec{\gamma }\) is
We see that even though the introduction of a Chern–Simons level will affect some physical aspects, it does not modify the type of states in the spectrum: again \(\delta _u,\delta _d\) correspond to the vector multiplets of the 4d subquivers depicted in Fig. 4b. What changes, however, is the number of tame chambers: because the symmetry group now does not include the \(Dih_4\) factor—as it is clear by inspection of the quiver—there is not the related chamber.
\(N_f=2\), \(k=0\)
When we include matter the situation is much richer, because we encounter the new feature of multiple commuting flows, each characterizing the spectrum in a different chamber of the moduli space. The relevant quiver is the one of \(dP_3\), engineering the SU(2) theory with two flavors, depicted in Fig. 5. It has adjacency matrix
The extended Weyl group is \({\tilde{W}}((A_2+A_1)^{(1)})\), which is generated by
In this case, there are four commuting evolution operators, given by Weyl translations of \({\tilde{W}}((A_2+A_1)^{(1)})\), acting on the affine root lattice \(Q\left( (A_2+A_1)^{(1)}\right) \) [30, 34]^{Footnote 2}. One has the three operators
satisfying \(T_1T_2T_3=1\), and finally
Let us consider the flow \(T_1\) first, given by
Its action on the BPS charges is the following:
We see that \(T_1\) generates infinite towers of hypermultiplets given by
These are the BPS states corresponding to two copies of the weakly coupled spectrum for the \(N_f=1\) theory in four dimensions, and correspond to the decomposition of the 5d quiver into two 4d subquivers for \(N_f=1\), as in Fig. 6. One can easily check that the towers of states are mutually local, and as \(n\rightarrow \infty \) they accumulate on the rays
which are indeed the vector multiplets for the 4d \(N_f=1\) subquivers. More precisely, the towers of hypermultiplets above are only half of the towers from \(N_f=1\) theory. To complete the picture, here we have to consider, like in the pure gauge case, the states constructed from right mutations: these are generated as before by powers of the inverse of the evolution operator, composed with an inversion \(\iota \):
The spectrum of the \(N_f=1\) theory in four dimensions also includes two quarks that for the subquivers in Fig. 6 are given by \(\gamma _5,\gamma _{4}+\gamma _{6},\gamma _2,\gamma _1+\gamma _3\). We see that we recover the quarks \(\gamma _5,\gamma _2\) as the states that are left invariant by \(T_1\), while the other quarks would be their complementary in the subquiver. We will see below how the remaining quarks can be recovered as the states that are fixed by a different flow.
\(T_2\) is given by
and acts on the BPS charges as
Reasoning as before, we find that the spectrum in this chamber is organized in two copies of the 4d \(N_f=1\) weakly coupled chamber, corresponding to the subquiver decomposition in Fig. 7, with vector multiplets
Finally, we have
with action
which gives another chamber organized as two copies of fourdimensional weakly coupled \(N_f=1\) depicted in Fig. 8, with vector multiplets
Before considering the evolution \(T_4\), let us make a remark. The picture above suggests that there exists a relation between the different flows in terms of permutations of the nodes of the quiver. Indeed, it is possible to check that we have the relations
From the point of view of the BPS spectrum, it is now clear that these three flows will generate the same spectrum up to relabeling of states, i.e. they will differ in what we call electric or magnetic in the field theory. Another interesting quiver automorphism is given by
which is known as halftranslation, because it satisfies \(R_2^2=T_2\) (halftranslations for \(T_1,T_3\) can be obtained by using equation (2.48)). Under this quiver automorphism, the BPS charges transform as
while of course \(R_2^{2n}=T_2^n\). Note that this generates the CPT conjugates of the towers of states as \(T_2\), while the states that are left fixed by the action of \(R_2\) are exactly the missing quarks from our analysis of \(T_2\), so that \(R_2\) generates the full spectrum of the two copies of \(N_f=1\) in the subquivers of Fig. 7.
Finally, the time evolution \(T_4\) is given by
and acts on the BPS charges as follows:
We can recognize in this chamber towers of states that accumulate to the same BPS ray \(\delta =\gamma _1+\dots +\gamma _6\), representing a multiplet with higher spin \(s\ge 1\). In this case, the fourdimensional interpretation is subtler and more interesting, and we postpone it to Sect. 5.3.
Discrete BPS Quiver Dynamics, Cluster Algebras and qPainlevé Equations
The translation operators acting on the BPS quivers we described so far can be regarded as time evolution operators for discrete dynamical systems arising from deautonomization of cluster integrable systems, naturally associated to the geometric engineering of the corresponding fivedimensional gauge theories. This allows to bridge between the BPS quiver description and classical results in the theory of qPainlevé equations, and actually inspired the reformulation of the BPS quiver analysis that we presented in the previous section. Indeed, the quivers studied in [18] are exactly the 5d BPS quivers appearing in Sect. 3. In this respect, the qPainlevé flows describe selfdualities (in the sense of section) of the 4d Kaluza–Klein theory obtained by reducing the 5d gauge theory on \(S^1\). We now turn to the study of all the examples we considered up to now from this perspective.
Cluster Algebras and Quiver Mutations
Let us first recall the notion of cluster algebra [35, 36], as well as the two types of cluster variables that will be used throughout the paper. The ambient field for a cluster algebra \({\mathcal {A}}\) is a field \({\mathcal {F}}\) isomorphic to the field of rational functions in \(n=\text {rk } {\mathcal {A}}\) independent variables, with coefficients in \({\mathbb {Q}}{\mathbb {P}}\), where \({\mathbb {P}}\) is the tropical semifield. The tropical semifield is defined as follows: starting with the free abelian group \(({\mathbb {P}},\cdot )\) with usual multiplication, the operation \(\oplus \) is defined in terms of a basis^{Footnote 3}\(\mathbf{u }\) of \({\mathbb {P}}\)
The cluster algebra \({\mathcal {A}}\) is determined by the choice of an initial seed. This is a triple \((Q,\varvec{\tau },\mathbf{y })\), where

Q is a quiver without loops and 2cycles, with n vertices;

\(\mathbf{y }=(y_1,\dots ,y_n)\) is an ntuple of generators of the tropical semifield \(({\mathbb {P}},\oplus ,\cdot )\) (which in general will not be independent generators, because \(\dim {\mathbb {P}}\le n\));

\(\varvec{\tau }\equiv (\tau _1,\dots ,\tau _n) \) is an ntuple of elements of \({\mathcal {F}}\) forming a free generating set: they are algebraically independent over \({\mathbb {Q}}{\mathbb {P}}\), and \({\mathcal {F}}={\mathbb {Q}}{\mathbb {P}}(\tau _1,\dots ,\tau _n)\).
