Exact WKB Analysis of N = 2 Gauge Theories

We study N = 2 supersymmetric gauge theories with gauge group SU(2) coupled to fundamental flavours, covering all asymptotically free and conformal cases. We re-derive, from the conformal field theory perspective, the differential equations satisfied by omega deformed instanton partition functions. We confirm their validity at leading order in one of the deformation parameters via a saddle-point analysis of the partition function. In the semi-classical limit we show that these differential equations take a form amenable to exact WKB analysis. We compute the monodromy group associated to the differential equations in terms of deformed and Borel resummed Seiberg-Witten data. For each case, we study pairs of Stokes graphs that are related by flips and pops, and show that the monodromy groups allow one to confirm the Stokes automorphisms that arise as the phase of the deformation parameter is varied. Finally, we relate the Borel resummed monodromies with the traditional Seiberg-Witten variables in the semi-classical limit.


Introduction
For some time now, we have been able to compute the low-energy effective action of N = 2 supersymmetric gauge theories in four dimensions. In [1,2], the solution for the low-energy theory was given in terms of an algebraic curve and an associated differential. Subsequent works have simplified and clarified many aspects of the Seiberg-Witten solution. The Seiberg-Witten curves may be intuitively pictured in terms of M-theory five-branes [3], and this geometric picture has inspired a description of class S theories in terms of punctured Riemann surfaces [4]. In a parallel development, it has also become possible to compute instanton contributions by invoking the powerful machinery of equivariant localization [5]. Of particular note, the calculation of the gauge theory partition function on S 4 via localization naturally incorporates these instanton sums [6]. All these developments were key to writing a dictionary between observables in four-dimensional gauge theories and those in two-dimensional conformal field theories: the 2d/4d correspondence [7].
The 2d/4d correspondence makes it possible to use the technology of conformal field theory to gain deeper insights into the behavior of N = 2 gauge theories. For instance, the Ω-deformed gauge theory partition function with a surface operator insertion maps to the meromorphic solution of a null vector decoupling equation [8,9]. Thus, an analysis of conformal blocks in two-dimensional conformal field theory yields information about surface operators in gauge theories. These conformal blocks can be viewed as solutions to Riemann-Hilbert problems specified by a differential equation with singularities and associated monodromies [10]. We expect this exact picture to be valid in gauge theory (and the field theory limit of topological string theory) [11].
In this paper, we study quantum chromodynamics with N = 2 supersymmetry and gauge group SU (2), and the corresponding Virasoro conformal blocks. In particular, we study the differential equation that the instanton partition function with surface operator insertion satisfies. This corresponds to an analysis of null vector decoupling equations in the presence of irregular blocks. The differential equations satisfied by correlators involving irregular blocks were described in [11][12][13]. The equations are exact in the Ω-deformation parameters ( 1 , 2 ), and provide for a map to standard gauge theory expressions for the Seiberg-Witten curve, including i corrections.
We then concentrate on the limit 2 / 1 → 0 [14], which is a large central charge limit in the conformal field theory. It has been shown in e.g. [15][16][17] that a WKB analysis of the null vector decoupling equations in this semi-classical limit reproduces the non-convergent 1 -expansion of the instanton partition function of the gauge theory. 6 There is a rich literature [20][21][22][23][24][25][26][27][28][29] on methods which may be used to enhance these results non-perturbatively. Using the exact WKB analysis, we study the resulting differential equations satisfied by the (ir)regular conformal blocks (equivalently, the 1 -deformed surface operator partition function). This allows us to compute the monodromy group of each of the differential equations as a function of (i) the parameters of the differential equations, and (ii) the Borel resummed monodromies that are properties of individual solutions. The monodromy group contains information about the instanton partition function with surface operator insertion, which is non-perturbative in 1 . In doing so, we provide the underlying exact picture [10] with a detailed description of how these beautiful and abstract mathematical constructs reduce to the more hands-on limiting analysis of N = 2 gauge theories to which we have become accustomed.
In this physical set-up, we apply the theorems of [29], thereby drawing on intuition from both gauge theory and the mathematical study of singular perturbation theory [27]. As a by-product, we add details to the WKB analysis and provide a calculation of the monodromy group of the differential equation in terms of deformed gauge theory data. For instance, we analyze the occurrence of a double flip, consisting of simultaneous single flips. Two different ways of splitting the double flip into two single flips give the same monodromy group and Stokes automorphism. Although we demonstrate this result in the context of N f = 4 theory, this is a new result in the exact WKB method and we believe it is valid in a more general context.
In [30], a WKB analysis of the Hitchin systems corresponding to circle compactifications of undeformed SU(2) gauge theories was undertaken. Our work may be viewed as an alternative route to the WKB analysis, which is closely related to [30] at zeroth order in 1 .
Our broader goal is to communicate the extreme generality of the correspondence between 1deformed N = 2 gauge theories -specifically, their instanton partition functions with surface operator insertions -and certain Schrödinger equations amenable to exact WKB analysis. As a first step, we show the extent to which the program applied to pure N = 2 super Yang-Mills in [31] generalizes to theories with matter.
We will now briefly present the structure of our paper. In section 2, we present a derivation of the null vector decoupling equation satisfied by the five-point conformal block with a light degenerate insertion, which has a null vector at level two. We apply the collision procedure of [32] to produce irregular conformal blocks and derive the null vector decoupling equations satisfied by the limit blocks. We then consider the semi-classical limit (of infinite central charge) of these differential equations. These equations will be the starting point for the exact WKB analysis of section 3. In this section, we briefly review the exact WKB approach, and in section 4 apply it to the calculation of the monodromy groups of our differential equations. We make contact with the standard undeformed Seiberg-Witten perspective in section 5 and end with comments and future directions for work in section 6. The appendices collect details regarding the derivation of the 2 -exact differential equations for the asymptotically free theories, and an independent check of the semi-classical differential equations via the saddle-point analyses of Nekrasov partition functions [34].

The Conformal Field Theory Perspective
In this section, we present the null vector decoupling equation satisfied by the five-point conformal block with one degenerate operator insertion. We then list the corresponding equations satisfied by irregular blocks that arise when punctures collide [32]. We study these equations within the framework of conformal field theory, and finally, exploit the fact that these conformal blocks also capture the i -deformed instanton partition function of N = 2 supersymmetric gauge theories in four dimensions with SU(2) gauge group and a varying number of flavours [5]. We thus lay the groundwork for further analysis of these partition functions, which will be non-perturbative in the deformation parameter 1 . For completeness, we provide the details of the derivation of all these equations in appendix A. We start our analysis by considering regular conformal blocks with four ordinary primary operator insertions on the sphere and one degenerate operator insertion with a null vector at level two, which remains light in the limit of large central charge. On the gauge theory side of the 2d/4d correspondence, this set-up corresponds to the conformal N f = 4 case. To get asymptotically free (lower N f ) theories, we sequentially collide primary operators on the sphere in such a way that they generate irregular conformal blocks [32]. The case of three flavours will correspond to one irregular block, the case of two flavours can correspond to either one or two irregular blocks, while a lower number of flavours corresponds to two irregular blocks in the conformal field theory. For all these collision limits, we give the corresponding null vector decoupling equations.

