Root systems, spectral curves, and analysis of a Chern-Simons matrix model for Seifert fibered spaces

We study a class of scalar, linear, non-local Riemann-Hilbert problems (RHP) involving finite subgroups of PSL(2,C). We associate to such problems a (maybe infinite) root system and describe the relevance of the orbits of the Weyl group in the construction of its solutions. As an application, we study in detail the large N expansion of SU(N) or SO(N) or Sp(2N) Chern-Simons partition function Z_N(M) of 3-manifolds M that are either rational homology spheres or more generally Seifert fibered spaces. It has a matrix model-like representation, whose spectral curve can be characterized in terms of a RHP as above. When pi_1(M) is finite (i.e. for manifolds M that are quotients of \mathbb{S}_{3} by a finite isometry group of type ADE), the Weyl group associated to the RHP is finite and the spectral curve is algebraic and can be in principle computed. We then show that the large $N$ expansion of Z_N(M) is computed by the topological recursion. This has consequences for the analyticity properties of SU/SO/Sp perturbative invariants of knots along fibers in $M$.


Introduction
Unless mentioned otherwise, the orbit space of Seifert fibered spaces in this article is assumed to be a sphere.

Scope
g denotes a semi-simple Lie algebra, and h its Cartan subalgebra, identified with R N . We shall study the probability measure over t " pt 1 , . . . , t N q P h: where the product ranges over positive roots of g, u is a positive parameter, and the partition function Z g pM q is such that´h dµptq " 1. We are interested in the A N , B N , C N , D N series of Lie algebras, in the regime where the rank N goes to 8. The model (1.1) arises in Chern-Simons theory with gauge group exppgq on a simple class of 3manifolds M called Seifert fibered spaces [38]: roughly speaking, these are S 1 -fibrations over a surface orbifold, here assumed to be topologically a sphere. (1.1) captures the contribution of the trivial flat connection in perturbative Chern-Simons theory, or equivalent, the evaluation of the LMO invariants of M in the weight system defined by g [37, 1, 2, 2, 3]. Moreover, the correlation functions in (1.1) compute the colored HOMFLY (in any representation of fixed size) of the knots going along the fibers of M , see § 4.4. The regime N Ñ 8 has aroused interest since it allows a rigorous definition of perturbative invariants ensuing from Z g N pM q and the colored HOMFLY of links in M , which should be related via geometric transitions to topological strings invariants in T˚M according to the physics literature [47], see § 1.4.
General arguments [14] show that the large N asymptotic expansion of models of the type (1.1) -once it is proven to exist using the tools of [15] -can be computed by the topological recursion of [22]. It then remains to compute the initial data pω 0,1 , ω 0,2 q of the recursion : ω 0,1 is related to the equilibrium measure of (2.1), also called spectral curve, and ω 0,2 is related to the large N limit of the 2-point correlation function. Both are characterized by a saddle point equation which takes the form of a linear but non-local Riemann-Hilbert problem (RHP) on a cut locus to determine. Addressing its solution is an important part of our present work. We shall devise in Section 3 a general method to construct the solution of a large class of RHP where the jumps are obtained by action of a finite subgroup of Möbius transformations. We actually build the -maybe infinite -monodromy group of the solution, and relate it to the Weyl group of a root system. The solution can be studied in more details by algebro-geometric means when this Weyl group is finite. This construction applies to the computation of pω 0,1 , ω 0,2 q relevant for (2.1), but has also its own interest and can be used as a tool in other problems.
Our work has two noteworthy consequences for knot theory.
Firstly, we obtain analyticity results on perturbative invariants of knots in manifolds different from the 3-sphere. The colored HOMFLY of links in S 3 are polynomials (after a suitable normalization) in q and q N . This is clear from skein theory, and can also be explained from quantum field theory perspective [48]. This implies that the coefficients in the Ñ 0 expansion while keeping q N " e u0 fixed, produce polynomials in e u0 . These properties are not expected to be true for invariants of links in other 3-manifolds. We show how to compute invariants of fiber knots in Seifert fibered spaces. Remarkably, the position of the singularities in the u " N -complex plane only depend on the ambient 3-manifold. They occur at the singularities of the family of spectral curves (parametrized by u) associated to (1.1). We conjecture more generally that for any link in a rational homology sphere M , the singularities of the perturbative invariants only depend on the ambient manifold M . More precise statements of the conjecture described in § 4.2.2.
Secondly, we find that the spectral curves and the perturbative invariants for the colored Kauffman perturbative invariants (associated to the B N , C N , D N Lie algebra) are closely related to the more conventional SUpN q invariants (A N´1 series), and we show they can all be computed by the topological recursion. The only difference is that the topological grading is not respected for the BCD cases. We think this has an interest, since very few was known so far on perturbative Kauffman invariants.

Outline and main results: matrix model
We establish in Section 2 the large N Ñ 8 behavior and asymptotic expansion of the partition function and moments of (1.1) for g " A N " supN`1q (Section 2). Some of the technical proofs are postponed to Appendix A. Section 3 is independent of the main body of the text: we introduce in (3.1) a class of linear non-local RHP, to which we associate a (maybe infinite) root system. Many information on the solution -and sometimes its full and explicit form -can be extracted from the analysis of this root system.
We review the geometry of Seifert spaces and Chern-Simons theory at large N in Section 4. An important geometric invariant of Seifert spaces is the orbifold Euler characteristic of their orbit space, here always assumed to be topologically a sphere : where a 1 , . . . , a m are integers prescribing the orders of extraordinary fibers. We also introduce: a " lcmpa 1 , . . . , a m q (1.3) Section 5-7 are devoted to the A N Chern-Simons matrix model for Seifert fibered spaces, and their extension to the Sp or SO Lie algebra is the matter of Section 8. Chern-Simons theory around the trivial flat connection depends on the single parameter: where σ P Q is another geometric parameter of the Seifert spaces. u 0 " N is sometimes called the string coupling constant. In Section 5, we apply the techniques of Section 3 to the construction of the spectral curves for Seifert spaces. We find in Theorem 6.1 that the finite quotients M " Z p zS 3 {H corresponding to χ ą 0 can be described in terms of an algebraic spectral curve S ãÑ CˆˆCˆ, whereas the spectral curve -if it exists -is never algebraic when χ ă 0. We will not insist on the cases χ " 0, which are resonant.

Proposition 1.1
The Chern-Simons matrix model for Seifert fibered spaces with χ ą 0 admits a spectral curve S of the form P px, yq " 0 for a u-dependent polynomial P whose Newton polygon is known. The coefficients on the boundary of the Newton polygon are known monomials in e´χ u{2a . Besides, the spectral curve comes with the action of a finite Weyl group on the sheets of x : S Ñ p C, as tabulated below.
Geometry pa 1 , . . . , a r q deg x P deg y P genus Weyl group Lpa 1 , a 2 q pa 1 , a 2 q a 1 a 2 a 1`a2 pa 1´1 qpa 2´1 q A a1`a2´1 This table assumes gcdpa 1 , a 2 q " 1 for the lens spaces.
We were able in Section 6: ‚ to determine completely the curves in the cases S 3 {D p`2 -corresponding to p2, 2, p´2q, in § 6.2 and 6.3.
‚ to determine the curves in the cases S 3 {E 6 (resp. S 3 {E 7 ) up to 2 (resp. 4) parameters fixed by complicated algebraic constraints.
Our spectral curve computations are supported by Monte-Carlo simulations of Alexander Weisse presented in Section 9. Bases on numerics, we also give conjectures about the spectral curve in the cases χ ă 0 (Conjecture 2.5).
Although the genus of the spectral curve can be quite large, it does not prevent them to cover in a simple way curves of a lower genus. The completely solved cases mentioned above and our computations leads us to: Conjecture 1.2 Assume χ ą 0. In the spectral curves describing the contribution of the trivial flat connection to the Chern-Simons partition function, if one eliminates x from the equations X " x a and P px, yq " 0, we obtain an equation RpX, yq " 0 describing a (in general : singular and with several components) curve of genus 0. This is true for the S 3 {D p cases. Some formulas and definitions met during those computations are collected in Appendices E-C.

Outline and main results: knot theory
In Section 7, we explain a practical consequence for knot theory: we can provide some informations on the analyticity properties of the coefficients (seen as functions of e u ) of the large N expansion of the colored HOMFLY polynomials of fiber knots in M , and of the Chern-Simons free energy. So far, they were only known to be analytic in a vicinity of u " 0 [26]. We prove in Section 7: The perturbative colored HOMFLY invariants of fiber knots in a Seifert fibered space with geometry S 3 {D p`2 for p even, defined initially as elements of Qrr ssrru 0 ss, are actually the u 0 Ñ 0 Taylor expansion of an element Qrr sspκ 2 qr a βpκqs, where: 2κ 1`1{p 1`κ 2 " e´u 0{p4σp 2 q , βpκq " pκ 2`1 qppp`1q´pp´1qκ 2 q κ 2 (1.5) u 0 ą 0 corresponds to κpu 0 q Ps0, 1r, and σ P Q introduced in (4.3) is a geometric parameter of the Seifert spaces. The singularities in the κ-complex plane occur when κ 4 P t1, pκ˚q 4 u where: κ˚" c p`1 p´1 (1.6) A result similar to Proposition 1.3, with u replaced by 2u, can be deduced from Section 7-8 for the perturbative colored Kauffman invariant of fiber knots in S 3 {D p`2 with p even. The situation is different from links in lens spaces, where the perturbative invariants are polynomials in e u [17], and thus had no singularities in the u-complex plane. Our work suggests the general conjectures: Conjecture 1.4 If M is a Seifert fibered space, the F pgq and the perturbative invariants of any link in M exist as a function of u 0 ą 0, and are real-analytic on the positive real line. Conjecture 1.5 Moreover, if χ ą 0, there exists a finite degree extension L of Qpe´χ u0{2σ q depending only on the ambient manifold and the series X P tA, BCDu of Lie algebra, such that, at least for fiber knots, the perturbative invariants colored in any representation are the Laurent expansion at u 0 Ñ 0 of elements of L.
From this perspective, for the Seifert space M " S 3 {D p with p even, one may ask for the geometric interpretation of the function κpu 0 q in (1.5), and of the location of the dominant singularity u0 " u 0 pκ˚q determined by (1.6).

Motivation from topological strings
The present work generalizes the analysis of the model r " 2 [29,17] relevant to study lens spaces, and the invariant of fiber knots in lens spaces are equal to invariants of torus knots in S 3 . Chern-Simons theory on M at large N is dual to type A open topological strings on T˚M [47], and through geometric transitions, this can sometimes be related to closed topological strings on another target space X M . This program has been completed for M " S 3 [28] and the lens spaces Z p2 zS 3 {Z p1 [29,17] and X M is obtained by fractional framing transformations from the resolved conifold in both cases. So far, it has remained elusive for general Seifert fibered spaces. Since we establish the existence of an algebraic spectral curve for all Seifert spaces with χ ě 0, this gives hope for the construction of their mirror geometries X M . This direction, and the connection with 5d gauge theories and quantum spectral curves, is under investigation [12].
A natural strategy to establish a duality to a closed string geometry would then be to prove that Gromov-Witten invariants of X M are also computed by the topological recursion, with same curve S. So far, the topological recursion is indeed known to compute Gromov-Witten invariants when X is a toric 3-fold Calabi-Yau [16,23], but this might be generalized in the future to a more general class of manifolds.

The matrix model 2.1 Equilibrium measures
We study the statistical mechanics of N particles, of position t 1 , . . . , t n P R N , with joint probability distribution: At this stage, a 1 , . . . , a r are arbitrary positive parameters. The dominant contribution to the partition function when N Ñ 8 should come from configurations t maximizing the probability density. It is reasonable to think that the empirical distribution: of such configurations will be close to a minimizer of the energy functional: among probability measures λ. E 0 is actually defined in R Y t`8u because of the singularity of the logarithm. It is a lower semi-continuous functional, so has compact level sets for the weak-* topology, therefore it achieves its minimum. We call equilibrium measure and denote λ eq any minimizer of E. It must satisfy the saddle point equation: there exist a constant C λeq such that with equality λ eq -everywhere. V λ eff ptq is the effective potential felt by a particle at position t " t i , taking into account the collective effect of all other t j 's distributed according to λ. Equilibrium measures are characterized by the property that the effective potential achieves its minimum on the locus of R where the particles accumulate. Classical techniques 6 of potential theory [43] show that the random measure L N 's limit points are equilibrium measures, and that: Z N " exp`´N 2 min E`opN 2 q˘(2.5) 6 The arguments of [43] were developed for the pairwise interaction ś iăj pt i´tj q 2 which has a zero at coinciding points, but it is easy to generalize to any interaction which has the same singularity and is smooth elsewhere.
Some qualitative properties of the equilibrium measures can be derived from their characterization: V grows sufficiently fast at infinity to ensure that equilibrium measures have compact support ; since V and the pairwise interactions are analytic (away from the singularity at x " y), it can be shown (see [15], generalizing [18] where´ln |t´t 1 | was considered) that equilibrium measures are supported on a finite number of segments, have a density which is analytic away from the edges, and is 1{2-Hölder at the edges. When the density vanishes exactly like a squareroot at the edges, it is said off-critical.
The question of uniqueness of µ eq has no easy answer. But, since the functional is quadratic, if the quadratic form over the space M 0 of differences of probability measures: is positive definite, then E is strictly concave and this guarantees uniqueness of its minimizer. Notice again that, a priori, Q takes values in R Y t`8u. Indeed, the singular part of Q is: and the right-hand side can be infinite if the measure ν is not regular enough. Here Frνs "´dνpxq e ikx is the Fourier transform of the finite mesure ν. As a matter of fact, it is enough to consider E and Q as functionals over measures with compact support, because V grows fast at infinity compared to the pairwise interactions.
Lemma 2.1 For any ν P M 0 : Q is definite positive whenever the function p2´rqγpkq`ř r m"1 γpa i kq is almost everywhere positive. In particular, it must be positive at k Ñ 8, which gives the necessary condition: In the model for Seifert fibered spaces, the a m are assigned integer values. We recognize in (2.9) the orbifold Euler characteristics of the orbit space, and the list of uples leading to χ ě 0 is finite. For r " 1 and r " 2, Q is obviously positive definite, and for the remaining r " 3 cases having χ ě 0, positivity can be checked by a direct computation.
Corollary 2.2 If a 1 , . . . , a r are integers, Q is definite positive iff 2´r`ř r m"1 a´1 m ě 0. In those cases, there exists a unique equilibrium measure.
The proof of Lemma 2.1 and Corollary 2.2 are presented in Appendix A.1 and A.2. We can also establish -see Appendix A.3 -some qualitative properties of equilibrium measures: Theorem 2.3 For any a 1 , . . . , a r ą 1, and any u ą 0, any equilibrium measureλ eq is supported on a single segment, and vanishes exactly as a squareroot at the edges (generic edge).

Corollary 2.4
If χ ě 0, since the equilibrium measure is unique, it must be invariant under t Þ Ñ´t. In particular, its support is a segment centered at the origin.
In Section 9, Weisse proposes a Monte-Carlo simulation to obtain the dominant configurations of probability densities like (2.1). It is observed that, for any value of a m integers and u ą 0, independently of the sign of χ, the empirical measure seem to have a unique limit point. It supports the Conjecture 2.5 For any a 1 , . . . , a r ą 1 and u ą 0, the equilibrium measure is unique, and its density away from the edges of the support is a real-analytic function of u ą 0.

Change of variable and saddle point equation
We introduce the exponential variables: a " lcmpa 1 , . . . , a r q,ǎ i " a{a i (2.10) The measure µ (2.1) on t P R N transforms into a measureμ on pRˆq N : and ζ a denotes a primitive a th -root of unity. λ eq denote an equilibrium measure forμ. It is the image via (2.10) of an equilibrium measure λ eq for µ. It is characterized by the saddle point equation: for some constant C, with equality on the support Γ Ď Rˆof λ eq . We shall rewrite the characterization of an equilibrium measure in term of its Stieltjes transform: W is a bounded, holomorphic function on CzΓ, such that: The contour of integration in (2.16) can be moved to pick up residues at rotations of x of order a: Since µ is invariant under pt 1 , . . . , t N q Ñ p´t 1 , . . . ,´t N q, in the case where the equilibrium measure is unique, it must have the same symmetry. This translates into the palindrome symmetry: When Q is definite positive and Γ is fixed, (2.18) characterizes W among the holomorphic functions in CzΓ satisfying (2.15) and whose discontinuity on Γ defines an integrable measure (see e.g. the argument of [14,Lemma 3.10]). When Q is not definite positive, we do not know how to address the question of uniqueness.