The variables \((\varvec{\tau },\mathbf{y })\) are called Acluster variables. We can alternatively define the seed as \((B,\varvec{\tau },\mathbf{y })\) in terms of the antisymmetric adjacency matrix B of the quiver.
Given these objects, the cluster algebra is the \(\mathbb {ZP}\)subalgebra of \({\mathcal {F}}\) generated recursively by applying mutations to the initial seed. A mutation \(\mu _k\) is an operation defined by its action on a seed:
where we defined \([x]_+=\max (x,0) \). It is clear from the above expression that the coefficients \(y_i\) represent an exponentiated version of the BPS charges \(\gamma _i\).
An alternative set of variables are the socalled Xcluster variables \(\mathbf{x }=(x_1,\dots ,x_n) \), taking values in \({\mathcal {F}}\). They are defined in terms of the Avariables as
and their mutation rules are the same as for coefficients, but with ordinary sum instead of semifield sum:
The Xcluster variables can be considered as coordinates in the socalled Xcluster variety, which is endowed with a degenerate Poisson bracket, with respect to which the Xcluster variables are logcanonically conjugated:
Given a convex Newton polygon \(\Delta \) with area S, it is possible to construct a quiver with 2S nodes describing a discrete integrable system in the variables \(x_i\) [22, 37]. Due to (3.7), in general the Poisson bracket is degenerate, as there is a space of Casimirs equal to \(\ker (B)\). For quivers arising in this way, the quantity
is always a Casimir. The system is integrable on the level surface
The number of independent Hamiltonians is the number of internal points of the Newton polygon. The set of discrete time flows of the integrable system is the group \({\mathcal {G}}_Q\) of quiver automorphisms^{Footnote 4}. We will in fact work with the extended group \(\tilde{{\mathcal {G}}}_Q\) that extends \({\mathcal {G}}_Q\) by the inclusion of the inversion operator \(\iota \). This operation reverses all the arrows in the quiver, and acts on the cluster variables as
while the variables \(\varvec{\tau }\) are invariant, consistently with the relation (3.5).
In [18] it was shown that it is possible to obtain qPainlevé equations by lifting the constraint \(q=1\), which amounts to the deautonomization of the system. This is no longer integrable in the Liouville sense, since the discrete Hamiltonians are no longer preserved under the discrete flows. The related equations of motion are wellknown qdifference integrable equations of mathematical physics, namely qPainlevé equations: the time evolution describes in this case a foliation, whose slices are different level surfaces of the original integrable system, see, for example, [38] for such a description of qPainlevé equations. These equations can be obtained geometrically by studying configurations of blowups of eight points on \({\mathbb {P}}^1\times {\mathbb {P}}^1\), or equivalently by configurations of nine blowups on \({\mathbb {P}}^2\). As in the case of differential Painlevé equations [39], this leads to a classification in terms of the space of their initial conditions, called in this context surface type of the equation, or equivalently by their symmetry groups due to Sakai [5], see Fig. 1. The former are given by an affine algebra, while the latter turns out to be given by the extended Weyl group of another affine algebra, which is the orthogonal complement of the first one in the group of divisors Pic(X), X being the surface obtained by blowing up points on \({\mathbb {P}}^1\times {\mathbb {P}}^1\).
It was further argued in [18] that the time evolution given by the deautonomization of the cluster integrable system, when written in terms of the cluster Avariables \((\varvec{\tau },\mathbf{y })\), takes the form of bilinear equations, so that we can identify the variables \(\varvec{\tau }\) with tau functions for qPainlevé equations. However, while the qPainlevé equations in terms of the Xcluster variables were derived for all the Newton polygons with one internal point in [18], their bilinear form was not obtained, except for the Newton polygon of local \({\mathbb {F}}_0\), corresponding to the qPainlevé equation of surface type \(A_7^{(1)'}\), and local \({\mathbb {F}}_1\) in [19], corresponding to \(A_7^{(1)}\) in Sakai’s classification.
In the next section, we review these two cases, before turning to the case of \(dP_3\), which corresponds instead to the surface type \(A_5^{(1)}\). In fact, this case is much richer, as it admits four commuting discrete flows: we will show that one of these reproduces the bilinear equations considered in [25, 26] for qPainlevé \(\text {III}_1\).
Pure Gauge Theory and qPainlevé \(\hbox {III}_3\)
Let us briefly review how qPainlevé equations are obtained from the quivers associated to local \({\mathbb {F}}_0\) and local \({\mathbb {F}}_1\), whose Newton polygons are depicted in Fig. 9a, b. These correspond to the pure SU(2) gauge theory with Chern–Simons level, respectively, \(k=0,1\).