The Five-Point Block
We study a conformal field theory with central charge We consider a five-point chiral conformal block Ψ with four primary operator insertions V α i and an insertion of a degenerate field Φ 2,1 (z) of the Virasoro algebra [9]: The degenerate field Φ 2,1 has conformal dimension ∆ 2,1 while the conformal dimensions of the generic primaries are denoted ∆ α i . We have chosen the degenerate insertion such that it remains light in the limit of large central charge 2 / 1 → 0. The degenerate field Φ 2,1 has a null vector at level two, and consequently satisfies the null vector condition where the operator T (z) is the holomorphic stress tensor of the conformal field theory. Using the operator product expansion between the stress tensor and the primary fields, the second term can be written as: : Imposing global conformal invariance allows us to express the derivatives with respect to z 1 , z 3 and z 4 in terms of the derivatives at z 2 and z. Then, setting the insertions to be at (z, 0, q, 1, ∞), the null vector decoupling equation takes the form The null vector decoupling on the five point conformal block was also studied in [9,33]. The conformal dimensions ∆ i of the primary fields V α i can be written in terms of the momenta α i as We further parameterize the momenta α i in terms of the four masses m i : As a function of the masses, the conformal dimensions are (2.9) In terms of these variables that are appropriate for comparison to the four dimensional gauge theory, the null vector decoupling equation for the N f = 4 theory takes the following form:

The Null Vector Decoupling Equations for Irregular Blocks
We now take limits of the five-point null vector decoupling equation (2.6) in which various primary operators V α i collide to form irregular conformal blocks of order one [32]. These limiting configurations are in direct correspondence with the i -deformed SU(2) gauge theories with N f < 4. We list below the null vector decoupling equations for each of these cases and refer to appendix A for a detailed derivation. A summary of these equations can also be found in [11]. N f = 3 : In this case, we have one irregular block of order one with a fourth order pole at z = 0. In the gauge theory variables, we take q → 0 and m 2 → ∞, keeping the dynamical scale Λ 3 = q m 2 finite. The resulting differential equation is: N f = 2 : There are two ways to reach the case with two flavours from the case with three flavours. One could decouple either the flavour with mass m 1 or one of those with masses m 3,4 . As shown in [30], these lead to inequivalent Hitchin systems and give rise to distinct differential equations.
Let us first consider the irregular block of order one with a third order pole at z = 0. This corresponds to decoupling m 1 . We refer to this as the asymmetric configuration and the associated null vector decoupling equation becomes: Alternatively, one can consider two irregular blocks of order one, with equal fourth order poles. This corresponds to decoupling m 3 while keeping m 1 and m 4 finite. We refer to this as the symmetric configuration and the associated null vector decoupling equation reads: N f = 1 : We consider two irregular blocks of order one with one fourth order pole and one third order pole. This corresponds to decoupling m 4 and the null vector decoupling equation takes the form N f = 0 : Finally, we consider the case with two irregular blocks of order one with equal third order poles. All masses have been decoupled and the null vector decoupling equation becomes This completes the list of six differential equations that we refer to throughout.

The Semi-Classical Limit
In the rest of our paper, we will concentrate on the limit 2 / 1 → 0, which is a large central charge limit. We keep the ratio of the mass parameters m i and the deformation parameter 1 fixed. In this limit, the primary insertions V α i are heavy, while the degenerate insertion Φ 2,1 is light. Thus, in this limit, the differential equation (2.6) simplifies, and we can drop the term proportional to ∂ z , while the terms proportional to the conformal dimensions ∆ α i grow large. To simplify the equation further, we must specify the leading dependence of the q-derivative of the five-point block on 2 .
To that end, we make the semi-classical 2 → 0 ansatz (2.14) We suppose that the q-derivative of the remaining function ψ(z, q) is sub-dominant in the small 2 / 1 limit, and observe that the leading dependence in 2 is only on the cross-ratio q of the heavy operators. We then define the quantity The parameter u is identified with the Coulomb modulus of the gauge theory up to shifts that depend on the masses. Substituting this parameterization into the null vector decoupling equation and taking the semi-classical limit 2 → 0 leads to the Schrödinger equation where the potential function Q has an 1 expansion which terminates at second order The coefficient functions are .

The Semi-Classical Irregular Blocks
The same type of ansatz (2.14) can be used in order to obtain the differential equations for the irregular blocks in the semi-classical 2 → 0 limit. The variable parameterizing the Coulomb modulus is now defined as 19) where Λ N f is the corresponding strong coupling scale of the N f < 4 gauge theory. As in the conformal case, the prepotential of the gauge theory will differ mildly from F . However, what is of importance to us is the pole structure of the functions Q k (z), and we choose a parameterization that descends naturally from the conformal theory and that allows for a simple presentation of the differential equations. In the following, we present all the asymptotically free cases: • N f = 3: The Schrödinger equation which governs the 1 -deformed gauge theory is given by (2.20) • N f = 2 (asymmetric realization): The differential equation in the semi-classical limit takes the form (2.23) • N f = 0: Finally, for the pure super Yang-Mills theory, the equation reads (2.24) We have thus obtained the differential equations which we analyze in detail in section 4.

The Exact WKB Analysis of Differential Equations
In this section, we review the exact WKB approach to the analysis of differential equations and apply it to the null vector decoupling equations in the semi-classical limit. We will carry out the exact WKB analysis with respect to the small parameter 1 . Our analysis will therefore be valid to zeroth order in 2 and non-perturbatively in 1 . Below, we briefly review the salient features of the exact WKB analysis and refer the reader to [27,29] for a more comprehensive treatment of the same.

