Quiver Asymptotics: $\mathcal{N}=1$ Free Chiral Ring

The large N generating functions for the counting of chiral operators in $\mathcal{N}=1$, four-dimensional quiver gauge theories have previously been obtained in terms of the weighted adjacency matrix of the quiver diagram. We introduce the methods of multi-variate asymptotic analysis to study this counting in the limit of large charges. We describe a Hagedorn phase transition associated with this asymptotics, which refines and generalizes known results on the 2-matrix harmonic oscillator. Explicit results are obtained for two infinite classes of quiver theories, namely the generalized clover quivers and affine $\mathbb{C}^3/\hat{A}_n$ orbifold quivers.


Introduction
The AdS/CFT correspondence gives an equivalence between four dimensional gauge theories and ten dimensional string theories [15]. Generalizations of the correspondence involve four dimensional quiver gauge theories [6] and a six-dimensional non-compact Calabi-Yau (CY) space in the transverse directions. The dictionary between N = 1 four-dimensional quiver gauge theories and the Calabi-Yau geometry has been developed, in the case of toric CY [9], using brane tilings [11,8,13].
Chiral gauge invariant operators, which are annihilated by supercharges of one chirality, play a central role in identifying the CY space for a given quiver gauge theory. These operators form a chiral ring and their expectation values are independent of the positions of insertion of the operators [3].
The combinatorics of the chiral ring at non-zero super-potential has been studied using generating functions and Hilbert series in the plethystic program [7,2,12]. The asymptotics of the counting formulae have also been studied [5,2,7,14]. These studies have primarily used one-variable methods appropriate for special cases of the asymptotics, although a few results in the multi-variable case are also available [5,14].
General results on the counting of chiral ring operators in the free limit of zero superpotential were obtained in [18,16]. The generating function for chiral operators in this free limit is an infinite product of inverse determinants involving the weighted adjacency matrix of the quiver diagram. This weighted adjacency matrix is a function of multiple variables associated with the edges of the quiver, and corresponding charges. The asymptotic behaviour of the free quiver counting in the limit of large charges naturally requires multi-variate complex analysis. In this paper, we will study the asymptotics of these generating functions using the methods of multi-variate asymptotics recently developed by Pemantle and Wilson [20,21]. These methods allow systematic algorithmic derivation of asymptotics associated to a rational generating function in several variables, and apply directly to the generating function of chiral operators.
We find that the asymptotic counting of the chiral operators for any free quiver gauge theory is given by a compact general formula (20). To make this general formula more explicit for different quivers requires, at present, symbolic computer tools. For two infinite classes of examples, we analytically derive the explicit asymptotic results. These two classes are the generalized clover quivers and the affine C 3 /Â n orbifold quivers. These results are the asymptotic formulas (24) and (39).
The rest of the paper is organized as follows. In Section 2, we define a thermodynamics of the chiral operators in a free quiver gauge theory, based on the counting problem of the chiral operators. We discuss associated phase transitions and a generalized Hagedorn hypersurface related to the asymptotic analysis. In Section 3, we present an adapted version of multivariate asymptotic techniques for our quiver gauge theory problem. In Section 4, we apply the developed method to two (infinite) classes of examples and obtain explicit results for the asymptotics. Section 5 discusses some possible directions for future studies.