Local \({\mathbb {F}}_0\) :
Let us consider first the cluster algebra associated to the quiver in Fig. 2. This corresponds to local \({\mathbb {F}}_0\). The group \({\mathcal {G}}_Q\) of quiver automorphisms contains the symmetry group of the qPainlevé equation qP\(\text {III}_3\) of surface type \(A_7^{(1)'} \), which is the semidirect product \(Dih_4\ltimes W(A_1^{(1)})\). It is generated by
The operator \(T_{{\mathbb {F}}_0}\) generates the time evolution of the corresponding qPainlevé equation, and is a Weyl translation on the underlying \(A_1^{(1)}\) lattice. From the adjacency matrix of the quiver
we see that the space of Casimirs of the Poisson bracket (3.7) is twodimensional. We take the two Casimirs to be
Therefore, the tropical semifield has two generators that we take to be the two Casimirs q, t. By fixing the initial conditions for the coefficients, consistently with equation (3.13), to be
one finds that the action of \(T_{{\mathbb {F}}_0}\) on the coefficients yields
while the tau variables evolve as
leading to the bilinear equations^{Footnote 5}
The actual qPainlevé equation is the equation involving the variables x. It takes the form of a system of two first order qdifference equations, or of a single secondorder qdifference equation, in terms of logcanonically conjugated variables
that satisfy
Their time evolution can be studied in a completely analogous way by using the mutation rules (3.6) for Xcluster variables, and leads to the qPainlevé \(\text {III}_3\) equation
Local \({\mathbb {F}}_1\): We now consider the Avariables associated to the local \({\mathbb {F}}_1\) quiver of Fig. 4a, engineering pure SU(2) SYM with 5d Chern–Simons level \(k=1\). The adjacency matrix is
The corresponding equation is the qPainlevé equation of surface type \(A_7^{(1)}\), which is a different qdiscretization of the differential Painlevé \(\text {III}_3\). The time evolution is given by
The Casimirs are now
Consistently with this relation, we choose the following initial conditions for the coefficients:
This yields the time evolution of the Casimirs
and of the \(\tau \)variables
The bilinear equations obtained in this way are
which are the same as the equations appearing in [19]
for the single tau function \(\tau _4\equiv \tau \). The identification is achieved by noting that
so that the first of our bilinear equations becomes
which coincides with (3.28) after using (3.25).
Super Yang–Mills with Two Flavors and \(\hbox {qPIII}_1\)
We now turn to consider the quiver associated to \(dP_3\), engineering the SU(2) theory with two flavors, depicted in Fig. 10. It has adjacency matrix
and Casimirs
that satisfy
As already discussed in Sect. 2.3, one has in this case four commuting time evolution operators, given by Weyl translations of \({\tilde{W}}((A_2+A_1)^{(1)})\), which act on the affine root lattice \(Q\left( (A_2+A_1)^{(1)}\right) \) [30, 34]. These time evolutions are not all associated to the same qPainlevé equation: the three operators
give rise to the qPainlevé equation qP\(\text {III}_1\) in the xvariables, and satisfy \(T_1T_2T_3=1\). On the other hand, the evolution
yields qPainlevé IV.
The time evolution of the Casimirs can be obtained easily from the Xcluster variables: it is
This is the counterpart of the fact that \(T_i\) are Weyl translations acting on the root lattice \(Q((A_2+A_1)^{(1)}) \). If \(\alpha _0,\alpha _1,\alpha _2\) are simple roots of \(A_2^{(1)}\), and \(\beta _0,\beta _1\) are simple roots of \(A_1^{(1)}\), the action of \(T_i\) as elements of the affine Weyl group is
where \(\delta =\alpha _0+\alpha _1+\alpha _2=\beta _0+\beta _1\) is the null root of \((A_2+A_1)^{(1)} \). From each one of these discrete flows, we can obtain bilinear equations for the cluster Avariables \(\varvec{\tau }\). Once we choose one of the flows as time, the other flows can be regarded as Bäcklund transformations describing symmetries of the time evolution.
Let us define the four tropical semifield generators to be \(q,t,Q_1,Q_2\), and the initial condition on the parameters to be
which means, in terms of the original parameterization of the Casimirs,
We now derive bilinear equations for the discrete flows of this geometry: the time evolution for \(T_1\) is
The action on the Casimirs is given by (3.37), that means
We then have
The time flow under \(T_2\) for the Acluster variables is
where the time evolution is given by
leading to the bilinear equations
In particular, from the flow \(T_2\) it is possible to reproduce the bilinear equations of [26], thus obtaining an explicit parameterization of the geometric quantities \(a_i,b_i\) coming from the blowup configuration of \({\mathbb {P}}^1\times {\mathbb {P}}^1\) in terms of the Kähler parameters of \(\text {dP}_3\).
The discrete flow \(T_3\) is not independent, being simply given by \(T_3=T_1^{1}T_2^{1}\), but we write it down for completeness:
leading to the bilinear relations
Super Yang–Mills with Two Flavors, qPainlevé IV and qPainlevé II
On top of the previous time evolutions giving rise to \(\hbox {qPIII}_1\) equations, there is another the time evolution \(T_4\) from a further automorphism of the \(dP_3\) quiver. This gives rise to the qPIV dynamics and has the following action on the Casimirs, dictated by (3.40):
On the tau variables, this amounts to
At first sight this seems to lead to cubic equations. However, by following the procedure explained in “Appendix B.2”, one obtains an equivalent set of bilinear equations
These provide a bilinear form for the qPIV equation, which to our knowledge did not appear in the literature so far.
qPainlevé II bilinear relations from “half” translations:
Given the root lattice \((A_2+A_1)^{(1)} \), there exists another time flow that preserves a \((A_1+A_1')^{(1)} \) sublattice only [40, 41]. It corresponds to the qPainlevé II equation, and it is given by
which is still an automorphism of the quiver in Fig. 10. Because \(R_2^2=T_2\), this flow is also known as halftranslation^{Footnote 6}. Its action on the Casimirs is a translational motion (i.e. a good time evolution) only on the locus \(a_0=q^{1/4} \), i.e. \(Q_+=1\), on which it acts as
corresponding to
Its action on the tauvariables reads
By setting \(Q_1=Q_2^{1}\equiv Q\), we obtain the bilinear equations
We see that in fact these equations are consistent under the further requirement \(Q=1\). This is because the third and fourth equations are obtained by simply applying \(T_2^{1}\) to the first and second one.
According to Sakai’s classification (see Fig. 1) and the analysis in [42], this flow correctly points to the Argyres–Douglas theory of \(N_f=2\) which is, in the fourdimensional limit, governed by the differential PII equation. It would be interesting to see if the relevant \(\tau \)function can be constructed from a 5d lift of the matrix model considered in [43].