The Exact WKB Method
The differential equations that we study can be written in the form of a Schrödinger equation: We allow the function Q to have an expansion of the form For the null vector decoupling equations that we study, the only non-zero coefficient functions are Q 0 , Q 1 and Q 2 . We choose a WKB ansatz for the solution to this differential equation, which takes the form with S(x, 1 ) expanded as a formal power series in 1 as Substituting this ansatz into the differential equation, we get recursion relations governing the coefficients S k We see that the initial conditions governing the system of recursion relations allow for two possible sets of solutions to these recursion relations, as S −1 = ± √ Q 0 . We also note the crucial feature that the zeroes of Q 0 , which we call turning points, introduce branch cuts on the Riemann surface Σ on which our differential equation and its exact solutions live. Thus, in our exact WKB treatment, we introduce a new manifoldΣ, which is a double cover of the Riemann surface, and we move between sheets as we pass branch cuts that emanate from turning points, or odd order poles. From hereon, we will distinguish the choice of WKB solution by attaching to it the subscript (±). We also observe that in the 1 -expansion of S(x, 1 ), the sets of odd and even coefficients are dependent. If we define we have the relation Putting all this together, we can write down a formal expression for the two linearly independent solutions to our differential equation: This formal expression should be understood as an analytic function of x multiplying an asymptotic series in 1 :

Borel Resummation
In the exact WKB approach, it is convenient to normalize wave-functions at distinguished points of the differential equation. As mentioned earlier, in addition to the singularities of the coefficient functions of the differential equations, their zeros (turning points) also play an important role. We will normalize our solutions with respect to the turning points, i.e. choose the starting point x 0 of the integration path to be a turning point t, Formal WKB solutions are generically divergent. To remedy this, we invoke Borel resummation: a technique that constructs an analytic function whose asymptotic expansion matches the formal WKB series. The Borel transformed series is defined as Next, define the function [29] Ψ( 1 ) = ψ 0 + θ dy e −y/ 1 ψ(y) , (3.13) where θ is the line connecting a point at which the series ψ(y) converges 7 -typically, a turning point -to the point at infinity at an angle θ. If this integral exists, Ψ( 1 ) is the requisite analytic function, called the Borel sum. Notice that the Borel sum contains an angular dependence. In order to understand this better, one must appreciate that Borel sums are typically defined only in regions of the complex 1 -plane, and not throughout. These regions are bounded by Stokes lines, defined by the condition and different Stokes regions are assigned different linear combinations of a given basis of analytic solutions to the differential equation, arrived at via Borel resummation. One of the key components of the exact WKB analysis is understanding how solutions in different Stokes regions are related by analytic continuation; these often go by the name of "connection formulae". However, before we 7 To be precise, this is true for Gevrey-1 series, which in our context corresponds to the following statement. If ψ k is the kth coefficient of the asymptotic series then the series is Gevrey-1 type if growth of ψ k is bounded by ψ k ≤ AB k k! for some constants A and B. If ψ k is a function of a continuous variable, say, x ∈ C then this condition applies to the supremum of ψ k (x) in a compact subset of C.
address this transition behaviour, we will find it necessary to endow Stokes lines with an orientation.
To this end, we adopt the convention that Stokes lines are oriented away (i.e. the arrow on the Stokes line is pointing away from a turning point) if along the Stokes line. Else, the arrow points towards the turning point. Three Stokes lines emanate from a first order zero of Q 0 (x), which is also referred to as a simple turning point. Thus one end of any Stokes line is at a turning point. The other end can either be at a singularity or at a turning point. When both the end points of a Stokes line in a given Stokes graph terminate at turning points then the corresponding Stokes graph is called "critical". 8

Connection Formulae
We are now in a position to state the connection formulae. For a Stokes graph which is not critical, consider two regions U 1 and U 2 separated by a Stokes curve Γ, and consider Ψ j ± to be the Borel sums of WKB solutions in each of the regions U j . The connection formulae for the Borel sums in different Stokes domains are given by: if Re if Re (3.17) In the above connection formulae, there is an ambiguity (±) that is fixed by noting that the turning point that Γ originates from serves as a point of reference. If the path of analytic continuation crosses Γ counter-clockwise as seen from the turning point, we pick the (+) sign, and if this path crosses Γ clockwise, we pick the (−) sign. Later in this section, we will write down the Stokes matrices that multiply wave-functions; these are equivalent to the above result. The global properties of solutions to the differential equations we consider are governed by the monodromy group and the Stokes phenomena around singular points. The monodromy group of these differential equations can be expressed entirely in terms of two sets of quantities: (a) the characteristic exponents at each singular point s k , and (b) the contour integrals of S odd around branch cuts. We now parameterize the characteristic exponents conveniently.
As a system of solutions to our differential equation, we consider the WKB solutions (3.9), and define the characteristic exponents as residues of the differential: (3.18) From the null vector decoupling equations we derived in the previous section, one can check that the residues M k are linear combinations of the mass parameters of the gauge theory. As the monodromy group computations will use WKB wave-functions (3.9), we relate the residues of S odd to our characteristic exponents as 9 Res S odd (x, ) Finally, upon exponentiating this contribution, we get the multiplier that affects WKB wavefunctions: Notice that ν + k = 1/ν − k , a fact that we will use repeatedly. Since the base point x 0 will not always be a turning point, the modified connection formulae can be obtained by a composition of the contour integrals. We find it convenient to use a matrix notation to exhibit the connection formulae. As an example, let us consider analytically continuing the Borel resummed wave-functions from Stokes region U 1 to Stokes region U 2 . As shown in figure  contour crosses a Stokes line that is directed inwards to a turning point as in figure 1 (A), we find the connection formula: In the above equation, we use the notation, where γ j is an oriented curve from the base point to the turning point t j . Along a contour that crosses a Stokes line which is directed outwards from a turning point as in figure 1 (B), we have the connection formula: In the above, the +(−) sign is chosen for counter-clockwise (clockwise) crossing of the contour from one Stokes region to the other, with respect to the turning point. For more complicated contours, it is important to take into account contributions from any branch cuts and/or singularities enclosed along the closed contour from the base point to the intersection point, the turning point and then back to the base point. As a simple example of this phenomenon, let us suppose the contour chosen happens to encircle a branch cut -say between t j and t i as in figure 2 -counter-clockwise. Here, Figure 2: Encircling branch cuts the curves γ i are those that define the parameter u i . The closed contour γ ji that encircles the branch cut has a contribution of the form where from the figure it is clear that One can see that although the u i by itself is dependent on the base point, the contour integral is independent of this choice.

Contour Encircling a Turning Point
Let us make another important preliminary point regarding the choice of cycles. In order to define the monodromy group, we first choose a base point and define a basis of closed loops that encircle just the singularities. In some of the cases we encounter, there are branch cuts between turning points and singularities. In such cases, we choose the contours to also include these turning points.
In order to prove that this is consistent with the usual definition of the monodromy group, let us consider a contour that only encircles the turning point, as shown in figure 3. If we choose to normalize the wave-functions at x 0 , the wave-functions undergo the following transformation as we travel along the path: Here we have associated the matrix −iσ 1 to the branch-cut crossing, which ensures that we remain on the same sheet of the Riemann surface. It can be easily shown that for any base point that one may choose, the answer is trivial as above. If we chose the turning point itself to be the base-point, u 1 = 1 and the matrix reduces to the identity matrix. Since the net result is simply the identity matrix, in order to calculate the monodromy matrix for the contour that encircles the singularity s, one may just as well compute the monodromy of the wave-functions around the cycle that encircles both the turning point t and the singularity s. We will make use of this repeatedly in those cases in which the branch cut extends between a turning point and a singularity. It is instructive to square this situation with the solution of a differential equation near an ordinary point. It is known that any solution of a differential equation can be written as a Taylor series in the neighbourhood of an ordinary point. The radius of convergence of this solution is at least as much as the distance from the chosen point to the nearest singularity. The Taylor series solution will clearly have trivial monodromy property. Although the WKB analysis assigns a special status to turning points, from the differential equation point of view the turning point is an ordinary point. Clearly, the branch cut and the Stokes lines emanating from a turning point are artefacts of the WKB approximation and the insertion of the matrix −iσ 1 restores the fact that the turning point is an ordinary point of the differential equation.