Thermodynamics of Chiral Ring in Free CY3 Quiver Theories
In this section, we will consider generating functions for the counting of chiral ring operators in free quiver gauge theories as generalizations of thermal partition functions in AdS/CFT. We will define a generalized Hagedorn hyper-surface. We will observe that it controls the asymptotics of these generating functions, a point which will be developed in more detail in subsequent sections.
In the context of the Anti-de-Sitter/Conformal Field Theory (AdS/CFT) correspondence, type IIB string theory on AdS 5 × S 5 is dual to N = 4 Super Yang-Mills (SYM) theory on four dimensional Minkowski space R 1,3 . Conformal field theories have a symmetry of scaling of the space-time coordinates. Their quantum states are characterized by a scaling dimension, which is the eigenvalue of a scaling operator. In the AdS/CFT correspondence, the scaling operator of the CFT corresponds to the Hamiltonian for global time translations in AdS [23]. The eigenvalues of this Hamiltonian are the energies of quantum states obtained from the quantum theory of gravity in Anti-de-Sitter space.
For free CFTs in four space-time dimensions, the scaling dimension for any scalar field is 1. For a composite field (also called composite operator), which is a monomial function of the elementary scalar fields, the scaling dimension is the number of constituent scalar fields.
The thermal partition function of the AdS theory is a function of β, the inverse temperature, given by (1) For a system with a discrete spectrum of energies, as in the case at hand, this is a sum over energy eigenvalues where a(E) is the number of states of energy E. For a free CFT, the partition function becomes where x = e −β , a r is the number of composite fields with r constituent elementary scalars and r is being summed over the natural numbers. In a case where we have multiple types of scalar fields, as in quiver gauge theories, the above partition function can be generalized to a multi-variable function where x i = e −β i and a r 1 ,r 2 ,··· is the multiplicity of composite operators with specified numbers r 1 , r 2 , · · · of the different types of scalar fields. The multivariate generating functions (4) are refined versions of the partition function (3). We will refer to r i as charges and the variables x i as fugacity factors for each field. It is convenient to introduce the following vector notation β = (β 1 , β 2 , · · · ), r = (r 1 , r 2 , · · · ), (r = |r|r, |r| = r 2 1 + r 2 2 + · · ·) and define We can then write Eq. (4) as The 1-variable partition function (3), for systems which have an exponential growth of the number of states a r ∼ e αr in the large r limit, has regions of convergence and divergence meeting at a critical β = α. The partition function converges for β > α and diverges for β < α. This type of behaviour occurs in string theory, where it is associated with the Hagedorn phase transition [10].
Similarly, in the multi-variable case, assuming that the function F (x 1 , x 2 , · · · ) has an exponential growth of the multiplicity factor a r 1 ,r 2 ,··· for large r i , then there is a hyper-surface separating convergence and divergence regions for the multi-variable partition function. This hyper-surface is given by the equation In the quiver examples we will be studying in this paper, there is indeed this type of exponential behaviour and a corresponding hyper-surface. This may be viewed as a generalized Hagedorn hyper-surface. This Hagedorn hyper-surface was studied in the case of a 2-matrix model in [1]. The 2matrix model problem is associated with a quiver consisting of a single node and 2-directed edges. We will revisit this model and consider the generalised Hagedorn hyper-surface for more general partition functions associated with quivers [18].
A quiver diagram is a directed graph G = (V, E) with a set V of nodes and a set E of directed edges; self-loops at a vertex are explicitly allowed. A quiver gauge theory has a product gauge group of the form a U (N a ), where each U (N a ) is associated with a node, and matter fields in the bi-fundamental representation of the gauge group are associated with the edges. The interactions between the matter fields are described by a superpotential W which is a gauge-invariant polynomial in the matter fields. In our study, we focus on the zero superpotential case W = 0.
The observables of quiver gauge theories are gauge invariant operators and their correlation functions. An interesting class of observables is formed by chiral operators, which form a ring, called the chiral ring. For the definition and properties of the chiral ring, see for example [3,12]. The space of chiral operators in the zero-superpotential limit is typically much larger than that at non-zero superpotential.
The generating function for the chiral operators in the large N limit in an arbitrary free (with zero superpotential W = 0) quiver gauge theory was derived in [17,18,16] and is given by where we have introduced fugacity factors x = (x 1 , x 2 , ..., x d ) and charges r = (r 1 , ..., r d ) associated with quiver edges. The weighted adjacency matrix of the quiver diagram has been denoted by The multivariate generating function (8) is an example of Eq. (6), i.e. a refined version of the partition function (3). The degeneracy a r is the number of chiral operators with charge vector r in the chiral ring of the free quiver gauge theory. In this paper, we will be interested in the asymptotic behaviour of a r for large (r 1 , r 2 , · · · , r d ).
In order to study this asymptotics, with the methods of [19], [20] and [21], it will be useful to write H as a ratio where In fact, we observe that, with the parameterization x = exp (−β), F (x) is convergent in the domain H(x) ≥ 0 for small enough (and positive) x i > 0 for all i . The boundary between the convergence and divergence domains is the phase transition hyper-surface, characterized by H(x) = 0. We will see some examples of this in Section 4. The asymptotic regime of the generating function is obtained by approaching the H = 0 hyper-surface inside the domain of convergence. Thus this hyper-surface controls the phase structure of the theory and also determines the leading asymptotic behavior of a r . This will be developed in Sections 3 and 4.