Summary of \(\hbox {dP}_3\) Bilinear Equations
To conclude this section, let us collect here all the flows we found for \(dP_3\) geometry together with the respective bilinear equations:
Solutions
In this Section we discuss how the solutions of the discrete flow of BPS quivers are naturally encoded in topological string partition functions having as a target space the toric Calabi–Yau varieties associated to the relevant Newton polygons. The corresponding geometries are given by rank two vector bundles over punctured Riemann surfaces. Let us recall that the BPS states of the theory are associated to curves on this geometry that locally minimize the string tension. More specifically, hypermultiplets are associated to open curves ending on the branch points of the covering describing the Riemann surface, while BPS vector multiplets are associated to closed curves^{Footnote 7}. The BPS states are then described in this setting by open topological string amplitudes with boundaries on those curves. The very structure of the discrete flow suggests to expand the \(\tau \) functions as grand canonical partition functions for the relevant brane amplitudes. Specifically, we propose that
where \(q=e^{\hbar }\), \(\hbar =g_s\) being the topological string coupling, \(Q_i\) the Calabi–Yau moduli and \(s_i\) the fugacities for the branes amplitudes associated to BPS states with intersection numbers \(m_i\) with the cycles associated to the \(Q_i\) moduli. These cycles represent a basis associated to the BPS state content of theory in the relevant chamber, the intersection numbers representing the Dirac pairing among them. It is clear from this that the expansion (4.1) for the tau function crucially depends on the BPS chamber. Moreover, distinct flows of the BPS quivers described in the previous sections correspond to bilinear equations in distinct moduli of the Calabi–Yau.
These bilinear equations are in the socalled Hirota form and turn out to be equivalent to convenient combinations of blowup equations [28, 44], which consist of many more equations, and suffice to determine recursively the nonperturbative part of the partition function, given the perturbative contribution [45].
In the following, we will mainly focus on the expansion of \(\tau \) functions in the electric weakly coupled frame which is suitable to geometrically engineer fivedimensional gauge theories. In this case, the \(\tau \) function coincides with the Nekrasov–Okounkov partition function.
Local \({\mathbb {F}}_0\) and \(\hbox {qPIII}_3\)
We first discuss the pure gauge theory case to gain some perspective. We’ll then pass to the richer, and so far less understood, \(N_f=2\) case. Let us point out here that the solution of PIII\({_3}\) in the strong coupling expansion was worked out in [9] in terms of the relevant Fredholm determinant (or the matrix model in the cumulants expansion). In [9] also the relevant connection problem was solved. Since we are interested in showing the classical expansion of the cluster variables, here we discuss the different asymptotic expansion in the weak coupling \(g_s\sim 0\).
In our favorite example, the local \({\mathbb {F}}_0\), the cluster variable \(x_2=G^{1}\), where G satisfies (see (3.20))
This can be written in terms of Nekrasov–Okounkov partition functions as
with
where Z is the full Nekrasov partition function for which we give explicit formulae in “Appendix A”. This case corresponds to the SU(2) pure gauge theory with Chern–Simons level \(k=0\), and we set \(u_1=u_2^{1}=u\), \(q_1=q_2^{1}=q\). The bilinear equations
turn into an infinite set of equations for Z:
where the coefficient for each power of s must vanish separately. Of course, most of these equations are redundant, but everything is determined by fixing the asymptotics, i.e. the classical contribution for the partition function. Selecting the term \(n+m=1\), for example, we can obtain the following equation for the \(t^0\) coefficient of Z (i.e. the perturbative contribution):
which is the qdifference equation satisfied by \(Z_{\text {1loop}}\). The term \(n+m=0\) allows us to determine the instanton contribution from the perturbative one in the following way:
We can express the above equation in terms of \(Z_{inst}\) as
which gives a recursion relation for the coefficients of the instanton expansion
For example, the oneinstanton term is fully determined just by the perturbative contribution:
which of course correctly reproduces the oneinstanton Nekrasov partition function.
It is possible to study the autonomous limit of the Xcluster variables by setting
and sending \(\hbar \rightarrow 0\). In this limit one can expand the Nekrasov partition function in the \(\Omega \)background parameter as
The behavior of the Fourier series in this limit is determined by a discrete version of the saddle point approximation [46]. Let us call \(n_*\) the saddle point for n: for given \(a,\eta \) the saddle point condition
can be satisfied only approximately. More precisely, denote by \(a_*\) the “true” saddle value, given by the condition
In general \(a+\hbar n_*\) will be able to approximate this value only up to \(\hbar \) corrections. We write this as
where the variable \(x\sim O(1)\) measures the offset between the true saddle and the approximate one. We review in “Appendix C” the computation of the leading and subleading order in the \(\hbar \rightarrow 0\) limit for the dual partition functions, first performed in [19]. The result is that
When we consider the qPainlevé transcendent, given by the cluster variable \(x_2^{1}\), the prefactor simplifies, so that it is given by a ratio of theta functions:
The above analysis can be easily extended to the local \({{\mathbb {F}}}_1\) case by making use of the results in [18]. We do not repeat it here.
qPainlevé \(\text {III}_1\) and Nekrasov Functions
In the case of the gauge theory with matter, let us first focus on the bilinear equations generated by the translation \(T_2\). As computed in the previous section, these are
Let us crucially note that these coincide with the bilinear equations studied in [26], eqs (4.54.8) after relabeling
and the identification
In the gauge theory, \(Q_1,Q_2\) parameterize the masses of the fundamental hypermultiplets through \(\hbar \theta _i=m_i\), where \(\hbar \) is the selfdual \(\Omega \)background parameter, while t is the instanton counting parameter. In terms of these, the time evolution is
so that the discrete time evolution shifts the gauge coupling while the masses stay constant. We can therefore write the bilinear equations as qdifference equations
It was shown in [26] that the above bilinear equations are solved in terms of the dual partition function for SU(2) SYM with two fundamental flavors. More precisely, in that paper it was shown that if we define
where Z is the Nekrasov partition function for the \(N_f=2\) theory, the \(\tau \)functions solving (4.25) can be written as
By using also \(\tau _4=T_2(\tau _2)\), \(\tau _1=T_2(\tau _5) \), we can add to these
Working in the same way as in Sect. 4.1, one can arrive at bilinear equations for Nekrasov functions, but differently from what happened in that simpler case, now one equation does not suffice to determine the nonperturbative contribution from the perturbative one: we have to use both the first and third equations of (4.25). The first equation takes the form
This leads to the following equation on the oneinstanton contribution:
Differently from what happened in the pure gauge case, one equation is no longer enough to determine the partition function, because we get two occurrences of the function with different shifts on the mass parameter \(Q_1\). We have to use the third equation of (4.25) that leads to
that gives an equation for the oneinstanton contribution
Putting the two equations together, and using the identities (A.10) and (A.11), we obtain the correct oneinstanton contribution
matching the one computed by instanton counting. One can go on and compute the higher instanton contributions in an analogous way. These two equations are enough to determine the nonperturbative contribution order by order in t, starting from the knowledge of the perturbative contribution, which is the \(t^0\) term.