Contours Encircling a Singular Point
Let us now consider the toy example, as shown in figure 4, where the contour encloses a singularity. 10 In order to determine the Stokes matrix at B, we need to know to which turning point the Stokes line is connected. Since this is irrelevant to the present discussion, we move on to consider the third intersection point C. This time the contour crosses a Stokes line going into t i , and the crossing is clockwise as seen from t i . Further, when this contour is completed using γ j , we see that a singularity is encircled counter-clockwise. Taking this into account, the Stokes matrix is Finally, at the fourth intersection point D, the contour crosses the Stokes line clockwise. In fact, it is very similar to the first intersection, except that now there is a singularity encircled. Consequently, the Stokes matrix is This concludes our brief review of the exact WKB analysis. We refer the reader to [27,29] for a more detailed discussion and further references.

The Applicability of the Exact WKB Analysis
The application of the exact WKB techniques depends on the precise differential equation under consideration. Before we apply the exact WKB method to the equations derived in the previous section, it is important to point out the subtleties in the applicability of this analysis. In the Schrödinger type differential equations listed in sections 2.3 and 2.4, the parameter 1 functions as the Planck's constant in the WKB approximation scheme. For our null vector decoupling equations, the potential has zeroth, first and second order terms in 1 . In order to apply the of a complete Stokes graph that has been zoomed into.
exact WKB techniques to the solution of the differential equation, the 1 -deformed potential must satisfy certain conditions. These consistency conditions not only ensure normalizability of the wave-functions at singularities but also are useful in proving Borel summability of the WKB wavefunctions.
The necessary conditions (eq. (2.8) and (2.9) in [29]) are: • If the leading coefficient Q 0 has a pole of order m ≥ 3, then the order of Q n≥1 at that pole should be smaller than 1 + m/2.
• If the pole of Q 0 (at, say z = z 0 ) is of order m = 2, then Q n =2 may have at most a simple pole there and Q 2 should have a double pole : It is easily checked that the potentials that appear in the various Schrödinger type differential equations in sections 2.3 and 2.4 satisfy these conditions.

Theorems on Stokes Automorphisms
Since all the equations listed in sections 2.3 and 2.4 satisfy the necessary conditions, the theorems proved in [29] using these conditions can be directly applied to our equations. There is however, an interesting exception and we will comment on it momentarily. In particular, the results of [29] include theorems on the Stokes automorphisms that relate WKB resummed monodromies with a given Borel resummation angle, to monodromies with another Borel resummation angle.
We will now list the relevant results from these theorems. Consider a closed curve γ on the double coverΣ of the Riemann surface Σ encircling either a singularity or a turning point. We then define the Voros symbol e Vγ as a formal power series using the integral The Borel sums of the Voros symbol are then defined as S ± [e Vγ ]. They satisfy the Stokes automorphism formula whereby we suppose a simple flip, with the critical Stokes cycle being denoted by γ 0 , and (γ 0 , γ) is the intersection number of the critical cycle with the cycle γ defining the Voros symbol. The resummations S ± are the Borel resummations of the Voros symbol on either side of (and close enough to) the critical graph. The intersection numbers are defined using the convention that, if the cycle γ 1 has the arrow pointing outwards in the positive x direction and the cycle γ 2 , which crosses γ 1 , with the arrow pointing towards the upper half-plane, then (γ 1 , γ 2 ) = +1. When we have Borel sums on either side of a pop rather than a flip, the Voros symbols (importantly, associated to closed cycles) are trivially related These two theorems govern the transformation of Voros symbols associated to closed cycles. In the next section we will frequently use results of these theorems to study global properties of our differential equations.
In the case of the conformal SU(2) gauge theory (with N f = 4 flavours) however, the extra assumptions of [29] are not always fully satisfied. In particular, in this case we find that pairs of Stokes graphs that are related by a simultaneous or double flip, excluded in [29]. When such a double flip occurs, we show that the formulae for the Stokes automorphisms derived for single flips compose without change to give the Stokes automorphism for the double flip. This is an extension of the results of [29]. We will discuss this case in detail in the next section.

The Monodromy Group
In this section, we study global properties of the differential equations derived in section 2. The differential equations are second order and hence have two linearly independent global solutions. The solutions undergo a monodromy as we analytically continue them around a singular point. The monodromies, defined up to a change of basis, form a group called the monodromy group. The monodromy group of the differential equations we consider can be expressed entirely in terms of two sets of quantities: (i) the characteristic exponents ν k at the singular points s k and (ii) the Borel resummed contour integrals of the WKB differential S odd around branch cuts, which we denote by u ij .
The connection formulae which relate the Borel resummed wave functions in the various Stokes regions are sufficient to completely determine the monodromy group associated to the relevant null vector decoupling equation. The Borel resummed exact WKB contour integrals depend on the Borel resummation angle, (equivalently, on the phase of 1 ) and undergo Stokes automorphisms as a function of these parameters. Thus, the expression of the monodromy group in terms of the resummed integrals varies, and we determine the explicit transformation rules as we pass through a critical graph. In this section, we calculate the monodromy groups, starting with the simplest case of zero flavours, with no regular singular points in the differential equation, and we end with the conformal case (N f = 4) which has four regular singular points.
We stress the fact that there is a dictionary between the Borel resummation angle θ, and the phase of the zeroth order differential which is determined by the phase of 1 in our set-up.
(See e.g. [29] for the details, which follow from the definition of the Borel sum.) We see that this dictionary is given a natural home in 1 -deformed N = 2 gauge theories. The formal dependence on the Borel resummation angle that induces the Stokes automorphism, has a physical counterpart in the dependence of all non-perturbatively resummed monodromies on the phase of the deformation parameter 1 .