Asymptotic analysis
We rely on the results of [15] to study the asymptotic expansion when N Ñ 8 of the model (2.11). We would like to compute the partition functionŽ N and the n-point correlators: where conn stands for "connected expectation value". Definition 2. 6 We say thatλ eq is off-critical if its density is nowhere vanishing on its support Γ, and it vanishes exactly like a squareroot at the edges of Γ.
Theorem 2.7 [15] Assume E is strictly concave andλ eq is off-critical. Then, we have an asymptotic expansion of the form:Ž The coefficients of expansion are real analytic functions of u ě 0, and W pgq n px 1 , . . . , x n q are holomorphic functions in pCzΓq n .
In particular, this confirms for Seifert spaces, by a different method, the analyticity of Chern-Simons free energies proved in general for rational homology spheres in [26] (see § 4.2). From (2.11), we have the basic relation:

Two point function
Definition 2. 8 We call 2-point function: It can be obtained formally from W pxq " W p0q 1 pxq by an infinitesimal variation of the potentialV : It also satisfies also a saddle point equation, which can be obtained by formally applying the variation of the potential to the saddle point equation (2.18) satisfied by W pxq. The result is that W p0q 2 px 1 , x 2 q is holomorphic in pCzΓq 2 , and has a discontinuity when x 1 P CzΓ and x 2 P Γ satisfying: The data of px, W pxqq is called the spectral curve. Together with the two-point function W p0q 2 px 1 , x 2 q, it provides the initial data for the the recursive computation of F pgq and W pgq n . We give in Section 3 general principles to solve the homogeneous linear equation, that will be put in practice to compute these data in the Seifert models (Section 4-5).
3 Algebraic theory of sheet transitions

The problem
The form taken by the saddle point equation (2.16) motivates a general study of homogeneous functional relations of the type: where: ‚ G is a finite subgroup of PSL 2 pCq, acting on the Riemann sphere by Möbius transformations (their classification is reminded in Appendix B).
‚ S is a generating subset of G, not containing id, and stable by inverse. pαpsqq sPS is a sequence of numbers in a number field K (for instance K " R), and we assume: ‚ Γ is a collection of arcs on the Riemann sphere such that gpΓq X Γ " H whenever g ‰ 1.Γ denotes the set of interior points of Γ.
‚ U is an open subset of p C containing Γ and stable under the action of G, and φ is a holomorphic function on U zΓ, that admits boundary values at any interior point of Γ. For simplicity, U will be C or Cˆhere.
This problem is usually supplemented with some growth prescription for φpxq at the edges of Γ and at the boundary of U . For instance, if U " Cˆ, one can ask for prescribed singular behavior at 0 and 8. This problem can also be studied with few modifications for φ " a section of a vector bundle over U , in particular for φ " a holomorphic 1-form.
This problem for φ " 1-form 7 actually appears in the study of equilibrium measures for models of the form (2.11) with arbitrary (analytic) pairwise interactionŘ. The dependence in the potentialV only affects the right-hand side of such an equation, which can be set to 0 by subtracting a particular solution, thus affecting the growth prescriptions for the solution φ we are looking for. In general, G is the Galois group of the equationŘpx, yq " 0, and may be complicated even for simpleŘ. It may not be finite, and the description of the orbit of Γ has to deal with the rich question of iterating functions in the complex plane. Here, we restrict to a subclass of model where the iteration problem is trivial, in the sense that G is a finite group of (globally defined) automorphisms of the Riemann sphere. The assumption that S is stable under inversion comes from the fact that pairwise interactions are symmetricŘpx, yq "Řpy, xq. One can go beyond this assumption and actually consider coupled linear systems of the form, see [14,11] for examples.
A complete, satisfactory solution of (3.1) would be a description, for any fixed Γ and α's, of an elementary basis of solutions which generate any solution of (3.1) with prescribed meromorphic or logarithmic singularities. So far, the non-trivial case for which this program has been achieved is G " Z 2 , i.e. G consists of the identity and an homographic involution ι, and only if Γ is a segment. This occurs in the Opnq matrix model, and n "´αpιq here. The solution was essentially found for all values of n by the second author [20,21] in terms of Jacobi theta functions. Apart from a few cases which reduce technically to the latter, it seems hopeless, even when G is finite or even cyclic, to find a complete solution in the above sense. It would be very interesting to solve any problem in which G » Z is a group of translation in the complex plane, and S consists of a generator and its inverse.
We now undertake the general study of (3.1). The outcome will be a method to decide if the solution of (3.1) can be expressed in terms of algebraic functions, and in this case, the answer can be in principle computed. It does not happen for generic α's, but it can actually lead actually to some explicit results for interesting models. The techniques leading to an algebraic solution of the Opnq model equation when n "´2 cospπbq for a rational number b [19] can be regarded as a special case of our construction. Obviously, the methods we describe also allows the multiplicative monodromies.
The theory will be applied to the Seifert models in Section 4, for which the Galois group is G " Z a .
Remark. If U " p C and φpxq is a function solution to (3.1) with meromorphic singularities at prescribed points in U , and if for instance Γ consists of finite numbers of arcs in U , it follows from the finiteness of the group G that φpxq has endless analytic continuation. This observation might provide another useful point of view for the computation of φpxq: if φpxq can be complicated, its Laplace transform on certain contours might have a simpler form.

Action of the group G algebra
Let K be a number field (here Q is enough) and letÊ " KrGs be the group algebra of G. It is a vector space 8 with a basis pê g q gPG indexed by elements of G, and endowed with a bilinear multiplication law: E can also be identified with the algebra of K-valued functions on G, with multiplication given by the left convolution: We denote p g q gPG the dual basis, i.e. g pvq "vpgq for any v P KrGs and g P G. Ifv PÊ, we call suppv " tg P G, vpgq ‰ 0u its support. We denote g¨Γ " g´1pΓq, and if φ is a function on p CzΓ, we associate to anyv PÊ the following function on U zp Ť gPsuppv g¨Γq: pv¨φqpxq " ÿ gPGv pgq φpgpxqq. (3.5) φpxq can be retrieved modulo holomorphic functions in U as the "part of pv¨φqpxq which has a discontinuity on Γ only". Indeed, for any g in the support ofv, is holomorphic for x P U . This piece of information stresses that it has no discontinuity on G¨Γ nor on its preimages.

Analytic continuation and algebraic rewriting
Let us denote:α If φ satisfies the functional relation (3.1), we can rewrite: Therefore, we can define the analytic continuation -denoted ϕ -of φ on two copies of U equipped with a coordinate x, and identified along the cut Γ. In the first copy, ϕpxq " φpxq, and in the second copy, ϕpxq "`pê id´α q¨φ˘pxq. Now, in the second copy, ϕpxq has cuts on Ť gPS g¨Γ. Actually, since φpxq itself is continuous across Ť gPG g¨σ, we deduce from (3.8) the functional relation 9 forv¨φpxq for any vectorv PÊ: @x P g¨Γ, pv¨φqpx`i0q " pv´ g pvqα¨ê g q¨φpx´i0q.
Therefore, gluing copies of U along the cuts encountered, we obtain a maximal (and maybe with infinitely many sheets) Riemann surface Σ on which ϕ is an analytic function. We may have chosen an initial vectorv 0 PÊ and started the same process with the function pv 0¨φ qpxq. We would obtain another analytic function ϕv 0 . We therefore need to study the dynamics of the linear maps inÊ: T g pvq "v´ g pvqα¨ê g (3.10) (3.10) was defined such that: @x P g¨Γ, pv¨φqpx`i0q " pT g pvq¨φqpx´i0q. (3.11) Thanks to id pαq " 2, we have g pT g pvqq "´ g pvq, andT g are involutive. More precisely, they are pseudoreflections, i.e. KerpT g`i dq is generated by a single vector, namelyê g .
Definition 3.1 We call group of sheet transitionsĜ the discrete subgroup of GLpÊq generated by pT g q gPG .
G is non-commutative since rT g ,T h s "αpgh´1qαp g bê g´ g bê h q. (3.12) However, if g, h P G such that gh´1 R S, we observe that rT g ,T h s " 0.Ĝ is in general infinite.
Question 1 Does there exists a non-zero vectorv 0 PÊ with finiteĜ-orbit ? If yes, can one classify the finite orbits, and find the orbits of minimal order ?
Indeed, for such vectors the function pv 0¨φ qpxq has a finite monodromy group around Γ. For instance, if we were looking for solutions φpxq with meromorphic singularities in the Riemann sphere away from Γ, this implies thatv 0¨φ pxq is an algebraic function, i.e. ϕv 0 is a meromorphic function defined on a compact Riemann surface. The order of the orbit is related to the degree of the algebraic function, and it is appealing to choosev 0 so that the degree is minimal. The nice thing about algebraic functions is that they can be efficiently identified by their divergent parts at a finite number of poles. And in our problem, there are by construction lots of symmetries between the sheets due to the existence of G, which can help to compute the solution.

Orbits and skeleton graphŝ
G acts transitively on the orbit of anyv 0 PÊ, but not freely. Let Gv 0 denote the stabilizer ofv 0 . It is a subgroup ofĜ, with the property:Ĝ We may construct the sketelon graph Gv 0 , whose vertices are elements of the orbits ofv 0 , and edges are tv,v 1 u decorated by an element g P G wheneverv 1 "T g pvq. The labels of the edges incident to a vertexv are actually the elements of the support ofv. The graph Gv 0 is isomorphic to the quotient of the Cayley graph ofĜ with generating set pT g q gPG , by the relation: @r,r 1 PĜ,r "r 1 ôr PĜv 0r 1 . (3.14) Following the procedure of § 3.3, we can analytically continuev 0¨φ pxq as a function ϕv 0 on a maximal Riemann surface Σv 0 . It is obtained from Gv 0 by: ‚ blowing a copy of U (denoted Uv) equipped with a coordinate x, at every vertexv of Gv 0 .
‚ for any edge tv,v 1 u decorated by g P G, opening a cut along x P g¨Γ in Uv and Uv1, and gluing them along the cut with opposite orientation.
The question 1 is then reduced to the description of the quotientĜ{Ĝv 0 , where in generalĜ is infinite. In this perspective, question 1 seems far from obvious. We will see in the next paragraph that we can use the action of a smaller (and in some cases, finite) group G, which is a reflection group and actually the Weyl group of a root system, in order to understand theĜ-orbits.

Construction of a root system
We endowÊ " KrGs with the scalar product pê g |ê h q " δ g,h . The left multiplication byα defines an endomorphism:Â pvq "α¨v (3.15) Sinceαpgq "αpg´1q, we have: @v,ŵ PÊ,Âpvq|ŵ "v|Âpŵq, (3.16) i.e.Â is symmetric. Therefore, we have a decomposition in orthogonal sum: Let us denote: ‚ π E , the orthogonal projection on E, and e g " π E pê g q for g P G. None of them can be zero.
Since pê g q gPG is a basis ofÊ, their projections pe g q gPG span E.
‚ A P GLpEq, the invertible endomorphism induced byÂ on E.
It can be computed as follows:T g pα¨vq "α¨v´pα¨vqpgqα¨ê g "α¨´v´pα¨vqpgqê g"α¨´v´pα¨v qpgq e g¯. Therefore: @g P G, @v P E, T g pvq " v´pα¨vqpgq e g . When studying the dynamics of pT g q gPG , we use unhatted notations for vectors in E. This is to remind that, if we want to come back to the dynamics of pT g q gPG , we have: We now introduce a symmetric bilinear form onÊ: xv,ŵy "Â pvq|ŵ 2 . (3.20) By construction, its restriction to E is non-degenerate. The projections e g have the properties: Sinceαp1q " 2, we have: @g P G, xê g ,ê g y " xe g , e g y " 1. (3.22) This bilinear form allows the rewriting: @g P G, @v P E, T g pvq " v´2xv, e g y e g , (3.23) which shows that T g are reflections in the quadratic space`E, x¨,¨y˘.
Definition 3.2 We call reduced group of sheet transitions the reflection group G Ď GLpEq generated by pT g q gPG .
The vectors˘e g are xorthogonaly to reflection hyperplanes, and their orbit by G then form of a root system R. G coincides with the Weyl group of R. If we choose a subset I Ď G so that pe g q gPI spans E, A " pApe g q|e h q g,hPI plays the role of a "Cartan matrix". We add quotes since it is not a priori a generalized Cartan matrix in the usual sense: off-diagonal elements might not be nonpositive. We also stress that the bilinear form x¨,¨y might not be definite -compared to standard definitions, we waive this condition when we speak of root systems. We have three more observations: Since the reflections are xisometriesy and xe g , e g y " 1, R is simply-laced.
Remark 3.4 If furthermore allαpgq are integers, R is crystallographic.
Proof. Indeed, assume that R can be decomposed in a disjoint union of two mutually xorthogonaly root systems: R 1 containing e id , and a second one R 2 . Consider G 1 " tg P G, e g P R 1 u. We claim that G 1 is a subgroup of G generated by S ; since we assumed initially that G is generated by S, this entails G 1 " G, thus R 2 " H.
To justify the claim, notice that 1 P G 1 . Then, S Y S´1 Ď G 1 since xe id , e ς y " xe id , e ς´1 y "αpςq ‰ 0 when ς P S, which means that e ς (and e ς´1 ) cannot be xorthogonaly to R 1 , so cannot belong to R 2 , hence must belong to R 1 . Eventually, if g P G 1 and ς P S, we have xe g , e ςg y "αpςq ‰ 0, so ςg P R 1 for the same reason. Since we assumed that S generates G, we conclude that R 2 contains no e g for g P G, so is empty. l We can now come back to the action ofĜ onÊ " KrGs. In the block decompositionÊ " E ' E 0 , it takes the form:T Therefore,Ĝ is isomorphic to a subgroup of the semi-direct product Hom K pE 0 , Eq G, where the group structure of the latter is: We will encounter examples where G is a finite Weyl group andĜ is the corresponding affine Weyl group. In general, it seems fairly complicated to describe completelyĜ, and we shall restrict ourselves to compute G.

Bonus: intertwining by the Galois group G
Because we are acting in a group algebraÊ " KrGs, we have more "symmetries" than just the Weyl group G. We first start with the observation -independent of the group structure of G -that we have a group homomorphism:

3.26)
i.e. an linear action of the group of permutations of G onÊ. Therefore, any action of a group G on G (i.e. a group homomorphism G Ñ SpGq) induces a linear action onÊ by composition with (3.26). Since they just permute the elements of the canonical basis, these actions are isometries with respect to the canonical scalar product. Now, let us take advantage of the group structure on G. G acts by right multiplication onÊ, and this leaves E and E 0 stable. We denote ε h the endomorphism ofÊ given by right multiplication by h´1: ε h pvq " h´1¨v (3.27) so as to have a left action of G onÊ. For any h P G, the ε h are isometries ofÊ as we have seen, but one can check with (3.20) that they are xisometriesy as well. To summarize: Remark 3.6 E carries a representation of G by xisometriesy, and a computation shows that this representation intertwines the generators of G: As a matter of fact, we see from the form (3.20) of x¨,¨y that for generic α's, this is the only possible action onÊ by xisometriesy.
Remark 3.7 If E contains at least one elementê g0 , by right multiplication it must contain all pê g q gPG , thusÊ " E, i.e. A is invertible. Similarly, no elementê g0 belongs to E 0 : if it was the case, E 0 would contain all pê g q gPG , which is not possible sinceα ‰ 0.
This explains that, when Ker A ‰ 0, φpxq will never have a finite monodromy group, but it does not prevent linear combinations pv¨φqpxq to have a finite monodromy group for well-chosenv. If G is non-commutative, G also act by left multiplication onÊ, but it is less interesting. This action is an isometry (for the canonical scalar product). E 0 remains stable under h¨iff h is a central element, and in this case, E is also stable.