Entropy
The logarithm of the degeneracy a r is the thermodynamic entropy S r := log a r . For convenience, sometimes we consider the leading term of the entropy, which we denote by S * r . Following general result (7), in the chiral ring of the free quiver gauge theories, the Hagedorntype transition can be seen as a result of the competition between the leading term of the entropy S * r obtained from the logarithm of the multiplicity and the temperature term −β · r in the generating function (8).
The generalized Hagedorn hyper-surface is given by Using Eq. (10), the critical couplings can be simply obtained as β i = ∂ i S * r . We will see explicit equations for this hyper-surface in some classes of examples in Section 4.

Method of Asymptotic Analysis
In this part we adopt a novel technique of asymptotic analysis of the multivariate generating functions, and apply it to the counting problem for the corresponding quiver gauge theories. First, we review some known material from the asymptotic analysis of multivariate generating functions and then in the second part, we present an ongoing development on the evaluation of some Hessian determinant specified at some critical points. In the third part, the phase structure of the quiver theories is explained and the relations between the entropy and critical couplings are discussed.

Multivariate Asymptotic Counting
In this section, we answer the question of asymptotic counting for the multivariate generating functions that appear in the chiral ring of the quiver gauge theories. We will not review the details of the proofs from multivariate asymptotic analysis in this article and only present the main result in the following. For a comprehensive presentation of such analysis see [19], [20] and [21].
We now briefly summarize the general results for asymptotics of multivariate generating functions obtained by Pemantle and Wilson [20], applied to our situation of interest. In the present paper, we encounter generating functions that only require the smooth point analysis of [20]. We present an adapted and extended version of these results, in four steps suitable for quiver gauge theories. In the next section we apply these results to some infinite classes of examples of quivers.
The basic steps of the analysis of [20] are as follows.
(i) We consider a generating function in d variables x = (x 1 , ..., x d ), where G and H are holomorphic in some neighbourhood of the origin, and H(0) = 0.
(ii) For a given r we find the contributing points to the asymptotics in direction parallel to r.
To do this we first find the critical points of H. A smooth critical point x * is a solution of the following set of equations From general theory [21] there is a solution to these equations that has only positive coordinates and which is a contributing point. Generically, this is the unique solution to the critical point equations. Furthermore all other contributing points, if they exist, lie on the same torus. with elements H ij = ∂ 2 φ ∂x i ∂x j . The diagonal and off-diagonal matrix elements can be written explicitly in terms of g: Thus we can write the Hessian matrix as The above holds for all values of the variables. We are only interested in the evaluation of the Hessian determinant at each contributing point, and this allows further simplification which we now carry out. The critical equation where we omitted the star for the critical points for simplicity. It implies the following relation for any j = 1, ..., d − 1: Putting this relation together with (12), we obtain Applying identity (17) to the Hessian matrix (14) we obtain the elements of the Hessian matrix evaluated at critical points, The computation of depends on the relative positions of the loops i, j and d in the quiver diagram.
(iv) The final step is to derive the asymptotic formula for a r from the critical points and Hessian determinant. The Cauchy integral formula yields where the torus T is a product of small circles around the origin in each coordinate and In the asymptotic regime |r| → ∞, the following smooth point asymptotic formula is obtained by Pemantle and Wilson, see Theorem (1.3) in [20]. We write simply x for x * (r).
The smooth point formula states that if G(x) = 0 then where x −r := d j=1 x −r j j . The expansion is uniform in the directionr := r/|r| provided this direction is bounded away from the coordinate axes.
We want to apply the above procedure in our case of interest. For any connected quiver with generating function (8), we have functions H(x) and G(x) as in Eq. (9). For x sufficiently close to the origin, G and H are holomorphic and H does not vanish at the origin.
Our next observation is the rediscovery of a folklore result in graph theory [4], that H(x) in Eq. (9) can be expanded graphically in terms of the loops in the quiver, where l 1 , l 2 , ..., l k are the loops which meet each node of the quiver diagram only once, d is the total number of the loops of the quiver diagram, and loops in the second sum are disjoint. To summarize, given a quiver diagram, one can easily find the function H and solve the critical equations to obtain the critical points. Then, by computing the Hessian determinant evaluated at critical point and inserting these results into Eq. (20), one can obtain the asymptotic for any quiver diagram. However, owing to the multidimensional nature of the problem, some of the computations, such as solving the critical equations and computing the Hessian determinant, require symbolic mathematical software. On the other hand, as we will present in this paper, alternatively, one can try to find an explicit analytic form of the asymptotic formula for some infinite classes of quivers, with the hope of finding a general analytic result for larger classes of quivers.