A Brief Summary of our Analysis
Throughout this section, we perform the calculation of the monodromy group in a strong coupling regime. In all the examples, we will plot the Stokes graphs emphasizing the connectivity of the graphs and the choice of branch cuts; we refer to [30] for various possible sequences of Stokes graphs. To illustrate the detailed coding of the monodromy group in terms of the characteristic exponents and the resummed monodromies, as well as the ambiguity of their formal expression in terms of the monodromies, we calculate the monodromy groups associated to two distinct Stokes graphs. Equating the invariants constructed from the monodromy groups of the two graphs gives us the Stokes automorphism relating the variables in each description. We will thus find concrete descriptions of the monodromy group, as well as the Stokes automorphisms that the exact WKB parameters undergo. The Stokes automorphisms must satisfy the theorems of [29] and this fact serves as a consistency check of our analysis.

Pure Super Yang-Mills
The semi-classical null vector decoupling equation corresponding to the case of pure super Yang-Mills theory has been discussed in detail in [31]. The description was mostly in terms of variables that resulted after mapping the sphere onto a cylinder, such that the differential equation became the Mathieu equation, and the monodromy group was coded in the Floquet exponent. Below, we perform an equivalent analysis on the sphere, which will prepare us to include flavours. A WKB analysis of the Mathieu equation can be found in [41,42] and further in [43] in the context of exact WKB and the 2d/4d correspondence.
The Stokes graph only depends on the leading potential term Q 0 (z). For the pure N = 2 super Yang-Mills theory, the zeroth order term is given by (2.24) (4.1) In figure 5, we exhibit various Stokes graphs in the strong coupling region of the pure super Yang-Mills theory. 11 We first draw the critical graph 5 that has a finite WKB line connecting the turning points t 1 and t 2 . The Stokes graphs we work with are related by a flip [30] about this finite WKB line.

The Monodromy Group
In this case, there is a single independent generator of the monodromy group and we choose the contour enclosing the singularity s 1 and the turning point t 1 to be it. Consider first the Stokes graph 5(A). The contour intersects two Stokes lines; the monodromy matrix is given by Note that the matrices are written from right to left as we go around the branch cut. The final matrix encodes the overall normalization factor as we return to the base point x 0 . The variable x 11 Using the form of the N f = 0 differential as in [30], we are within the strong coupling region if we make the choice Λ = 1 and u = 1/2. A series of conformal transformations and rescalings relate the differential presented here and the one presented in [30]. At the end of this series of transformations, we are led to the choice of parameters presented in the caption of figure 5.
critical graph (A) (B) Figure 5: The two Stokes graphs of the N f = 0 case that are related by a simple flip. We also exhibit the contour used to calculate the monodromy matrix. These graphs were obtained with the parameters Λ 0 = e −i π 4 and u = −1 + i, with the critical graph observed at θ = π.
which appears there is identified with the overall monodromy around the branch cut connecting s 1 and t 1 .
We now turn to the second Stokes graph 5(B). We see that the contour intersects four Stokes lines, including two lines arising from the flip. The monodromy matrix is given by We have denoted the variables in Stokes graph 5(B) by variables with tildes since they correspond to a different Borel resummation. The monodromy matrix M A,s 1 must be equivalent to the monodromy matrix calculated on the basis of graph (B), since the monodromy (equivalence class) is a property of the exact solutions on the Riemann surface Σ.

The Stokes Automorphism
Above, we have the explicit expressions for the monodromy matrices for the two Stokes graphs. The independent Stokes variables are given by x and u 21 in graph 5(A) and the tilde-variables in graph 5(B). Using this notation, we calculate the conjugation invariant traces of the two monodromy matrices: Requiring that the traces of the two monodromy matrices match leads to the map between the parameters appearing in the two graphs: This agrees with the Stokes automorphisms derived in [31]. This is also consistent with the general analysis in [29]. Let us expand on this briefly: the two Stokes graphs lie on either side of the t 1 − t 2 flip in the critical graph 5. Since the t 1 − t 2 cycle corresponding to u 12 has zero intersection number with itself, the variable u 12 is unaffected by the flip. However, the x variable changes because the contour around the branch cut has intersection number 1 with the t 1 − t 2 cycle.

One Flavour
The Stokes graphs corresponding to the differential in the case of one flavour are determined by the corresponding zeroth order differential (2.23): (4.7) From the form of the differential, one can see that in the z-plane, there are three turning points and two irregular singularities (at z = 0 and z = 1). The WKB triangulations are given in figure 61 of [30]. We consider a particular pair that are separated by a flip 12 and draw only the corresponding Stokes graphs. The two Stokes graphs correspond to a flip about the t 2 − t 3 finite Stokes line in the critical graph (see figure 6).

The Monodromy Group
We proceed to calculate the monodromy group for both the Stokes graphs. There are two irregular singularities in the graphs and one expects two independent generators of the monodromy group. We choose the two corresponding generators of the monodromy group as shown in figure 6. The contour around the singularity s 2 is treated in much the same way as the irregular singular point in the N f = 0 case, while the singularity s 1 behaves slightly differently. Let us first consider Stokes graph 6(A) and calculate the monodromy matrices; we find (4.8) 12 These are the first and the third out of the six triangulations given in figure 61 of [30].
critical graph (A) (B) Figure 6: The critical graph, the pair of Stokes graphs related by a flip and the contours that define the monodromy group for the N f = 1 case. The parameters chosen were Λ 1 = 2, u = −1/2, and m 1 = 1, and the critical graph was observed at θ = π.
The matrix element on the extreme left in the second monodromy matrix is the naive WKB monodromy around the branch cut connecting t 3 and s 2 . This contribution x satisfies the relation, x u 12 ν 1 = 1 . Let us now turn to the Stokes graph 6(B). The monodromy matrices are given by (4.10) As before, we definex = 1 u 12 ν 1 .

The Stokes Automorphism
Now that we have the two sets of monodromy matrices, we calculate the traces of the two sets and obtain Tr (M B,s 2 ) = 1 ν 1 (ũ 21 ) + (ũ 12 +ũ 13 +ũ 23 )ν 1 .  Since there is more than one generator of the monodromy group, one can calculate higher-order invariants by calculating traces of products of the matrices. Using the Stokes automorphism, one can check that the trace of the products also coincide, thus confirming the identification of the monodromy group.

Two Flavours
In this section, we consider the SU(2) gauge theory with two flavours. We concentrate on the asymmetric configuration. The zeroth order potential function is given by (2.21) (4.14) In the z-plane, the quadratic differential has three singularities, and three turning points. One of these is an irregular singularity at z = 0. As before, we work in a strong coupling limit, where u Λ 2 2 . We consider the critical graph (see figure 7), and by a flip about the t 1 − t 3 finite WKB line, obtain the two Stokes graphs, as shown in the figure.An important difference from the earlier cases is that we have regular singularities at s 1 and s 2 .
critical graph (A) (B) Figure 7: The critical graph and the Stokes graphs for the N f = 2 case. While plotting the figures, we used a potential that is conformally equivalent to (4.14). We set Λ 2 → i, u → 1 2 , m 3 → 0, m 4 → −2. The two Stokes graphs presented were observed at θ = 2π 3 and θ = 3π 4 .