Orbits: description and finiteness
We can now reap the rewards of our algebraic discussion: Corollary 3.8 There exist a non-zero vectorv PÊ with finiteĜ-orbit iff R is finite. Then,v has a finite orbit iffv P E. l Since R is crystallographic -for integer α's -, simply-laced and irreducible, if we assume that it is finite, it must be of ADE type. In this paragraph, we assume it is the case. If F rIs Ď E is a subspace generated by a subset I of the roots, we denote H F rIs its xorthogonaly subspace, and: It is made of the elements of E xorthogonaly to the roots in the set I and only to them. For instance, F rRs " 0, and H 0 " E, whereas H 1 0 is the complement of the union of the reflection hyperplanes: its connected components are the Weyl chambers. In general, the connected components of H 1 F rIs define cones of dimension dim E´dim F rIs, and provide a partition of E indexed by subsets of simple roots. Actually, pH 1 F 1 q F 1 for F 1 Ď F provides a partition of H F . We can give a complete description of the G-orbits of an element v P E: The stabilizer of v P H 1 F is the subgroup of G generated by the reflections associated to the roots r which belong to F . The connected components of H 1 F are in bijection with points in G{G v , and there is exactly one point of the G-orbit of v in each of them. The G-orbit of v spans E. l The type of orbits are therefore classified by the parabolic subgroups of G, themselves classified up to conjugacy by subsets of a set of simple roots for R (see e.g. [27,Chapter 2]). Three types of orbits are remarkable: ‚ If v belongs to H 1 0 (i.e. belongs to one of the Weyl chambers), the skeleton graph G v is isomorphic to the Cayley graph of G with set of generators pT g q gPG .
‚ If v is colinear to a root, the set of vertices of the skeleton graph is isomorphic to the set of roots R.
‚ Orbits of small order are obtained when v is a non-zero element of H F rIs where I is the set of simple roots minus one of them. H F rIs is then one-dimensional. In order to obtain orbits of minimal orders, we have to choose the simple root to delete such that RrIs with a Weyl group of maximal order. Then, we call RrIs a maximal sub-root system, and any generator v˚of H F rIs a maximal element.
We insist that, once an element v 0 P E giving rise to a G-orbit is chosen, we are actually interested in theĜ-orbit ofv 0 "α¨v 0 " Apv 0 q in order to describe the analytic continuation ϕv 0 of a solution to the functional equation (3.1). We introduce: M " tv PÊ,v " Apv˚q, v˚maximal elementu, (3.30) the set of non-zero vectors inÊ whose orbit has minimal order. It is obviously stable under the action ofĜ, but more interestingly, as a consequence of § 3.6, it is also stable by right multiplication by elements of G.

Enlarging the root system
If we waive the restriction that the quadratic form be non-degenerate, we may define "root systems" larger than R, which will contain more information on the full group of sheet transitionsĜ. Given (3.24), if E 1 0 is any subspace of E 0 (the orthogonal of E for the usual scalar product inÊ), E 1 " E 1 0 'E is stable under action of pT g q gPG . So, the orthogonal projection of pê g q gPG onto E 1 , and its images under pT g q gPG still belong to E 1 , and define a "root system" R 1 (depending on the choice of E 1 0 ). We can also take E 1 "Ê, and define a "root system"R. For practical purposes, the guideline is to include as much vectors as possible in E 1 , keeping in mind that we eventually would like to describe theirĜ-orbit.

More information on the Riemann surface
In this paragraph, we assume thatv 0 PÊ is chosen to have a finite orbit, and we want to gain some information on the Riemann surface Σ " Σv 0 on whichv 0¨φ pxq can be analytically continued to a function ϕv 0 .
For simplicity, we assume that U is the Riemann sphere except a finite number of points away from Γ where φ can have meromorphic singularities. So, in the construction of Σ from the skeleton graph Gv 0 , we can blow a Riemann sphere Cv (instead of just U ) at each vertexv of Gv 0 . The outcome is a compact Riemann surface Σ, equipped with a branched covering x : Σ Ñ p C. Its degree d is the number of vertices in G, i.e. the order of the orbit ofv 0 . Lemma 3.10 Let |Γ| be the number of connected components of Γ. The genus of Σv 0 is: Proof. By construction, the branched covering x : Σ Ñ p C has simple ramification points at the edges of the cuts. Each cut has two extremities, and g¨Γ is a cut in p Cv iff g is in the support ofv. Remember also that this cut is identified with g¨Γ inT g pvq. Hence a total of |Γ|ˆřv PG |suppv|. The announced result is then deduced from Riemann-Hurwitz formula: 2´2g " 2d´|Γ|´ÿ vPG |suppv|¯(3.32) l ϕv 0 is a meromorphic on Σ. So, there exists a polynomial equation of degree d in y " ϕ: Here are some general principles to grasp more information, and hopefully compute P.
p1q We usually want to solve the problem for φpxq with prescribed singularities when x Ñ 0 or 8.
In other words, we know in each sheetv P G the leading term in the Puiseux expansion of ϕv 0 pxq are related at leading order when x Ñ 0 or 8 in Cv. This fixes the Newton polygon of P, and the coefficients on its boundary. p2q To go further, one may give names (say c j 's) to the first few subleading coefficients in the Puiseux expansion of φpxq when φ or x go to 8. Then, one can identify all coefficients in P in terms of c j 's, just by writing: and finding what is the Puiseux expansion of the product in the right-hand side at x Ñ 0 or 8.
p3q If the orbit ofv 0 has some extra symmetries, i.e. if there exists ψ P AutpÊq leaving the orbit stable, this imposes some symmetries for the polynomial P. Notice that, since the vectors of a given orbit must span E, and since T g are xisometriesy, such a ψ must also be an xisometryy. For instance, it could happen (but it is not automatic) that the right multiplication by an element h P G leaves the orbit stable.
p4q If the solution itself has a priori some symmetries (like (2.19)), they should also reflect on P.
p5q We know what is the ramification data of x on Σ: the solutions of Ppy, xq " B y P py, xq " 0 must all be of the form py b , gpx b qq for some g P G and x b an edge of Γ, and they must also satisfy to We have derived the properties that must satisfy the analytical continuation ϕv 0 of pv 0¨φ q when φ is a solution of (3.1). A priori, these are not sufficient conditions, and one has to check that ϕv 0 satisfying the above constraints indeed provide a solution of the initial problem via (3.6). In particular, if we are given a collection of functions pψvq with the correct branching structure and asymptotic behavior, it is not at all automatic that: does not depend onv. This is nevertheless a property of the solution we are looking for. So, we only hope that imposing enough necessary constraints will lead to a finite number of possible polynomial equations, that can be browsed to meet more subtle constraints like positivity of the spectral density, or behavior at u Ñ 0, and hopefully find a unique Ppx, yq. In a practical problem -like Seifertexistence and unicity a priori is guaranteed, so we may conclude to the identification of the solution if the necessary conditions described above single out of unique Ppx, yq " 0.

Geometry of Seifert fibered spaces
We review a number of facts about the geometry of Seifert fibered spaces [44].

Definition
One first defines the standard fibered torus with slope b{a: It is a cylinder r0, 1sˆD 2 , so that p0, zq gets identified with p1, e 2iπb{a zq, and the whole space is seen as a S 1 fibration over the disk D 2 , for some order a ě 1. A closed 3-manifold is a Seifert fibered space if it admits a foliation by S 1 , so that each leaf (also called fiber) has a tubular neighborhood isomorphic to a standard fibered torus. Two famous examples of Seifert fibered spaces are: ‚ the lens space Lpa, bq " S 3 {Z p . It can be realized by considering S 3 " tpz 1 , z 2 q P C 2 , |z 1 | 2| z 2 | 2 " 1u and the identifications pz 1 , z 2 q " pζ a z 1 , ζ b a z 2 q. We need to assume a and b coprime for the space to be smooth.
‚ the Poincaré sphere, which can be described in several ways. It is the space of configurations of a icoashedron in R 3 . It is also obtained by identifying opposite faces of a dodecahedron (the icosahedron's dual). It is the only non-trivial integer homology sphere with finite fundamental group (= the binary icosahedral group).

Classification
Let M be a Seifert fibered space. All but a finite number number of fibers are ordinary, i.e. they have order a " 1. We denote a 1 , . . . , a r ě 2 the orders of exceptional fibers. Identifying in M all points of the same leaf, one obtains the orbit space O. Ordinary fiber project to a smooth point in O, whereas an exceptional fiber of order The classification depends on the topology of O and the orientability of M . In this article, we always assume that O is topologically a sphere and M is orientable. The classes of oriented Seifert fibered spaces (modulo orientation and fiber preserving maps) are in one-to-one correspondence with uples of integers pb; a 1 , b 1 ; . . . ; a r , b r q, a i , b i characterize the neighborhood of exceptional fibers, and the integer b tells how exceptional fibers sit together in the global geometry. Changing orientation has the effect: pb; a 1 , b 1 ; . . . ; a r , b r q ÝÑ pb´r; a 1 , a 1´b1 ; . . . ; a r , a r´br q.
Therefore, the integer: is an invariant of orientable Seifert fibered spaces. If there are r ě 3 exceptional fibers, a 1 , . . . , a r themselves are topological invariants of M . This is not true anymore in presence of one or two exceptional fibers, i.e. for r ď 2 there exist homeomorphism which do not preserve fibers and change pa 1 , a 2 q. r " 1, 2 give lens spaces, and since they are well-understood from the point of view of Chern-Simons theory [29,17], we shall assume r ě 3 in the following.

Orbifold Euler characteristic
Another important invariant, as we have seen in the matrix model, is the orbifold Euler characteristic of the orbit space: M has a finite fundamental group iff χ ą 0. As we have already seen, for r ě 3 the only possible exceptional fiber data in this case are p2, 2, pq, p2, 3, 4q and p2, 3, 5q. The corresponding Seifert fibered are precisely the quotients of S 3 by the free action of a group of isometries. The outcome is that, up to isomorphism, M " Z n zS 3 {H, with H is a finite subgroup of SUp2, Cq, namely a cyclic group of order n 1 (leading to lens spaces), or a binary polyhedral group: ‚ dihedral group of order 4p, giving exceptional fibers p2, 2, pq -labeled by D p`2 as regards the ADE classification of subgroups of SOp3, Rq.

Fundamental group and homology
The fundamental group of an orientable Seifert fibered spaces whose orbit space is a sphere is generated by c i going around the i-th exceptional fiber (1 ď i ď r), and a central element c 0 with relations: For r ě 3, one can show that π 1 pM q is finite iff χ ą 0 (a fortiori it requires exactly r " 3). Another natural question is to ask for Seifert fibered spaces which are integer (resp. rational) homology spheres, i.e have trivial (resp. trivial up to torsion) H 1 pM q but are not S 3 . The answer is that a 1 , . . . , a r must be pairwise coprime. If this constraint is satisfied, then M is a rational homology sphere, with: and there exists a unique b, b 1 , . . . , b r such that aσ "˘1, i.e. such that M is an integer homology sphere. The Poincaré sphere is the unique case with finite fundamental group, its data is: and for this reason it is the most interesting geometry treated in this article, but also the most cumbersome among the χ ą 0 cases . . . For other values of pb; a 1 , b 1 ; a 2 , b 2 ; a 3 , b 3 q, one obtains the Brieskorn spheres.

Avatars of Chern-Simons theory
As a matter of fact, (2.1) first appeared in [4] for the Lê-Murakami-Ohtsuki invariant [37] on Seifert spaces. If M is a closed, framed 3-manifold obtained by surgery on a link in S 3 , the LMO invariant Z LMO pM q is a graph-valued formal power series in [37]. Its relation to the Kontsevich integralwhich is a universal formal series of finite type invariant -and Aarhus integral for rationally framed links was exposed in [1,2,3], see also [4]. For any choice of compact Lie algebra g, it can be evaluated to a g-dependent, formal power series in . In particular, the evaluation with the Lie algebras of the series A N , B N , C N or D N yields up to normalization ln Z u N LMO as an element of ´2 Qrr 2 ssrru 0 ss with: This is no more than a repackaging of LMO invariants, which is a weaker invariant than LMO [45], and gives the geometric foundation of the quantities we shall compute in this article. LMO can be considered as a mathematical definition of the perturbative expansion of the Chern-Simons partition function: where the path integral runs over g-connections A modulo small gauge transformations. The saddle points of the action are the flat connections, and in principle, Z g CS should be given by the sum over all flat connections of its perturbative expansions. When M is a rational homology sphere, the LMO invariant is tailored to capture the contribution of the trivial flat connection. In particular, we have for the A N series: and for any of the X P tB N , C N , D N u series: where F X g P Qrru 0 ss are the Chern-Simons free energies. It is known [26] hat F A g has a finite radius of convergence independent of g, i.e. can be seen as the power series expansion at u 0 Ñ 0 of an analytic function.
Problem 1 Describe the singularities of F X g considered as a function of u 0 .
Chern-Simons theory is a cornerstone in quantum topology, because of Witten's discovery [46] that the expectation value @ Tr R exp`¸K A˘D with respect to the Chern-Simons measure (in principle computed by a path integral), is an invariant of framed knots in M , called "Wilson loops". Depending on the 3-manifold, there are several ways to define rigorously those invariants, as functions of q, as elements of the Habiro ring, or as formal series, see e.g. the review [7].

Link invariants in the 3-sphere
When K is a knot in M " S 3 , Wilson loops turn out to be Laurent polynomials in: q " e (4.12) and they coincide with R-colored HOMFLY polynomials. When R is the fundamental representation of g " supN`1q, this retrieves the HOMFLY-PT polynomial [24], and for g " sup2q, this is the Jones polynomial [30]. When R is the fundamental representation of a Lie algebra in the BCD series, the Wilson line is related to the Kauffman polynomial [34]. Reshetikhin and Turaev [41] later provided the foundation for the rigorous TQFT construction of those invariants. The HOMFLY-PT of a link in S 3 satisfies skein relation, which allows to reduce the computation of HOMFLY-PT of any link to the computation of HOMFLY-PT by resolution of crossings. Besides, the R-colored HOMFLY of a link can be realized as the HOMFLY-PT of a link obtained by taking parallels in a R-dependent way [40]. The Kauffman polynomial also satisfy a more general skein relation, but in general the colored HOMFLY polynomial of a link is not known to reduce to the HOMFLY of a related link. It is well-known that F A g and the coefficients of a given power of in the colored HOMFLY are entire functions of e u (see e.g. [17]).