Some Infinite Classes of Quivers
Having introduced and discussed a general procedure of the asymptotic methods for quiver diagrams, in this section, we implement these methods in two infinite classes of examples and obtain explicit analytic results for the entropy and phase structure of these quiver gauge theories.

Generalized Clover Quivers
As the first class of examples, we consider a generating function of the following form: with This is the Generalized Clover Quiver class, see Fig. 1. It is interesting to notice that the d-Kronecker quivers, consisting of d loops, shown in Fig. 2, have also the same generating function.
It is easy to observe that the critical points are x * = ( r 1 R , r 2 R , ..., r d R ), with R = d i=1 r i . In the asymptotic limit all r i → ∞ while the ratio r i /R is kept fixed and bounded away from Hence we can choose a polydisk with radii f i slightly more than λ i . Then for all k ≥ 2, i f k i < 1 and so each factor in the product defining G is analytic. Thus F = G/H has the desired form as a quotient of analytic functions in an appropriate polydisk.
To compute the Hessian, we observe that x i is linear and so the second partial derivatives are zero. Thus from (18) we have By changing to the variables r i /r d the determinant of this matrix is easily computed to be By implementing the above explicit formula for the determinant of the Hessian matrix, in Eq. 20, it is straightforward to obtain the asymptotic result. By using Eq. (20), for the asymptotics of a r of the generating function (22), in the "central region", as R → ∞ and r i /r j (for each i, j = 1 to d) bounded away from zero, we obtain the following explicit asymptotic formula, The entropy of the generalized clover quiver is obtained in the following Shannon form: • Univariate Case In this part we characterize a special type of quiver whose asymptotics cannot be studied with the methods above. This is the one-variable case of the generalized clover quiver and is called the Jordan Quiver. In fact, the generating function of this type of quiver is the generating • l 1 l 2 Figure 4: Bi-Clover Quiver function of the (integer) partitions and derivation of its asymptotics is a classical problem in analytic combinatorics. The reason that the asymptotic method of this paper does not apply is that the relevant singularities in the one-variable case occur at all possible roots of unity, 1 − x i = 0, and each factor in the product contributes to the asymptotic, while for example in the two-variable case the exponential order of the contribution of 1 − x − y = 0 is higher than that of 1 − x 2 − y 2 = 0, etc.
All the cyclic graphs with no multiplicity (multiple edges) can be reduced to a quiver consisting of a single vertex and single loop. This loop variable is a product of edge variables for the cyclic graph. The simplest example of this class is the two node graph with two oppositely directed edges. The asymptotics for this class of quivers follows the asymptotics of partitions.