The Monodromy Group
In order to calculate the monodromy group, we first consider the Stokes graph 7(A) and determine the generators

(4.15)
In the above, x is the naive WKB monodromy around the branch cut connecting t 1 and s 3 . This contribution satisfies the relation, u 23 ν 1 ν 2 x = 1 . A similar calculation for Stokes graph 7(B), gives us the following monodromy matrices for circling the singularities

(4.19)
These equations illustrate a recurring feature: the traces of the monodromy matrices around regular singular points will always be given by the critical exponents, with no u ij monodromy factors entering the expression. This is because the Stokes lines are either all going in or coming out at such regular singular points. As a result, the relevant Stokes matrices are all either upper triangular or lower triangular, respectively. This leads to the trivial nature of the trace. The irregular singularity, on the other hand, has non-trivial structure even at the level of the simple traces.
Matching the traces between the graphs 7(A) and 7(B) leads to the Stokes automorphism relations,  This is once again as expected from the general results of [29] and the intersection numbers between the various cycles. As a consistency check on the monodromy matrices, we have also computed the traces of products of matrices, and a similar analysis as above confirms the Stokes automorphisms (4.20).

Three Flavours
We move on to the SU(2) theory with three flavours. The Seiberg-Witten differential is There are four turning points and three singularities on the z-plane. The two Stokes graphs in figure 8 are related by a flip about the t 2 − t 3 finite line in the critical graph.

The Monodromy Group
For Stokes graph 8(A), we find the generators of the monodromy group: (4.23)

The Stokes Automorphism
As in the previous examples, the Stokes automorphism can be obtained by comparing the invariants built out of the monodromy matrices. The trace of the monodromy around the irregular singular point s 3 is non-trivial and it is important that we express it in terms of independent Stokes variables. The variables are constrained by the relation 24) and similarly for theũ variables. We solve for u 34 using this relation and choose the independent variables to be u 12 and u 23 . In terms of these variables, we find that (1 + u 23 + u 12 u 23 ) .  As a check of our monodromy matrices, we computed the traces of the products of the monodromy matrices. These imply the same Stokes automorphism as above.

The Conformal Theory
We consider Stokes graphs in a strong coupling region of the conformal SU(2) theory. The zeroth order potential has four regular singular points and four turning points.
In particular, we consider the Stokes graphs corresponding to two out of the six triangulations in figure 74 of [30] (see figure 9(A) and (C)). 13 It can be seen that the two Stokes graphs are related by a double flip, a simultaneous flip about the t 1 − t 3 and t 2 − t 4 finite WKB lines in the critical graph. We realize the double flip as two alternative sequences of two single flips. We provide the corresponding intermediate graphs after the single flips and perform the calculation we have familiarized ourselves with by now.
Let us first consider the flip from Stokes graph 9(A) to 9(B ) via the t 1 − t 3 flip. The relevant Stokes graphs and contours that generate the monodromy group are given in figure 10.

The Monodromy Group
For Stokes graph 9(A), using the contours as shown in figure 10(A), the monodromy matrices are given by (4.29) 13 Our graphs are topologically equivalent to those appearing in [30]. Next we consider the Stokes graph 9(B ) and move along the contours C k around the singularities as shown in figure 10(B ). We compute the monodromy matrices:

(4.30)
We now implement the t 2 − t 4 flip to go from Stokes graph 9(B ) to graph 9(C). The relevant Stokes graphs and contours are given in figure 11. Notice that the base point of the contours in figure 11 is different from that used in figure 10, however, this is irrelevant because the monodromy group is independent of the choice of a base point. The monodromy matrices for the (B ) graph are given (4.31) Finally, we consider Stokes graph 9(C) and calculate the generators of the monodromy group (4.32)

The Stokes Automorphism for the Double Flip
The monodromy matrices obtained above by encircling the singularities in both the pairs of graphs in the sequence of flips have standard traces, since all the singularities are regular. Hence, in order to compute the transformation of Voros symbols, we compute the traces of products of monodromy matrices. We express the traces in terms of the variables u 13 and u 34 in the t 1 − t 3 flip and in terms of the variables u 42 and u 21 in the t 2 − t 4 flip. Since the entries of the matrices are a bit cumbersome, we merely present the results; here the variables in a given Stokes graph are denoted with the appropriate superscript: This gives us the Stokes automorphism relations.
Upon composing the Stokes relations from the two single flips, we obtain the desired Stokes relation for the double flip from Stokes graph (A) to (C). If we consider a counter-clockwise loop encircling both the branch cuts and the four singularities in graphs (A), (B ) and (C), we get the relation, There is an analogous condition that is given by Using these and the Stokes automorphisms for the sequence of flips, we obtain the following relations for the independent variables of the Stokes graphs (A) and (C): (4.37) Because the double flip is composed of single flips, each taking place within their own arena, the final result for the double flip is a composition of the result for single flips [29]. So far, we have implemented the double flip via a sequence of two single flips, (A)→ (B )→ (C) as in figure 9. The double flip can equivalently be implemented by a different sequence of two single flips, (A)→ (B)→ (C) as shown in figure 9. We have checked that this results in the same Stokes automorphism relations as were arrived at earlier. This is further confirmation of the rules for computing the monodromy groups and of our resolution of the double flip into two single flips.

Pops
Finally, we consider two Stokes graphs that are related by a pop rather than a flip, in the conformal gauge theory. We concentrate on the situation depicted in figure 12 (see appendix B for more details). This corresponds to the degenerate triangulations in figure 74 of [30]. The pop is expected to give rise to a trivial Stokes automorphism for our closed loop Voros symbols. We first consider Next we compute the monodromy matrix as we traverse a closed contour that goes around the singularity s 2 Around the singularity s 3 and s 4 , we find (4.40) In the above, the set of matrices clubbed together by an underbrace gives the rule for crossing a Stokes line that runs through a branch cut into the other sheet and approaches the turning point. We repeat the above exercise for Stokes graph 12(B) and the monodromy matrices around the singularities in this case are given by  Again, it can be checked that the monodromy matrices have standard traces in both the graphs. Similarly it can be checked that the product of matrices from both the graphs have the same traces. Thus, we have a consistency check on our calculation, which is the triviality of the Stokes automorphism acting on the Voros symbols of graphs related by a pop (3.34).

The Gauge Theory Perspective
We have computed the monodromy groups associated to the differential equations governing the instanton partition function with surface operator insertion in terms of the exponents ν i characterizing the singularities, and the Borel resummed monodromies u ij . The characteristic exponents are readily calculated in terms of the masses of the gauge theory using the explicit expression of the differential equation. In this section, we further relate the exact WKB parameters u ij with the Seiberg-Witten periods a and a D , which are leading order approximations. As a result, we obtain the monodromy groups in terms of the (deformed, resummed) gauge theory data. Next, we present ideas on how to exploit this information to obtain non-perturbative corrections to the prepotential.