Link invariants in rational homology spheres
When K is a knot in M ‰ S 3 , Wilson loops can always be defined as a power series in in perturbative Chern-Simons around a chosen flat connection, but cannot in general be upgraded to a function of q and q N (see [46,25]). When M is a rational homology sphere, one can formulate a skein theory for the HOMFLY-PT invariant of a link K in M considered as a formal series in and u 0 " N . Skein theory then determines the colored HOMFLY for any link L, if one knows the value of the HOMFLY-PT for a set of basic knots representing the conjugacy classes of π 1 pM q [31].
Two interesting and open questions are: Problem 2 What is the value of HOMFLY-PT for basic knots in a given rational homology sphere ?
Problem 3 Consider the coefficient of a given power of . Is it the power series expansion of an analytic function of q N " e u0 when u 0 Ñ 0 ? What are the singularities in the complex plane of this function ?
In this article, we study the Wilson loops (in any representation R of fixed size) for the knots going along the extraordinary fibers of Seifert spaces. Our method gives in principle a way to compute the coefficients of the power series in as functions of e u0 for Seifert spaces with χ ě 0, and provide some partial answers to Problem 1 and 3 : the perturbative invariants are algebraic functions of e u0 , but in general not entire functions. For instance, in the case of S 3 {D p`2 with p even, we could push the computation to the end and describe precisely the algebraic function field in which the perturbative invariants sit (see Theorem 7.4). For all S 3 {D p , we can also show that the perturbative invariants have no singularity for u 0 on the positive (resp. negative) real axis if σ ą 0 (resp. σ ă 0), and Conjecture 2.5 would imply this is also true for all Seifert fibered spaces.
The fiber knots only form a subset of the basic knots: we are missing the knots going along a meridian of the exceptional fibers. At present, it is not known how to rewrite their Wilson lines as observables in the model (2.1). We nevertheless propose the following: In this case, L would be an invariant of the ambient 3-manifold. In Theorem 7.4, we show for S 3 {D p`2 that L contains Qpu, κ 2 , βq{I, where I is generated by:

Exact evaluations
Exact evaluations of the Chern-Simons path integrals are rare. By "exact evaluation", we mean the reduction to a finite-dimensional sum (over dominant weights of g) or integral (over the real Cartan subalgebra h of g). Seifert fibered spaces [44] are one of the few classes of non-trivial 3-manifolds for which it has been performed so far, and the contribution of the trivial flat connection takes the form (2.1): where C g is a known prefactor given in [38]. Actually, (2.1) can be derived in various ways, either in the realm of LMO or of TQFT. Seifert spaces turn out to be tractable either because they can be obtained by rational surgery on a very simple link in S 3 (see Figure 1), and TQFT behaves well under surgery ; or because they carry a U p1q action and localization of the path integral occurs. Here is a schematic account of the history of those exact formulae: ‚ For g " sup2q or sop3q and M a Seifert integer homology sphere, Lawrence and Rozansky [36] have used the Reshetikhin-Turaev construction to rewrite Z g CS as a 1-dimensional integral (2.1), including contributions of all flat connections.
‚ Mariño generalized their derivation to any simply laced-Lie algebra g and Seifert rational homology spheres M [38].
‚ Bar-Natan [4] has computed the LMO invariant of Seifert rational homology spheres, via the Kontsevich integral.
‚ Beasley and Witten [6] have developed a non-abelian localization method, allowing the computation of the contribution of isolated flat connections 10 Then, correlation functions of Schur polynomials for the measure (2.1) can be interpreted in terms of Wilson loops along exceptional fibers [5].
‚ Källen [32] derives the same results, building on earlier work of [33] on a supersymmetric version of Chern-Simons theory.
‚ Blau and Thompson developed a diagonalization technique, first for U p1q bundles over smooth surfaces [9], then for U p1q bundles over orbifolds [10], allowing the computation of the full Chern-Simons partition function. As a particular case, they retrieve the earlier results on Seifert rational homology spheres.

Correlators and Wilson loops
We review the interpretation of the correlators of the model (2.1) in terms of Wilson loops. [5] tells us that the holonomy operator U am along the exceptional fiber of order a m -on the Chern-Simons side -gets identified with diagpe t1{am , . . . , e t N {am q " S a{am -on the matrix model side, with the notations of (2.1). Therefore, the Wilson loop in representation R is equal to: where ch R is the character of R, i.e. the Schur polynomial indexed by R. We prefer to work in the power-sum basis of the representation ring, and with connected observables: where: The H's and the W's are related by a change of basis: to extract H R for a representation R corresponding to a Young diagram with less n rows, we need to compute W n 1 with n 1 ď n.
Recalling a " lcmpa 1 , . . . , a r q, we define the n-point correlators of the matrix model as: so that the W n 's can be read from the coefficients of the expansion of W n px 1 , . . . , x n q in Laurent series when x 1 , . . . , x n Ñ 8. If a 1 , . . . , a r are not coprime, the expansion of W n also records expectation values of fractional powers of the holonomy along fibers, which do not have a clear interpretation in knot theory. In a perturbative expansion, we have a decomposition of formal power series in u of the form: Later, we shall consider only certain linear combinations of rotations of W pgq n , namely: forv i in a certain set V of vectors in Z a . If we denote the discrete Fourier transform: we have the expansions when x i Ñ 8: i¯¨W pgq n px 1 , . . . , x n q " ÿ k1,...,kně0 and when x Ñ 0: i¯¨W pgq n px 1 , . . . , x n q " p´1q n ÿ k1,...,kně0 In the latter, we have used that S is distributed like S´1. Therefore, knowing (4.19) forv i P V will only give access to the coefficients of expansion ofW pgq n in x´m i with pm mod aq such that there existŝ v P V with non-zero F m rvs.

Remark on formal series versus asymptotic series
Our point of view is to consider the Chern-Simons matrix model (2.1) for u " u 0 {σ " N {σ ą 0. We thus have to assume that 0 ă q " e ă 1 if σ ą 0, or q ą 1 if σ ă 0. The correlators W n px 1 , . . . , x n q of the matrix model are then defined as functions of u, N and q. Our point of view is to analyze the asymptotic expansion of the correlators when N Ñ 8 for a fixed value of u ą 0 and x 1 , . . . , x n P CzR. When the equilibrium measure of the matrix model has one cut Γ Ď R˚(a property guaranteed by Lemma 2.3 when χ ě 0) and is off-critical, the results of [15] ensure that we have an asymptotic expansion when N Ñ 8 of the form: where now W pgq n px 1 , . . . , x n q is a holomorphic function of x 1 , . . . , x n P CzΓ and of u ą 0. Its Laurent expansion when x i Ñ 8 and power series expansion when u 0 " uσ Ñ 0 retrieves the formal series of (4.18). This approach has the extra benefit to provide W pgq n as function of u 0 , hence to allow analytic continuation in u 0 , and thus to address Problem 3 concerning the singularities in u 0 .
Given the results of Section 2 for χ ě 0, off-criticality boils down to checking that the density of the equilibrium measure remains positive in the interior of its support. We already know this is true for any χ ě 0 provided u is small enough. We did this check for all values of u ą 0 in the cases p2, 2, pq since we have an explicit expression for W p0q 1 pxq. For the remaining cases with χ ě 0, such an expression is not available because of algebraic complexity, so we were not able to check: Conjecture 4.2 For χ ě 0 (except p2, 2, pq and r ď 2 already known), off-criticality (and thus (4.23)) holds for all values of u ą 0.
We checked numerically this conjecture (see Section 9), but we could not find an a priori, potentialtheoretic argument ruling out zeroes of the density in all cases χ ě 0. We will assume (4.2) to continue with our reasoning. Nevertheless, all propositions and theorems stated in the text are independent of this assumption.

Origin of the measure
The key feature of the model (2.1) is the interaction: where the product runs over α " positive roots of the Lie algebra. This is a pairwise interaction between t j for the ABCD series of Lie algebras. From a geometric perspective [46,10], (4.24) is essentially the Ray-Singer torsion of Seifert fibered spaces. From the LMO perspective [4], (4.24) arises from the evaluation of the wheels in the weight system g: and the decomposition of the Lebesgue measure dX over the real Lie algebra g R in terms of the Haar measure dU on exppgq and the Lebesgue measure dt on t: dX " C g dU dt |detpad tq| " C g dU dt ź αą0 pα¨tq 2 for some constant C g .

Generalizations
We describe generalizations of (2.1), whose study is out of scope of this article.

Non-trivial flat connections
In exact evaluations, the partition function is in general obtained as a sum of terms identified with contributions of the different flat connections. The contribution of the trivial flat connection respect the full Weyl symmetry of t and correspond to (2.1) up to a known prefactor. The contribution of other reducible flat connections is the analogue of (2.1) with a potential V breaking the Weyl symmetry [38], in a maximum of aS pieces. More precisely, the t j in this case are partitioned: and the term V pt j q is replaced by: And, there may exist residual terms corresponding to irreducible flat connections [36,38]. Since the measure in (2.1) is now complex, we cannot apply stricto sensu the arguments of asymptotic analysis raised in Section 2 and [15]. Nevertheless, we can take the saddle point equation (2.18) with complex valued right-hand side as a starting point, and compute the corresponding spectral curve with the methods of Section 3. The only difference in the result is a rescaling of the Newton polygon, and now the coefficients inside the Newton polygon depend on the collection filling fractions l " N . This dependence is in general transcendental, since the I are periods of the 1-form ln y d ln x on the spectral curve. For lens spaces, this analysis has been explicitly performed in [29].

Orbit space of any topology
For Seifert fibered spaces whose orbit space O is a Riemann surface of genus h, (4.24) appears to a power 1´h (half the usual Euler characteristics of O) [10]. For h ě 2 the corresponding partition function would be ill-defined for t i integrated over R. But, in [36,38], the formula as an integral over t is actually derived from a sum over dominant weights of g, by an Euler-MacLaurin type formula and analytical continuation in . In other words, the original expression is a sum over discrete t's where, among other details, hyperbolic functions are replaced by their trigonometric analogue, and the walls of the Weyl chamber are excluded. When h " 0, we can add the wall contribution since it is 0, and arrive to an integral over t. When h ě 2, the correct formula is the discrete sum, with pairwise interactions between the t i 's behaving like |t i´tj | 2´2h when t i Ñ t j . We remark that the same kind of sums appear in the partition function counting simple coverings of surfaces of genus h (simple Hurwitz numbers) [13]. Since t i 's now attract each other -but belong to a lattice -the large-N asymptotic analysis could be very different from the repulsive case treated so far, and it is not clear how to adapt our techniques to this case. For instance, it is already not obvious that the asymptotic expansion (4.23) holds, even for a small value of u.
5 Spectral curve and 2-point function : inhomogeneous part

The spectral curve
Let a 1 , . . . , a r be integers. We have established in § 2.2 that the spectral curve satisfies -on top of growth constraints -the functional relation: Here Γ is a subset of Rˆto determine with the solution. The first step is to get rid of the right-hand side, and the way to achieve this depends whether χ " 0 or not. Then, we arrive to the problem presented in Section 3, with Galois group G " Z a . We denote it additively, and pê 0 , . . . ,ê a´1 q is the canonical basis ofÊ " ZrGs. The sheet transitions are ruled by the vector: It is easy find a particular solution of (5.1) which has no discontinuity on Γ, and subtracting it to W pxq we find that: satisfies to the homogeneous equation bringing us back to Section 3: The price to pay with (5.3) is that φpxq now has a logarithmic singularity, but we can turn into a meromorphic singularity by setting: The functional equation for Y is now multiplicative, but it does not make much difference from the point of view of Section 3. Ifv PÊ, we write: We keep the same notation but the context should make clear if the action ofÊ should be additive (on φ) or multiplicative (on Y ). (5.2) for φpxq translates into: Let us introduce the parameter: The growth conditions (2.15) on φpxq imply: where n 0 rvs " ř a´1 l"0v plq and n 1 rvs " ř a´1 l"0 lv l .

χ " 0
If χ " 0, we can find a particular solution of (5.1) containing ln 3 x. Since we prefer to avoid this type of singularities, we take another route. The function satisfies the homogeneous equation: The analytic properties of W pxq imply that: ‚ The palindrome symmetry implies φ 2 pxq`φ 2 p1{xq " 0.
Since W pxq and φ 1 pxq are continuous in CzΓ, the integration in (5.11) do not depend on the paths. And the period of W pxq{x is 1 since the equilibrium measure has total mass 1. So we have the extra conditions: ‚¸Γ W pxq dx 2iπx " 1.

The two-point function
Let us introduce: which is a holomorphic function of x 2 P p CzΓ, and a holomorphic 1-form in x 1 P p CzΓ. The functional equation (2.25) for W p0q 2 px 1 , x 2 q implies, for x 1 PΓ and x 2 P p CzΓ: The strategy is to get rid of the right-hand side. We first look, for any integer k ě 1, for a particular solution h k pxq of the linear equation with right-hand side x k´1 1 dx 1 . Then, ř kě1 x´k 2 h k px 1 q solves the linear equation with right-hand side: One can first try to find a solution of the form h k pxq " c k x k´1 dx. When the Fourier coefficient F k rαs vanishes, such rational solutions do not exist, but one can always find a solution of the form h k pxq " c k x k´1 pln j k xq for a minimal value of j k . Since in discrete Fourier space, the convolution bŷ α becomes a multiplication by Frαs, there are precisely dimpKerÂq " a´dim E values of pk mod aq for which a logarithm is needed (we use the notations of § 3). All in all, we can construct a particular solution for Hpx; x 0 q for the primitive of the two-point function of the form: HereĈ pjq " ř iPZaĈ pjq piqê i represents a vector inÊ, and with our notations: i.e.Ĉ pjq acts on the first variable andê 0 on the second variable (its action is the identity). Then, we introduce which solve the homogeneous linear equation. The price to pay is that ω p0q 2 | hom is a p1, 1q form having the singularities of´d x2 H part px 1 ; x 2 q. Therefore, for anyv 1 ,v 2 PÊ, the singular part of: where we have: and higher B pjq piq can be obtained by similar formulae. Then, if we are given some initial vectorv 0 , we deduce that pv 0 bv 0 q¨ω p0q 2 | hom can be analytically continued as a bidifferential pω p0q 2 qv 0 continued on the Riemann surface Σv 0ˆΣv0 , and it has -on top of the log singularities 11 -double poles without residues at points so that x a " x a0 for x 1 , x 2 in the sheet Cv 1ˆCv2 labeled by two vectorsv 1 ,v 2 of the orbit ofv 0 .
Definition 5.1 The coefficients of the double poles are recorded in the matrix of singularities: whose rows and columns are labeled by elementsv 1 andv 2 in the orbit ofv 0 , and whose entries are vectors inÊ.

Remark on the matrix of singularities
The easiest way to produce a meromorphic form with double poles at x a " x a 0 is to consider: 11 We do not insist on the log singularities ofv b2 0¨ω p0q 2 | hom : in the Seifert cases, it turns out thatB pjq pi;v 1 ,v 2 q " 0 whenever j ě 1 andv 1 ,v 2 have finite orbits, so they are absent. Then, is also meromorphic on Σ 2 v0 with double poles, but admits as matrix of singularities: where J is the matrix full of 1's. If it is possible to chooseB so thatB sp becomes sparse, the computation of ω p0q 2 | sp px 1 , x 2 q is facilitated. To summarize the relation with the two-point function: As a matter of fact, the structure of a suitableB carries information on how to lift the action of Z a by rotations on p C, to the total space Σv 0 of the covering x : Σv 0 Ñ p C. This will be exemplified in the case by case study, in particular for the geometry S 3 {D p .
In general, x 1 and x 2 may not play a symmetric role in ω p0q 2 | sp px 1 , x 2 q (whereas they do in ω p0q 2 by construction). This is nevertheless the case when the entries of B 6 Spectral curve and 2-point function: case study 6

.1 General results
We study in detail the sheet dynamics of a solution of the homogeneous equation in the Seifert models, characterized by: Following § 3.5, we construct a simply-laced, irreducible root system R on E: E " Im A, A "α¨(6.2) Since the coordinates ofα are integers, R is crystallographic. We remind (Corollary 3.8) that the groupĜ " xT l , l P Z a y has non-zero finite orbits iff R is finite. We also remind the definition of the orbifold Euler characteristics of Seifert spaces χ " 2´r`ř r m"1 1 am .
Proof. Let n 0 rvs " ř a´1 l"0v plq be the sum of coordinates. According to (3.10)-(6.1), we have: Assume there existv ‰ 0 such that the set n 0 rĜ¨vs Ď Z is finite. If χ ‰ 0, it cannot be contained in t0u, and up to considering´v instead ofv, we may assume it contains a positive integer. Then, letŵ PĜ¨v such that n 0 rŵs is maximal. Then, n 0 rŵs ą 0, so there exists j P v0, a´1w such that wpjq ą 0. By maximality and the relation: we deduce that χ ě 0. As a corollary, if χ ă 0, the set n 0 rĜ¨vs is never finite forv ‰ 0, so the orbits themselves cannot be finite. l In the remaining of the text, we assume χ ě 0. So, the spectral curve is uniquely defined (Corollary 2.2), has the palindrome symmetry (Corollary 2.4), and the cut Γ consists of one segment (Theorem 2.3). We will see that R is finite in all cases χ ě 0. The resulting root systems are presented in Figure 2. We have written a short maple program computing the orbits under pT l q l of an initial vectorv 0 , which was very useful to guess vectors with small finite orbits -before we were able to construct them without involving a guess, as presented below. The left action of G onÊ discussed in § 3.6 is generated by the shift operator: which intertwines the generators as: The shift (or its powers) will be useful to decompose further the G-orbits. (2) the order of exceptional fibers ; (3) a " lcmpa 1 , . . . , a r q, which is the dimension of the group algebrâ E " ZrGs ; (4) the simply-laced, crystallographic root system we have found according to Section 3, denoted according to its ADE classification ; (5) the number of roots ; (6) the order of the Weyl group G ; (7) the minimal order of aĜ-orbit. We included for completness the case of lens spaces treated in [17,29], and set a 1,2 " a 1`a2 .