Let us finally note that all these bilinear equations could be written as lattice equations on \(Q((A_2+A_1)^{(1)})\) by noting that all the various tau functions can be obtained starting from a single one, let us say \(\tau _1\), since we have
so that it is possible to introduce, following [34], the tau lattice
In this notation the original tauvariables can be denoted by
and the time flows are integer shifts of the indices of the tau function (4.37). However, not all the flows are compatible with the instanton expansion: from (3.47) and (3.55) we see that the natural expansion parameter for the solution of the \(T_1\) and \(T_4\) flows are, respectively, \(Q_1Q_2\) and \(Q_2/Q_1\).
In fact, the usual Nekrasov expansion, as defined in “Appendix A” by a converging expansion in t, can only solve the equations for \(T_2\), which have t as time parameter: this is because if we try to solve the other equations iteratively by starting with the perturbative contribution as defined in equation (A.7), there is no region in parameter space where all the multiple qPochhammer functions with shifted arguments entering the bilinear equations have converging expressions simultaneously. To find a solution, one should find an analogue of the perturbative partition function (A.7) which is of order zero, not in t, but rather in the appropriate time parameter, solving the order zero of the bilinear equations. This indeed corresponds to an expansion of the topological string partition function (4.1) in the corresponding patch in the moduli space in the Topological Vertex formalism [3].
A preliminary analysis shows that, on top of the evolution in the mass parameters, comparing with the solution in terms of Nekrasov functions, we see that consistency requires also that
To see why this must hold, one has to consider tau functions related by time evolutions of the flows \(T_1,T_4\). For example, the action on the flow \(T_1\) on the solutions \(\tau _1,\tau _3\) (the same considerations would hold by considering the other tau functions):
Equation (4.40) is consistent with the interpretation of the flows \(T_1,T_3,T_4\) as the Bäcklund transformations of \(T_2\).
Degeneration of Cluster Algebras and FourDimensional Gauge Theory
In the previous sections we saw how the cluster algebra structure yields the qdifference equations satisfied by the partition function of the theory, and produces the spectrum of the theory in a weakly coupled chamber by a systematic application of the generalized mutation algorithm. Further, it was observed in [17] that the BPS quivers describing the purely fourdimensional theory (with all KK modes decoupled) are contained in the fivedimensional one as subquivers with two fewer nodes: roughly, one of the additional nodes is the fivedimensional instanton monopole, while the other corresponds to the KK tower of states. From the point of view of cluster integrable systems and qPainlevé equations, this was already realized in [18]. Graphically, to go from the 5d theory to the 4d one, one “pops” two nodes of the quiver.
We now show how it is possible to explicitly implement the operation of deleting the two nodes that brings the fivedimensional quiver to the fourdimensional one, at the level of the full cluster algebra, so that we recover the fourdimensional description of the BPS states. From the gauge theory point of view, the fourdimensional limit \({\mathbb {R}}^4\times S^1_R\rightarrow {\mathbb {R}}^4 \) is obtained by taking the radius of the fivedimensional circle \(R\rightarrow 0\). More precisely, one has to scale the Kähler parameters in such a way that the KK modes and instanton particles decouple from the BPS spectrum. This limit is usually achieved by implementing the geometric engineering limit [47, 48], and takes the form
We see that this limit amounts to sending
Because this limit involves
while the other Casimir is still given by a product of cluster variables, we can already see that it is unlikely for this limit to be able to reproduce cluster algebra transformations. Another way to see this is the case, consider the relation between X and Acluster variables for the case of local \({\mathbb {F}}_0\):
Because of this, if we implement the limit \(t\rightarrow 0,q\rightarrow 1\) by using the geometric engineering prescription, the tau functions, which are given in terms of fivedimensional Nekrasov partition function, will simply go to their fourdimensional limit. Then, no Xcluster variable has an interesting limit. In fact, this is instead the continuous limit of the corresponding Painlevé equation, in which the qdiscrete equations become differential equations (see, for example, [21] for the explicit implementation of the limit on the 5d Nekrasov functions).
We will now show how to instead implement the limit \(q\rightarrow 1\), \(t\rightarrow 0\) for the cases we considered in this paper: local \({\mathbb {F}}_0\), \({\mathbb {F}}_1\), and \(\text {dP}_3\) (respectively, 5d pure gauge theory without and with Chern–Simons term, and the theory with \(N_f=2\) hypermultiplets), in such a way that the cluster algebra structure of the quiver is preserved: in particular we will see that:

The mutations of the fivedimensional quiver degenerate to those of the fourdimensional one in terms of the reduced set of variables;

The qPainlevé time flows (or a subflow, in the case of \({\mathbb {F}}_1\)), which were given by automorphisms of the fivedimensional quivers, degenerate to appropriate sequences of mutations and permutations which are automorphisms of the fourdimensional ones.
Of course, the recipe taken to implement these limit is quite general, and we have no reason to expect it not to work for the other cases. We will implement the limit on the Xcluster variables, because they carry no ambiguity related to the choice of coefficient/extended adjacency matrix. The fourdimensional cluster Avariables can then be obtained from the Xcluster variables using the adjacency matrix as usual. However, because we are implementing this limit on the Xcluster variables, we do not have an explicit expression in terms of Nekrasov functions for the limiting system.