The Seiberg-Witten Variables
We start by relating the monodromies u ij to more standard Seiberg-Witten data. In the following figures, we mark only the turning points and the singularities on the Riemann surface, and identify theα andβ cycles of the genus one Seiberg-Witten curve. For the conformal case [35], the cycles are identified as in figure 13. The cycles have a smooth limit when the masses are set to zero, i.e. when the turning points coincide with the singularities. Further, in [35], it was explicitly checked that the prepotential of the conformal theory, obtained by calculating the period integrals with this choice of cycles, matches the results from equivariant localization methods. Once this identification is made, it is possible to go down in the number of flavors sequentially, each time identifying theα andβ cycles, until we finally reach the N f = 0 theory, where we obtain agreement with the results of [31].
The Conformal Theory The Stokes automorphisms we have derived for the conformal theory then imply the following relations for the gauge theory variables between the Stokes graphs 9(A) and 9(C): (e a ) A = (e a ) C , We will comment on the meaning of these relations after we list the cycles and appropriate gauge theory variables for the asymptotically free cases.

Three Flavours
From figure 14, for the three flavours case, we find the relation u 12 ν 1 ν 2 = e a , u 23 ν 2 ν 3 = e a D . One can check that these precisely coincide with the Stokes automorphisms obtained for the pure super Yang-Mills case in [31].

The Non-Perturbative Prepotential
In this subsection, we restrict ourselves to some preliminary remarks on how to exploit the results we have obtained on the monodromy group to find non-perturbative corrections to the prepotential (parameterized in terms of the given exact WKB monodromy data). We note that the expressions e a( 1 ) and e a D ( 1 ) used above (where we have now rendered explicit the 1 dependence) refer to the exact WKB, Borel resummed 1 perturbation series. At leading order in 1 , the first term in the expansion matches the Seiberg-Witten periods a(0) and a D (0). In the 1 -deformed theory, these are corrected as a perturbation series in 1 . In the exact WKB approach, these perturbation series are Borel resummed, but the result depends on the region of Borel resummation, and the resummed expressions e a( 1 ) depend on the phase of Borel resummation, or equivalently, on the phase of 1 . Thus, the above identifications undergo the Stokes automorphisms of which we have discussed many an example. Note that when we refer to such expressions, we have in mind that they are valid with a given resummation angle.
The monodromy group itself, and in particular the gauge invariant traces of products of monodromy matrices, are exact invariants of the solution to the differential equations. In [31], it was proposed in the context of the theory with zero flavours to solve the equation for the Borel resummed period a( 1 ) in terms of the invariant quantity a exact , determined by the exact periodicity (or Floquet exponent, or monodromy) of the exact solution to the (Mathieu) differential equation. This solution is valid at a particular resummation angle, and a is identified with a exact , up to nonperturbative corrections. Near a point in moduli space where a perturbative expansion is possible, like the weak, magnetic or dyonic point, we can then solve the relation in terms of a transseries [28]. On the other hand, since in this case a D ( 1 ) is independent of the Borel resummation angle, we can posit a exact D = a D ( 1 ) (independently of the specification of the resummation angle). The variable u determining the derivative of the prepotential is known as a function of a, and therefore as a function of a exact . Inverting this relation and calculating the dual period integrals allows one to calculate non-perturbative corrections to the prepotential F.
Thus, one application of the identifications above is to attempt to integrate up the nonperturbative relation between the period a and the parameters coding the exact monodromy group, towards non-perturbative corrections to the prepotential F. The encompassing case in which to execute this program is the conformal N f = 4 theory.

Conclusions
In our work, we filled in many details of the connection between conformal field theory and fourdimensional gauge theory which was made handily available starting from the matching of partition functions and correlators in [9]. We applied the technology developed in [11][12][13]32] to study in greater detail SU(2) super Yang-Mills theories with a varying number of flavours. We thus provided a complete list of i -deformed differential equations satisfied by the five-point conformal block with surface operator insertion, in the irregular conformal block limit. Subsequently, we took a semiclassical limit and analyzed the resulting differential equations with exact WKB methods. We used this technology to give a detailed parameterization of the monodromy groups in terms of Voros symbols and external conformal dimensions or, in the language of gauge theory, in terms of Borel resummed 1 -deformed Seiberg-Witten periods and masses. The Borel resummed variables depend intrinsically on the resummation angle (and the phase of 1 ), and we illustrated how to bridge this ambiguity using Stokes automorphisms, with at least one illustration for each number of flavours.
The Stokes automorphisms we obtained are consistent with the general theorems proved in [29]. The one subtlety arose in the case of the conformal gauge theory: in this case, we analyzed a pair of Stokes graphs that were related by a simultaneous flip along two independent finite WKB lines. In such a case, we showed that the resulting Stokes automorphisms could be obtained by treating the double-flip as a sequence of single flips. We demonstrated consistency of this approach by checking that the final Stokes automorphisms between the Stokes graphs were independent of the order in which the single flips were taken.
We believe it is instructive to develop these most elementary of N = 2 models in still further detail. There are many avenues to explore. As was mentioned in the previous section, it should be possible to calculate non-perturbative corrections in 1 to the prepotential, by suitably generalizing the analysis that was done for the case of pure super Yang-Mills [31]. Similarly, it should be informative to match the cluster algebra description of Stokes automorphisms of [29] to the cluster algebra description of the spectrum of BPS states [36]. (See also e.g. [29,37,38].) In the present work, we studied general properties of the Borel resummed wave-functions of the null vector decoupling equations in the semi-classical limit, without focusing on the specific form of the wave-function. An interesting direction would be to study in greater detail the dependence of the five-point conformal block on the insertion point of the degenerate operator. This should yield non-trivial information about the gauge theory in the presence of a surface operator. Most interestingly, perturbative and non-perturbative corrections in 2 can now be studied from the differential equation point of view.
All these projects have straightforward extensions, both to the N = 2 theory, where the Riemann surface is of genus one [7], as well as to higher rank conformal gauge theories, the simplest of which is super Yang-Mills theory with SU(N ) gauge group and 2N fundamental flavours. As explained in Section 5, one not only needs the description of the monodromy group, but also requires a local expansion of the period integrals in terms of the Coulomb moduli in order to carry out the goal of calculating corrections to the prepotential that are non-perturbative in 1 . For the N = 2 theory with gauge group SU(2), the null vector decoupling equation for the toroidal block has been well studied in the semi-classical limit (see e.g. [16,45,46]) and the instanton series for the prepotential has been resummed in terms of modular functions [47]. One can therefore attempt to study the monodromy group along the lines of the present paper and hope to carry out the proposal put forward in Section 5. For higher rank N = 2 theories, there has been much progress on the gauge theory side in resumming the instanton expansion for the prepotential using modular anomaly equations in deformed gauge theories with arbitrary gauge group [48,49]. It remains an open problem to reproduce these successes using conformal field theory methods and to do the corresponding WKB analysis.
For the higher rank (and undeformed) SQCD theories, the instanton series has been resummed in a special locus with Z N symmetry [52,53]. From the CFT approach to the problem, one has to work with Toda theory [50] and the corresponding null vector decoupling equations in Toda have been analyzed recently in [51]. In the semi-classical limit, such differential equations can also be derived using a saddle point analysis of the Nekrasov integrand [34] and the resulting deformed Seiberg-Witten curve. It would be interesting to analyze these higher order differential equations using exact WKB methods and make contact with the general approach to these systems using spectral networks [54,55].