Seifert manifold
We may wonder how the geometry of Z n 1 zS 3 {H is reflected in the (reduced) group of sheet transitions and in the equilibrium measure. Here are three curious observations: ‚ a " dimÊ " |G| coincides with the Coxeter number of R. We do not know how to interprete this coincidence.
‚ the ADE Dynkin diagram associated to H always contains as a subdiagram the ADE Dynkin diagram associated to R. In particular, they are both E 8 for the Poincaré sphere, and this is the only case of equality. In the other cases, it looks like a symmetry breaking of a bigger Weyl symmetry associated to the ADE label of H occuring by restriction to the contribution of the trivial flat connection. This probably can be made concrete by physics dualities between p2, 3, 6q 6 A 5 30 6 p A 5 Figure 3: For Seifert manifolds with χ " 0, the group of sheet transitions is an affine Weyl group.
Chern-Simons theory on S 3 {H and another gauge theory with gauge group associated to the ADE Dynkin diagram of H.
‚ p2, 2, pq with p even is the only case whereĜ is finite. It is also the only case where a is also the order of an exceptional fiber, and it is tempting to speculate that this is responsible for the finiteness ofĜ, for a reason yet unknown to us.
If V is a complete orbit generated by some vectorv 0 , all moments µrTr S k s can be retrieved from the Puiseux expansion of the analytical continuation ϕv 0 pxq ofv 0¨φ pxq at a point x Ñ 0 or 8 in some sheet of Σv 0 . From the case by case analysis, we find that: Lemma 6.3 In all χ ě 0 cases, the orbits of finite minimal order are complete.
and we recognize the Cartan matrix of the root system A p , with β's playing the role of simple roots. ThereforeĜ " G is the symmetric group in p`1 -in particularĜ is finite.

Spectral curve
This implies that φpxq "ê 0¨φ pxq is already an algebraic function, and it is actually the only case in our list where this happens. The orbit ofê 0 consists ofv 0 " ř p´1 l"0 p´1q l`1ê l andv 1 j " p´1q jê j for j P Z p , its order is p`1. Since it containsê 0 , this orbit is complete. This is actually a minimal orbit, since the maximal sub-root system of A p is A p´1 , and: The skeleton graph is a star with p branches connecting to the central vertexv 0 (Figure 4). Therefore, the Riemann surface Σv 0 has genus 0. We denote y the function which is equal tov¨Y pxq when x belongs to the sheet labeled byv PĜ¨v 0 . We find that: ypzq "´p z p{2´κ qpκz p{2`1 q pκz p{2´1 qpz p{2`κ q (6.13) where 2κ 1`1{p 1`κ 2 " e´u {4p 2 :" c 2 (6.14) is the only possibility meeting the growth requirements (5.9)-(5.10), so it is the correct solution. The equilibrium measure is extracted as: for the branches of Y which meet on the positive real axis. The support of the equilibrium measure µ eq is r1{γ, γs with γ " xpz`q ą 1 such that dxpz`q " 0. We find: By consistency, we must choose κpuq as the unique solution of (6.14) such that 0 ă κpuq ď 1 when u ě 0.

Two-point function
Particular solutions of the linear equation with right-hand side x k´1 dx are all rational H k pxq " c k x k´1 dx with: This results in the residue vector:Ĉ p0q "ê 0`ÿ PZp p´1q `1 p`1ê (6.18) Then, we compute the matrix of singularities takingv 0 " ř jPZp p´1q `1ê as initial vector. We found above the other vectors in the orbit arev 1 j " p´1q jê j for j P Z p . We find the following decomposition of the matrix of singularity:B p0q "BJ`ÿ PZp p´1q ê Πp q (6.19) We have defined:B so that the subtraction ofBJ leaves us with a sparse matrix B p0q sp . We find that the Πp q are permutations matrices of the sheets: where Eŵ 1,ŵ2 are the elementary matrices whose only non-zero entry is a 1 at the intersection of the row labeledŵ 1 and column labeledŵ 2 .
We found previously that the curve Σv 0 is rational, with uniformization variable z. Therefore, ω p0q 2 | sp pz 1 , z 2 q must be a rational form in z 1 and z 2 , and the only possibility compatible with its matrix of singularity is: dpζ p zqdz 0 pz 0´z q 2 (6.22) But we learn more from the structure of B p0q sp : indeed, if we denote z pŵq pxq the values of z in the sheet w above the point x, we must have @ŵ PĜ¨v 0 , @ P Z p , z pŵq pζ p xq " ζ j p z pΠp qpŵqq pxq (6.23) Therefore, the matrices Πp q permuting the sheets tell us how the action of ζ p P Z a in the x-plane lifts to an action of Z a in the z-plane. We thus have gained, through the computation of the two-point function, a precise description of the monodromy of the function zpxq that is useful in our problem. Putting all results together, since we observe thatB "Ĉ p0q , there is no contribution from the fundamental 2-form of the 2 nd kind of the x-plane, and we have: The dynamics are generated by:α The subspace E has rank p`1, and is generated by the orthogonal projections e l on E of the canonical basisê l : e j " ε´jpe 0 q (6.27) Out of them, we can construct of basis of E: β j " p´1q j´1 pe j`ej`1 q, p1 ď j ď p´1q, β p " e 0 , β p`1 " e p . (6.28) The Cartan matrix of the root system R in this basis reads: We recognize the Cartan matrix of the root system D p`1 , where the β's play the role of the simple roots, labeled as in Figure 5.

Minimal orbits
A maximal sub-root system is obtained by deleting β 1 from R: it is of type D p . The minimal order of an orbit is therefore: d " |WeylpD p`1 q| |WeylpD p q| " 2pp`1q (6.30) Figure 5: Dynkin diagram of the root system D p`1 .
Vectors having such an orbit can be found of the formv˚"α¨v˚with v˚PÊ orthogonal to the chosen maximal sub-root system: @j P v2, p`1w, xv˚, β j y " 0 (6.31) By decomposing v˚on the basis of β's, we get a linear system which can be solved, and we find: We compute the image of the β's under right multiplication byα, and deduce the image ofv˚, giving a generator for a line of vectors inÊ with minimalĜ-orbit: v˚"ê 1`êp`1 (6.33) Since any shift of the latter is also a vector with minimal orbit, and we may choosê v 2 " ε 1 pv˚q "ê 0`êp (6.34) as initial vector. The orbit ofv 2 is depicted in Figure 6. It consists of p´ε´1q j pv 2 q for j P Z 2p and˘v 0 " ř 2a3´1 l"0 p´1q l`1ê l . The discrete Fourier transform F k rv 2 s vanishes iff k mod 2p " p, hence the orbit is complete. We observe that all vectors in this orbit are fixed points of ε p 1 , i.e. that pv¨φqpxq is a actually a function of x 2 . So, instead of Σv 2 , we consider its quotient inducing the covering x 2 : Σ 1v 2 Ñ p C. Its skeleton G 1 v2 is the quotient of Figure 6 by ε p , i.e. a star with p edges connecting to the same central vertexv 0 . Genus counting by Riemann-Hurwitz indicates that Σ 1v 2 is rational : we must have a parametrization z Þ Ñ px 2 pzq, ypzqq, such that the possible values of y for a fixed x 2 pzq are pŵ¨φqpx 2 pzqq forŵ P G 1 v2 . These functions can have poles and zeroes only when x 2 Ñ 0, 8, and their leading behavior is prescribed by (5.9)-(5.10). This fixes completely the coefficients of a polynomial equationPpx 2 pzq, ypzqq " 0. By uniqueness, it is enough to exhibit a solution with the correct leading behavior at x Ñ 0 and 8. Here is one: ypzq "´p z p´κ qpκz p`1 q pz p κ´1qpz p`κ q where κ is the function of u already encountered in (6.14): The equilibrium measure can be retrieved by: Since all components ofŵ in the orbit considered are˘1, (6.37) gives the density of the equilibrium measure independently of the choice of branch zpxq, provided the sign in prefactor is arranged to have a positive measure. The support of the equilibrium measure is of the form r1{γ, γs, where γ " xpz`q ą 1 depends on u: which turns out to be the squareroot of (6.16). By consistency, κ " κpuq should be chosen so that 0 ă κ ď 1 when u ě 0, hence it is the same function occuring in the D p , p even cases. See Figure 7.
We can directly check that the equilibrium measure is off-critical for any u ą 0. Figure 6: The skeleton graph G (see § 3.4) generated byv 2 . An edge label j indicate that the two sheets are connected via ζ´j 2p¨Γ . Its quotient G 1 " G{ε 6 coincide with the subgraph only made of v 2 and p´ε 1 q j v 0 for j P v0, p´1w.

Two-point function
A rational solution to the linear equation with right-hand side x k´1 dx exist whenever pk mod 2pq is even or equal to p, and that a single logarithm is needed in the other cases. After some algebra, we pê `p´ê q (6.40) Taking as initial vectorv 0 " ř PZ2p p´1q `1ê , the other vectors in the orbit are´v 0 and˘v 1 j " p´1q j pê j`êj`p q for j P Z 2p , and we find that the matrix of singularityB p0q is already sparse, namelŷ B " 0. It can be decomposed as:B with the following permutation matrices: Notice that the curve ±v 0 is not rational, but it covers a rational curve with uniformization variable z by a degree 2 covering. In a given sheet of the basis, the two choices of the squareroot correspond to a lift to a sheet of Σv 2 labeled by˘v. Since p is odd, the matrix of singularities is anti-invariant with respect to the involutionv Ñ´v on one line. Therefore, ω p0q 2 | sp pz 1 , z 2 q cannot be expressed as a rational form in z 1 and z 2 .

p2, 3, 3qŘ
ps, s 1 q " s`s 1 s 2´s s 1`s12 , a " dimÊ " 6, χ " 1 6 (6.44) The generator of the dynamics is:α " 2ê 0´ê1`ê3´ê5 . (6.45) The subspace E has dimension 4. The orthogonal projection ofê 0 on E is: and using the shift operator, the projections of the other vectors of the canonical basis are e j " ε´jpê 0 q. We observe that: β 1 " e 0 , β 2 " e 1 , β 3 " e 2 , β 4 "´e 4 (6.47) is a basis of E. Equivalently, E can be characterized: The Cartan matrix of the root system R read in the basis of β's is: We recognize the Cartan matrix of the root system D 4 (see Figure 5), where β's play the role of the simple roots. They can be represented in the more standard form: where the δ's now form an xorthonormaly basis of E: This basis is convenient for computations.

Minimal orbits
The A 3 root system generated by I " tβ 1 , β 2 , β 4 u forms a maximal sub-root system of D 4 . Since the δ's from a xorthonormaly basis, it is easy to see that H F rIs (the subspace of E orthogonal to I) is spanned by: v˚" 1 2`δ 1`δ2`δ3`δ4˘" 1 12`ê 0`2ê1`ê2`5ê3´2ê4`5ê5˘.

(6.51)
Multiplying on the left byα, we find an elementv˚PÊ with a minimal orbit: v˚"α¨v˚"ê 3´ê4`ê5 , (6.52) and the order of its orbit is: d " |WeylpD 4 q| |WeylpA 3 q| " 8. (6.53) We choose as initial data a shift of (6.52), namelyv 1 "´ê 0`ê1`ê5 . Its orbit is shown in Figure 8, it consists of the vectors: and their opposite, and one can check that the orbit is complete.

Spectral curve
There exists a polynomial Ppx, yq of degree: deg y P " 8 (6.55) such that y "ŵ¨φpxq forŵ P G¨v 1 are the solutions of Ppx, yq " 0. Let us describe its symmetries: ‚ The orbit is stable under the xisometryy ε 3 . Hence: Ppx, yq " 0 iff Pp´x, yq " 0. But contrarily to § 6.3, the orbit is not pointwise invariant under ε 3 , so it will not be simpler to study the quotient by the involution x Ñ´x.
Since Γ is a single segment, we count (Lemma 3.10) a genus g " 5 for the Riemann surface obtained from the skeleton graph. Since a rational parametrization does not exist, there is no shortcut and we have to determine the coefficients of Ppx, yq " 0. It must have the Puiseux expansions at x Ñ 0 and They give rise to the slopes pn 0 rŵs,´1q and pn 0 rŵs, 1q in the Newton polygon of P -overall sign chosen so that they leave the polygon on their left -, and the leading coefficients in (6.56) give the roots of the slope polynomials. We observe that all leading coefficients are roots of unity, up to a homogeneity factor of c n0rŵs (resp. c´n 0 rŵs ) on the facets with x Ñ 0 (with x Ñ 8). Absorbing this homogeneity factor in the definition, the slope polynomials are thus cyclotomic: We deduce that: deg x P " ÿ wPĜ¨v2 n0rŵsą0 n 0 rŵs " 8 (6.58) and the slopes must be arranged in a sequence such that they form the boundary of a convex polygon. Starting with South-East steps -belonging to facets with x, y Ñ 0 -and following the counterclockwise order, this sequence reads:¨¨Ñ p1,´1q Ñ p1,´1q Ñ p2,´1q Ñ p2, 1q Ñ p1, 1q Ñ p1, 1q Ñ p0, 1q Ñ p0, 1q (6.59) p´1, 1q Ñ p´1, 1q Ñ p´2, 1q Ñ p´2,´1q Ñ p´1,´1q Ñ p´1,´1q Ñ p0,´1q Ñ p0,´1q Ñ¨¨L et us put the lexicographic order on the monomials x i Y j : first compare the degree in x, then the degree in y. We infer from (6.59) that the minimal monomial appearing in P is x 4 , and we can normalize P so that it appears with coefficient 1. Then, following the slope polynomials along (6.59) determines completely the coefficients of P on the boundary of the Newton polygon, see Figure 9.
With this normalization, the ε 3 symmetry of the orbit and the palindrome symmetry of our solution imply: Ppx, yq " Pp´x, yq " y 8 x 8 Pp1{x, 1{yq. (6.60) If we setx " x 2 andP px, yq " Ppx, yq, it remains to find 7 coefficients inside the Newton polygon of r P : r Ppx, yq " " py 8`1 qx 2`p y 7`y qpc 2x`C 1x 2`c2x3 q`py 6`y2 qpC 2x`C3x 3´c4x4 q (6.61) Figure 9: Case p2, 3, 3q -Newton polygon of Ppx, Y q. Once the coefficient 1 at the bottom is fixed, all the coefficients in blue are determined by the slope polynomials. A rule of thumb to find the homogeneity factors of c: on the left (resp. right) of the picture -the facets x Ñ 0 (resp. x Ñ 8)we take a c (resp. c´1) at each step left (resp. right).
If the C's were generic, the curve of equation Ppx, yq " 0 would have genus 17. But according to (3.31), our solution must achieve genus 5, and the C's should be determined as functions of c. First, we try to reduce the number of parameters. Let us give names to the first few moments of the equilibrium measure: The M 's are function of c and their definition implies M k Ñ 0 when c Ñ 1. By construction we have: Expanding this polynomial when x Ñ 0, we can identify the C's in terms of the M 's, and the form (6.61) also imposes constraints on the M 's. We find that all M 's and C's can be eliminated except M 1 and M 2 .
If M 2 and M 3 were generic, the curve of equation r Ppx, yq " 0 would have genus 13, but the solution we look for has genus 5. It means that the curve has 8 singular points, and they will come in two orbits under the symmetries px, yq Ñ px´1, yq and px, y´1q. Let us exploit the symmetries by setting: ξ "x`1{x, η " y`1{y (6.65) and eliminatex and y to get a polynomial equation Qpξ, ηq " 0: This curve should have two singular points, i.e. there exists pξ 1 , η 1 q and pξ 2 , η 2 q such that: Solving explicitly these algebraic constraints is complicated, but we do expect that they select a finite number of solutions for M 1 and M 2 . Eventually, the (hopefully unique) appropriate branch should be chosen by requiring that the coefficients of u Ñ 0 expansion are rational numbers.