From local \({\mathbb {F}}_0\) to the Kronecker Quiver
Recall the expression for the Casimirs in terms of the cluster variables:
Let us say that we want to decouple the nodes 3,4 on the corresponding quiver, so that we remain with the Kronecker quiver with nodes 1,2 (the red quiver in Fig. 3a). We need then to implement the limit
which, as we argued above, is different from the geometric engineering limit. We then have to take
We are interested in the expressions for the mutations at the remaining nodes after decoupling, as well as for the qPainlevé translation. For the mutations, this case is very simple: the limit takes the form
We see that the mutations for \(x_1,x_2\) do not involve the variables \(x_3,x_4\), so that no limit is actually necessary, and in fact they are already in the form of mutations for the Kronecker subquiver. Further, these mutations preserve the limiting value of \(x_3,x_4\). In fact, the choice of the subquiver is completely arbitrary: by this limiting procedure we can consider any of the Kronecker subquivers of the quiver in Fig. 2. The limit is less trivial on the qPainlevé flow:
Again the limiting value of \(x_3,x_4\) is preserved, while in terms of operations of the Kronecker quiver the qPainlevé flow becomes
which is an automorphism of the Kronecker quiver.
Local \({\mathbb {F}}_1\)
We can proceed and take the analogous limit for local \({\mathbb {F}}_1\), for which
We again focus on the Kronecker subquiver with nodes 1,2, and set
The limiting behavior of the mutations is now
which again yields the correct limiting behavior. The qPainlevé flow does not have a good limiting behavior: however, its square does, since
as we saw above.
Local \(\text {dP}_3 \)
This case is much more interesting, because we get different decoupling limits, and only one of them is similar to the usual fourdimensional limit, involving \(t\rightarrow 0\). These are related to the presence of different discrete flows. We consider as usual \(T_2\) first, which we have already seen to be related to the usual weakly coupled/instanton counting picture. In analogy to what was done in the previous cases, since
we take the limit
by taking a limit on two of the cluster variables. Looking at the quiver for this case, we recognize that the subquiver with vertices 2,3,4,6 (or equivalently 1,3,5,6) gives the BPS quiver of the fourdimensional \(N_f=2\) theory, as in Fig. 11. We will focus on the former case.
Because we are “popping out” the nodes 1,5 from the quiver, we want to achieve this by implementing the limit (5.19) directly on the cluster variables. By studying the expressions for the Casimirs (3.32), (3.33), we find
so that we want to study the limit
Taking the limit on the mutations, we obtain
and similarly for the other mutations \(\mu _3,\mu _4,\mu _6\) that all degenerate to the mutations of the fourdimensional quiver. The discrete flow also has a “good” limit:
If we call the variables after taking the limit
we have that the limit of the discrete flow is
which is an automorphism of the 4d quiver. We can follow the same logic for the other discrete flows \(T_1,T_3,T_4\): we will from now on discuss only the limits on the discrete time flows, because those on the mutations are rather simple and given by essentially the same computations as above. In the first case the flow is
so that the natural guess for the right limit to consider is \(Q_+\rightarrow \infty \), in analogy with the previous case. By looking at the Casimirs, we arrive to the conclusion that we can either decouple the nodes 1,3 or 4,6, producing the 4d \(N_f=2\) subquivers in Fig. 12.
The limit we want to implement is then
which gives
which is the same 4d quiver automorphism as for \(T_2\), up to permutations of the nodes. The time evolution \(T_3\) is characterized by
so that the natural limit on the Casimirs is
This can be achieved by decoupling the nodes 3,5 or 2,6, keeping the subquivers depicted in Fig. 13.
Choosing the former one for concreteness, we want to compute the limit
on the discrete evolution \(T_3\). This is given by
We see that in all the cases that yielded the time evolution of qPainlevé \(\text {III}_1\), the degeneration of the time flow produces the same automorphism of an appropriate subquiver. It remains to study the flow \(T_4\), which yielded a qPainlevé IV time evolution, characterized by
In terms of the Casimirs \(b_0,b_1\), this leads to
To achieve this without affecting the Casimirs \(a_0,a_1,a_2\) we have to decouple either the nodes 2,5, or the nodes 3,6, or the nodes 1,4, giving the subquivers in Fig. 14, and we will consider the first option, given by the limit
Here, something similar to what happened when we studied the degeneration of the qPainlevé \(\text {III}_3\) associated to local \({\mathbb {F}}_1\) happens: recall that in that case \(T_{{\mathbb {F}}_1}\) did not have a good degeneration limit as an automorphism of the subquiver, but rather its square did. We observed that this was related to the \({\mathbb {Z}}_2\)periodicity of the action of \(T_{{\mathbb {F}}_1}\) on the BPS charges. What happens here is that not \(T_4\), but rather \(T_4^3\) has a good action after taking the limit, in particular only for \(T_4^3\) it is true that
consistently with the limit.
The resulting subquiver is the oriented square with arrows of valency one and no diagonals with adjacency matrix
The corresponding fourdimensional gauge theory has been already classified in [14] as Q(1, 1) and shown to correspond to \(H_3\), which is the Argyres–Douglas limit of the \(N_f=3\) with SU(2). All this is consistent with the reductions of the Sakai’s table in Fig. 1.
The symmetry type of the fivedimensional SU(2) \(N_f=2\) gauge theory is \(E_3^{(1)}\). The reduction of the \(T_1\), \(T_2\) and \(T_3\) flows corresponds to the reduction \(E_3^{(1)} \rightarrow D_2^{(1)}\), the latter being the symmetry type of the fourdimensional SU(2) \(N_f=2\) gauge theory. The reduction of the \(T_4\) flow corresponds to the reduction \(E_3^{(1)} \rightarrow A_2^{(1)}\), the latter being the symmetry type of the \(H_3\) theory. According to Sakai’s classification (see Fig. 1) and the analysis in [42], this flow correctly points to the Argyres–Douglas theory of \(N_f=3\) which is, in the fourdimensional limit, governed by the differential PIV equation.