• Bivariate Case
In the bivariate case (d = 2), the generating function is . The asymptotics for a rs as r + s → ∞ and r/s, s/r are bounded away from zero, can be obtained as a special case of the computations above. Let λ = r/(r + s) ∈ (0, 1). This yields the first order asymptotic (r + s) (r+s) r r s s r + s rs .
This is uniform in λ as long as it stays in a compact subinterval of (0, 1) (alternatively, the slope r/s lies in a compact interval of (0, ∞) -note that r/s = λ/(1 − λ)). In particular for the main diagonal r = s, corresponding to λ = 1/2, we obtain The exact value of G at the critical point is not completely explicit, being given by an infinite product. It is a positive real number greater than 1, since each factor satisfies those same conditions. The minimum value of G(λ) over all λ occurs when λ = 1/2 and equals the reciprocal of the Pochhammer symbol (1/2; 1/2) ∞ . This has the approximate numerical value 3.46275. The value of G(λ) approaches ∞ as λ → 0 or λ → 1.

Phase Structure
We start with the simplest example of the class, which is the bivariate clover quiver. Following the discussion in section (3.2), the phase transition line in this example is 1 − x 1 − x 2 = 0 and the phase diagram is depicted in Fig. 5. In the unrefined case x 1 = x 2 , we obtain the critical coupling β * = log 2. In the unrefined case of the generalized clover quiver, we have β * = log d. Similarly the phase diagram of the other examples in this class is a hyper-plane obtained from H(x 1 , ..., x d ) = 1 − d j=1 x j = 0. Using Eq. (25), up to leading order, the entropy and couplings on the critical hypersurface are obtained as Notice that critical points obtained as above are the same as the solutions of critical Eqs. (12) in the case of generalized clover quivers.

• Multivariate Cases
The cyclic graphs with multiplicity are the generalized clover quiver where the number of the loops in the quiver is determined by the number of the cycles in the cyclic graph. Their generating functions reduce to the generating functions of the multivariate linear generating function. This class appears in the generating function of C 3 and conifold (C) quiver gauge theories, as the trivariate and quadravariate cases, respectively.
• Conifold C This is an special case of the generalized clover quiver with four variables. The determinant of the adjacency matrix of the conifold is where l 1 , l 2 , l 3 , and l 4 are product of edge variables in the conifold quiver, see section (2) of [18]. By direct computation, the critical points and Hessian determinant with the choice of l d = l 4 are where R = 4 i=1 r i . The asymptotic and the dominant terms in entropy can be obtained from the result for the generalized clover quiver, where • Hirzebruch F 0 and del Pezzo dP 0 (C 3 /Z 3 ) As other examples of this class we can mention Hirzebruch F 0 and del Pezzo dP 0 . The generating function of Hirzebruch F 0 is the 16 loop variable case of the generalized clover quiver and del Pezzo dP 0 is the generalized clover quiver with 27 loop variables. Their asymptotics can be obtained as a special cases of Eq. (24).

Affine C 3 /Â n Orbifold Quivers
The next inf inite class of examples consists of the affine C 3 /Â n orbifold quiver theories, see Fig. 6. The first observation is that by Eqs. Thus the generating function of this quiver can be written as First, we choose g function or x d as the central loop denoted by x c . We denote the other loops in the quiver by x i for i = 1, ..., n. We claim that the critical points and Hessian determinant evaluated at critical points are where r i /r c and r c /r i are fixed and bounded away from zero as r i , r c → ∞. The proof of the critical points satisfying H = 0, follows from the following identity for elementary symmetric function e i (r 1 , ..., r n ), r n−i c e i (r 1 , ..., r n ).
For the proof of the other critical equations, r × ∇ log H = 0, first observe that and denoting the g function of the quiver with n surrounding loops by g n , we observe that ∂ j g n = −g n−1 . Notice that critical points obtained as above are the same as the solutions of critical equations Eqs. (12), presented in Eq. (32).