Two-point function
A rational solution of the linear equation with right-hand side x k´1 dx exists for k mod 6 P t0, 2, 3, 4u, otherwise we have a log. The outcome of the computation of the residue vectors of the particular solution for the primitive of the two-point function is: e order the vectors of the orbit described in (6.54) as follows: w 1 "v 2 ,ŵ 2 "v 1 ,ŵ 3 " ε 3 pv 1 q,ŵ 4 "´v 0 w 5 "v 0 ,ŵ 6 "´ε 3 pv 1 q,ŵ 7 "´v 1 ,ŵ 8 "´v 2 in order to write down the matrix of singularitiesB p0q . Substracting a matrix filled with: we find the sparse part given in Appendix C. It has only 0,˘1 entries, and there are atmost 4 non-zero entries in the same row.

Minimal orbits
Deleting the root β 6 generates a maximal sub-root system, of type D 5 . Therefore, the order of the minimal orbit is: d " |WeylpE 6 q| |WeylpD 5 q| " 51840 1920 " 27 (6.75) A vector with minimal orbit can be obtained asv˚"α¨v˚where v˚is xorthogonaly to all β's except β 6 . Using the matrix (6.74) containing the xβ i , β j y, we can write down those equations and solve them. We find a generator: v˚" 1 3`2 a 1`3 a 2`4 a 3`6 a 4`5 a 5`4 a 6" 1 36`1 0ê 0`5ê1`ê2`2ê3`ê4`5ê5`1 0ê 6´7ê7`ê8`1 4ê 9`ê10´7ê11˘. (6.76) and thus:v˚"ê 0`ê6´ê7`ê9´ê11 (6.77) 12 Let us give some explanation about the introduction of the β's. The matrix of scalar products xê j ,ê k y can be directly read offα, and it is easy to check that it is definite positive. This implies that R is a finite root system, so must be of the form ADE. We already know the rank dim E " 4 here, so it does not leave much choice for R. Then, there is a part of guess to define a basis pβ j q j which puts the Cartan matrix in standard form.
Since Γ is a segment, we find a Riemann surface of genus 46.

Spectral curve
The formulae applied to the orbit (6.78) determine the list of slopes of the Newton polygon of P, and the list of slope polynomials: slope polynomial p3,´1q ξ 2´1 p2,´1q ξ 3`1 p1,´1q ξ 6`1 p0,´1q pξ`1qpξ 2`1 q 2 " ξ 5`ξ4`2 ξ 3`2 ξ 2`ξ`1 p´1,´1q ξ 6´1 p´2,´1q ξ 3´1 p´3,´1q ξ 2`1 (6.79) We can then read off the coefficients on the boundary of the Newton polygon (see Figure 11). The degree in x is given by: deg x P " The residual´ε 1 symmetry and the palindrome symmetry give: Ppx, yq "´y 27P p´x, 1{yq " y 27x6P p1{x, 1{yq (6.83) The proportionality coefficients have been found by checking the coefficients on the boundary of the Newton polygon. This gives us a polynomial involving 32 unknown coefficients inside the Newton polygon: r Ppx, yq (6.84) here the¨¨¨are completed by symmetry (6.83). For generic C's, the equation r Ppx, yq " 0 defines a curve of genus 102, but the solution we look for achieves genus 46. By construction, we have: To determine the remaining M k 's in terms of c, one has in principle to impose the existence of enough singular points -modulo the symmetries px, yq Ñ p´x, 1{yq and p1{x, 1{yq -to lower the genus down to 46. This yields algebraic equations, that we were not able to write down given the length of the expressions. We expect -although we cannot prove before writing them down -that these equations have a finite number of solutions, and among those, one can hope -again without proof -that there is a unique one such that M k P Opuq when u Ñ 0 and such that the coefficients of Taylor expansion in u are rational numbers. The solution we seek for has all this property, and since we proceeded by necessary conditions, we know that at least one such solution exists.

Two-point function
We can find a rational solution of the linear equation with right-hand side x m´1 dx for m mod 12 P t0, 3, 4, 6, 8, 9u , and a solution with a log otherwise. We find residue vectors: Figure 11: Case p2, 3, 4q -Newton polygon of Ppx, yq. Actually, only the coefficients of x 6j y k "x j y k are non-zero, and they satisfy the symmetries (6.83).
whereê 1 j "ê 6´j`ê6`j . On the orbit described above, we find that the plain part of the matrix of singularities is:B p´1q ê (6.87) and the sparse part is a 27ˆ27Ê-valued matrix stored in a maple file available on request. Its entries are 0,˘1 and in the matrixB p0q in front of eachê j there are exactly 11 of them per row (or column) are non zero. Some 1 and´1 could cancel to make a 0, so we guess that the curve is a 12-fold cover of a rational curve, which should actually be the quotient curve by the symmetries px, yq Ñ pζ 6 x, yq and px, yq Ñ p1{x, 1{yq.
(6.91) in which the Cartan matrix of R reads: We recognize the Cartan matrix of the root system E 8 , and the β's play the role of the simple roots labeled as in Figure 12. The 240 roots generated by the β's under action of the Weyl group G are the elements v P E of the form: v P Zrpβ j q j s Y pZ`1{2qrpβ j q j s, xv, vy " 1, The simple roots can be represented in the standard form: where the δ now form an xorthonormaly basis, expressed in terms of the canonical basis pê k q k in Appendix E.1. Figure 12: Dynkin diagram of the root system E 8 6.6.1 Minimal orbit I " tβ i 1 ď i ď 7u generates a maximal 13 sub-root system, which is of type E 7 . We can find a vector of minimal orbit asv˚"α¨v˚where v P E is xorthogonaly to all β's except β 8 . It is easy using (6.94) to get the decomposition of a generator on the xorthonormaly basis of δ's: We find that the orbit is stable by the shift ε 1 and the involutionv Þ Ñ´v. This allows the decomposition of O into ε 1 orbits generated by the following vectors, and their opposites. The size of the ε 1 orbits are displayed in the second column.
The 2˚in the last line means that ε 1 pv 0 q "´v 0 , so we do not need to include the ε 1 orbit of´v 0 when describing the other part of the orbit viav Ñ´v. One can check that the orbit is complete.

Spectral curve
There exists a polynomial equation Ppx, yq " 0 of degree 240 in y, whose solutions are theŵ¨Y pxq forŵ P G¨v˚. The symmetries ε 1 andv Ñ´v or the orbit, as well as the palindrome symmetry of our solution, imply: We obtain according to Lemma 3.10 a Riemann surface of genus 1471. The computation of n 1 rŵs for all elementsŵ of the orbit described above gives the list of slopes and slope polynomials: slope polynomial p6,´1q ξ 5`1 p5,´1q ξ 6´1 p4,´1q ξ 15´1 p3,´1q pξ 10´1 q 2 " ξ 20´2 ξ 10`1 p2,´1q pξ 15`1 q 2 " ξ 30`2 ξ 15`1 p1,´1q ξ 30´1 p0,´1q pξ`1q 2 pξ 2`ξ`1 q 3 pξ 4`ξ3`ξ2`ξ`1 q 5 (6.100) Since all slopes are of the form pn,˘1q, the degree of the slope polynomials give the multiplicity of the corresponding slope belonging to the facet x Ñ 0. Those of the facets x Ñ 8 form an identical set of slopes and slope polynomials thanks to the palindrome symmetry. We learn that P must have degree 540 in x and its minimal monomial is x 270 . Choosing its coefficient to be 1 and following the slope polynomials then determine the coefficients on the boundary of the Newton polygon of P (Figure 13). Enforcing the symmetries yields: degxP " 18 (6.101) and:P px, yq "x 18P p1{x, yq " y 240P px, 1{yq (6.102) We count that P depends on 801 independent coefficients in the interior of the Newton polygon. If they were generic, the plane curve of equationPpx, yq " 0 would have genus 2949 (this is the total number of points in the interior of the Newton polygon), but our solution must achieve genus 1471. By construction, we have:P px, yq "x 9 ź wPO py´ŵ¨Y pxqq,x " x 30 (6.103) If we write the expansion at x Ñ 0 of our solution: then plug it inPpx, yq " 0 and use the information -shape and symmetries (6.102) -on its Newton polygon, we find thatPpx, yq can entirely be expressed in terms of the 3 moments M 6 , M 10 , M 15 , which are however not specified.

Two-point function
We can find a rational solution of the linear equation with right-hand side x m´1 dx for m mod 30 P t0, 6, 10, 12, 15, 18, 20, 24u , and a solution with a log otherwise. We find residue vectors: whereê 1 j "ê 15´j`ê15`j , whileĈ p1q is a vector having huge rational coefficients with huge denominator 2274168727208063179297410229829 that we do not reproduce here. Nevertheless, the 240ˆ240 matrix of singularity for the orbit described above is not so complicated. We identify its plain part as: B " ÿ PZ30ê (6.106) and its sparse part has entries P t´3,´2,´1, 0, 1u. And in each row of B p0q sp pjq, there are exactly 184 non-zero entries. This is larger than the covering of degree 30ˆ4 " 120 that we get by dividing out by the symmetries. x Ñ 1{x and y Ñ 1{y. Only monomials of the form x 30j y k can appear. 6.7 p2, 2, 2, 2qŘ ps, s 1 q " 1 ps`s 1 q 2 , a " 2, χ " 0 (6.107) The dynamics are generated byα " 2pê 0´ê1 q. The matrix model is actually the Op2q model [35] with a log-square potential, its saddle-point equation is: W px`i0q`W px´i0q´2W p´xq " a 2 ln x u (6.108) G is found to be the affine Weyl group p A 1 , whileĜ » Z 2 is the Weyl group A 1 , and up to rescaling the finite orbit is˘v 0 "˘pê 1´ê2 q and has order 2.
Since (6.108) is very simple, we present a direct solution. The simplest particular solution of (6.108) involves a log-cube, so we prefer to introduce as explained in § 5.1.2: which solves, for any x Ps1{γ, γr: Thus, pφ 2 pxq´φ 2 p´xqq 2 has no discontinuity, and given the growth and symmetries of phi 2 pxq, it must be of the form:`φ 2 pxq´φ 2 p´xq˘2 " Since φ 2 pxq has a discontinuity on r1{γ, γs only, the numerator must be a perfect square, hence an answer of the form for some constants β and β 1 . Notice that this function is odd/odd with respect to the symmetries x Ñ´x/x Ñ x´1, because the determination 14 . The result can be integrated once: and the answer is of the form: where C E and C F are constants to determine, and we introduce the elliptic integrals Erks " Er1; ks "ˆ1 0 dt c 1´k 2 t 2 1´t 2 E 1 rks " Er a 1´k 2 s (6.115) and: Since lim xÑ8 W pxq " 1, we must have lim xÑ8 φ odd 1 " 0. Since we have the asymptotic behavior when x Ñ 8: Frx{γ; γ 2 s "´iK 1 rγ 2 s`Op1{xq we must impose a relation between C E and C F : Next, we have to match the first equation of (6.110). Using the functional relations: Erpx`i0q{γ, γ 2 s`Erpx´i0q{γ, γ 2 s " 2Erks Frpx`i0q{γ, γ 2 s`Frpx´i0q{γ, γ 2 s " 2Krks (6.118) we deduce the value: where all complete elliptic integrals are evaluated at k " γ 2 , except for E evaluated at a 1´γ´4. Eventually, the position of the cut γ ą 1 is determined implicitly by the normalization of the spectral density:ˆγ 1{γ φ 1 px 1´i 0q´φ 1 px 1`i 0q 2iπ x 1 dx 1 " 1 (6.120)

2-point function
We find residue vectors:Ĉ Takingv "ê 0´ê1 as initial vector, the matrix of singularity can be decomposed: and we findB p1q "B p2q " 0. Therefore, if we introduce a uniformization variable for the curve: the two-point function reads: ω p0q 2 | hom pz 1 , z 2 q " dz 1 dz 2 pz 1´z2 q 2´1 2 dx 2 pz 1 q dx 2 pz 2 q px 2 pz 1 q´x 2 pz 2 qq (6.124) 6.8 P " p3, 3, 3qŘ ps, s 1 q " 1 s 2`s s 1`s12 . (6.125) The dynamics are generated by:α " 2ê 0´ê1´ê2 . (6.126) E has dimension 2, and it is spanned by the orthogonal projections of pê 0 ,ê 1 ,ê 2 q on E: The Cartan matrix of the root system R " t˘e 0 ,˘e 1 ,˘e 2 u in the basis pa 1 , a 2 q " pe 0 , e 1 q is: We recognize the Cartan matrix of the root system A 2 , where pa 0 , a 1 q plays the role of the simple roots, and its Weyl group G is the symmetric group in 3 elements. Deleting a 0 (or a 1 ) gives a maximal sub-root system, of type A 1 . A generator of H ta0u is: (6.131)

Its image is:v˚"
Apv˚q "ê 0´ê2 , (6.132) The orbit ofv˚is a cycle of order d " 3, the two other vectors begin´ê 0`ê1 and´ê 1`ê2 . We may also consider the "root system"R. Its "Cartan matrix" in the canonical basis pê 0 ,ê 1 ,ê 2 q is:Â henceR is the affine root system of type p A 2 . By construction, its Weyl group isĜ: it is therefore the symmetry group of the tiling of the plane by equilateral triangles.

Spectral curve
We set: which solves, for any x Ps1{γ, γr, the linear equation: The discussion of § 6.8 tells us that φ 2 pxq´φ 2 pζ´1 3 xq can be analytically continued to an analytic function y a Riemann surface with 3 sheets. The value of y in the 2 other sheets is φ 2 pζ 3 xq´φ 2 pxq and φ 2 pζ´1 3 xq´φ 2 pζ 3 xq, and its singularities follow from those of φ 2 pxq listed in § 5.1.2. We therefore have a polynomial equation P px, yq " 0, but the symmetry tells us that, if px, yq is a solution, then pζ 3 x, yq is a solution, and p1{x,´yq is a solution. This information, and the order of singularities of y, prescribe the following form: P px, yq " px 3`x´3´γ3´γ´3 q 3 y 3`P 1 px 3`x´3 qy`P 0 px 3`x´3 qpx 3´x´3 q (6.136) where P 1 and P 0 are degree 2 polynomials.

2-point function
We find the residue vectors: The matrix of singularities for the orbit ofv˚can be decomposed as: and we findB p1q "B p2q " 0.
6.9 P " p2, 4, 4qȒ ps, s 1 q " 1 s 2`s12 . (6.139) The dynamics are generated by:α " 2ê 0´ê1´ê3 . (6.140) E has dimension 3, and it is spanned by the orthogonal projections of pê i q 0ďiď3 on E: A basis of E is given by pa 1 , a 2 , a 3 q " pe 0 , e 1 , e 2 q, and the Cartan matrix of the root system in this basis is: (6.142) hence R is the root system of type A 3 . Its Weyl group G is isomorphic to the symmetric group in 4 elements. Deleting a 1 (or a 3 ) gives a maximal sub-root system, of type A 2 . A generator of H ta2,a3u is: v˚" 1 4`3 a 1`a2`a3˘" 1 8`3ê 0`ê1´ê2´3ê3˘.

(6.143)
Its image is:v˚" Apv˚q "ê 0´ê3 P M, (6.144) The orbit ofv˚is a cycle of order d " 4, containing the vectors´ê 0`ê1 ,´ê 1`ê2 and´ê 2`ê3 . We may also consider the "root system" R. Its "Cartan matrix" in the canonical basis ofÊ is: henceR is the affine root system of type p A 3 , andĜ is the corresponding affine Weyl group.

2-point function
We find the residue vectors: The matrix of singularities for the orbit ofv˚can be decomposed in a plain matrix and a sum of permutation matrices: 6.10 P " p2, 3, 6qȒ ps, s 1 q " s`s 1 s 2´s s 1`s12 . (6.151) The dynamics are generated by:α " 2ê 0´ê1´ê5 . (6.152) E has dimension 5, and is generated by the orthogonal projections of pê i q 0ďiď5 on E: i"0 vpiq " 0. a i " e i´1 for 1 ď i ď 5 defines a basis of E, and the Cartan matrix of the root system R reads in this basis: We recognize the Cartan matrix of the A 5 root system. Deleting a 1 (or a 5 ) gives a maximal sub-root system, of type A 4 . A generator of H tpa2,a3,a4,a5qu is: v˚" 1 36 Its image is:v˚" Apv˚q "ê 0´ê5 (6.156) The orbit ofv˚is a cycle of order d " 6, containing the vectors p´ê i`êi`1 q for i P Z 6 . We may also consider the "root system" R. Its "Cartan matrix" in the canonical basis ofÊ is: henceR is the affine root system of type p A 5 , and G is the corresponding affine Weyl group.