Conclusions and Outlook
In this paper, we studied the discrete flows induced by automorphisms of BPS quivers associated to Calabi–Yau geometries engineering fivedimensional quantum field theories. We showed that these flows provide a simple and effective way to determine the BPS spectrum of these theories, producing at the same time a set of bilinear qdifference equations satisfied by the grand canonical partition function of topological string amplitudes. In the rank one case these are known as qPainlevé equations and admit in a suitable region of the moduli space solutions in terms of Nekrasov–Okounkov, or free fermions, partition functions.
A very attractive feature of this approach is that a simple symmetry principle—the symmetry of the BPS quiver—provides strong constraints on the BPS spectrum and contains a rich and deep set of information which goes well beyond the perturbative approaches to the same theories. Indeed, one can show that the nonperturbative completion of topological string via a spectral determinant presentation, arising in the context of the topological string/ spectral theory correspondence, arises naturally as solution of this system of discrete flow equations [9]. Moreover, some of the flows associated to the BPS quiver directly link to nonperturbative phases of the corresponding gauge theory, as we have seen for a particular flow of the local \(dP_3\) geometry which describe a \((A_1,D_4)\) Argyres–Douglas point, see Sect. 5.3. A discussion of the relation between Painlevé equations and Argyres–Douglas points of fourdimensional gauge theories can be found in [42] based on the class \({{{\mathcal {S}}}}\) description of these theories [49].
It is also very interesting that a fully classical construction, the cluster algebra associated to the BPS quiver, contains information about the quantum geometry of the Calabi–Yau. Indeed, the zeroes of the \(\tau \)functions of the cluster algebra provide the exact spectrum of the associated quantum integrable system, as it was shown in [9] for the local \({\mathbb {F}}_0\) geometry corresponding to relativistic Toda chain [50], and in [51, 52] for its fourdimensional/nonrelativistic limit. Further evidence we provide in this paper is that the Xcluster variables of the 5d quiver flow in the 4d limit to the Xcluster variables of the corresponding fourdimensional BPS quiver, which are known to be related to the Voros symbols, i.e. exponential of the exact quantum WKB periods of the fourdimensional integrable system [53]. The fact the we find that standard topological string—or equivalently 5d gauge theory in the selfdual \(\Omega \)background \(\epsilon _1+\epsilon _2=0\) rather than in the Nekrasov–Shatashvili [54] background \(\epsilon _1=0\)—provides a quantization of the Calabi–Yau geometry is not fully surprising from the view point of equivariant localization. Indeed, the difference between the two cases resides in a different choice of oneparameter subgroup of the full toric action, and therefore contains the same amount of information, although under possibly very nontrivial combinatorial identities. A first instance of this phenomenon was discussed from the mathematical perspective in [55]. For the case at hand, the nontrivial relation between the two approaches is encoded in a suitable limit of blowup equations [43, 56,57,58,59].
There are several directions to further investigate.
Let us notice that, with respect to the framework of [13, 31], to define a BPS chamber one should set the precise order of the arguments of the central charges \(Z(\gamma _i)\) for all the charges \(\gamma _i\) in the spectrum. While our method efficiently computes the spectrum, at least in the tame chambers, it doesn’t point yet to a precise definition of the corresponding moduli values. This is because we still miss a link with the relevant stability conditions. Our method relies on the existence of patches in the moduli space where the topological string partition function allows finite radius converging expansions^{Footnote 8}. Let us notice that clarifying this point would prepare the skeleton of the demonstration that Kontsevich–Soibelman wallcrossing would be equivalent to the discrete equations (qPainlevé and higher rank analogues) we obtain. Moreover, the chambers we do compute are “triangular” in the sense of [60]. In this paper, it is shown that similar chambers exist for all the class \({{{\mathcal {S}}}}[A_1]\) theories: while multiple affinizations are generically involved, these coincide with the ones computed with our methods, at least in the examples we work out explicitly.
We have seen that when the Calabi–Yau geometry admits several moduli there are various inequivalent flows for the same BPS quiver, but only few of them have a realization in terms of weakly coupled Lagrangian field theories. In some cases the other flows correspond simply to Bäcklund transformations mapping solutions one into the others. This is the case, for example, for the fluxes \(T_1, T_2, T_3\) discussed in Sect. 3.3. We expect that the full solution to this system of equations will be given in terms of suitable expansions of the topological vertex^{Footnote 9} [3], while its nonperturbative completion should be given by the spectral determinant of the corresponding \(N_f=2\) spectral curve. In other cases the flows are intrinsically nonperturbative, like the flux \(T_4\) discussed in Sect. 5.3. It would be interesting to characterize the solutions of these flows in terms of supersymmetric indices of fourdimensional gauge theories [63,64,65].
We expect that the full refined topological string or equivalently the gauge theory in the full \(\Omega \) background is captured by the quantum cluster algebra. The bilinear equations in this case are expected to have a direct relation to the Ktheoretic blowup equations [24].
We have also shown that the Xcluster variables correctly reproduce the ones of the fourdimensional BPS quivers under a suitable scaling limit. It would be very interesting to further explore the relation of our results with the ones on exact WKB methods and TBA equations [56, 66, 67], possibly extending these methods to the qdifference/5d case. For the class S theories, an important rôle should be played by the group Hitchin system [68], in the perspective of its quantization [69, 70]. An interesting open question is the nature of the quantum periods in five dimensions and their relation to the cluster variables appearing in the study of qPainlevé equations.
In the fourdimensional case, the Painlevé/gauge theory correspondence [42] extends also to nontoric cases, corresponding to isomonodromic deformation problems on higher genus Riemann surfaces, see [71, 72] for the genus one case. These have a 5d uplift in terms of qVirasoro algebra [73] and matrix models [74, 75] whose BPS quiver interpretation would be more than welcome. Also the higher rank extension of BPS quiver flows and the associated taufunctions is to be explored in detail. As a first example, one can consider SU(N) Super Yang–Mills, whose spectral determinant in matrix model presentation was presented in [76]. In the one period phase, this satisfies Nparticle Toda chain equations. The corresponding cluster integrable system is discussed in [19]. More in general, our method should extend beyond the rank 1 case and qPainlevé systems, pointing to more general results about topological string partition functions and discrete dynamical systems.