Conclusions and Forthcoming Research
In this work we introduced a new method from analytic combinatorics to study the asymptotic limit of the gauge invariant chiral operators in quiver gauge theories. In fact, we adapted recent results in the multivariate asymptotic analysis of generating functions to study the asymptotic limit of generating functions associated to quiver diagrams. Then, we obtained an explicit asymptotic formula for two infinite classes of examples. Our next step is to continue in this direction and consider more classes of examples, e.g. infinite classes of orbifolds such as C 3 /Z n and C 3 /A n , ADE class, L a,b,c , Y p,q Sasaki-Einstein spaces, etc.
In this work we formulate a novel approach to the problem of asymptotic counting of multi-trace chiral operators in an arbitrary free quiver gauge theory. One of our ambitious future goals is to formulate the solution of the problem in a straightforward way such that one can read the explicit asymptotic formula from the adjacency matrix of the quiver. In other words, our final goal is to obtain general results for general quiver gauge theory with arbitrary quiver diagram, in which the asymptotic of any quiver can be written in terms of the geometry of the graphs, edges, vertices and loops of the graph.
In this study, we did not discuss the quivers with multiplicity, i.e. multiple edges between two nodes, except the Kronecker quiver which is reducible to the non-multiple generalized clover quiver. The (irreducible) multiple quivers can be considered with similar methods to those used in this article, and their asymptotic results can be obtained as a straightforward generalization of the current results.
In the asymptotic analysis of this work, we only considered the first (leading) asymptotic term in the asymptotic series. The current level of technique in analytic combinatorics for the asymptotic counting of multivariate generating functions [22] allows for computation of the higher correction terms and subleading contributions in the asymptotic analysis of quiver gauge theories. Using such result, one can find higher order corrections.
In this work, the matter content of the gauge theory is restricted to matter in bifundamental representations, however, one can consider fundamental matter and quiver gauge theories with flavours. The generating functions for counting gauge invariant operators in quivers with flavours are studied and obtained in [16]. Thus, it is straightforward to generalize the method of this work to consider quivers with flavours and obtain the asymptotic counting of chiral operators in these quiver theories. One of the specifications of this study was to consider the zero superpotential limit and using the advantages of having a general formula for the generating functions of the multi-trace chiral operators. However, as we mentioned, the W = 0 sector of the quiver gauge theories and the counting of the chiral operators have been studied vastly. The generating function technology, such as Hilbert series, plethystic exponential/logarithm introduced and applied successfully in this context. Thus, it is desirable to study the asymptotic of the multivariate generating functions in this sector, by means of the multivariate asymptotic analysis in analytic combinatorics.
In this work, we mostly discussed the mathematical aspects of the asymptotic counting of the holomorphic gauge invariant chiral operators in free quiver theories. Certainly, more physics oriented discussions, towards the physical applications of the obtained results in terms of the entropy and phase structure of the quiver gauge theories and their implications for the full quiver gauge theory and black holes, are highly desirable.

Acknowledgement
The research of S.R. is supported by the STFC consolidated grant ST/L000415/1 "String Theory, Gauge Theory & Duality" and a Visiting Professorship at the University of the Witwatersrand, funded by a Simons Foundation grant to the Mandelstam Institute for Theoretical Physics. A.Z. is supported by research funds from the Centre for Research in String Theory, School of Physics and Astronomy, Queen Mary University of London, and National Institute for Theoretical Physics, School of Physics and Mandelstam Institute for Theoretical Physics, University of the Witwatersrand. We are deeply grateful to Robert de Mello Koch for valuable discussions and collaboration in the early stages of this work.