General result
It was shown in [14] that the 1{N asymptotic expansion of multidimensional integrals of the form (2.1) is governed by the topological recursion introduced in [22].
We denote S the spectral curve : it is the Riemann surface on whichv¨W pxq (for a certain set of v) has been analytically continued on C. We denote ϕ the analytic continuation. S is equipped with two points z˘P Σ, and a covering x : Σ Ñ p C, such that xpz˘q " γ˘(the endpoints of the support of the equilibrium measure) are simple branchpoints. We introduce the 1-form on Σ defined by: It has simple zeroes at z˘.
Definition 7.1 Let ι be the deck transformation, defined in the neighborhood of z˘: it is the holomorphic map such that xpιpzqq " xpzq and ιpzq ‰ z for z ‰ z˘.
The spectral curve is also equipped with a bidifferential ω p0q 2 having (among possible other singularities) a double pole at coinciding points with coefficient 1 and no residues. pS, ω p0q 1 , ω p0q 2 q is the initial data needed to compute the large N expansion of all correlators. Definition 7.2 We define the recursion kernel: The recursion kernel has a simple pole at z " z˘. are known, this formula provides an algorithm to compute the Chern-Simons free energies, and the perturbative expansion of the colored HOMFLY of fiber knots in Seifert spaces. For lens spaces, the equation of the spectral curve was a polynomial in e u , and that implied that the perturbative invariants were polynomials in e u . In particular, they did not have any singularity for u in the complex plane.
The situation is different for the other Seifert spaces χ ą 0, since the interior coefficients of the equation of the spectral curves are in general algebraic function of e u . We have seen for instance in the S 3 {D p case that the spectral curve was polynomial in κpcq given by (6.14). It would be interesting to give an interpretation of this "algebraic change of variable" as a non-trivial mirror map in the context of topological strings (u should be in correspondence with some Kähler parameter).
We will now derive a few formulas and consequences of our results for S 3 {D p . In particular, in § 7.2-7.3 we obtain hypergeometric formulas for the planar limit of the HOMFLY-PT of fiber knots generalizing those given in [17] for torus knots. We also establish in § 7.2.2 arithmetic properties of the all-genus perturbative invariants when p is even.

Disk invariants
The spectral curve is given in (6.13) The expansion when z Ñ 8 (i.e. x Ñ 8 in the sheet ř j p´1q j`1ê j ) is of the form: lnp´ypxqq "´u a 3 ÿ mě0 xTr U a3pm`1{2q y p0q x´a 3pm`1{2q (7.4) We compute by Lagrange inversion: xTr U ppm`1{2q y p0q "´p u Res x ppm`1{2q pζ´2 {p q 2m`1 (7.5) where we have defined a new coordinate ζ " z´p {2 , which is such that x p{2 P Opζ´1q in the regime we are interested.
Each term can be explicitly computed and gives a Laurent polynomial in κ that can be put in hypergeometric form: For instance: for the knot K 2 going along an exceptional fiber of order 2. Similar formulas involving sum of two polynomial hypergeometric can be found for @ Tr U kp{2 D for any k P N. Those formulae can be seen as generalizations of the large N limit of Rosso-Jones formula for torus knots [42].
Theorem 7.5 The endpoints of the singularity locus of u 2 B u F pgq -seen as a function of κ -occur at κ 2 " t0,˘1,˘pκ˚q 2 u. A more precise description arise follows from (7.31).
Two-point function. The two point function was given in (6.24):
No new singularity occur in the κ-plane.
Combining all results, we find that @ Tr U k1¨¨¨T r U kn D belongs to Qpκ 2 qr a βpκqs. Moreover, the denominators can only contain powers of pκ 4´1 q and pκ 4´κ4 q. We remind the relation (6.36) between κ and the Chern-Simons variable u " N : Therefore, the list of singularities of the perturbative colored HOMFLY in the u-complex plane occur when pu mod 8iπp 2 Zq is equal to: where: Sppq " pp`1q lnpp`1q`pp´1q lnpp´1q (7.27) This list is complete, except that u " 0 must be removed since by construction, we know that invariants are Op1q when u Ñ 0. The derivatives can be explicitly computed, and we find: Besides, the three rational functions ofz 1{2 appearing in this expression are odd if m is odd (resp. even if m is even). It is clear from (7.29) that the singularities of J m,˘a s a function of κ only occur when the singularities of lnxpzq as a function ofz approach the cutz P rz´,z`s. The quantities u 2 B u F pgq are linear combinations of J m,˘w ith coefficients in Qrκ 2 , e˘ζs, and the result should be even in ζsincez`" e ζ andz´" e´ζ play a symmetric role. And, we remind thatxpz`qxpz´q " 1. Therefore: u 2 B u F pgq P Qpκ 2 qr a βpκqs¨ln 2x pz`q`Qpκ 2 qr a βpκqs a β 2 pκq¨lnxpz`q`Qpκ 2 qr a βpκqs (7.31) where: a β 2 pκq " sinh ζ " a pκ 4´1 qppp`1qκ 4´p p´1qq κ 2 (7.32) The location of singularities of the free energies can easily be deduced. l 7.3 S 3 {D p , p odd

Disk invariants
The spectral curve is given in (6.35). Checking the behavior (5.9)-(5.10) shows that z Ñ κ˘1 {a3 corresponds to x˘1 Ñ 8 in the sheet labeled by´pê 0`êa3 q. Therefore, the expansion at those points of ln ϕ in terms of the variable x, reads: By Lagrange inversion, we find: After a change of variable ζ " z p , this residue can be explicitly computed: Since κ is a function of u independent of the parity of p, we observe the relation: p2,2,p evenq (7.33) In particular, the planar limit of the HOMFLY-PT of K 2 is obtained by taking 2m`1 " p in (7.33). The computation of even powers is very similar, and we get: In particular, the planar limit of the HOMFLY of the knot following the fiber of order p is obtained by taking m " 1: Planar HOMFLY´PTpK p q " pp1´κ 2 puqq 2uκ 2{p puq`κ´2 puqp2`pq`2´p˘ (7.35) 8 Generalization to SO and Sp Chern-Simons

The model
For more general gauge groups, Chern-Simons theory on Seifert spaces is described by the following measure on the Cartan subalgebra -identified with R N : where the product ranges over positive roots. For the A series, we had: Now, we would like to address the BCD series: They correspond to the Lie algebras: We will explain the main points of the analysis of those models, and skip the details which are very similar to the already studied A series.

First rewriting
Let us perform the change of variable s i " e ti{a with a " lcmpa 1 , . . . , a r q, and put the measure (8.3) in the form: where the potential is: The terms U G are absent for the A series, and are equal to: And, the pairwise interaction is: for the A series, and:Ř px, yq "Ř A px, yqŘ A´? xy, 1 ? xy¯( 8.10) for the BCD series.

Second rewriting
The BCD matrix model can also be rewritten: with: Rps i , s j q " The potential for this model is obtained by including the part of the diagonal terms i " j which are regular: V tot,G pxq " NV pxq`δV G pxq " N a 2 pln xq 2 2t`ÿ mǓ G pxǎ m q,Ǔ G pxq " U 0 pxq´U G pxq (8.13) with: U 0 pxq " 1 2 lnpx´x´1q (8.14) and U G were defined in (8.8). In other words: The expressions (8.11)-(8.12) are prepared in order to write down Schwinger-Dyson equation (see § 8.5), while the more compact expression (8.6) is appropriate to study equilibrium measures.

Observables and symmetries
We observe in (8.3) that t i individually is distributed like´t i . Therefore, the joint probability density vanishes at t i "˘t j , and to find the dominant configuration of (8.3) when N Ñ 8 of (8.3) via potential theory, it is enough to restrict to even configurations. One can then show uniqueness and off-criticality of the equilibrium measure at least when χ ě 0 (see Appendix A.4). We assume χ ě 0 in all what follows.
The naive definition of the n-point correlators is:

Tr
x i x i´e T{a ı conn (8.16) with T " diagpt 1 , . . . , t n q. In order to emphasize later the relation between the A series and G P tB, C, Du series, it is convenient to introduce the symmetrized correlators: By symmetry, we have: . . , 1{x n q " p´1q n W G|n px 1 , . . . , x n q (8.18) and thus for the leading order lim N Ñ8 N´1 W G|1 pxq " W G pxq: Stricto sensu, the existence of an all-order asymptotic expansion of the form: does not follow from the results of [15] since the potential has a log singularity at t i " 0. It is nevertheless likely that the argument can be generalized for this model. In the remaining, we will take the asymptotic expansions (8.20) as starting point 15 , with W rks G|n px 1 , . . . , x n q being analytic functions of n variables x 1 , . . . , x n away from a cut r1{γ, γs. Compared to the expansion (4.18) for the A series, since the log term of the potential contains a 1{N corrections (the log terms coming from i " j),W G|n is not expected to have an expansion of parity p´1q n in N .
Our goal is to compute the coefficients of (8.20). We will first write linear functional equations W  (8.20) and equating the discontinuities of the two sides of the Schwinger-Dyson equations. For this purpose, we will show that the symmetrized correlators satisfy a set of loop equations, which are solved by a version of the topological recursion adapted to the grading in (8.20). This is performed by adapting the tools of [14] to take into account the presence of a symmetry.

Spectral curves
We first have a look at the equilbrium measure in the model (8.3). We denote Γ G puq the support r1{γ, γs of the equilibrium measure, emphasizing its dependence in the parameter u. We find, for any x P Γ BCD puq. The saddle point equation for W G pxq is obtained as (2.16) was obtained from (2.11).
Since it concerns only the leading order when N Ñ 8, the only term relevant in the potential is that involving V A . Notice that here, the joint probability density is singular when x i Ñ x j and x i x j Ñ 1. The second singularity is also relevant since configurations with x i x j " 1 can occur with x i , x j P Γpuq. We find, for any x P Γ G puq: In terms of the symmetrized correlator, we therefore find, for any x P Γ G puq: For χ ě 0, this equation characterizes the equilibrium measures together with the condition: This is shown by repeating the proof of Corollary 2.2 using the strict concavity proved in Appendix A.4. for the BCD series. Therefore, we deduce, if χ ě 0: In other words, the spectral curve are the same in the B, C, or D series. They are obtained from the spectral curve of the A series upon replacement of u by 2u, for which we can use the results of Section 5-6.

Two-point function
Following the strategy of § 2.4, we can derive the saddle point equation for the leading order of the 2-point correlator. We find, for any x P Γ G puq and x 2 P CzΓ G puq: where B 0 px, x 2 q "´xx 2 {px´x 2 q 2 . This implies the saddle point equation for the symmetrized 2-point function: Since B 0 px 1 , x 2 q " B 0 px´1 1 , x´1 2 q, the right-hand side is a symmetric function of x and x 1 , which is transformed into its opposite when px, x 1 q Ñ p1{x, x 1 q. Let us compare (8.25) with the saddle point equation of the two-point function for the A series: for any x P Γ A p2uq and x 2 P CzΓ A p2uq, Since Γ G puq " Γ A p2uq and this equation characterizes the two-point function with appropriate growth condition at 8 in the cases χ ě 0, we deduce: Remark. The two-point function W r0s A|2 px 1 , x 2 ; 2uq plays the role of a fundamental solution in the following sense: the function x Þ Ñ¸d x2 2iπ f px 2 q W r0s A|2 px 1 , x 2 q for any choice of function f px 2 q holomorphic near the contour of integration which avoids the cut, is the solution of the functional equation on the cut Γ G puq: for a function φ holomorphic on CzΓ G puq satisfying appropriate growth conditions at 8. It is convenient to introduce:

First correction to the spectral curve
In the A series, the first correction to the spectral curve was of order 1{N 2 . In the BCD series, because the potential V tot contains an extra term of order 1{N , the first correction is W G|1 pxq " N W G pxq`W r1s G|1 pxq`op1q, and is obtained by formally writing down the N -dependent saddle point equation for V tot , and collecting the term of order 1{N . We find, for any x P Γ G puq: Given the expression of the right-hand side (8.30), it can be decomposed in partial fraction expansion: We first add the diagonal term i " j to obtain ř i,j and substract it, which has for effect to cancel the : this is a consequence of vanishing of order β " 2 of the joint probability density (8.11) when s i Ñ s j . Adding the terms i " j also gives extra contributions which can be put in the form of a perturbation of the potential. Then, by symmetry between the two terms in the ř i,j in (8.33), we find: The first term is the expectation value of an analytic function in a neighborhood of the cut, so can be rewritten in terms of a contour integral of W 2 . If we rewrite µ " ř i¨¨¨‰ in terms of W 1 , we should keep in mind that δV 1 pξq has poles at ξ " 1 which is on the cut. All in all, we find the equation: The first double integral could be perform by picking up residues at η " x and η " 0, but this expression is enough for our purposes. For comparison, the first Schwinger-Dyson equation of the model of the A series in terms of its correlators can be written: (8.39) (8.37) only differs from (8.39) by addition of terms involving η´1, due to the structure of (8.12), and a potential V tot,G instead of V .

. . . and its symmetrized form
Our next goal is to derive Schwinger-Dyson equation involving the symmetrized correlators only: We denote A 1 pxq and A 2 pxq the terms involving W G|1 and W G|2 in (8.37). We observe that: Therefore, we have: W G|1 pη´1q x´η (8.42) Therefore: Similarly, let us defineǎ 0 " a and consider a summation index α " pm, m q with m P t0, . . . , ru and P Zǎ m , and We compute: Since the sum over α contains both c α and c´1 α , we find: We observe that in (8.43) (resp. (8.45)) the second term -call it S 1 pxq, resp. S 2 pxq -is the negative part of the Laurent expansion of x´4A 1 pxq (resp. x´4A 2 px´4q) at x " 0, whereas the first term is regular at x " 0. Since we have the equation: this implies that S 1 pxq`S 2 pxq " 0. Then, summing (8.43) and (8.45) gives: tot,G pηq x´η Ă W G|1 pηq " 0 (8.47) In this form, we recognize the Schwinger-Dyson equation of the correlators of the model in the A series, in the potential V tot given by (8.13).

Higher equations
Let δ x be the insertion operator: it computes B λ"0 under the perturbationV pξq ÑV pξq´λ N 1 x´ξ . We have the properties: δ x W G|n px 1 , . . . , x n q " W G|n`1 px, x 1 , . . . , x n q, δ x pNV 1 pyqq "´1 px´yq 2 (8.48) Higher Schwinger-Dyson equations involving W G|n`1 , W G|n , . . . , W G|1 can be derived by applying the operator δ x2¨¨¨δxn to (8.37). One can also define an insertion operator adapted r δ x adapted to symmetrized observables: it computed B λ 1 "0 under the perturbationV pξq ÑV pξq´λ 1 N`x x´ξ`´x´1 x´1´ξ˘. It has the property: A word of caution: the symmetries (8.18)- (8.19) are only valid when the potential (and its perturbation) are invariant under x Ñ x´1. Therefore, if we make any use of this relation to transform the Schwinger-Dyson equation, only the application of symmetrized insertion operators r δ x is allowed.