Notes
 1.
This is an example of a cluster algebra structure that we will introduce more thoroughly in Sect. 3.
 2.
For the action on the roots, see Sect. 3.3.
 3.
We allow for the free abelian group to have dimension less than n, as it will typically the case for us.
 4.
To be more precise, the discrete time flows are given by a subgroup \({\mathcal {G}}_\Delta \subset {\mathcal {G}}_Q\), of automorphisms preserving the Hamiltonians. These are given by spider moves of the associated dimer model that we are not introducing here. They are very specific mutation sequences that in this work we will rather view as Weyl translations acting on an affine root lattice. For the cases that we will be concerned with in this paper, the two groups coincide and we can forget about the distinction.
 5.
We follow the usual convention that one overline denotes a step forward in discrete time, while one underline denotes a step backwards.
 6.
Other halftranslations can be analogously defined from \(T_1\) and \(T_3\).
 7.
Let us notice that in the 5d theories on a circle, one finds in general also “wild chambers” with multiplets of higher spin which can be reached via wallcrossing from the “tame” ones. It would be interesting to realize these higher spin multiplets as curves on the spectral geometry.
 8.
See also the comments at the end of the Sect. 4.2 on this point.
 9.
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Acknowledgements
We would like to thank M. Cirafici, D. Fioravanti, M. Semenyakin, and A. Marshakov for interesting discussions. Many thanks also to M. Del Zotto and P. Gavrylenko for a careful reading of the manuscript and interesting comments. The research of G.B. and F.DM. is partly supported by the INFN Iniziativa Specifica ST&FI and by the PRIN project “Nonperturbative Aspects Of Gauge Theories And Strings”. The research of A.T. is partly supported by the INFN Iniziativa Specifica GAST and by the PRIN project “Geometria delle varietà algebriche”.
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Appendices
qSpecial Functions and Nekrasov Functions
The 5d instanton partition function [77, 78] for \({\mathcal {N}}=1\) 5d SU(N) SYM with \(N_f\) fundamental flavors and Chern–Simons level \(k=0,1,\dots ,N1\) is given by a sum over Ntuples if partitions \(\varvec{\lambda }(\lambda _1,\dots ,\lambda _N)\) with counting parameter t:
The Chern–Simons factor is given by
while the matter and gauge contributions can all be written in terms of the single building block
in the following way:
The perturbative contribution is given by the following:
Here \(\left( u_i/u_j;q_1,q_2 \right) _\infty \) is the multiple qPochhammer symbol, defined by
In all the formulae above, the following notations are used, in terms of the fourdimensional gauge theory parameters:
An important property of the double Pochhammer symbol that has to be used repeatedly when solving the bilinear equations, is the following:
The full partition function Z(u; t) is given by \(Z=Z_{cl}Z_{\mathrm{1loop}} Z_{inst}\).
qPainlevé III and IV in Tsuda’s Parameterization
We provide here the choice of parameters for the cluster algebra that reproduces the qPainlevè \(III_1\) and IV equations of [30]. It turns out for this to be useful to introduce
Note that in [30] the Casimirs have a geometric meaning in terms of points blownup on \({\mathbb {P}}^1\times {\mathbb {P}}^1\).
qPIII
Let us consider first the case of qPIII. We choose as basis for the tropical semifield four independent Casimirs and parameterize in terms of them the \(y_i\)’s in such a way that (3.32) and (3.33) are satisfied, together with the correct time evolution (3.37). We choose as independent Casimirs \(a_1,a_2,q,b_0\). To match with [30], we also have to make a different choice for the time evolution parameter q,
Set
The time evolution is the following (we only write the relevant \(\tau \)variables):
leading to the bilinear equations
These differ from the bilinear equations of [30] by different overall factors of the RHS, so in principle it would seem that they are different bilinear equations. However, they are still equivalent to qPIII. If we define
we get the system of firstorder qdifference equations
which is the qPIII equation appearing in [30].
qPIV
The action of \(T_4\) on the tau function is the following:
At a first glance, these seem trilinear, rather than bilinear, equations. However, take linear combinations of (B.11) in such a way that the first term on the RHS cancels out:
We see that the second equation is now bilinear! We can repeat this procedure to obtain three bilinear equations from (B.11):
We can make these three secondorder equations in three variables by using the (3.57):
These are the bilinear equations for qPIV, with the parameterization of [30].
qPainlevé Transcendents and Autonomous Limit
Recall from Sect. 4.1 that the solution G(t; q, u, s) to the qPainlevé \(\text {III}_3\) equation
was expressed in terms of cluster Xvariables as \(G=x_2^{1}\). In terms of dual Nekrasov functions, this translates to
The dual partition functions here are defined as
where Z is the partition function of pure fivedimensional SU(2) super Yang–Mills. It is possible to study the autonomous limit of this object, following the discussion of [19] for the tau functions. One sets
sending \(\hbar \rightarrow 0\). In this limit, one can write the partition function as
so that we want to take the “semiclassical” (actually autonomous) \(\hbar \rightarrow 0\) limit of the dual partition functions, keeping the leading and first subleading term. Let us review the argument: in [7] it was shown that the saddle point of Z is the Seiberg–Witten Aperiod: let us denote this by \(a_*\). It is such that
The sum over n will be dominated by terms that are close to this saddle point: let us denote by \(n_*\) the value of n such that \(a+\hbar n\) is closest to \(a_*\): in other words, set
To expand \(Z_0^D\) around such value of n, we set \(n=n_*+k\) and we sum over k. We also invert the relation for \(n_*\),
in order to parameterize everything in terms of the UV Cartan parameter a and the IR Aperiod \(a_*\). The resulting saddle point expansion for \(Z_0^D\) is
where we defined the IR coupling constant
An analogous computation can be done for the halfinteger Fourier series:
When we take the ratio of these two expressions, the overall factors simplify, giving
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Bonelli, G., Del Monte, F. & Tanzini, A. BPS Quivers of FiveDimensional SCFTs, Topological Strings and qPainlevé Equations. Ann. Henri Poincaré 22, 2721–2773 (2021). https://doi.org/10.1007/s00023021010343
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