A The Null Vector Decoupling with Irregular Blocks
In this section, we derive in detail the null vector decoupling equations that involve irregular conformal blocks. These equations were summarized in a different form in [11]. Our final results are listed in section 2.

A.1 One Irregular Puncture
The case of N f = 4 is standard and is described in section 2 in sufficient detail. We turn to the conformal block involving an irregular puncture. This involves rendering one flavour very massive.
From the gauge theory point of view, the flavour decoupling is carried out by taking the limit: The parameter Λ 3 has mass-dimension one and is the strong coupling scale of the SU(2) gauge theory with N f = 3. As shown in [13], in the conformal field theory this involves a collision of the regular conformal block at z = q and z = 0 and leads to an irregular conformal block at z = 0. We now consider the five point conformal block with the insertion of the degenerate field Φ 2,1 (z): The Ward identity for this chiral conformal block can be obtained by considering the five point block relevant for the conformal N f = 4 and scaling the wave-function in the following way [11,13]: On the rescaled wave-function ψ 3 (z), the q-derivative is traded for a Λ 3 -derivative. Combining these, and taking the decoupling limit, we obtain the null vector decoupling equation for the N f = 3 theory: The quartic pole at z = 0 in the OPE between the stress tensor and the irregular block explains our notation: we denote such an irregular block by I (4) (0).
We can tune the two scales suitably in order to obtain the null vector decoupling equations for the remaining gauge theories of our focus.
A.2.1 N f = 2: symmetric realization We set Λ = − Λ = Λ 2 , in which case we obtain the null vector decoupling equation: We set Λ = Λ 1 and Λ → 0 such that Λm 4 = − Λ 2 1 4 is a finite combination. In other words, the relevant conformal block of interest is (A.14) The coefficient of the cubic pole at z = 1 is tuned and related to that of the quartic pole at z = 0. Such a conformal block satisfies the null vector decoupling equation: As for the earlier case with N f = 2 in the asymmetric realization, in order to obtain finite results, one has to rescale the wave-function ψ(z, Λ, Λ) = Λ      gauge-theoretic considerations. This will serve as a consistency check of the 2d/4d correspondence, and our calculations.
In this section, we explain how to derive differential equations starting from saddle-point difference equations, valid for any SU(N ) theory with N f = 2N fundamental hypermultiplets. We specialize to the case of the conformal SU (2) theory (N f = 4) and consider various decoupling limits that give asymptotically free theories with fewer fundamental hypermultiplets.
Our analysis begins with the Nekrasov partition function, and considers its saddle-points in the limit 2 → 0, with 1 held constant and finite [34,39,40,44]. The result of this analysis is the 1 -deformed Seiberg-Witten equation: Here, q is the instanton counting parameter. The gauge polynomial P (x) is of degree N and encodes the Coulomb moduli of the gauge theory. The flavour polynomial M (x) is of degree 2N , and we choose to factorize it into two pieces: where A(x) and D(x) are degree N polynomials. As should be evident, this decomposition is far from being unique. Different decompositions can be mapped to the different ways in which the flavour symmetry is realized in the type II construction using two NS5 branes and a stack of D4 branes. One can associate the number of semi-infinite D4 branes on each side of the NS5 branes with the degree of the polynomials A(x) and D(x). We now peel off a factor of A(x) from the function y(x) and express the remainder as the ratio of some rational function Q(x) as There are other ways to perform the split but this one will reduce, in the 1 → 0 limit, to the correct M-theory curve. The result of these decompositions gives us the Baxter T Q-relation A(x)Q(x) + qD(x − 1 )Q(x − 2 1 ) − (1 + q) P (x)Q(x − 1 ) = 0. (C.4) We now trade Q(x) for its Fourier transform [34] Ψ(t) = x∈Γ Q(x) e −x/ 1 , (C. 5) and arrive at the differential equation of order N : where in the above equation, the Fourier transform effectively sends x → − 1 t∂ t .
C.1 N f = 4 We specialize to the case of the conformal SU(2) gauge theory, with N f = 4. The matter polynomials we start with are The gauge polynomial is Substituting this into the difference equation and taking the Fourier transform as described earlier leads to a second order differential equation. We would like this differential equation to be of the Schrödinger form, i.e. with no linear derivative terms. This may be achieved by peeling off an appropriate factor. 14 The resulting differential equation is C.2 N f = 3 The N f = 3 case is obtained by decoupling one of the masses; we choose here to send m 2 → ∞, while simultaneously sending q → 0 such that the combination Λ 3 = q m 2 remains finite: We identify Λ 3 to be the strong coupling scale in the SU(2) gauge theory with N f = 3. The difference equation takes the form (C.14) 14 The terms we peel off are proportional to the products of square roots of eigenfunctions of the monodromy at 0, 1 and q.
The polynomial functions are given by 16) while the gauge polynomial is The differential equation can be obtained in a similar fashion as in the conformal case by taking the Fourier transform as in (C.5). The analysis leading to the Schrödinger type equation can be repeated as before and we obtain the differential equation: The u S we use in the saddle-point analysis must be shifted in order to make contact with the form of the potential in the text (2.20) and the shift is given by The shift of the Coulomb modulus is accompanied by a global rescaling m 1 → m 1 2 and Λ 3 → Λ 3 2 . After these shifts and rescalings, the non-zero potential functions Q m (t) are given by C.3 N f = 2: asymmetric realization There are two distinct cases to be considered when N f = 2. These correspond to the fashion in which we decouple fundamental matter. We first consider the case when m 1 → ∞ and Λ 3 → 0 with Λ 2 2 = m 1 Λ 3 finite . its form requires a series of conformal transformations and rescalings before it can be literally compared with the form in (2.24); for convenience, we reproduce these transformations below: (C.44) After this sequence of transformations, we get the potential as in the bulk of the paper (2.24): (C.45) Thus, we completed the derivation of the null vector decoupling equations (at 2 = 0) from the purely gauge theoretic instanton partition function.