Loop equation and recursive formula
The Schwinger-Dyson equation (8.47) and their analogue is now similar to those treated in [14,Section 3]. If one insert the expansion (8.21) in the Schwinger-Dyson equations and study the discontinuity of both sides, one can prove by recursion on pn, kq that Ă W rks G|n px 1 , . . . , x n q defined for x i P CzΓ G puq, satisfies the linear functional equation: if pn, kq ‰ p1, 0q, p2, 0q, p1, 1q, and (8.22)-(8.25)-(8.30) in the first three cases. One can then establish for pn, kq ‰ p1, 0q, p2, 0q, p1, 1q that Ă W rks G|n diverges polynomially when x i approaches the edges of the cut Γ G puq, but is regular at other points, including x i " 1. This contrasts with the singularity at x " 1 occuring for pn, kq " p1, 1q, which was due to the log singularity in the potential.
From (8.50) by Schwarz principle, we deduce that W rks G|n px 1 , . . . , x n q can be analytically continued as a meromorphic function on the spectral curve Σ. The spectral curve is equipped with a covering x : Σ Ñ p C, and with a local involution ι which exchanges the two sheets at the vicinity of the ramification points. The initial CzΓ G puq is identified with the physical sheet in this covering. We still denote γ˘1 the preimages ot the two endpoints of the cut in the physical sheet. We introduce the differential n-forms on the spectral curve: Ă W rks G|n pz 1 , . . . , z n q "˜Ă W rks G|n pxpz 1 q, . . . , xpz n qq`δ k,0 δ n,2 xpz 1 q´xpz 2 q˘2¸n They are meromorphic with poles at all preimages of the endpoints of the cut Γ G puq in Σ. We also introduce the Cauchy kernel: where G G is given by (8.29). Then, following [14, Section 3.7], the Schwinger-Dyson equation -and its higher versions -can be written for any n, k, with the convention that Ă W rks G|n " 0 for k ă 0: G||J|`1 pιpzq, z IzJ q " r P rks G|n pz; z I q`δ r P rk´1s G|n pz; z I q (8.53) where r P rks G|n pz; z I q is a holomorphic quadratic differential in z in a neighborhood of the image of Γ G puq in Σ, and we have written apart the term which has a simple pole at xpzq " 1: δ r P rk´1s G|n pz; z I q " G|n pξq¯(8.54) There are two differences with the setting of the topological recursion presented in [14]. Firstly, we have an extra term δ r P rks G|n pz; z I q which contains a pole a xpzq " 1. This pole thus occurs away from the branchpoints. Secondly, since the potential was perturbed by a term of order 1{N , we have all possible powers of N 2´k for k ě 0 in the expansion of Ă W G|n -instead of just N 2´2g´n for g ě 0. We can nevertheless proceed as usual to solve the abstract loop equations. We use the notations: ∆f pzq " f pzq´f pιpzqq, Sf pzq " f pzq`f pιpzqq (8.55) so that we can write: Then, we may recast (8.53) as:

Markov chain Monte Carlo
The probability measure describes a statistical system of N classical particles with coordinates t i P I Ă R, and Z N "´I N dµ N is its partition function. We are interested in the equilibrium distribution of the particle coordinates, µ eq , which can be estimated from the large-N limit of the empirical measure The phase space of such a many-particle system is far too large to be summed up completely or to be explored by naively selecting random configurations. Instead, the standard simulation technique is a Monte Carlo algorithm with importance sampling [8]. It is based on a suitable Markov chain process which converges to the measure dµ N pt 1 , . . . , t N q " p eq pt 1 , . . . , t N q dt 1¨¨¨d t N . A Markov chain is a stochastic process in which a set of random variables tt i u is updated to a new configuration tt i u 1 with a probability W that depends only on tt i u and tt i u 1 , but not on the earlier history of the process,¨¨Ý The process converges to the distribution dµ N , if the transition probability W fulfills the condition of detailed balance, p eq ptt i uqW ptt i u Ñ tt i u 1 q " p eq ptt i u 1 qW ptt i u 1 Ñ tt i uq . W ptt i u Ñ tt i u 1 q " minˆ1, p eq ptt i u 1 q p eq ptt i uq˙. (9.5) For large N the probability ratio in the above expression is a ratio of two very small numbers. Since this is difficult to handle numerically, we rewrite dµ N in terms of an energy function Ept 1 , . . . , t N q, and use the energy difference of two configurations, ∆E " Eptt i u 1 q´Eptt i uq, to evaluate the transition probability W ptt i u Ñ tt i u 1 q " minp1, expp´∆Eqq. In our simulations we use local updates, i.e., we propose a random change of one coordinate x j . If this change lowers the energy, ∆E ă 0, we always accept the update. If the change increases the energy, we accept the update only with probability expp´∆Eq. We loop over all coordinates and repeatedly try updates for each of them. After a certain "warm-up" period the Markov process usually reaches equilibrium with a stable coordinate distribution L N , which we can measure in a histogram. We call the successful Metropolis update of N coordinates a Monte Carlo step. For the results presented below we typically performed 10 4 warm-up steps and 10 6 measurement steps.  Figure 14: Equilibrium density µ eq of the statistical model for torus knots with pp, qq " p2, 3q and interaction strength u " 0.5. The data is given in terms of the variablesx i " exppt i q.

Torus knots
The reliability of the Monte Carlo approach can be tested by first considering the special case r " 2 of the general model (2.1), which has been studied earlier in the context of torus knots, .
Here, p and q are two coprime integers labeling the knot, and u is a free parameter. The equilibrium distribution µ eq of this model is determined by the equations [17] x " e p1{p`1{qqu{2 z´1 {qˆ1´e´u z 1´z˙1 {p , (9.9) where the boundaries of the domain ofx " expptq (the branch points) are given by the two zeros of dx{x. In Figure 14 we compare this exact result to the numerical data obtained by simulating N " 200 particles. The agreement is almost perfect: only the edges of the distribution are slightly softened due to the finite number of particles.  and χ is also expected to have an influence on the properties of the statistical model (2.1). The long distance behavior of the pairwise interaction Rpt 1 , t 2 q " Rpt 1´t2 q is determined by χ, logpRpdqq " χ|d|{2 for d Ñ 8 . (9.12) For χ ě 0, Rpt 1 , t 2 q is always repulsive and strictly convex, and a unique equilibrium measure µ eq exists and can be calculated. For χ ă 0, unfortunately, the analytic methods cannot be applied, and it was unclear whether a unique µ eq exists and how it looks like. We therefore performed numerical simulations for both situations, χ ě 0 and χ ă 0, in the case r " 3. The statistical model (2.1) then specializes to Rpt 1 , t 2 q " sinhrpt 1´t2 q{p2a 1 qs sinhrpt 1´t2 q{p2a 2 qs sinhrpt 1´t2 q{p2a 3 qs sinhrpt 1´t2 q{2s , where a i are three integer parameters and u is real and positive. The Monte Carlo simulations with N " 200 particles suggest that a unique equilibrium measure µ eq exists independently of the sign of χ. In Figure 15 we show data for the parameter sets pa 1 , a 2 , a 3 q " p2, 3, 5q, p2, 3, 97q and p11, 11, 11q. The first set fulfills the condition χ ą 0 and leads to a unique equilibrium distribution µ eq . However, the simulations converge equally well for the other parameter sets with χ ă 0. The shape of the resulting equilibrium distributions differs only marginally from the first case. The small oscillations near the edges of the distributions and the softening of the edges are caused by the finite particle number. For N Ñ 8 we expect smooth distributions which at the boundaries of the domain vanish like a half-integer power of x.
We also studied the scaling of µ eq with increasing u, i.e., with decreasing strength of the potential V ptq. For χ ą 0 the interaction´logpRpt 1 , t 2 qq is always repulsive and the expansion of the particle distribution is limited by the potential (and thus depends on u). For χ ă 0 the interaction is repulsive at short distance but attractive at long distance. Therefore, for large enough u, it is plausible that the support of the particle distribution reaches a limit which is independent of u. In Figure 16 we illustrate the scaling of µ eq ptq for the three cases p2, 3, 5q, p2, 3, 6q and p11, 11, 11q, which correspond to positive, vanishing and negative χ, respectively. For χ ą 0 and large enough u the width of the distribution scales like u, whereas it seems to approach a constant for χ ă 0. For χ " 0 the scaling seems to be weaker than the first guess ?
u. This could be due to logarithmic corrections.
9.3.2 The case p2, 2, evenq In this and the remaining sections we compare analytic results for the model (9.13) with numerical data. For the case p2, 2, pq with even p the characteristic χ ą 0 and the equilibrium density µ eq is given by dµ eq " p 2 dx 2iπ ux logˆy px`i0q ypx´i0q˙, (9.14) wherex and y are solutions to the equations (see § 6.2): xpzq " z z p´κp κ p z p´1 , ypzq "´p z p{2´κp{2 qpκ p{2 z p{2`1 q pκ p{2 z p{2´1 qpz p{2`κp{2 q , In Figure 17 we compare this result to simulation data for N " 200 particles and u " 1. The agreement is excellent. Note that we plot the densities as a function of t rather thanx. : Simulation data obtained with N " 200 particles compared to exact equilibrium density for u " 1 and pa 1 , a 2 , a 3 q " p2, 2, p evenq.

The case p2, 2, oddq
For the case p2, 2, pq with odd p the equilibrium density, dµ eq "˘p 2 dx iπ ux logˆy px`i0q ypx´i0q˙, (9.17) can be obtained analytically from ypzq "´p z p´κp qpκ p z p`1 q pz p κ p´1 qpz p`κp q , where 2κ p`1 1`κ 2p " e´u {p4p 2 q ,x " exppt{p2pqq . In Figure 18 we compare the analytical result to simulation data for N " 200 particles and u " 1. Again, the agreement is excellent. The case p2, 3, 3q admits only a partial analytical solution. The equilibrium density again corresponds to a branch cut, dµ eq "˘a 2 dx iπ ux logˆy px`i0q ypx´i0q˙, (9.20) but the equations defining the Riemann surface contain two parameters m 2 and m 3 whose dependence on the interaction c :" expr´u{p2a 2 qs as yet could not be determined (see § 6.4). In terms of the variables z " y`1{y and w "x`1{x withx " x 2 , the branch points fulfill: A " c 4 z 4`c4 z 3´2 c 4 z 2´5 c 4 z´6c 4`2 c 2 m 2 z 3`1 2c 2 m 2 z 2´1 2c 2 m 2 ź 40c 2 m 2`c 2 wz 3´3 c 2 wz´2c 2 w´6cm 3 z 2`2 4cm 3´4 m 2 2 z 8m 2 2`2 m 2 wz´4m 2 w´w 2 z´w 2`3 z`6 " 0 , dA{dz " 0 . Taking w and m 2 as adjustable parameters, we can easily solve these equations for z and m 3 and fit the solution of Apz, wq " 0 to the numerical data. For all values of u we can perfectly match the simulation results, see Figure 19 below. The corresponding parameters m 2 and m 3 are given in the legend.  Figure 19: For pa 1 , a 2 , a 3 q " p2, 3, 3q the parameters m 2 and m 3 in Eq. (9.21) can be adjusted such that the analytical results (lines) fit the numerical data (crosses) very well. We simulated N " 200 particles with interaction u as given in the legend.

The gauge groups SO and Sp
In Section 8 the generalization of the statistical model (2.1) to gauge groups SO and Sp was studied, and it was shown in § 8.24 that in the limit N Ñ 8 the corresponding equilibrium distributions µ eq for parameter u are given by the SU data with parameter 2u. One noticeable difference between the SO and Sp densities (BCD series) and the SU density (A series) is a finite-size correction taking the form of a Dirac peak of order Op1{N q at t " 0 (see (8.32)).
To verify these predictions, which were derived for the convex interaction χ ą 0, and to analyze whether they hold for χ ď 0, we performed numerical simulations for various parameter sets pa 1 , a 2 , a 3 q and increasing particle numbers N " 50, . . . , 400. In Figure 20 we show equilibrium densities for three parameter sets pa 1 , a 2 , a 3 q with positive, zero and negative χ. The densities for the B, C, and D cases at u " 0.5 agree very well with the data for the A case at u " 1. Only in the vicinity of t " 0 we observe the predicted finite-size correction. The total mass of the Dirac peak is negative for the cases B and C, and positive for the case D, but its absolute amplitude is the same for all BCD cases. In Figure 21 we analyze the scaling with increasing particle number N . Clearly, doubling N reduces the δ-peak by a factor 1{2. These observations are in agreement with (8.15)-(8.32), which predicts for χ ą 0 the following masses m for the Dirac peak:  Figure 20: Equilibrium density for the BCD series at u " 0.5 compared to data for the A series at u " 1.0. The curves agree very well, except for a finite-size correction " δptq{N , which is negative for cases B and C, and positive for case D. We simulated N " 200 particles with interaction parameters pa 1 , a 2 , a 3 q as given in the titles.  Figure 21: Finite-size scaling of the equilibrium density for case B with parameters pa 1 , a 2 , a 3 q " p2, 2, 2q and u " 0.5. As expected, the δ-singularity at t " 0 vanishes like 1{N .

Conclusion
The matrix models considered in this work can reliably be simulated with Monte Carlo methods. The numerical data has been used to verify analytic results and to gain further insight into cases that elude an analytic solution.

A.3 Proof of Theorem 2.3
Proof. The effective potential is: V λeq ptq " t 2 2u´ˆΨ pt´t 1 q dλ eq pt 1 q (A.14) where: Ψptq " p2´rq lnˇˇsinhpt{2qˇˇ`r We compute:´4 Ψ 2 ptq " 2 sinh 2 pt{2q`r ÿ m"1´1 a 2 m sinh 2 pt{2a m q´1 sinh 2 pt{2q" 2 sinh 2 pt{2q`r We have introduced f am ptq " 2sinhpt{2q´2a m sinhpt{2a m q. If a m ě 1, we have f ptq ą 0 ensuing from f p0q " 0 and: f 1 ptq " coshpt{2q´coshpt{2a m q ě 0 (A.17) Therefore, Ψ is a concave function, hence V λeq is a convex function for any fixed u ą 0. This implies that it must achieve its minimum C λeq in a connected set, i.e. the support must be a segment. We claim that the density of λ eq vanishes exactly like a squareroot at the edges. By analyticity arguments, we know that it behaves " c|t´α| p2k`1q{2 at any edge α of the support, for some integer k and irrelevant constant c ą 0, but we claim k " 0. Indeed, the effective potential V λeq behaves like pt´αq p2k`3q{2 when t Ñ α from the exterior of the support. However, from the expression, the effective potential is C 2 in the variable t not belonging to the support, and: 1 u´ˆΨ 2 pt´t 1 qdλ eq pt 1 q (A.18) Since we have shown that Ψ is concave: which is only compatible with k " 0.
In case the equilibrium measure is unique, λ eq is even, so its support is of the form r´δ, δs, and after the change of variable, we find that the support ofλ eq is Γ " r1{γ, γs. In case the equilibrium measure is not unique, and denote rδ´, δ`s the support of a given equilibrium measure λ eq . Denote λé q the image of λ eq under t Þ Ñ´t. The characterization of equilbrium measures shows that any convex combination βλ eq`p 1´βqλé q is an equilibrium measure. By the previous argument its support rδ´, δ`s Y r´δ`,´δ´s is connected, therefore δ´ď 0 ď δ`, i.e. the support of any equilibrium measure contains 0. l

A.4 For SO and Sp gauge groups
We have to repeat the analysis of § A.1 for the quadratic form: Q BCD rνs "´¨dνptqdνpt 1 q " p2´rq lnˇˇsinh`t`t Since ν is even, its Fourier transform is real and therefore, Q BCD is definite positive iff (A.6) is strictly convex. In that case, one can prove repeating § A.3 that the equilibrium measure for the BCD series is supported on 1 segment and behaves exactly as a squareroot at the edges.

B Finite subgroups of PSL 2 pCq
We are studying functional relations with the values of φpx`i0q and φpx´i0q on both sides of a cut σ are related to values of φ at gpxq, where g is a Möbius transformation, and the g's involved generate a finite subgroup of Aut p C " PSL 2 pCq. The list of possibilities, and has an ADE classification: up to conjugation, G must be either cyclic, dihedral, or one or the three polyhedral groups.
‚ Z n , cyclic group of order n. Generated by r n pzq " e 2iπ{n z -A n case.
‚ D 2n , dihedral group of order 2n. Generated by r n and ιpzq " 1{z -D n case.
‚ A 4 , tetrahedral group, order 12. It is also the alternate subgroup of S 4 . It is generated by r 2 and pzq " i z´1 z`1 (order 3) -E 6 case.
‚ S 4 , octahedral group, order 24. It is generated by r 4 and  -E 7 case.